Mathematical Physics, Analysis and Geometry - Volume 9

Mathematical Physics, Analysis and Geometry (2006) 9: 1–21 DOI: 10.1007/s11040-005-3896-z

© Springer 2006

Semiclassical Weyl Formula for a Class of Weakly Regular Elliptic Operators LECH ZIELINSKI LMPA, Centre Mi-Voix, Université du Littoral, BP 699, 62228 Calais Cedex, France. e-mail: [email protected] IMJ, Mathématiques, case 7012, Université Paris 7, 2 place Jussieu 75251, Paris Cedex 05, France (Received: 21 April 2004; in final form: 19 January 2005) Abstract. We investigate the semiclassical Weyl formula describing the asymptotic behaviour of the counting function for the number of eigenvalues in the case of self-adjoint elliptic differential operators satisfying weak regularity hypotheses. We consider symbols with possible critical points and with coefficients which have Hölder continuous derivatives of first order. Mathematics Subject Classification (2000): 35P20. Key words: spectral asymptotics, semiclassical approximation, Weyl formula, elliptic operator, pseudodifferential operator

1. Introduction Since the papers of Chazarain [2] and Helffer and Robert [3], the semiclassical spectral asymptotics have been investigated in numerous works (cf. the monographs [2, 8, 11, 12]). Let us assume that for h > 0 the differential operators Ah = a w (h, x, hD) are self-adjoint in the Hilbert space L2 (Rd ) and a(h, x, ξ ) =
hn an (x, ξ ) holds with sufficiently regular symbols an (n = 0, 1, . . .). If the real number E satisfies E < lim inf a0 (x, ξ ), |x|+|ξ |→∞

(1.1)

then the spectrum of Ah is discrete in ]−∞; E] for h small and we can ask whether the behaviour of the counting function N (Ah , E) (i.e. the number of eigenvalues smaller than E counted with multiplicities) for h → 0 is given by the semiclassical Weyl formula N (Ah , E) = (2π h)−d |Ea0 | + O(hµ−d ),  where µ > 0 and |Ea0 | = a0 (x,ξ )<E dx dξ is the volume of the region Ea0 = a0−1 (] − ∞; E[) = {(x, ξ ): a0 (x, ξ ) < E}.

(1.2)

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LECH ZIELINSKI

Applying a development of the microlocal analysis based on the ideas of Hörmander [5], (1.2) can be proved with µ = 1 under the hypothesis a0 (x, ξ ) = E ⇒ ∇a0 (x, ξ ) = 0.

(1.3)

It is natural to ask two related questions: (1) what can be said when the condition (1.3) is violated? (2) what can be said for operators with nonsmooth coefficients? These questions were first considered by Ivrii [9] for Schrödinger operators and the recent work Ivrii [10] contains a large scope of results obtained by a development of the multi-scale analysis (the condition (1.3) corresponds to the microhyperbolicity condition of the monograph [8]). Our paper presents a different method of treating the problem and the aim is to obtain estimates expressed exclusively in terms of the volume of suitable regions determined by a0 in the phase space. More precisely instead of (1.2) we consider the formula N (Ah , E) = (2π h)−d |Ea0 | + O(h−d )REa0 (hµ ), where REa0 (hµ ) =

 dx dξ.

(1.4)

(1.4 )

E−hµ
In the case of smooth coefficients the formula (1.4) was proved with µ < 23 (cf. Levendorskii [12]) by using a development of the approximate spectral projector method of Tulovskii and Shubin [13] with the improvement of Hörmander [6]. In this paper we show that (1.4) still holds for µ ∈ [ 23 ; 1[ without any additional hypotheses on the symbol. More precisely: our proof is based on an idea of integrations by parts for which there makes no difference whether ∇a0 or ∇ 2 a0 vanishes or not. This seems to us the main difference between our method and the approach of the multi-scale analysis (the results of [10] depend strongly on the properties of the Hessian matrix ∇ 2 a0 ). The question of the minimal regularity of coefficients is the only reason to state our results under the ellipticity assumption. Indeed, this assumption is used only to recover the asymptotic behaviour of the counting function in the case of nonsmooth coefficients from the asymptotics associated with a suitable family of regularized operators. In the case of operators with smooth coefficients we start by Theorem 2.2 and recover (1.4) in a standard way (i.e. in the framework of tempered variation symbolic calculus of [4]). Moreover, the regularity properties of coefficients are really essential only for x such that (x, ξ ) ∈ Ea 0 with a certain E0 > E (i.e. the behaviour of coefficients for other values of x can be more general). For this reason, instead of tempered variation hypotheses, we assume that coefficients are bounded for sake of simplicity.

SEMICLASSICAL WEYL FORMULA

3

Finally we remark that under the hypothesis (1.3) we can obtain (1.2) with µ = 1 for operators with coefficients which have first order derivatives Hölder continuous of exponent r (for a certain r > 0, cf. [14]) or more general: continuous with the modulus of continuity |log |x − y||−1 (cf. [10]). Formulation of the results. Let r¯ ∈]0; 1] and let U be an open set of Rd . We denote B r¯ (U) = {a ∈ L∞ (U): ∃C > 0, |a(x) − a(y)|
C|x − y|r¯ for x, y ∈ U} and for r ∈]1; 2], B r (U) = {a ∈ L∞ ∩ C 1 (U): ∂xj a ∈ B r−1 (U) for j = 1, . . . , d}. Let m ∈ N∗ and for ν, ν¯ ∈ Nd , |ν|, |¯ν |
m we consider real-valued functions aν,¯ν = aν¯ ,ν ∈ L∞ (Rd ). We define  aν,¯ν (x)ξ ν+¯ν (1.5) a0 (x, ξ ) = |ν|,|¯ν |
m

and assume that a0 (x, ξ )  c|ξ |2m − C holds for certain constants C, c > 0. For h > 0 let Ah be the quadratic form defined for ϕ, ψ ∈ C0m (Rd ) by  (aν,¯ν (hD)ν ϕ, (hD)ν¯ ψ), Ah [ϕ, ψ] =

(1.6)

(1.7)

|ν|,|¯ν |
m

where (·, ·) is the scalar product of L2 (Rd ) and (hD)ν = (−ih)|ν| ∂ ν /∂x ν . The ellipticity hypothesis (1.6) ensures the fact that Ah is bounded from below and its closure defines a self-adjoint operator Ah . Our main result is THEOREM 1.1. Let Ah be defined as above, let r ∈]1; 2] and aν,¯ν ∈ B r (Rd ). If E satisfies (1.1) and h0 > 0 is small enough, then the spectrum of Ah is discrete in ] − ∞; E] for h ∈]0; h0 ]. Moreover (1.4) holds if µ < 2r/(2 + r). A general plan is the following. In Section 2 we define regularized operators Ph and use the Fourier transform to express suitable functions f˜h (Ph ) by means of µ the evolution group Ut = eitPh / h . In Section 3 we describe an approximation of Ut giving a pseudodifferential approximation of f˜h (Ph ) with correct asymptotic properties. The correct asymptotic behaviour of the approximation is obtained in Section 4 by means of simple integrations by parts. A similar idea is used to estimate the difference between f˜h (Ph ) and the approximation, completing the proof in Section 6. At the end we remark that combining the results of [15] obtained under the condition (1.3), it is possible to formulate the following generalization of Theorem 1.1:

4

LECH ZIELINSKI

THEOREM 1.2. Let Ah be as above and denote CEa = {(x, ξ ) ∈ R2d : a(x, ξ ) = E

and ∇ξ a(x, ξ ) = 0}.

Let r¯ ∈]0; 1], r ∈]1; 2] and assume aν,¯ν ∈ B r¯ (Rd ) ∩ B r (U), where U is an open set satisfying CEa ⊂ U × Rd . If E satisfies (1.1), then (1.4) holds with µ < min{¯r , 2r/(2 + r)}.

2. Regularized Problem The hypothesis µ < 2r/(2 + r) allows us to find a real number δ satifying µ < rδ

and

1/2 < δ < 2/(2 + r).

(2.1)

The conditions (2.1) ensure that κ = 1 − δ − µ/2 > 1 − δ(1 + r/2) > 0.  Let γ ∈ C0∞ (Rd ) satisfy γ (x) dx = 1 and γ (−x) = γ (x). We define  aν,¯ν (y)γ (h−δ (x − y))h−δd dy, aν,¯ν ,h (x) =  ph (x, ξ ) = aν,¯ν ,h (x)ξ ν+¯ν .

(2.2)

(2.3) (2.4)

|ν|,|¯ν |
m

Then according to the results of [14], the hypothesis aν,¯ν ∈ B r ensures |aν,¯ν (x) − aν,¯ν ,h (x)|
Chδr , |∂xα aν,¯ν ,h (x)|
Cα (1 + hδ(r−|α|) ) and we can find h0 > 0 such that the operators  Ph± = (hD)ν aν,¯ν ,h (x)(hD)ν¯ ± hµ (I − h2 )m

(2.5) (2.6)

(2.7)

|ν|,|¯ν |
m

have discrete spectrum in ] − ∞; E] and Ph−
Ah
Ph+ for h ∈]0; h0 ]. Therefore due to the min–max principle, we obtain Theorem 1.1 from THEOREM 2.1. If (2.1) holds, then (1.4) holds with N (Ah , E) replaced by N (Ph± , E). We fix E  ∈ R such that ph (x, ξ )  E  and the half-line [E  ; ∞[ contains the spectrum σ (Ph± ) for h ∈]0; 1]. Let 1I[E  ;E] : R → {0, 1} denote the characteristic function of the interval [E  ; E] ⊂ R. Thus 1I[E  ;E] (Ph± ) denotes the spectral projector of Ph± on [E  ; E] and N (Ph± , E) = tr1I[E  ;E] (Ph± ).

5

SEMICLASSICAL WEYL FORMULA

If bh is a polynomially bounded function of (x, ξ ) ∈ R2d , then we write Bh = bh (x, hD) if Bh is the operator acting on ϕ ∈ C0∞ (Rd ) according to the formula   dξ ixξ/ h e bh (x, ξ ) dy e−iyξ/ h ϕ(y). (Bh ϕ)(x) = (2π h)d Further on l ∈ C0∞ (R2d ) is a fixed real-valued function such that l = 1 on Ea01 = a0−1 (] − ∞; E1 [) for a certain E1 > E. We denote Lh = l(x, hD) and L∗h denotes the adjoint of Lh in L2 (Rd ). Then it suffices to prove THEOREM 2.2. If (2.1) holds, then  ± ∗ tr Lh 1I]−∞;E[ (Ph )Lh = ph (x,ξ )<E

dx dξ l(x, ξ )2 + O(h−d )REa0 (hµ ). (2.8) (2π h)d

Indeed, since ph = a0 + O(hδr )(1 + |ξ |2 )m holds due to (2.5), there is h0 > 0 p such that l = 1 on Eh = ph−1 (] − ∞; E[) for h ∈]0; h0 ] and  p dx dξ = |Ea0 | + REa0 (Chδr ). |E h | = ph (x,ξ )<E

Therefore (2.8) implies tr Lh 1I]−∞;E[ (Ph± )L∗h = (2π h)−d |Ea0 | + O(h−d )REa0 (hµ )

(2.8 )

and the following estimate N (Ph± , E) = tr Lh 1I[E  ;E] (Ph± )L∗h + O(1) was proved in [14]. Thus we can conclude that (2.8) implies (1.4). The main idea of the proof of Theorem 2.2 is similar as in [14]. Further on we drop the index h. In particular, we write simply L, p instead of Lh , ph and we abbreviate Ph± = P . We introduce a family of functions f˜h ∈ C ∞ being an approximation of 1I[E  ;E] similarly as in [14], i.e. satisfying 1I[E  +hµ ;E−hµ ]
f˜h−
1I[E  ;E]
f˜h+
1I[E  −hµ ;E+hµ ]

(2.9)

and the estimates of derivatives |(f˜h± )(k) (λ)|
Ck h−kµ . Clearly it suffices to prove (2.8) with f˜h± (P ) instead of 1I[E  ;E] (P ). ∞ µ Let us abbreviate f˜h± = f˜h and introduce fh (t) = −∞ dλ e−iλt/ h f˜h (λ). Then µ denoting Ut = eitP / h we find the expression  ∞ dt f (t) tr LUt L∗ tr Lf˜h (P )L∗ = µ h 2π h −∞ and in Section 3 we describe a sequence of pseudodifferential operators    itp/ hµ n ◦ t qN¯ ,n (x, hD), QN¯ ,t = e 0
n
N¯

6

LECH ZIELINSKI

being an approximation of LUt . We require QN¯ ,t |t=0 = L, hence  t  t d ˜ N¯ ,t−τ Uτ LUt − QN¯ ,t = dτ (QN¯ ,t−τ Uτ ) = − dτ Q dτ 0 0 with iQN¯ ,t P d . Q˜ N¯ ,t = QN¯ ,t − dt hµ

(2.10)

It is easy to see that |fh (t)|
Cn (1 + |t|)−n holds for every n ∈ N and to obtain (2.8) it suffices to prove PROPOSITION 2.3. Let N¯ ∈ N. Then we can find QN¯ ,t satisfying   ∞ dt dx dξ 2 ∗ fh (t) tr QN¯ ,t L = l + O(h−d )REa0 (hµ ) µ d −∞ 2π h p<E (2π h) and

  tr Q˜ N¯ ,t−τ Uτ L∗ 
hκ N¯ /2−5d (2 + |t|)CN¯ ,

(2.11)

(2.12)

˜ N¯ ,t is given by (2.10), κ = 1 − δ − µ/2 > 0, CN¯ > 0 is a constant large where Q enough, (t, τ ) ∈ R∗ × R and τ/t ∈ [0; 1].

3. Description of the Approximation Let s ∈ R. We define S s writing b ∈ S s if and only if b = (bh )h∈]0;h0 ] is a family of functions such that for every α, β ∈ Nd one has the estimate |∂ξα ∂xβ bh (x, ξ )|
Cα,β h−s−|β|δ (1 + |ξ |)C−|α| .

(3.1)

Due to (2.6) we have p, ∂xj p ∈ S 0 , ∂xα p ∈ S |α|δ−rδ ⊂ S |α|δ−µ if |α|  2 and it is easy to check that  p˜ ν (x)(hD)ν = p(x, ˜ hD) (3.2) P = |ν|
2m

holds with p(x, ˜ ξ) = p − p˜ ∈ S −1 ,


|ν|
2m

p˜ ν (x)ξ ν belonging to S 0 . Moreover

∂xj (p − p) ˜ ∈ S −1+δ(2−r) ⊂ S 2δ−1−µ .

(3.3)

We denote

 −µ/2 ∂ξj p(x, ξ ) if j ∈ {1, . . . , d},  h if j = 0, pj (x, ξ ) = 1   hδ−µ ∂ p(x, ξ ) if − j ∈ {1, . . . , d} x|j |

(3.4)

7

SEMICLASSICAL WEYL FORMULA

s s writing b ∈ S(0) if and only if b = (bh )h∈]0;h0 ] ∈ S s and and we define S(0) s S(N supp bh ⊂ supp l for all h ∈]0; h0 ]. Next we define ) by induction with respect
s to N ∈ N writing b ∈ S(N +1) if and only if b = −d
j
d bj pj holds with some s s bj ∈ S(N ) for j ∈ {0, ±1, . . . , ±d}. It is easy to see that b ∈ S(N ) if and only if

b=



¯

¯

bβ,β¯ h−|β|µ/2+(δ−µ)|β| (∇ξ p)β (∇x p)β

¯ d β,β∈N ¯ |β|+|β|
N

with some symbols bβ,β¯ ∈ S s satisfying supp bβ,β¯ ⊂ supp l and

(∇ξ p)β =



¯

(∇x p)β =

(∂ξj p)βj ,

1
j
d

¯

(∂xj p)βj

1
j
d

for β = (β1 , . . . , βd ), β¯ = (β¯1 , . . . , β¯d ) ∈ Nd . Thus it is clear that s+˜s s s˜ ˜ ˜ b ∈ S(N ) , b ∈ S ⇒ bb ∈ S(N ) ,

b∈

s ˜ S(N ), b

∈

S(s˜N˜ )

⇒ bb˜ ∈

(3.5)

s+˜s S(N . +N˜ )

(3.6)

Let j, k ∈ {1, . . . , d}. We observe that ∂ξj pk = h−µ/2 ∂ξj ∂ξk p ∈ S µ/2 , ∂xj p−k = hδ−µ ∂xj ∂xk p ∈ S (µ−δ)+2δ−µ = S δ and due to δ  1/2 > µ/2 we have also ∂xj pk = h−µ/2 ∂xj ∂ξk p ∈ S µ/2 ⊂ S δ , ∂ξj p−k = hδ−µ ∂ξj ∂xk p ∈ S µ−δ ⊂ S µ/2 . s it is easy to see that Using this observation and the definition of S(1) s+µ/2

s ⇒ ∂ξj b ∈ S(0) b ∈ S(1)

s+δ , ∂xj b ∈ S(1)

and by induction with respect to N ∈ N we obtain s+µ/2

s+δ s b ∈ S(N ) ⇒ ∂ξj b ∈ S(N −1) , ∂xj b ∈ S(N ) .

(3.7)

s Moreover for N ∈ N
\ {0} we define Sˇ(N ) as the set of symbols b that can be s written in the form b = 0
j
d bj pj with bj ∈ S(N −1) for j ∈ {0, . . . , d}. Finally for a smooth function bt (x, ξ ) we denote    h|α|−µ µ µ α itp/ hµ α ∂ (b e ∂ p) ˜ . (3.8) P˜N¯ bt = e−itp/ h ∂t (bt eitp/ h ) − t x |α|+1 ξ α!i ¯ |α|
N

8

LECH ZIELINSKI

s LEMMA 3.1. Let κ = 1 − δ − µ/2 > 0. If b is independent of t and b ∈ S(N ), then  t n bn (3.9) P˜N¯ b = 0
n
N¯ s−κ s−nκ ˇ s−κ ˇ s−κ ˇ s−nκ holds with b0 ∈ S(N ) ⊂ S(N +1) , b1 ∈ S(N +2) and bn ∈ S(N +n) ⊂ S(N +n+1) for ¯ n ∈ {2, . . . , N}. Proof. In the first step we check that



˜ + b0 = ih−µ (p − p)b

1
|α|
N¯

h|α|−µ α s−κ ∂ (b∂xα p) ˜ ∈ S(N ). α!i |α|+1 ξ

(3.10)

To begin we observe that s |α|  2 ⇒ hδ|α|−µ ∂xα p˜ ∈ S 0 ⇒ bhδ|α|−µ ∂xα p˜ ∈ S(N )

(3.11) s+|α|µ/2

˜ ∈ S(N ) and applying (3.7) |α| times we obtain ∂ξα (bhδ|α|−µ ∂xα p) s−κ|α| h|α|−µ ∂ξα (b∂xα p) ˜ = h|α|(κ+µ/2) ∂ξα (bhδ|α|−µ ∂xα p) ˜ ∈ S(N )

, i.e. (3.12)

if |α|  2. Next we observe that s−κ ˜ ∈ S µ−1 ⊂ S −κ ⇒ h−µ (p − p)b ˜ ∈ S(N h−µ (p − p) )

(3.13)

and to complete the proof of (3.10) it remains to show s−κ ˜ ∈ S(N h1−µ ∂ξj (b∂xj p) ).

(3.14)

s δ−µ ∂xj (p − p) ˜ ∈ S δ−1 ⊂ S 0 due However bhδ−µ ∂xj p = bp−j ∈ S(N +1) and h to (3.3), hence s bhδ−µ ∂xj p˜ ∈ S(N +1) .

(3.15)

s−κ ˜ ∈ S(N Using (3.15) and (3.7) we find h1−δ ∂ξj (bhδ−µ ∂xj p) ) , i.e. (3.14) holds. Next we consider n ∈ {1, . . . , N¯ }. Then we have  β¯ bn = h|α|−(n+1)µ bβ,β¯ (∇ξ p)β ∂ξ (b∂xα p) ˜ (3.16) β
|n|
|α|
N¯ ¯ β+β
α

with bβ,β¯ ∈ S 0 for β, β¯ ∈ Nd . More precisely: in the case |β| < n we can express ¯ p where α(k) ¯ ∈ Nd are such bβ,β¯ as a linear combination of terms 1
k
n−|β| ∂ξα(k)
¯ = α, that |α(k)| ¯  2 for k ∈ {1, . . . , n − |β|} and β + β¯ + 1
k
n−|β| α(k) ¯ + 2(n − |β|)
|α|, i.e. implying |β| + |β| ¯
|α| + |β| − 2n. |β|

(3.17)

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SEMICLASSICAL WEYL FORMULA

In the case |β| = n the symbols bβ,β¯ are constant and (3.17) still holds. As before we consider first the case |α|  2 and observe that (3.7) ensures β¯

¯

s−κ|α|

˜ ∈ S(N ) hκ|α|+|β|µ/2 ∂ξ (bhδ|α|−µ ∂xα p)

.

(3.18)

However, (3.17) implies the inequality ¯ κ|α| + |β|µ/2
(1 − δ − µ/2)|α| + (|α| + |β| − 2n)µ/2 = (1 − δ)|α| − nµ + |β|µ/2, hence (3.18) implies β¯

s−κ|α| ˜ −|β|µ/2 (∇ξ p)β ∈ S(N h(1−δ)|α|−nµ+|β|µ/2 ∂ξ (bhδ|α|−µ ∂xα p)h +|β|)

(3.19)

s−κ|α| s−nκ s−nκ and |β|
n
|α| ⇒ S(N +|β|) ⊂ S(N +n) ensures bn ∈ S(N +n) if n  2. s−2κ If n = 1 then the terms of (3.16) corresponding to |α|  2 belong to S(N +1) ⊂ s−κ Sˇ(N +2) . It remains to consider the case n = |α| = 1 and to check s−κ ˜ ξj p ∈ Sˇ(N h1−2µ b∂xj p∂ +2) .

˜ ξj p = hκ pj bhδ−µ ∂xj p˜ and (3.20) follows from (3.15). But h1−2µ b∂xj p∂

(3.20) 2

PROPOSITION 3.2. Let κ = 1 − δ − µ/2 > 0 and N ∈ {0, 1, . . . , N¯ }. Then we can find  t n qN◦¯ ,n (3.21(N )) qN¯ ,N,t = 0
n
N −κ , such that qN◦¯ ,0 = l, qN◦¯ ,1 ∈ S(0) −(n−1)κ/2

qN◦¯ ,n ∈ Sˇ(n) and

for n ∈ {2, . . . , N}



P˜N¯ qN¯ ,N,t =

t n q˜N◦¯ ,N,n

(3.22(N ))

(3.23(N ))

N
n
N +N¯

holds with −nκ/2 q˜N◦¯ ,N,n ∈ Sˇ(n+1)

¯ for n ∈ {N, . . . , N + N}.

(3.24(N ))

0 , then Lemma 3.1 with b = l Proof. If N = 0, qN¯ ,0,t = qN◦¯ ,0 = l ∈ S(0) −κ ◦ . Next we assume that the ensures (3.23(0)), (3.24(0)) and moreover q˜N¯ ,0,0 ∈ S(0) statement of Proposition 3.2 holds for a given N
N¯ − 1 and using the induction hypothesis (3.23(N )) to express P˜N¯ qN¯ ,N,t we find

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LECH ZIELINSKI

P˜N¯ qN¯ ,N +1,t = P˜N¯ (t N +1 qN◦¯ ,N+1 ) + P˜N¯ qN¯ ,N,t

 = t N (N + 1)qN◦¯ ,N +1 + q˜N◦¯ ,N,N + t N +1 P˜N¯ qN◦¯ ,N+1  + t n q˜N◦¯ ,N,n . N +1
n
N +N¯

To cancel the term with t N and to obtain (3.23(N + 1)) we take qN◦¯ ,N +1 = −q˜N◦¯ ,N,N /(N + 1), −κN/2

belonging to Sˇ(N +1) by the induction hypothesis (3.24(N )), i.e. (3.22(N + 1)) −κ ◦ ˜N◦¯ ,0,0 ∈ S(0) . holds. Moreover in the first step we obtain qN,1 ¯ = −q −κN/2 ◦ ∈ S(N +1) , we can express t N +1 P˜N¯ q ◦¯ Using Lemma 3.1 with b = q ¯ N ,N+1

as claimed, i.e. (3.24(N + 1)) holds due to

−κN/2 − κ max{1, n}
−κ(N + 1 + n)/2.

N ,N+1

2

4. Integrations by Parts To show the estimate (2.11) we introduce the notation  dx dξ itp(x,ξ )/ hµ e b(x, ξ ) Jt (b) = (2π h)d

(4.1)

for b ∈ C0∞ (R2d ) and begin by LEMMA 4.1. Let n ∈ N. s (a) If b ∈ S(N ) then  t k Jt (bk,n ) t n Jt (b) =

(4.2(n))

0
k
n s holds with some bk,n ∈ S(max{0,N −n}) for k ∈ {0, . . . , n}. s s ˇ (b) If b ∈ S(N ) , then (4.2(n)) holds with bk,n ∈ S(max{0,N −1−n}) . Proof. (a) Reasoning by induction we assume that the statement holds for a given N ∈ N. In order to show that the statement still holds for N + 1 instead
s s of N we consider b ∈ S(N . Then b = −d
j
d bj pj with bj ∈ S(N ) and +1) t n Jt (b0 p0 ) = t n Jt (b0 ) can be expressed in a suitable way due to the induction hypothesis. Then the integration by parts gives  ihµ/2 ∂ξj bj if j ∈ {1, . . . , d}, (4.3) tJt (bj pj ) = Jt (b˜j ) with b˜j = ihδ ∂x−j bj if − j ∈ {1, . . . , d}. s Since b˜j ∈ S(N ) holds due to (3.7), (4.3) implies (4.2(1)). Reasoning by induction with respect to n ∈ N∗ we obtain (4.2(n)).

11

SEMICLASSICAL WEYL FORMULA


s s µ/2 ˜ (b) If b ∈ Sˇ(N ∂ξj bj ∈ 0
j
d bj pj with bj ∈ S(N ) and bj = ih +1) , then b = s S(N −1) . To complete the proof of (4.2(1)) it remains to use the expression of tJt (b0 ) described in (a). If n  2 then to obtain (4.2(n)) as claimed it suffices to use moreover the expression of t n−1 Jt (b˜j ) described in (a). 2 Proof of the estimate (2.11). To begin we observe that   dx dξ itp/ hµ ∗ e q l = t k Jt (qN◦¯ ,k l). tr QN¯ ,t L = ¯ ¯ N , N ,t d (2π h) ¯ 0
k
N

0 Using Lemma 4.1 with n = k − 1 and b = qN◦¯ ,k l ∈ Sˇ(k) for k ∈ {2, . . . , N¯ }, we find  t k Jt (bN¯ ,k ) tr QN¯ ,t L∗ = Jt (l 2 ) + 1
k
N¯ 0 . Changing the order of integrals we find with some bN¯ ,k ∈ S(0)  ∞  dx dξ 2 ˜ dt 2 fh (t)Jt (l ) = l (fh ◦ p) µ (2π h)d −∞ 2π h

(4.4)

and supp (f˜h − 1I[E  ;E] ) ⊂ [E  − hµ ; E  + hµ ] ∪ [E − hµ ; E + hµ ] ensures (f˜h − 1I[E  ;E] )(p(x, ξ )) = 0 ⇒ E − hµ
p(x, ξ )
E + hµ , i.e. the integral (4.4) can be written as  dx dξ 2 p l + O(h−d )REh (hµ ), d (2π h) p<E where

(4.5)

 p

REh (hµ ) =

E−hµ
dx dξ
REa0 (2hµ ).

Next we use t k fh (t) = (ihµ )k (Fhµ f˜h(k) )(t) to write   ∞ dt dx dξ k f (t)t J (b ) = b (ihµ )k (f˜h(k) ◦ p) ¯ h t N ,k µ d N¯ ,k 2π h (2π h) −∞

(4.6)

0 and we complete the proof of (2.11) observing that hkµ f˜h(k) (λ) = O(1), bN¯ ,k ∈ S(0) and k  1 ⇒ supp f˜h(k) ⊂ [E − hµ ; E + hµ ] ∪ [E  − hµ ; E  + hµ ] allow us to p 2 estimate (4.6) as before by O(h−d )REh (hµ ).

To show (2.12) we denote V = {(t, τ ) ∈ R∗ × R: τ/t ∈ [0; 1]} and for s ∈ R s s writing b ∈ S˜(0) if and only if b = (bh,t,τ )(h,t,τ )∈]0;h0 ]×V satisfies we define S˜(0) β |∂ξα ∂x,y bh,t,τ (x, ξ, y)|
Cα,β h−s−|β|δ (1 + |t|)C

for every (α, β) ∈ Nd × N2d and supp bh,t,τ ⊂ (supp l) × Rd .

(4.7)

12

LECH ZIELINSKI

s ˜s For N ∈ N we define S˜(N ) by induction with respect to N writing b ∈ S(N +1) if and only if  bj (x, ξ, y)pj (x, ξ ) (4.8) b(x, ξ, y) = −d
j
d s holds with some bj ∈ S˜(N ) for j ∈ {0, ±1, . . . , ±d}. s If b ∈ S˜(N then the formula )  dy dξ i(x−y)ξ/ h+i(t−τ )p(x,ξ )/ hµ (Opt,τ (b)ϕ)(x) = e bh,t,τ (x, ξ, y)ϕ(y) (4.9) (2π h)d s ˜ s+µN/2 and a result defines the operators Opt,τ (b) of trace class. Indeed, S˜(N ) ⊂ S(0) proved in [14] allows us to find a constant C¯ > 0 such that ¯

Opt,τ (b)tr
h−s−µN/2−5d (2 + |t|)C .

(4.10)

˜ ξ )) and Similarily as in [14] we write QN¯ ,t P = Opt,0 (qN¯ ,N¯ ,t (x, ξ )p(y, Q˜ N¯ ,t = Opt,0 ((P˜N¯ qN¯ ,N¯ ,t )(x, ξ ) + rN¯ ,t (x, ξ, y)) with −itp(x,ξ )/ hµ

rN¯ ,t (x, ξ, y) = e

(N¯ + 1)



1

(4.11)

¯

dσ (1 − σ )N r˜N¯ ,σ,t (x, ξ, y),

(4.12)

0

where r˜N¯ ,σ,t (x, ξ, y)   hN¯ +1−µ  α itp/ hµ α p(x (q e )(x, ξ )∂ ˜ + σ (y − x), ξ ) . ∂ = N¯ ,N¯ ,t x N¯ +1 α! ξ i ¯ |α|=N +1

(4.13) Using the form of qN¯ ,n,t in (4.13) we obtain the expressions of the form considered in the proof of Lemma 3.1 and applying (3.12), (3.19) with |α| = N¯ + 1 we find
N¯ n ◦ −κ N¯ ◦ t rN¯ ,n with rN,n ∈ S˜(n) . Thus similarly as in [14] we can conclude rN¯ ,t = 2n=0 ¯ −κ N¯ /2 such that COROLLARY. There exist q˜N◦¯ ,n ∈ S˜(n)

Q˜ N¯ ,t−τ =



t n Opt,τ (q˜N◦¯ ,n ).

(4.14)

0
n
2N¯ s If Yτ ∈ B(L2 (Rd )) for τ ∈ R and b ∈ S˜(N ) , then we denote

Jt,τ (b, Y ) = tr Opt,τ (b)Uτ Yτ

(4.15)

SEMICLASSICAL WEYL FORMULA

and using this notation we can write (4.14) in the form  t n Jt,τ (q˜N◦¯ ,n , L∗ ). tr Q˜ N¯ ,t−τ Uτ L∗ =

13

(4.16)

0
n
2N¯

To complete the proof of Theorem 2.2 it suffices to prove s PROPOSITION 4.2. Let b ∈ S˜(N ) and n ∈ N. If K = K(N, n) ∈ N is large enough, then  Jt,τ (bk,n , Yk,n ) (4.17) t n Jt,τ (b, L∗ ) = 1
k
K

holds with some operators Yτ,n,k ∈ B(L2 (Rd )) satisfying the estimates Yt,k,n 
s (2 + |t|)CN,n and symbols bk,n ∈ S˜(max{0,N −n}) for k ∈ {1, . . . , K}. Indeed, it remains to show (2.12) and using Proposition 4.2 with b = q˜N◦¯ ,n ∈ −κ N¯ /2 we can write (4.16) in the form S˜(n)   Jt,τ (bn,k , Yn,k ) tr Q˜ N¯ ,t−τ Uτ L∗ = 0
n
2N¯ 1
k
KN¯ −N¯ κ/2 with bn,k ∈ S˜(0) and Yt,n,k 
(2 + |t|)CN¯ for k ∈ {1, . . . , KN¯ }. Then writing |Jt,τ (bk,n , Yk,n )|
Opt,τ (b)tr Yτ,n,k  we obtain (2.12) using ¯ (4.10) with N = 0 and s = −Nκ/2.

5. Auxiliary Properties The proof of Proposition 4.2 is given in Section 6 and it uses the following idea: the integrations by parts of Lemma 4.1 can be considered as a special case of the formal identity tr [Z, Tj ] = 0. Further on the operator Tj is the operator of multiplication by −hµ/2−1 xj if j ∈ {1, . . . , d} and Tj = ihδ ∂x−j if −j ∈ {1, . . . , d}. Moreover, we are going to use the notation P0 = I and Pj = [ih−µ P , Tj ] if j ∈ {±1, . . . , ±d}. Therefore we have Pj = p˜ j (x, hD) with  −µ/2 ∂ξj p(x, ˜ ξ ) if j ∈ {1, . . . , d}, h if j = 0, p˜ j (x, ξ ) = 1  δ−µ ˜ ξ ) if − j ∈ {1, . . . , d}. h ∂x−j p(x,

(5.1)

(5.2)

For s ∈ R we define  s writing B ∈  s if and only if B = (Bh )h∈]0;h0 ] holds with Bh = bh (x, hD) and b = (bh )h∈]0;h0 ] ∈ S s .

14

LECH ZIELINSKI

LEMMA 5.1. Let κ = 1 − δ − µ/2 as before. Then  Pk Bj,k [ih−µ P , Pj ] = hκ

(5.3)

−d
k
d

holds with some Bj,k ∈  0 . Proof. To begin we observe that for j, k ∈ {1, . . . , d} the symbols ˜ bj,k = −hδ−µ/2 ∂xk ∂ξj p, bj,−k = ∂ξk ∂ξj p, ˜

b−j,k = −h2δ−µ ∂xk ∂xj p, ˜

b−j,k = hδ−µ/2 ∂xk ∂ξj p˜

belong to S 0 . Therefore for j ∈ {±1, . . . , ±d} we have the following expression of the Poisson brackets   ˜ p˜ j } = h1−µ (∂xk p∂ ˜ ξk p˜ j − ∂ξk p∂ ˜ xk p˜ j ) = hκ p˜ k bj,k . {h1−µ p, 1
k
d

1
±k
d

˜ ξα p˜ j and h|α|−µ−κ ∂ξα p∂ ˜ xα p˜ j belong to S 0 for |α|  2, we can Since h|α|−µ−κ ∂xα p∂ find bj ∈ S 0 such that  [ih−µ P , Pj ] − hκ (p˜ k bj,k )(x, hD) = hκ bj (x, hD). 1
±k
d

 0 we obtain (5.3) with Taking Bj,k = bj,k (x, hD) for k =  ((p˜ k bj,k )(x, hD) − Pk Bj,k ). Bj,0 = bj (x, hD) +

2

1
±k
d

Let L(S(Rd )) denote the space of linear continous operators on S(Rd ) (the Schwartz space of rapidly decreasing functions). It is not difficult to check that Ut ∈ L(S(Rd )) for every t ∈ R and moreover t → Ut is strongly continuous R → L(S(Rd )) (i.e. t → Ut ϕ is continuous R → S(Rd ) for every ϕ ∈ S(Rd )). Further on we use the notation Yt (B) = Ut BUt∗

(5.4)

for t ∈ R and B ∈ L(S(R )). We define Y as follows: we write Y ∈ Y if and only if there is N ∈ N such that Y = (Yt )t∈R has the form N   dwsw,k,t Ytsw,k,1 (Bk,1 ) · · · Ytsw,k,N (Bk,N ), (5.5) Yt = d

k=1

[0;1]N

where Bk,n ∈  0

for (k, n) ∈ {1, . . . , N}2 ,

sw,k,n ∈ [−N ; N ] for (w, k, n) ∈ [0; 1]N × {1, . . . , N}2 , sw,k,t ∈ C and |sw,k,t |
(2 + |t|)N for (w, k, t) ∈ [0; 1]N × {1, . . . , N} × R.

15

SEMICLASSICAL WEYL FORMULA

We observe that taking sw,k,n = sw,k,t = 1 we obtain (Yt (Bk,1 ))t∈R ∈ Y and it is not difficult to see that Y is an algebra containing the operators  0 . ◦ ∈ Y such that LEMMA 5.2. There exist Yj,k  ◦ Pk Yt,j,k . Yt (Pj ) = Ut Pj Ut∗ = Pj + hκ

(5.6)

−d
k
d

Proof. Using (5.3) we can express  d Yt (Pj ) = Yt ([ih−µ P , Pj ]) = hκ Yt (Pj )Yt (Bj,k ). dt −d
k
d

(5.7)

We denote E = s−C(R; L(S(Rd ))2d+1 ) and write  ∈ E if and only if : t → (t) = (j (t))−d
j
d ∈ L(S(Rd ))2d+1 holds with t → j (t) strongly continuous R → L(S(Rd )) for j ∈ {0, ±1, . . . , ±d}. Let us introduce a linear operator M: E → E by the formula  t  dτ k (τ )Yτ (Bj,k ), (5.8) (M)j (t) = 0

−d
k
d

where Yτ (Bj,k ) are as in (5.7). Then integrating (5.7) we find that ˜ (t) = hκ (M)(t) + hκ (t)

(5.9)

˜ ˜ j (t))−d
j
d given by holds with (t) = (j (t))−d
j
d , (t) = ( j (t) = Yt (Pj ) − Pj ,  t  ˜ j (t) = Pk dτ Yτ (Bj,k ).  −d
k
d

0

Iterating (5.9) we find



(t) = hN κ (M N )(t) +

˜ h(n+1)κ (M n )(t)

(5.10)

0
n
N −1

and using the induction with respect to n ∈ N∗ we check that  t  dτ k (τ )Yτ,j,k,n (M n )j (t) = 0

−d
k
d

˜ holds with Yt,j,k,n belonging to Y. Thus taking (t) as before we find  ˜ = Pk Y˜t,j,k,n (M n )(t) −d
k
d

with some Y˜t,j,k,n belonging to Y. It is clear that to obtain (5.6) it suffices to write (5.10) with N  1 + 1/κ, which ensures the fact that h(N −1)κ (M N )j (t) belong to Y. 2

16

LECH ZIELINSKI

◦ LEMMA 5.3. If B ∈  0 , then there exist Y˜j,k ∈ Y such that  ◦ Pk Y˜t,j,k . [B, Yt (Pj )] = hκ

(5.11)

−d
k
d

Proof. Using (5.6) we can express the left hand side of (5.11) as  ◦ ◦ ([B, Pk ]Yt,j,k + Pk [B, Yt,j,k ]). [B, Pj ] + hκ −d
j
d ◦ ∈ Y by the formula Since h−κ [B, Pj ] ∈  0 we can define Y˜j,0  ◦ ◦ ◦ Y˜t,j,0 = h−κ [B, Pj ] + [B, Yt,j,0 ]+ [B, Pk ]Yt,j,k −d
j
d ◦ ◦ and we obtain (5.11) taking Y˜t,j,k = [B, Yt,j,k ] for k ∈ {±1, . . . , ±d}.

LEMMA 5.4. If Y ∈ Y, then there exist Yj,k ∈ Y such that  Pk Yt,j,k . [Yt , Pj ] = hκ

2

(5.12)

−d
k
d

Proof. In the first step we consider the case Yt = Yt (B) with B ∈  0 . Using (5.6) to express Y−t (Pj ) we find  [Yt (B), Pj ] = Ut [B, Y−t (Pj )]Ut∗ = hκ Yt (Pk )Y˜t,j,k , (5.13) −d
k
d ◦ Ut∗ belonging to Y. Using (5.6) once more to express Yt (Pk ) with Y˜t,j,k = Ut Y˜−t,j,k in (5.13), we obtain (5.12) in the case Yt = Yt (B). If Yt = Yt (B1 )Yt (B2 ) then using the result of the first step it is easy to express

[Yt , Pj ] = [Y (B1 ), Pj ]Yt (B2 ) + Yt (B1 )[Yt (B2 ), Pj ] as claimed in the general statement. Similarly we find that the statement holds for finite products Yt = Yt (B1 )Yt (B2 ) · · · Yt (Bn ) and consequently for all Y ∈ Y. 2 + ˜− , Yt,j,k ∈ Y such that LEMMA 5.5. Let Y ∈ Y. Then there exist Y˜t,j  + − + hκ t Pk Y˜t,j,k . [Yt , Tj ] = Y˜t,j

(5.14)

−d
k
d

Proof. Using (5.1) and (5.6) we find  1 dsYts (Pj ) = tPj + hκ t Yt (Tj ) − Tj = t 0

with some Yt,j,k belonging to Y.

 −d
k
d

Pk Yt,j,k

(5.15)

17

SEMICLASSICAL WEYL FORMULA

As before we consider first the case Yt = Yt (B) with B ∈  0 . Using (5.15) we can express [Yt (B), Tj ] = Ut [B, Y−t (Tj )]Ut∗ as  Ut ([B, Pk Y−t,j,k ])Ut∗ . (5.16) Yt ([B, Tj ]) − tYt ([B, Pj ]) − hκ t −d
k
d

Since [B, Tj ], [B, Pj ] ∈  0 , the first and the second term of (5.16) belong to Y. Moreover Ut ([B, Pk Y−t,j,k ])Ut∗ = Yt ([B, Pk ])Y t,j,k + Yt (Pk )Y t,j,k ,

(5.17)

where Y t,j,k = Ut Y−t,j,k Ut∗ and Y t,j,k = Ut [B, Y−t,j,k ]Ut∗ belong to Y. Thus to complete the proof of (5.14) in the case Yt = Yt (B) it suffices to write the second term of (5.17) with Yt (Pk ) expressed as in Lemma 5.2. Finally we observe that using Lemma 5.3 we can reason similarly as at the end of the proof of Lemma 5.4, i.e. deduce (5.14) for a general Y ∈ Y from the special 2 case Yt = Yt (B). + − , Yt,j,k ∈ Y such that LEMMA 5.6. Let Y ∈ Y. Then there exist Yt,j  + − + hκ t Pk Ut Yt,j,k . tPj Ut Yt = [Ut Yt , Tj ] + Ut Yt,j

(5.18)

−d
k
d

Proof. Let Yt,j,k be as in (5.15). Then we have [Ut , Tj ]Yt = (Yt (Tj ) − Tj )Ut Yt = tPj Ut Yt + hκ t



Pk Ut Y t,j,k (5.19)

−d
k
d

with Y t,j,k = Ut∗ Yt,j,k Ut belonging to Y. Next we use (5.14) to write  + − + hκ t Yt (Pk )Ut Y˜t,j,k Ut [Yt , Tj ] = Ut Y˜t,j

(5.20)

−d
k
d

and to obtain (5.18) we observe that [Ut Yt , Tj ] is the sum of (5.19) and (5.20), i.e. 2 it remains to write Yt (Pk ) in (5.20) as in the assertion of Lemma 5.2. Let us define Sˆ s writing b = (bh )h∈]0;h0 ] ∈ Sˆ s if and only if for every n ∈ N we have (bh (x, ξ )(1+|x|2 +|ξ |2 )n )h∈]0;h0 ] ∈ S s . Finally we define Yˆ as the set of operators Y ∈ Y of the form (5.5) where Bk,k , sw,k,t , sw,k,k are as before and for every k ∈ {1, . . . , N} there is k  ∈ {1, . . . , N} such that Bk,k = (bh,k,k (x, hD))h∈]0;h0 ] holds with (bh,k,k )h∈]0;h0 ] ∈ Sˆ 0 . COROLLARY. Lemma 5.5 still holds if Y is replaced by Yˆ and the estimate Yt 
(2 + |t|)C ˆ holds if (Yt )t∈R ∈ Y.

18

LECH ZIELINSKI

6. End of the Proof It remains to prove Proposition 4.2. We will prove s ∗ ˆ PROPOSITION 6.1. Let b ∈ S˜(N ) and Y ∈ Y. If KN ∈ N is large enough, then we can write  Jt,τ (bk+ , Yk+ ) + hκ tJt,τ (bk− , Yk− ), (6.1) tJt,τ (b, Y ) = 1
k
KN + − + s ˜s ˆ where bk− ∈ S˜(N ) , bk ∈ S(N −1) , Yk , Yk ∈ Y for k ∈ {1, . . . , KN }.

Let us check that Proposition 4.2 follows from Proposition 6.1. To begin we observe that (6.1) still holds without Jt,τ (bk− , Yk− ) (i.e. we can assume that bk− = 0). Indeed, having written (6.1) we can use Proposition 6.1 again to express tJt,τ (bk− , Yk− ) and as the result we find a formula similar to (6.1) with h2κ instead ¯ of hκ . After n¯ iterations we can replace hκ by hnκ and for n¯  1/κ we have s− nκ ¯ s s nκ ¯ − h bk ∈ S˜(N ) ⊂ S˜(N −1) . Therefore we obtain (4.17) with n = 1, bk,1 ∈ S˜(N −1) , Yk,1 ∈ Yˆ for k ∈ {0, . . . , K1 }. Using the induction with respect to n ∈ N we s ˆ obtain (4.17) with bn,k ∈ S˜(max{0,N −n}) and Yn,k ∈ Y. The proof of Proposition 6.1 uses s ˜+ ˜ s LEMMA 6.2. If bj ∈ S˜(N ) then there exists bj ∈ S(N ) such that

[Opt,τ (bj ), Tj ] = (t − τ )Opt,τ (bj pj ) + Opt,τ (b˜j+ ),

(6.2)

where we denoted (bj pj )(x, ξ, y) = bj (x, ξ, y)pj (x, ξ ). s ˜− ˜ s LEMMA 6.3. If bj ∈ S˜(N ) then there exists bj ∈ S(N +1) such that

Opt,τ (bj )Pj = Opt,τ (bj pj + hκ b˜j− ).

(6.3)

The proofs of Lemma 6.2 and 6.3 will be described at the end of the section. Proof of Proposition 6.1. Using the induction we assume that the statement holds for a fixed N ∈ N and we are going to show that the analogical statement ˆ holds with N + 1 instead of N . Thus we consider
tJt,τ (b, Y ) with Y ∈ Y and s s b ∈ S(N +1) . Let bj ∈ S˜(N ) be such that b = −d
j
d bj pj . Since the term tJt,τ (p0 b0 , Y ) = Jt,τ (b0 , tY ) can be counted between the terms of the right-hand side of (6.1), it remains to fix j ∈ {±1, . . . , ±d} and consider tJt,τ (bj pj , Y ). To begin we write tJt,τ (bj pj , Y ) in the form (t − τ )tr Opt,τ (bj pj )Uτ Yτ + τ tr Opt,τ (bj pj )Uτ Yτ .

(6.4)

Then (6.2) allows us to write the first term of (6.4) as tr [Opt,τ (bj ), Tj ]Uτ Yτ − Jt,τ (b˜j+ , Y )

(6.5)

19

SEMICLASSICAL WEYL FORMULA

and (6.3) allows us to write the second term of (6.4) as tr Opt,τ (bj )τ Pj Uτ Yτ − hκ τ Jt,τ (b˜j− , Y ).

(6.6)

Moreover due to (5.18), the first term of (6.6) can be written as  tr Opt,τ (bj )Pk Uτ Yτ,j,k tr Opt,τ (bj )[Uτ Yτ , Tj ] + Jt,τ (bj , Yj+ ) + hκ τ −j
k
d

(6.7)

and summing up the first term of (6.5) and (6.7) we find tr [Opt,τ (bj ), Tj ]Uτ Yτ + tr Opt,τ (bj )[Uτ Yτ , Tj ] = tr [Opt,τ (bj )Uτ Yτ , Tj ] = 0. It is clear that the terms Jt,τ (b˜j+ , Y ), Jt,τ (bj , Yj+ ) and hκ τ Jt,τ (b˜j− , Y ) = hκ tJt,τ ( τt b˜j− , Y ) (the terms appearing respectively in (6.5), (6.7) and (6.6)) have the form of terms indicated in the right hand side of (6.1) (with N + 1 instead of N ). To complete the proof we express the remaining terms of (6.7) in the form hκ τ tr Opt,τ (bj )Pk Uτ Yτ,j,k = hκ tJt,τ (bj,k , Yj,k ), s where bj,k ∈ S˜(N +1) are found by using Lemma 6.3 with bk instead of bj .

2

Proof of Lemma 6.2. We consider first the case j > 0. Then the integral kernel of [Opt,τ (bj ), Tj ] is  dξ µ (yj − xj )hµ/2−1 ei(x−y)ξ/ h+i(t−τ )p(x,ξ )/ h bj (x, ξ, y) (x, y) → d (2π h) (6.8) and integrating by parts, we can rewrite it as  dξ µ ei(x−y)ξ/ h+i(t−τ )p(x,ξ )/ h (h−µ/2 (t − τ )∂ξj pj bj + ihµ/2 ∂ξj bj ). d (2π h) s Thus (6.2) holds with b˜j− = ihµ/2 ∂ξj bj ∈ S˜(N ). To show (6.2) with j < 0 we observe that the differentiation of (4.9) gives

Dj Opt,τ (bj ) = Opt,τ ((h−1 ξj + (t − τ )h−µ ∂xj p)bj + i∂xj bj ).

(6.9)

However writing (4.9) with Dj ϕ instead of ϕ we can integrate by parts in the integral with respect to y and obtain the standard result Opt,τ (bj )Dj ϕ = Opt,τ ((h−1 ξj + i∂yj bj )ϕ. s Using (6.9) and (6.10) we obtain (6.2) with b˜j− = ihδ (∂xj + ∂yj )bj ∈ S˜(N ).

(6.10) 2

20

LECH ZIELINSKI

Proof of Lemma 6.3. Since Pj = Pj∗ = p˜ j (x, hD)∗ is a differential operator of order 2m − 1, we can use (6.9) to compose Opt,τ (b) with (hD)ν and we obtain the standard composition formula Opt,τ (b)Pj = Opt,τ (bj p˜ j + hκ b˜j ),

(6.11)

where (bj p˜ j )(x, ξ, y) = bj (x, ξ, y)p˜ j (y, ξ ), b˜j (x, ξ, y) =



h|α|−κ α ∂ b (x, ξ, y)∂ξα p˜ j (y, ξ ). |α| α! y j i 1
|α|
2m−1

(6.12)

s It is easy to check that b˜j ∈ S˜(N ) and to complete the proof it remains to show the s  ˜ ˜ existence of bj ∈ S(N +1) such that

Opt,τ (bj p˜ j ) = Opt,τ (bj pj + hκ b˜j ).

(6.13)

To show (6.13) we write p˜ j (y, ξ ) − p˜ j (x, ξ ) =



 (xk − yk )

1
k
d

1

dσ ∂xk p˜ j (x + σ (y − x), ξ ). (6.14)

0

s We have bj,0 = hδ−1 (p˜¯j − pj )bj ∈ S˜(N ) due to (3.3) and using (6.14) we find  (xk − yk )h−δ bj,k bj (pj − pj ) = h1−δ bj,0 + 1
k
d s where bj,k ∈ S˜(N ˜ j ∈ S˜ δ . ) holds due to ∂xk p Integrating by parts as in the proof of Lemma 6.2, we find

Opt,τ ((xk − yk )bj,k ) = Opt,τ (h1−µ (t − τ )∂ξk pbj,k + ih∂ξk bj,k ). −κ To complete the proof we observe that h1−µ ∂ξk ph−δ bj,k = hκ bj,k pk ∈ S˜(N +1) and −κ 1−δ 2 h ∂ξk bj,k ∈ S˜(N ) due to (3.7).

References 1. 2. 3. 4. 5.

Chazarain, J.: Spectre d’un hamiltonien quantique et mécanique classique, Comm. Partial Differential Equations (1980), 595–644. Dimassi, M. and Sjöstrand, J.: Spectral Asymptotics in the Semiclassical Limit, London Math. Soc. Lecture Note Ser. 268, Cambridge Univ. Press, Cambridge, 1999. Helffer, B. and Robert, D.: Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques, Ann. Inst. Fourier, Grenoble 31(3) (1981), 169–223. Helffer, B. and Robert, D.: Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Anal. 53(3) (1983), 246–268. Hörmander, L.: The spectral function of an elliptic operator, Acta Math. 121 (1968), 173–218.

SEMICLASSICAL WEYL FORMULA

6. 7. 8. 9. 10.

11. 12. 13. 14. 15.

21

Hörmander, L.: On the asymptotic distribution of the eigenvalues for pseudodifferential operators in Rn , Arkiv Mat. 17 (1979), 297–313. Hörmander, L.: The Analysis of Linear Partial Differential Operators, Vol. 1–4, SpringerVerlag, Berlin, 1983, 1985. Ivrii, V.: Microlocal Analysis and Precise Spectral Asymptotics, Springer-Verlag, Berlin, 1998. Ivrii, V.: Sharp spectral asymptotics for operators with irregular coefficients, Internat. Math. Res. Notices (2000), 1155–1166. Ivrii, V.: Sharp spectral asymptotics for operators with irregular coefficients, I: Pushing the limits and II: Domains with boundaries and degenerations, Comm. Partial Differential Equations 28 (2003), 83–102 and 103–128. Levendorskii, S. Z.: Asymptotic Distribution of Eigenvalues of Differential Operators, Math. Appl., Kluwer Acad. Publ., Dordrecht, 1990. Robert, D.: Autour de l’approximation semi-classique, Birkhäuser, Boston, 1987. Tulovskii, V. N. and Shubin, M. A.: On the asymptotic distribution of eigenvalues for pseudodifferential operators in Rn , Math. USSR Sbornik 21 (1973), 565–583. Zielinski, L.: Semiclassical Weyl formula for elliptic operators with non-smooth coefficients, In: Oper. Theory Adv. Appl. 153, Birkhäuser, Basel, 2004, pp. 321–344. Zielinski, L.: Semiclassical distribution of eigenvalues for elliptic operators with Hölder continuous coefficients, Colloq. Math. 99 (2004), 157–174.

Mathematical, Physics, Analysis and Geometry (2006) 9: 23–52 DOI: 10.1007/s11040-005-9000-x

#

Springer 2006

The PDEs of Biorthogonal Polynomials Arising in the Two-Matrix Model MARCO BERTOLA1,2 and BERTRAND EYNARD3 1

Centre de recherches mathe´matiques, Universite´ de Montre´al, CP 6128, succ. centre ville, Montre´al, Que´bec, Canada H3C 3J7. e-mail: [email protected] 2 Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke W., Montre´al, Que´bec, Canada H4B 1R6. 3 Service de Physique The´orique, CEA/Saclay Orme des Merisiers, F-91191, Gif-sur-Yvette Cedex, France. e-mail: [email protected] (Received: 8 August 2005; in final form: 9 August 2005) Abstract. The two-matrix model can be solved by introducing biorthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of biorthogonal polynomials (called windows) satisfy polynomial ODEs as well as deformation equations (PDEs) and finite difference equations (DE) which are all Frobenius compatible and define discrete and continuous isomonodromic deformations for the irregular ODE, as shown in previous works of ours. In the one matrix model an explicit and concise expression for the coefficients of these systems is known and it allows to relate the partition function with the isomonodromic tau-function of the overdetermined system. Here, we provide the generalization of those expressions to the case of biorthogonal polynomials, which enables us to compute the determinant of the fundamental solution of the overdetermined system of ODE + PDEs + DE. Mathematics Subject Classifications (2000): 34M50, 33C50, 35Q15. Key words: biorthogonal polynomials, two-matrix models.

1. Introduction The two-matrix model and biorthogonal polynomials have recently witnessed a renewed interest due to the hope that the asymptotics could be found from a RiemannYHilbert approach similar to that used in one-matrix models [1Y9]. Our recent papers have shown some unexpectedly rich algebraic structure associated to the biorthogonal polynomials for two polynomial potentials [8, 10, 11]. We recall that the two-matrix model is defined as the ensemble of pairs of Hermitian matrices of size N, with the measure 1
1 ½trV1 ðM1 ÞþV2 ðM2 Þ
M1 M2  e
h dM1 dM2 ZN

ð1:1Þ

24

MARCO BERTOLA AND BERTRAND EYNARD

where dM1dM2 is the standard Lebesgue measure (product of Lebesgue measures of all real components of M1 and M2), V1 and V2 are two polynomials (called the potentials) of degree d1 + 1 and d2 + 1: V1 ðxÞ ¼

dX 1 þ1 k¼1

uk k x ; k

V2 ð yÞ ¼

dX 2 þ1 k¼1

vk k y k

and Z N is the partition function Z 1 Z N ¼ e

h ½trV1 ðM1 ÞþV2 ðM2 Þ
M1 M2  dM1 dM2

ð1:2Þ

ð1:3Þ

The two-matrix model has a wide range of applications in physics, in particular Euclidean 2d gravity i.e., the statistical physics of a random surface [12Y14]. The 2-matrix model can be solved with the help of two families of biorthogonal polynomials
n ðxÞ ¼ xn þ . . . ;

n ðyÞ ¼ yn þ . . .

which are such that Z Z 1 dxdy
n ðxÞm ðyÞe

h ½V1 ðxÞþV2 ðyÞ
xy ¼ hn nm :

ð1:4Þ

ð1:5Þ

It was shown in [10], that a sequence of d2 + 1 consecutive p n’s (or d1 + 1 consecutive s n’s) obeys a closed ODE with respect to the Fspectral parameter` x (or y) 0 1 0 1 N
d2 ð xÞ N
d2 ð xÞ d B C B C .. .. ð1:6Þ

h @ A ¼ D1 ðxÞ@ A; . . dx N N ð xÞ N ð xÞ as well as deformation equation (PDEs) with respect to infinitesimal variations of the coefficients of the potentials V1, V2 and difference equations with respect to the choice of the consecutive elements (DE). The aim of this article is to provide for an explicit expression of said ODE and PDEs, which is similar in spirit to the work done in [15, 16] (the DE are already very simple and can be found in [10]). Here we follow the same logical lines that were followed in our previous [7]. We should remark that an explicit expressions for the ODE is given in [10] but is not convenient for practical purposes. For instance, it is not at all obvious how to compute the trace of the matrix D1 ðxÞ from the expression written in [10]. N We expect that all the spectral invariants of D1(x) will play an essential roˆle in the future developments on the basis that, in the one-matrix model [3, 7], the corresponding invariants allow us to put in direct relation the partition function of the model with the isomonodromic tau-function in the sense of [6, 17, 18].

25

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

Indeed it is well known that the one-matrix model is a subcase of the two-matrix model where one of the potentials is Gaussian. The expressions contained in this paper allow us to easily compute the first invariants (traces) of the ODEs and PDEs, thus allowing us to evaluate the determinant of a fundamental system of solutions of the Frobenius compatible system. Hopefully, they will prove useful for the higher invariants as well. NOTATIONS 1.1. In the following we will often consider semi-infinite matrices. If M is a semi-infinite matrix we denote

Y M+ the strictly upper triangular part of M, Y Mj the strictly lower triangular part of M, Y Mk the kth diagonal above the main diagonal if k U 0, and theªkªth Y Y Y Y

diagonal below the main diagonal if k r 0, Mþ0 ¼ Mþ þ 12 M0 ; M
0 ¼ M
þ 12 M0 ; MU the upper triangular part (including the main diagonal), Mr the lower triangular part (including the main diagonal).

We introduce the following semi-infinite matrices: L is the shift matrix ij ¼ iþ1; j :

ð1:7Þ

 is the projector on the subspace spanned by the first N + 1 basis vectors N

Nþ1-times
zfflfflfflfflfflffl}|fflfflfflfflfflffl{   ¼ diag 1; 1; . . . ; 1 ; 0; . . . :

ð1:8Þ

N

We have the simple properties (L has only a right inverse)  ¼  ; N

N
1

t  ¼ 1
; 0

t t  ¼   ; N

ð1:9Þ

N
1

t ¼ 1;

 ¼ 0; 0

t  ¼ 0: 0

ð1:10Þ

Finally, the notation for the folded matrices is slightly changed from [10] in order to make it more consistent (in particular, the location of certain over/underscripts). 2. Biorthogonal Polynomials: Definitions and Properties It is convenient to introduce the wave functions and their FourierYLaplace transforms, related to the biorthogonal polynomials (1.5) by the relations n ð xÞ

1 1 ¼ pffiffiffiffiffi
n ðxÞe

h V1 ðxÞ ; hn

1 1 n ð yÞ ¼ pffiffiffiffiffi n ð yÞe

h V2 ðyÞ hn

ð2:1Þ

26

MARCO BERTOLA AND BERTRAND EYNARD

n

ð yÞ ¼

Z dxe

1
h xy

n ð xÞ;

 n ð xÞ ¼

Z

1

dye
h xy n ðyÞ:

ð2:2Þ

The unspecified contour of integration can be chosen in d1 homologically distinct ways for the functions n ðyÞ ðd2 ways for n ðxÞÞ, as explained in [8, 19]. We also introduce the semi-infinite wave vectors   t t ¼ ð ; ; . . . Þ ; ¼ ð  ;  ; . . . Þ ;  ¼ ; ; . . . ;   0 1 0 1 0 1 1 1 1   ð2:3Þ  ¼ 0 ; 1 ; . . . : 1

In order to express the ODEs that these wave functions satisfy, we define the following Fwindows_ of consecutive wave functions of dimension d2 + 1 (resp. d1 + 1): ¼ð N N

¼

N
d2 ; . . . ;



; N
1

NÞ

N
1 ;

t

t

 ¼ ðN
d1 ; . . . ; N
1 ; N Þ

; 

;...; N

ð2:4Þ

N

Nþd1
1

  N  ¼  N
1 ;  N ; . . . ;  Nþd
1 :

;

2



N



ð2:5Þ

N

The two windows in each of the pairs ;  and ;  (each constituted of N N windows of the same size) are called Fdual windows`, as explained in [10]. It is shown in [8, 10, 11] that Y similarly to the ordinary orthogonal polynomials Y the semi-infinite wave vectors obey x ¼ Q; 1

1



h@x  ¼ P; 1

1

y ¼ Pt ; 1

y ¼ Pt ;

1

1



h@y  ¼ Qt ; 1

1

xðxÞ ¼ Q

1

1


h@y  ¼ Q; 1

1

ð2:6Þ

1


h@y  ¼ Pt ; 1

ð2:7Þ

1

where the semi-infinite matrices P and Q are of finite band size. Component-wise the above equations translate into x

n ð xÞ

¼  ðnÞ

nþ1

þ

d2 X

k ðnÞ

n
k ð xÞ;

ð2:8Þ

k¼0

yn ð yÞ ¼  ðnÞnþ1 þ

d1 X

k ðnÞn
k ð yÞ;

ð2:9Þ

k¼0

xn ð xÞ ¼  ðn
1Þn
1 þ

d2 X k¼0

k ðn þ kÞnþk ðxÞ;

ð2:10Þ

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

y

ð yÞ ¼  ðn
1Þ n

þ n
1

d1 X

k ðn þ kÞ

nþk

ð yÞ

27

ð2:11Þ

k¼0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where  ðnÞ ¼ hnþ1 =hn . Since they implement the multiplication and differentiation by the spectral parameters x or y, the matrices P and Q obey the Heisenberg relation ½P; Q ¼
h1

ð2:12Þ

and they have the properties [8, 10, 11] Pþ ¼ V10 ðQÞþ ;

Q
¼ V20 ðPÞ
;

P0 ¼ V10 ðQÞ0 ;

ð2:13Þ

Q0 ¼ V20 ðPÞ0 ;

ð2:14Þ

Pn;n
1 ¼  ðn
1Þ ¼ V10 ðQÞn;n
1

h Qn;nþ1 ¼  ðnÞ ¼ V20 ðPÞn;nþ1

h

n ;  ðn
1Þ

nþ1 :  ðnÞ

ð2:15Þ ð2:16Þ

The relations (2.7) can be Ffolded` [10] onto the finite windows defined in (2.4, 2.5) using the recursion relations (2.6) so as to give finite-dimensional ODEs:

h@x  ¼ D1 ðxÞ; N

N

N

N

N N

h@y  ¼  D 2 ðyÞ;




h@y  ¼ D2 ð yÞ; N

N

N

N

ð2:17Þ

N N

h@x  ¼  D1 ð xÞ;


N

N

where D1 ðxÞ and D1 ðxÞ ðresp. D2 ðyÞ and D2 ðyÞÞ are matrices of dimension d2 + 1 N N (resp. d1 + 1) whose entries are polynomials in x (resp. y) of degree at most d1 (resp. d2). They enjoy the duality property ([10], Thm 4.2): N

N

N

D1 ðxÞA ¼ AD1 ðxÞ; N

N

N

N

N

D2 ðyÞB ¼ BD2 ðyÞ;

ð2:18Þ

N

N

where A and B are the ChristoffelYDarboux pairing matrices 0

0... B  N B d2 ð N Þ . . . A¼B .. @ . 0

0 1 ð N Þ .. . d2 ðN þ d2
1Þ

1
 ðN
1Þ C 0 C C; .. A . 0

ð2:19Þ

28

MARCO BERTOLA AND BERTRAND EYNARD

0

0... B N B d1 ð N Þ . . . B¼B .. @ .

1
 ðN
1Þ C 0 C C: .. A .

0 1 ð N Þ .. . d1 ðN þ d2
1Þ

0

N

ð2:20Þ

0

N

The two C-D pairing matrices A and B are the only nonvanishing blocks in the semi-infinite matrix commutators:




 t resp: P ;  ; ð2:21Þ Q;  N
1

N
1

where  is the projector defined in (1.8). These matrices play an essential roˆle N
1 in the computation of the relevant spectral statistics of the model inasmuch as they enter directly in the kernels used in the Dyson-like formulas [20]

N N 0 0 0 0 ðx
xÞK11 ðx; x Þ ¼ ðx ÞAðxÞ ¼ ðx Þ Q;  ðxÞ; N

1

N
1 1



N N 0 0 0 0 ðy
yÞK22 ðy; y Þ ¼ ðy ÞBð yÞ ¼ ðy Þ Q;  ðyÞ: N

1

N
1 1

ð2:22Þ

2.1. DEFORMATION EQUATIONS (PDEs) The deformations of the biorthogonal polynomials (derivations w.r.t the coefficients of the two potentials) are represented by semi-infinite finite band matrices: h@uK  ¼ UK ;
1

1

h
@uK  ¼
UK ; 1

1

h@uK  ¼ UKt ;
1

1

h
@uK  ¼
UKt ; 1

1

h@vJ  ¼
VJ ;
1

1

h@vJ  ¼ VJ ;
1

ð2:23Þ

1

h@vJ  ¼
VJt ;
1

1

h@vJ  ¼ VJt ;
1

ð2:24Þ

1

where the semi-infinite matrices UK, VJ are related to the matrices P, Q by the equations UK :¼


1 K Q ; K þ0

1 VJ :¼
PJ
0 : J

ð2:25Þ

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

29

which can be folded onto the windows using Equation (2.6) as explained in [10] h@uK  ¼ U K ðxÞ ;
N

N

h@vJ  ¼
VJ ð xÞ ;


N

N

N

N N

N

h@uK  ¼
 U K ðxÞ;


N

NN

e K ð yÞ; h@uK  ¼  U


e J ð yÞ; h@vJ  ¼
V


e K ð yÞ; h@uK  ¼
U


e J ðyÞ;
h@vJ  ¼ V

N

N

N

ð2:26Þ

N N


h@vJ  ¼  V J ðxÞ;

N N

N

N

N

N

N

ð2:27Þ

N

N

where U K ðxÞ and U K ðxÞ are matrices of dimension d2 + 1 whose entries are polyN N nomials in x of degree at most k, and V J ðxÞ and V J ð xÞ are matrices of dimension N d2 + 1 whose entries are polynomials in x of degree at most Jd1 (and similar statement for the tildematrices with d1 replacing d2 and viceversa). They satisfy [10]: N

N

N

N

U K ðxÞA þ A U K ðxÞ þ
h@uK A ¼ 0;

ð2:28Þ

N

N

N

N

N

V J ðxÞA þ A V J ðxÞ

h@vJ A ¼ 0:

ð2:29Þ

N

The expressions of the matrices UJ, VK in terms of P and Q can be found with the same notation in [8] but are essentially well known since [21] in a slightly different context and with different notation.

2.2. DEFORMATIONS WITH RESPECT TO The --deformations of the biorthogonal polynomials (derivations w.r.t. -) are also represented by semi-infinite matrices of finite band size which we can write as follows:
2 @
h  ¼ H ; h 1

1

e:
2 @
h  ¼ H h 1

1

ð2:30Þ

The relationship with the matrices P and Q is easy to derive but since we could not literature (except in special cases [9]), we give it hereafter. From R xyfind it in the e
h 1 ðxÞt1 ðyÞ ¼ 1 we promptly obtain et ¼ 0: H
QP þ H

ð2:31Þ

30

MARCO BERTOLA AND BERTRAND EYNARD

Moreover, we have   e
V2 ðPt Þ ; ðH
V1 ðQÞÞþ ¼ 0 ¼ H

ð2:32Þ

þ

upper

lower zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ et
V2 ðPÞ ¼ 0; H
V1 ðQÞ þ ðV1 ðQÞ
QP þ V2 ðPÞÞ þ H

ð2:33Þ

  e
V2 ðPt Þ ¼
1 ðV1 ðQÞ
QP þ V2 ðPÞÞ ; ðH
V1 ðQÞÞ0 ¼ H 0 0 2

ð2:34Þ

ðH
V1 ðQÞÞ
¼ ðQP
V1 ðQÞ
V2 ðPÞÞ
;

ð2:35Þ

  e
V2 ðPt Þ ¼ ðPt Qt
V1 ðQt Þ
V2 ðPt ÞÞ ; H


ð2:36Þ




so that we have 1 H ¼ V1 ðQÞþ þ ðQP þ V1 ðQÞ
V2 ðPÞÞ0 þ ðQP
V2 ðPÞÞ
2 ¼ V1 ðQÞþ0 þ ðQPÞ
0
V2 ðPÞ
0 ;

ð2:37Þ

e ¼ V2 ðPt Þ þ ðPt Qt Þ
V1 ðQt Þ : H þ0
0
0

ð2:38Þ

It is also interesting to consider another derivation of (2.37), from scaling properties. Consider the change of variables x 7! x and y 7! y , it gives hÞnm ð Þnþ1 hn ðfuK g; fvJ g;
Z x  ¼ dxdyn
n ; fuK g; fvJ g;
h   m

 m





ve1 ðxÞ þ ve2 ðyÞ
xy ; y e
h ; fuK g; fvJ g;
h e

ð2:39Þ

where Ve1 ðxÞ :¼  V1

x 

;


 y Ve2 ð yÞ :¼  V2 ;

e
h ¼ 
h:

ð2:40Þ

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

This immediately implies the identities x     
n x; uK 1
K ; vJ 1
J  ; 
h ¼ n
n ; fuK g; fvJ g;
h ; 
    1
J  y 1
K n n y; uK  ; vJ  ; 
h ¼ n ; fuK g; fvJ g;
h ; hn



  uK 1
K ; vJ 1
J  ; 
h ¼ ð Þnþ1 hn ðfuK g; fvJ g;
hÞ;

    x; uK 1
K ; vJ 1
J  ; 
h x  nþ1 n
1 ¼  2
2 n ; fuK g; fvJ g;
h ;      n y; uK 1
K ; vJ 1
J  ; 
h
 nþ1 y n
1
¼ 2  2 n ; fuK g; fvJ g;
h :

31

ð2:41Þ ð2:42Þ ð2:43Þ

n

In infinitesimal form, from @j¼ ¼1 and @ ¼ ¼1 we obtain ! X X nb
1 þ x@x
ðK
1ÞuK @uK þ vJ @vJ þ
h@
h ðxÞ ¼ 0; 2 1 K J

ð2:44Þ

ð2:45Þ

ð2:46Þ

! X 1 þ nb X ðJ
1ÞvJ @vJ þ
h@
h ðxÞ ¼ 0; þ uK @uK

2 1 K J

ð2:47Þ

! X 1 þ nb X
ðK
1ÞuK @uK þ vJ @vJ þ
h@
h ðyÞ ¼ 0;
2 1 K J

ð2:48Þ

! X X nb
1 uK @uK
ðJ
1ÞvJ @vJ þ
h@
h ðyÞ ¼ 0; þ y@y þ 1 2 K J

ð2:49Þ

where nb :¼ diagð0; 1; 2; 3; 4; . . .Þ. Using Equation (2.23) we immediately obtain that
H ¼

X X hðnb
1Þ

PQ
ðK
1ÞuK UK
vJ VJ 2 K J

¼


X hðnb þ 1Þ X
þ uK UK þ ðJ
1ÞvJ VJ 2 K J

ð2:50Þ

32

MARCO BERTOLA AND BERTRAND EYNARD

X hðnb þ 1Þ X
vJ VJ þ ðK
1ÞuK UK þ 2 X X hðnb
1Þ
¼ ðJ
1ÞvJ VJ ;
PQ
uK UK
2

et ¼

H

which, using Equations (2.15), (2.16), is identical to Equation (2.37). We also get the identity X X
PQ ¼ JvJ VJ þ KuK UK

h b n: J

ð2:51Þ

ð2:52Þ

K

This last relation is nothing but the expression of the scaling properties of the biorthogonal polynomials and will prove itself very useful later on. 2.3. LADDER RECURSION RELATIONS (DEs) We give here only the main statements without proofs and refer the reader to [10] for further details. Let us introduce the sequence of companion-like matrices aðxÞ N and aN ðxÞ of size d2 + 1 2 3 0 1 0 0 .. 6 7 6 0 7 0 . 0 að xÞ :¼ 6 ð2:53Þ 7; N U d2 ; N 4 0 5 0 0 1
d2 ð N Þ ðx
0 ð N ÞÞ 1 ðNÞ   
 ðNÞ ðNÞ ðNÞ 2 3 x
0 ð N Þ 1 0 0 6 ðN
1Þ 7 6
1 ðNþ1Þ 7 .. 6 N 0 . 07  ð N
1 Þ 6 7; NU1; að xÞ :¼ 6 ð2:54Þ 7 .. 6 7 . 0 0 15 4
d2 ðNþd2 Þ 0 0 0  ðN
1Þ N

and also the analogous sequence of matrices bðyÞ and bðyÞ of size d1 + 1 N

2

0

1

6 6 bð yÞ :¼ 6 N 4

0 0

0 0 

2 6 6 N 6 bð yÞ :¼ 6 6 6 4


d1 ð N Þ ðNÞ y
0 ð N Þ  ðN
1Þ
1 ðNþ1Þ  ðN
1Þ

.. .


d1 ðNþd1 Þ  ðN
1Þ

0 .. . 0


1 ð N Þ ðNÞ

1

0

0

..

0 0

0 0

0 .

0

3

0 1

7 7 7; 5

ðy
0 ð N ÞÞ ðNÞ

NUd1 ;

ð2:55Þ

3

7 7 07 7; 7 7 15 0

NU1:

ð2:56Þ

33

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

The equations in (2.6) imply the following lemma: N

N

LEMMA 2.1. The sequences of matrices a; b and a; b implement the shift N [ N N N + 1 and N [ N j 1 in the windows of quasi-polynomials and FourierYLaplace transforms in the sense that N

Nþ1

N

N

Nþ1

N

að xÞ ¼  ðxÞ;

ð xÞ ¼  ð xÞ a;

bð yÞ ¼  ðyÞ;

ð yÞ ¼  ð yÞ b

NN

Nþ1

NN

Nþ1

ð2:57Þ ð2:58Þ

and in general  ¼

Nþj

 ¼

Nþj

N

N

Nþj Nþj
1

a    a;

¼ 

b    b;

¼ 

Nþj
1

Nþj
1

NN

N

NN

a

Nþj Nþj
1

b

N

   a; N

   b;

ð2:59Þ ð2:60Þ

N

where ;  here denote a window in any of the FourierYLaplace transforms defined in Equation (2.2). It was proven in [10] that the differentialYdeformationYrecursion relations (2.17), (2.26), (2.57) are compatible and admit a common fundamental solution. 2.4.

FOLDING

Important remark. From this point on, we focus on the wave functions

n(x) and their dual n ð xÞ, but everything being said can be immediately extended to the wave functions n(y), n ð yÞ by interchanging the roˆles of the spectral parameters, the potentials and the matrices P and Q. We recall that the notion of Ffolding_ is the following: to express any quasipolynomial n(x) as a linear combination of d2 + 1 fixed consecutive quasipolynomials Njj, j = 0, . . . d2 with polynomial coefficients. We now provide a way of computing the folding which is different from (but equivalent to) the one used in [10] where the ladder matrices were used instead. Let N (i.e., a window) be fixed; we seek to describe the folding of the infinite ð xÞ onto the window at N by means of a single matrix F ð xÞ of size wave vector  1 N V  (d2 + 1) and with polynomial entries such that

8n ;

n ð xÞ ¼

N X

F n;k ð xÞ

k ð xÞ:

ð2:61Þ

k¼N
d2 N

In fact, it is more convenient to think of F as a V  V matrix with only a vertical N band of width d2 + 1 of nonzero entries (with column index in the range N j

34

MARCO BERTOLA AND BERTRAND EYNARD

d2, . . . , N). In order to describe as explicitly as possible the matrix F , we first N introduce a convenient notation to express the band matrices P, Q: Q ¼  þ

d2 X

j tj ;

P ¼ t  þ

j¼0

d1 X

j j ;

ð2:62Þ

j¼0

where L is the shift matrix defined in (1.7), and j, bj,  here above denote the diagonal matrices   j :¼ diagj ð0Þ; . . . ; j ðnÞ; . . . ; j :¼ diag j ð0Þ; . . . ; j ðnÞ; . . . ;  :¼ diagð ð0Þ; . . . ;  ðnÞ; . . .Þ: ð2:63Þ It is known that the existence of biorthogonal polynomials satisfying (1.5) is equivalent to saying that (n) m 0, On. Therefore, we assume that  is invertible. Equations (2.13) imply that d2

d

d2 t ¼ vd2 þ1 ðt  Þ 2 ;

ð2:64Þ

or, more transparently, d2 ðnÞ ¼ vd2 þ1  ðn
1Þ     ðn
d2 Þ:

ð2:65Þ

Let us now define the following semi-infinite matrices: 1 


1 

d2

ðQ
xÞ; vd2 þ1   B :¼ 1

t 
1 ðQ
xÞ:

A :¼ 1


0

ð2:66Þ

We note that A is strictly upper triangular while B is strictly lower triangular. When acting on the semi-infinite wave vector ð xÞ by definition we have 1 ðQ
xÞ ¼ 0 so that 1
 ð2:67Þ ¼ Bþ   ¼ A; 1

1

1

0

1

Notice that 1j A and 1j B are invertible because they are upper (resp. lower) triangular with ones on the main diagonal. Notice also that the matrix Q j x has a left inversej 1 vd2 þ1

 d2 ð1
AÞ
1 
1  ðQ
xÞ ¼ 1

ð2:68Þ

j The fact that Q j x has left inverse (Q j x)j1 L is not in contradiction with (Q j x) V = j1 0. Indeed ((Q j x)j1 L (Q j x)) V m (Q j x)L ((Q j x) V) because both left-hand and righthand sides are infinite sums and the order of summation cannot be exchanged.

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

35

and a right inverse ðwrite 1
B ¼ 0 þ t 
1 ðQ
xÞÞ and multiply by L on the left and by 0 (1 j B)j1Lt j1 on the right):   ðQ
xÞð1
BÞ
1 t 
1 ¼ 1 ð2:69Þ but they do not coincide (one is upper and the other is lower triangular). 2.4.1. Upper Folding We start our construction of the folding matrix F by first looking at the Fupper` N folding, i.e., the part of F with first index greater than N. N We introduce the following notations: N

M

 :¼ 1
;

M

N

M

 :¼ 
 ¼ 
 ¼  ; N

N

N

M

ð2:70Þ

N

and we remark the following formulas which hold since B is strictly lower triangular N

N

 B  ¼ 0;

N

N

 B  ¼ B:

N

ð2:71Þ

With these formulas in mind we compute
 N N N N ¼ B ¼ B þ        1

1

1

N

N

N

N

N

N




N

N



N

¼  B  þ  B  ¼  B  þ  B  B  þ  B  N1

1

N

N1

N

N

N1

N

N

1

N

¼ ð1 þ BÞ B  þ  B  B  ¼ ð1 þ BÞ B  þ B2   N1

¼ ½iterating r
times¼

1

r X

N

N1

N

Bj  B   þ Brþ1  :

j¼0

N 1

1

1

ð2:72Þ

Given that B is lower triangular (in a sloppy sense it is nilpotent), the remainder in the formula above is a vector whose nonzero components are only those with index greater than N + r + 1. Since in our problems we are only folding finite band matrices and we will never need arbitrarily high indices, we can disregard the remainder and send r Y V (i.e., we take an inductive limit). The result is then N

N


1  ¼ ð1
BÞ  B : 1

N1

ð2:73Þ

Finally note that, since the matrix B has only d2 bands below the main diagonal, we have N

N

 B   ¼ 0: N N
d2

ð2:74Þ

36

MARCO BERTOLA AND BERTRAND EYNARD

In other words, the expression in Equation (2.73) contains only the quasipolynomials N
d2 ; . . . ; N . Therefore we have achieved the first part of our computation N

N


1  F ¼ ð1
BÞ  B ; N

ð2:75Þ

N

which can be simplified further as follows: N

N


1  F ¼ ð 1
BÞ  B  N

N


1 N 

 ¼
ð1
BÞ  t 
1 ðQ
xÞ N

 N
1 ¼
ð1
BÞ
1 t 
1  ðQ
xÞ N

¼
ð1
BÞ


1 


1

t




1  t
1 

N
1



  ðQ
xÞ  
ð1
BÞ   Q
x   N N
1 N


  N
1 ¼
ð1
BÞ
1 1
B
 
ð1
BÞ
1 t 
1 Q;  N 0 N
1

  N
1 ¼

ð1
BÞ
1 t 
1 Q;  : ð2:76Þ N

N
1

Concluding this part, we have N




1  t
1 

N
1

F ¼

ð1
BÞ N



N

Q;  : N
1

ð2:77Þ

2.4.2. Lower Folding Similarly to the case before we have now N
d2
1

 A 

N
d2
1

¼ 0;

 A 

N
d2
1 N
d2
1

¼A 

N
d2
1

ð2:78Þ

and, hence, we promptly obtain along the same lines (we set for convenience M = N j d2 j 1) M

 ¼ A ¼ AþA 1 M 1 M M
1 M M 1  M M ¼ AþA AþA M

1

M M

M

1

M

M 1

¼ ð1 þ AÞ  A   þA2   ¼ ½iterating r-times ¼

r X j¼0

1

M

M 1

M

Aj  A   þArþ1  : M

1

M 1

ð2:79Þ

37

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

For the same reason given above, we disregard the remainder term and thus we obtain M


1  ¼ ð1
AÞ  A : M1

ð2:80Þ

1

M

Once more this expression contains only the quasipolynomials within the window at N. Therefore M


1 F ¼ ð1
AÞ  A : MN

ð2:81Þ

M

Once again, this expression can be simplified further as follows: M


1  F ¼ ð 1
AÞ  A  M N

M

 d2 M ð1
AÞ
1  
1  ðQ
xÞ

1

¼


vd2 þ1 M  d2 M 1 ¼
ð1
AÞ
1 
1   ðQ
xÞ vd2 þ1 N
1   M 1 d2 ¼
ð1
AÞ
1 
1  ðQ
xÞ   þ vd2 þ1 N
1

1
1 
1 d2 ð1
AÞ   Q
x;  þ vd2 þ1 N
1 M

¼
ð1
AÞ
1 ð1
AÞ  þ N
1

þ

1 vd2 þ1

M

¼
 þ N
1

ð1
AÞ


1 
1

 

d2





Q;  N
1

  1 d2
1
1 ð1
AÞ   Q;  :

vd2 þ1

N
1

ð2:82Þ

i; j for N j d2 r i, j r N is the d2 + 1 identity matrix on Finally, we remark that F N the selected window so that, putting Equations (2.77), (2.82) together, we have

N

M

F ¼ F þ F þ N

M N

¼
ð1
BÞ

N


1 

N

t 
1





Q;  þ N
1

1 vd2 þ1



 d2 ð1
AÞ
1 
1  Q;  : N
1

ð2:83Þ We can summarize the above computations into the single formula


 1
1 
1 d2
1  t
1  F¼ ð1
AÞ  
ð1
BÞ   Q;  : N vd2 þ1 N
1

ð2:84Þ

38

MARCO BERTOLA AND BERTRAND EYNARD

Using this folding operator, we can immediately find the folded version of any other operator, viz, if L is any semi-infinite matrix with finite band size, its folded counterpart is M

L ! LðxÞ ¼  L F N :

ð2:85Þ

N

Note that Y strictly speaking Y formula (2.85) defines a semi-infinite matrix with only a nonzero square block of size d2 + 1 located on the main diagonal = N
d2 ; . . . ; N: Lð xÞij  0; i; j 2

ð2:86Þ

By abuse of notation, we will denote by the same symbol the (d2 + 1)  (d2 + 1) matrix corresponding to the nonzero block. In this fashion we will not distinguish between tracing over the finite or the semi-infinite matrix (with some advantages which will appear later). 2.5. DUAL FOLDING For completeness we report the formulas for the folding of the dual wave vector  onto the window N j 1 r n r N + d2 j 1. The computation is completely 1 parallel to the above using now the matrices (remember that the dual wave vector is a row-vector and, hence, we must act on the right) d

ð
1 Þ 2 A ¼ 1

ðQ
xÞ : vd2 þ1 d2
1

t
1

B ¼ 1
ðQ
xÞ  ;

With this in mind, it is straightforward to obtain

N
2 N
2   ¼   B  ð1
BÞ
1 ¼

Q;  t 
1 ð1
BÞ
1 ; 1 N
2

 1

Nþd2
1



N
1

ð2:88Þ

Nþd2
1


1  A  ð1
AÞ

¼

1 Nþd2
1 N
1

¼


N
d2
1

N

N
1

N
2

1

ð2:87Þ

þ

1 vd2 þ1



Q; 

N
1


1 d2   ð1
AÞ
1 ;

ð2:89Þ

N
2

F ¼   B  ð1
BÞ
1 1

N
2





t 
1 ð1
BÞ
1 þ ¼ Q;  N
1

 1 
1 d2
1 :   ð1
AÞ

vd2 þ1

ð2:90Þ

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

39

3. Folded Deformation Matrices U K ðxÞ N

From Equation (2.77) and using the folding relation (2.85) we obtain the desired formula for the deformation matrices U K (the folded version of K1 QKþ0 ), indeed N (recall M = N j d2 j 1): M

M

N

N

M


KU K ¼  QKþ0 F N ¼  QKþ0 F N

M

M

M

N

¼  QKþ0 F þ  QK F N ¼

N

N

N

M

M

M

K  Qþ0  N N M

K

N
1

M

N

N

K


 Q 
Q ð1
BÞ N

M

M

N
1

N
1

N

N


1  t
1 





Q;  N
1

¼  QKþ0 
 QK
0 
N




1  t
1  K K Q; 

 Q
x ð1
BÞ   N N
1

  M
xK ð1
BÞ
1 t 
1 Q;  N N
1

M M M N
1 M QK
xK N
1 K K ¼  Qþ0 
 Q
0 
 Q; 
xK  N N N N Q
x N N
1 N
1

M M M N
1 M N
1 ¼  QKþ0 
 QK
0 
 WK ð xÞ Q; 
xK  N N N N N N
1 N
1


 M M M N
1 1 ¼  QKþ0 
 WK ðxÞ Q;  þ xK
QKN;N  ; 2 N N N N
1 N
1 M

ð3:1Þ

where we have defined WK ð xÞ :¼

QK
xK : Q
x

ð3:2Þ

This formula generalizes similar formulas for the one-matrix model appearing in [7] which allowed us to perform explicit computations linking the isomonodromic tau-function and the partition function of the model (see Conclusion). 3.1. TRACE FORMULAS We compute the trace of the deformation matrices by noticing that the last term in (3.1) has zero trace because WK(x) commutes with Q and because of the cyclicity of the trace





M 0 0 M 0 Tr  WK ðxÞ Q;  ¼ Tr WK ðxÞ Q;   ¼ Tr WK ð xÞ Q;  ¼ 0; ð3:3Þ N

n
1

n
1 N

n
1

40

MARCO BERTOLA AND BERTRAND EYNARD

where Tr means the trace of the (finite rank) semi-infinite matrix. Therefore d2 X 1 K : ð3:4Þ Q
K tr U K ðxÞ ¼ xK
1=2QKN; N þ 2 N
m; N
m N m¼1 Now, the deformation equations imply the deformation equation for the normalization coefficients of the biorthogonal polynomials, 1 K ¼

h@uK ln hN ð3:5Þ Q K N; N which immediately implies ! 2 1 k
h dm¼1 hN
m x þ @uk ln K 2 hN
 1 k
h Z N2 ; ¼
x þ @uk ln K 2 Z N
d2 Z Nþ1

tr U K ð xÞ ¼
N

where we have used the fact that [20] N
1 Y hj ; Z N ¼ CN

ð3:6Þ

ð3:7Þ

j¼0

with Cn dependent only on N but independent on the Vis. 4. Differential System D1 ðxÞ N

From the formulas for the deformation equations we can derive the formula for the differential equation. Recall that P is of finite band size and that we have
 d1 X 1 K 0 t K uKþ1 Qþ0 þ Q0 : ð4:1Þ P ¼ P
1 þ PU ¼ P
þ V1 ðQÞU ¼   þ 2 K¼0 Using (2.13) and (2.14), we get P ¼ t  þ u1 1


d1 X K¼1

KuKþ1 UK þ

d1 1X uKþ1 QK0 : 2 K¼1

ð4:2Þ

Since DðxÞ is nothing but the folded version of P and the folding is linear, we N1 have d1 d1 X M 1X D1 ðxÞ ¼ t F N þ u1 1d2 þ1
uKþ1 KU K ðxÞ þ uKþ1 qK ; ð4:3Þ 2 N N N N0 K¼1 K¼1 where  and qK are the diagonal matrices N

N0

N ¼ diagð ðN
d2
1Þ; . . . ;  ðn
1ÞÞ;   qKN0 ¼ diag QKN
d2 ; N
d2 ; . . . ; QKN; N :

ð4:4Þ

41

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

We leave to the reader to check that the first term is M

M

M

M

t t t  F ¼  þ F N N N N N M N 0 0 1
d2
1 ðN
1Þ  ð MÞ B d2 ðN
1Þ 1 B B C . .. ¼@ AB @  ðN
1Þ 0



x
0 ðN
1Þ d2 ðN
1Þ

..

. ...

1 Þ
dðN
1 ðN
1Þ 2 C 0 C C; .. A .

1

0 ð4:5Þ

where the second matrix in the product is nothing but the inverse of the ladder matrix a ð xÞ. Using the expression (3.1) for the folded matrices U K and with N
1 N simple manipulations one obtains (M = N j d2 j 1).

N M M M N
1 M 0 D1 ðxÞ ¼ V1 ðQÞ 
 W ðxÞ Q;  þ V10 ðxÞ  þ t F N ; ð4:6Þ N

N

N
1

N

N
1

N

where W(x) is the semi-infinite matrix (polynomial in x and Q): W ð xÞ ¼

V10 ðQÞ
V10 ð xÞ : Q
x

ð4:7Þ

In a totally explicit fashion, we have the expression of the matrix D1 ð xÞ: N

0 B B D1 ð x Þ ¼ B @ N

V10 ðQÞN
d2 ; N
d2

 .. . 0 

0 0 0 0

B þ@

 ð MÞ ..

V10 ðQÞN
d2 ; N
1 .. . V10 ðQÞN
1; N
1 0 1

0 .. . 0 V10 ð xÞ

1 C C Cþ A

C A

.  ðN
1Þ

0 d
1 ðN
1Þ 2

B B B @

d2 ðN
1Þ

1

0 0



x
0 ðN
1Þ d2 ðN
1Þ

 ðN
1Þ d2 ðN
1Þ

1

0

..

. 

W ð xÞMþ1; N
1 B ..
@ . W ðxÞN; N
1

... ...

0 .. .

1 C C C
A

1 W ðxÞMþ1; Nþd2
1 CN .. AA; . W ð xÞN; Nþd2
1

which is the exact analogue of the very explicit formula appeared in [7].

ð4:8Þ

42

MARCO BERTOLA AND BERTRAND EYNARD

Notice that (using (2.13) and (2.14)), if k U l, we have 0

V1 ðQÞN
k; N
l ¼
PN
k; N
l ¼ k
l ðN
lÞ which allows us to rewrite 0 0 ðN
d2 Þ B B  ð N
d2 Þ D1 ð x Þ ¼ B B N @ 0 0


ð4:9Þ

. . . d2
1 ðN
1Þ .. .. . . .. . 0 ðN
1Þ ...  ðN
1Þ

1

0 .. . 0 V10 ð xÞ

C C Cþ C A

 ð MÞ  d2 ðM
1Þ 0

d2
1 ðN
1Þ B 0 B .. @ .

... ...

0 ðN
1Þ
x

1  ð N
1Þ C 0 Cþ .. A .

...

0 0

W ð xÞN
d2 ; N
1 B .. j@ . W ðxÞN; N
1

0 1

. . . W ðxÞN
d2 ; Nþd2
1 CN .. AA: . ... W ð xÞN; Nþd2
1

ð4:10Þ

4.1. TRACE FORMULA Using the previous formulas for the deformation matrices we can promptly obtain the trace of the matrix D1 . N

tr D1 ðxÞ ¼ V10 ðxÞ þ N

d2 X

0 ð N
k Þ
 ð M Þ

k¼1

d2
1 ðN
1Þ d2 ðN
1Þ

ð4:11Þ

and, using Equation (2.13), we have d2 ðN
1Þ ¼

d2 Y

 ðN
1
kÞ vd2 þ1

ð4:12Þ

k¼1

and d2
1 ðN
1Þ ¼

dY 2
1

 ðN
1
kÞ vd2 þ vd2 þ1

k¼1

d2 X

! 0 ðN
jÞ ;

ð4:13Þ

j¼1

therefore tr D1 ðxÞ ¼ V10 ðxÞ
N

vd2 : vd2 þ1

ð4:14Þ

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

43

Note the remarkable fact that the trace of D1 does not depend on N. We also point N out that this result cannot be obtained directly from the formula contained in [10]. 4.2. FOLDED DEFORMATION MATRICES V J ðxÞ N

We first consider the deformations w.r.t. vJ, 0 r J r d2 because they have simpler expressions and allow us to compute the deformation w.r.t. vd2 þ1 by the folding of (2.52). The semi-infinite deformation matrix is the finite band size matrix VJ ¼


1  J P
0 : J

ð4:15Þ

Its folded counterpart, as per Equation (2.85), is (for brevity in the formulas we set M := N j d2 j 1)   M M
J V J ðxÞ ¼  P J
0 F N ¼  P J
0  F N N

N

N

M

¼ P

 J

N

M


0  N

N

M

FN þ  PJ  FN N

M

 M M M M ¼  P J
0  þ  P J ð1
AÞ
1  A  : N

N

N

ð4:16Þ

M

From expression (2.66), for A we have the relation M


1 t d2 M M 1 
1 t d2      ðQ
xÞ ¼
 ðQ
xÞ vd2 þ1 M vd2 þ1 N
1

    M 1 1 d2 d2 ¼

1 t ðQ
xÞ   þ 
1 t Q;  vd2 þ1 v N
1 N
1 d2 þ1

M 1 
1 t d2 ¼
ð1
AÞ  þ   Q;  : ð4:17Þ vd2 þ1 N
1 N
1 1

A¼
M

Plugging (4.17) into (4.16), we obtain M

M

M

M

N

N

N

N
1

J J
JV J ðxÞ ¼  P
0 
 P  þ N

þ

1 vd2 þ1

M


1 
1

J

 P ð 1
AÞ

 

N

d2



Q;  : N
1

ð4:18Þ

The last deformation matrix is obtained from Equation (2.52) by also recalling that d2 þ1 QF N ¼ xd2 þ1 F N is
ðd2 þ 1Þvd2 þ1 V d2 þ1 N

¼ x D1 ð xÞ þ N

d2 X J¼1

JvJ V J þ N

dX 1 þ1 K¼1

KuK U K

hdiagðN
d2 ; . . . ; N Þ N

ð4:19Þ

44

MARCO BERTOLA AND BERTRAND EYNARD

4.2.1. Trace We will compute first the traces of the deformation matrices for J r d2, whereas the case J = d2 + 1 will be obtained by tracing the identity (4.19). Hereafter, the notation tr denotes the finite-dimensional trace for the folded matrices (of dimension (d2 + 1)  (d2 + 1)) while Tr denotes the trace of a semi-infinite matrix. Bearing this in mind, we compute

 N
d2
1 N
d2
1 J J
J tr V J ð xÞ ¼ 1=2Tr P 
Tr P  þ N N
1 N


 1
1 
1 d2 J Tr P ð1
AÞ   Q;  þ vd2 þ1 N
1


 N
1 X 1 1
1 
1 d2 J k J ¼ 1=2PN;N
P þ Tr P ð1
AÞ   Q;  : 2 j¼M j; j vd2 þ1 N
1 ð4:20Þ Let us focus our attention on the last term. First, notice that 1 vd2 þ1 ¼

 d2 Tr P J ð1
AÞ
1 
1  ½Q; N
1  1 vd2 þ1

 
1  d2 Tr P J 1
Ae 
1  ½V20 ðPÞ
x; N
1 ;

where Ae is the following strictly upper triangular matrix  d2 Ae :¼ 1

1  ðV20 ðPÞ
xÞ:

ð4:21Þ

ð4:22Þ

Indeed, for J r d2, the difference is the trace of a strictly upper triangular matrix. Then we have

 
1  d2 0 J
1 e   V2 ðPÞ
x;  Tr P 1
A N
1

¼ Tr P J




1  d2 0
1 1
Ae   ; V2 ðPÞ
x  : N
1

ð4:23Þ

Let us denote by yj = yj (x), j =1, . . . , d2 the solutions of algebraic equation = x and define the following polynomials in y:

V20 (y)

Sj ð yÞ :¼

V20 ð yÞ
x : y
yj

We have, in particular, Sj (yj) = V200(yj).

ð4:24Þ

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

We also define the following strictly upper triangular matrices:    Rj :¼ 1

1  P
yj :

45

ð4:25Þ

We are now going to use the following identity, whose proof can be found in the Appendix: d2  
1  d2 X 1
Ae 
1  ¼ j¼1

1   ð1
RJ Þ
1 
1 : V 002 yj

ð4:26Þ

Using (4.26), we have 


1  d2 0 1 J
1 Tr P 1
Ae   ; V 2 ð PÞ
x  vd2 þ1 N
1 ¼

h i 1   Tr P J ð1
RJ Þ
1 
1 ; V20 ðPÞ
x  N
1 V200 yj

d2 X j¼1

¼

d2 X

1   Tr P J ð1
RJ Þ
1 
1 ðV20 ðPÞ
xÞ 
N
1 V200 yj

j¼1




¼

d2 X 00 j¼1 V 2

d2 X j¼1




¼1

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{   V 0 ðPÞ
x 1   Tr P J ð1
RJ Þ
1 
1  P
yj 2  þ P
yj N
1 V200 yj

d2 X j¼1

¼

d2 X j¼1




¼


   
1
1 1 V 0 ð PÞ
x t
1   Tr P J 2    þ   1
R P
y  j j P
yj 0 N
1 V200 yj

d2 X 1 1   Tr P J Sj ðPÞ 
  Tr PJ SJ ðPÞt  
00 00 N
1 N
1 V2 yj V j¼1 2 yj

d2 X 00 j¼1 V 2

d2 X j¼1

1   Tr P J ðV20 ðPÞ
xÞð1
RJ Þ
1 
1   N
1 yj

   1   Tr P J SJ ðPÞ P
yj 1
Rj
1 
1   0 N
1 yj

d2 X 1 1   Tr P J SJ ðPÞ
  Tr P J Sj ðPÞ  00 y 0 0 V200 yj V j¼1 2 j

  
1  P
yj 1
Rj 
1  

N
1

ð4:27Þ

46

MARCO BERTOLA AND BERTRAND EYNARD

Notice that if N > 2d2 the PNj 1 in the last trace is irrelevant and therefore that trace is independent of N. Assuming N > 2d2, we obtain can carry on with our computation 1 vd2 þ1 ¼

 d Tr P J ð1
AÞ
1 
1  2 ½Q; N
1  d2 X    1 1   Tr P J SJ ðPÞ
  Tr  P
yj 1
Rj
1 
1 PJ SJ ðPÞ 00 00 0 0 V2 yj j¼1 V 2 yj

d2 X j¼1

¼

d2 X ykj    1   Tr P J SJ ðPÞ
  Tr  P
yj 1
Rj
1 
1 SJ ðPÞ
00 y 0 0 V V200 yj j¼1 2 j

d2 X j¼1




d2 X j¼1

¼

d2 X ykj    1   Tr P J SJ ðPÞ
  Tr  P
yj 1
Rj
1 
1 Sj ðPÞ
00 00 0 0 V2 yj j¼1 V 2 yj

d2 X j¼1




d2 X j¼1

¼

d2 X j¼1




     1   Tr  P
yj 1
Rj
1 
1  PJ
yJj Sj ðPÞ 0 V200 yj

     P J
yjJ 1   Tr  P
yj 1
Rj
1 
1  P
yj Sj ðPÞ P
yj 0 V200 yj

d2 X yJj      1   Tr P J Sj ðPÞ
  Tr  P
yj 1
Rj
1 
1 Sj ðPÞ
Sj ðyi Þ
00 y 0 0 V V200 yj j¼1 2 j

d2 X

d2 X   
1 yJj Tr P
yj 1
Rj 
1  
0

j¼1

j¼1

  P J
yJj 1   Tr  P
yj Sj ðPÞ: P
yj 0 V200 yj

ð4:28Þ Recalling that  ¼ 0, we continue 0

¼

d2 X

d2   X 1 1   Tr P J Sj ðPÞ
  Tr  P J
yjJ Sj ðPÞ
00 0 0 V 002 yj j¼1 V 2 yj

j¼1 d2 X

  yjJ   
1
1   Sj ðPÞ
Sj yj   Tr  P
yj 1
Rj   P
yj
00 P
yj 0 j¼1 V 2 yj   d2 d2 X X yjJ yjJ   Sj ðPÞ
Sj yj   Tr Sj ðPÞ
  Tr  P
yj ¼ 00 00 P
yj 0 0 j¼1 V 2 yj j¼1 V 2 yj ¼

d2 X j¼1

¼

d2 X j¼1

d2 X yjJ yjJ      Tr Sj ðPÞ
  Tr  Sj ðPÞ
Sj yj 00 00 0 0 V 2 yj V 2 yj j¼1

yjJ Tr  0

¼

d2 X j¼1

yjJ

ð4:29Þ

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

47

With Equation (4.29), we can finally state d2 N
1 X 1 J 1X PJj; j þ yjJ PN;N
2 2 j¼M j¼1


JtrV J ð xÞ ¼ N

ð4:30Þ

Using Equation (4.30) and tracing (4.19), one can obtain the following formulas for the traces:
Qd2  H yJ V 002 ð yÞ h j¼1 N
j h
þ dy J 0 trV J ð xÞ ¼
2 @vJ ln ; J ¼ 1; . . . ; d2 hN V 2 ð yÞ
x N ! Qd2
h j¼1 hN
j þ ¼ trV d2 þ1 ðxÞ ¼
@vd2 þ1 ln 2 hN N
 I V 002 ðyÞ yd2 þ1 d2 1 þ dy
h
N
0 d2 þ 1 V2 ðyÞ
x 2 vd2 þ1

ð4:31Þ

4.3. DETERMINANT OF THE FUNDAMENTAL SYSTEM It was proven in [10] that the following overdetermined set of differential equations for a (d2 + 1)  (d2 + 1) matrix  is Frobenius compatible: N

ðxÞ;

h@x  ðxÞ ¼ D1 ð xÞ

ð4:32Þ

h
@uK  ðxÞ ¼ U K ðxÞ ðxÞ;

ð4:33Þ

ð xÞ;

h@vJ  ðxÞ ¼ V J ðxÞ

ð4:34Þ

að xÞ ð xÞ ¼  ð xÞ:

ð4:35Þ

N

N

N

N

N

N

N

N

N

N

N

Nþ1

The biorthogonal polynomials constitute one column of  ðxÞ, while the other d2 N can be also explicitly obtained as in [8]. From (4.31) it follows that the dependence on the coefficients of V2 of the determinant of the fundamental system is

 dX 2 þ1 dvJ
h@vJ ln det  ðxÞ hdV ln det  ð xÞ :¼

ð4:36Þ N

¼

N

J¼1 dX 2 þ1


 dvJ tr V J ðxÞ

J¼1

N

48

MARCO BERTOLA AND BERTRAND EYNARD

0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Qd2 I dX 2 þ1 hN
j y J V 002 ð yÞ j¼1 @ A
¼

hdV ln dvJ dy þ hN J V20 ð yÞ
x J¼1
 d2

h N
dlnðvd2 þ1 Þ: 2 Now one can directly check that
I  I V 002 ðyÞ y J V200 ðyÞ dyðV2 ð yÞ
xyÞ 0 ¼ @vJ dy : V2 ðyÞ
x J V 0 ðyÞ
x

ð4:37Þ

ð4:38Þ

Therefore, as a consequence of Equations (4.14), (4.31) and (3.6), we find that any matrix fundamental solution to the compatible system of equations satisfies the equations
 vd2 ; ð4:39Þ h@x ln det  ð xÞ ¼
V10 ðxÞ þ
N vd2 þ1




hdu ln det  ð xÞ
N

¼ du

 hdv ln det
ð xÞ N 2 0

h

V1 ð xÞ þ ln 2

Qd2

m¼1 hN
m

hN

!! ;

ð4:40Þ




d2 2
N

¼

hdv 4ln@vd2 þ1

3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Qd2 I 00 V ð y Þ h m¼1 N
m A 5; þ dyðV2 ð yÞ
xyÞ 0 2 hN V2 ðyÞ
x ð4:41Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 h2N hN
d2
1 ðxÞ
1 ¼ det aðxÞ ¼ vd2 þ1 ð
Þd2 þ1 det  ð xÞ N Nþ1 N hN þ 1

ð4:42Þ

and, hence, we have finally the complete formula

d2 det  ðxÞ ¼ ðvd2 þ1 ÞN
2 ð
1ÞNðd2 þ1Þ N

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qd2 m¼1 hN
m  hN




I 1 V200 ðyÞ  exp
V1 ðxÞ þ dyðV2 ð yÞ
xyÞ 0 : h
V2 ðyÞ
x ð4:43Þ

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL

49

Note that the contour integral just returns the sum of the critical values of V2( y) j xy. We know from [10] that a joint solution of the dual system of overdetermined N equations for the dual window denoted by  ðxÞ has the property that N

N

 ð xÞA ðxÞ ¼ C;

ð4:44Þ

N

where C is an invertible matrix which does not depend on any of the potentials or n or - or x and can be conveniently normalized to unity (see [8] for an explicit construction). Therefore

1 N   N 
1 : ð4:45Þ det  ðxÞ ¼ det A det  ðxÞ N

5. Dual Folded System For completeness, we add the formulas for the relevant folded operators for the N dual window  ¼ ðN
1 ; . . . ; Nþd2
1 Þ. Since the steps are essentially the same, we give only the results (here we set L = N + d2 j 1) N

N
1

N
1

N
1

L

L

L

KU K ¼  QKþ0 
½Q; N
1 WK ðxÞ  þ
 N
1 1 þ xK
QKN
1; N
1  ; 2 N N

N
1

N
1

L

L

L

ð5:1Þ N
1

JV J ¼  PJ
0  þ  A ð1
AÞ
1 P J  ; 0

V10 ðxÞ B N B  ðN
1Þ D1 ð x Þ ¼ B @ 0 0

ð5:2Þ

L

L

0 0 ð N Þ .. .

... ... .. .

0 d2
1 ðLÞ .. .

1 C C C
A

...  ð L
1Þ 0 ðLÞ 0 1 0 . . . 0  ðN
1Þ 0 ð N Þ
x C  ð LÞ B B0 C
B C
. .. A d2 ðL þ 1Þ @ 0 . . . 0 . . . 0 d2
1 ðLÞ 1 0 W ðxÞN
d2 ; N
1 . . . W ðxÞN
d2 ; Nþd2
1 NB C .. ..
A@ A: . .   ... W ð xÞN; Nþd2
1 W xN; N
1

ð5:3Þ

50

MARCO BERTOLA AND BERTRAND EYNARD

6. Conclusion We have given the most explicit construction for the matrices describing the differentialYdeformationYdifference folded system. This has allowed us to determine explicitly the determinant of the fundamental solution of the system in terms of the partition function of the model and the two potentials. The final expression is quite simple in comparison with the complexity of the computation, especially for the traces of the deformation matrices V J . N It is our hope (in fact, it is our plan) that these formula be used to relate explicitly the partition function of the two-matrix model to a (suitably defined) isomonodromic tau-function. Indeed the system (4.32)Y(4.35) can (and should) be regarded as a monodromy-preserving set of infinitesimal (4.33, 4.34) and finite (4.35) deformation equation for the ODE (4.32) which has an irregular, degenerate singularity at infinity. As such it is envisionable that one can define the associated notion of tau-function although there are technical difficulties due to the degeneracy at infinity. In the similar case of the ODE + PDE + DE for the orthogonal polynomials in the one-matrix model a similar approach has given very satisfactory results [3, 7]. Appendix PROOF OF FORMULA (4.26)

We need to prove formula (4.26) d2  
1  d2 X 
1  ¼ 1
Ae j¼1

 1    1
Rj
1 
1 ; V 002 yj

ðA:1Þ

which can be written more transparently as  
1  d2 d2 
1  
1  ðV20 ðPÞ
xÞ ¼

d2 X 00 j¼1 V 2


1
1 1 
1      P
yj  : yj

ðA:2Þ

We remark for better understanding that if the matrices L, (V20 (P) j x) were invertible, this would simply amount to the partial fraction expansion of 1/ (V20 (P) j x). Multiplying both sides by the invertible matrix (1 j Ae) on the left and recalling the definition (4.22) of Ae: ! d2 
1 d2 
1 d2 X   1
1   ðV20 ðPÞ
xÞ 1
Rj 
1    ¼   00 y V j¼1 2 j

THE PDEs OF BIORTHOGONAL POLYNOMIALS ARISING IN THE TWO-MATRIX MODEL



¼ 
1 

51

!  
1
1 1 V20 ðPÞ
x    P
yj 1
Rj   00 P
yj j¼1 V2 yj
 d2 X   1 V20 ðPÞ
x t
1    þ    P
yj 00 P
yj 0 j¼1 V2 yj

d2 d2 X

 d2 ¼ 
1 

ð1
Rj Þ
1 
1 Þ

ðA:3Þ

Noticing now that 

d2 V20 ðPÞ
x 
1   ¼ 0; P
yj 0

we continue ! 1 V 0 2 ðPÞ
x t
1      00 P
yj j¼1 V 2 yj
! d2 
1 d2 X 1 V 0 2 ð PÞ
x   1
 ¼   00 P
yj 0 j¼1 V 2 yj
  d2  d2 ¼ 
1  1
 ¼ 
1  ;

d2 
1 d2 
1 d2 X   ¼  

0

where we have used that ð
1 Þ

d2

ðA:4Þ

 ¼ 0. The identity is thus proved. 0

Acknowledgements M.B. was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) Grant No. 261229-03, B.E. was supported in part by the EC IHP Network FDiscrete geometries: from solid state physics to quantum gravity`, HPRN-CT-1999000161.

References 1. 2.

Adler M. and Van Moerbeke, P.: String-orthogonal polynomials, string equations and 2-Toda symmetries, Comm. Pure Appl. Math. J. 50 (1997), 241Y290. Adler, M. and Van Moerbeke, P.: The spectrum of coupled random matrices, Ann. of Math. (2) 149 (1999), 921Y976.

52

MARCO BERTOLA AND BERTRAND EYNARD

3.

Bleher, P. and Its, A.: Semiclassical asymptotics of orthogonal polynomials, RiemannYHilbert problem, and universality in the matrix model, Ann. of Math. (2) 150(1) (1999), 185Y266. Deift, P., Kriecherbauer, T., McLaughlin, K. T. R., Venakides, S. and Zhou, Z.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335Y1425. Deift, P., Kriecherbauer, T., McLaughlin, K. T. R., Venakides, S. and Zhou, Z.: Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), 1491Y1552. Fokas, A., Its, A. and Kitaev, A.: The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), 395Y430. Bertola, M., Eynard, B. and Harnad, J.: Partition functions for matrix models and isomonodromic tau functions, J. Phys. A 36 (2003), 3067Y3083. Bertola, M., Eynard, B. and Harnad, J.: Differential systems for bi-orthogonal polynomials appearing in two-matrix models and the associated RiemannYHilbert problem, Comm. Math. Phys. 243 (2003), 193Y240. Kapaev, A. A.: The RiemannYHilbert problem for the bi-orthogonal polynomials, J. Phys. A 36 (2003), 4629Y4640. Bertola, M., Eynard, B. and Harnad, J.: Duality, biorthogonal polynomials and multi-matrix models, Comm. Math. Phys. 229 (2002), 73Y120. Bertola, M., Eynard, B. and Harnad, J.: Duality of spectral curves arising in two-matrix models, Theor. and Math. Phys. 134(1) (2003), 27Y32. Kazakov, V. A.: Ising model on a dynamical planar random lattice: exact solution, Phys. Lett. A 119 (1986), 140Y144. Daul, J. M., Kazakov, V. and Kostov, I. K.: Rational theories of 2D gravity from the twomatrix model, Nuclear Phys. B 409 (1993), 311Y338, hep-th/9303093. Di Francesco, P., Ginsparg, P. and Zinn-Justin, J.: 2D gravity and random matrices, Phys. Rep. 254 (1995), 1. Bauldry, W.: Estimates of asymmetric Freud polynomials on the real line, J. Approx. Theory 63 (1990), 225Y237. Bonan, S. S. and Clark, D. S.: Estimates of the Hermite and the Freud polynomials, J. Approx. Theory 63 (1990), 210Y224. Jimbo, M., Miwa, T. and Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients I, Physica D 2 (1981), 306Y352. Jimbo, M. and Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients II, III, Physica D 2 (1981), 407Y448; ibid., D 4, 26Y46 (1981). Bertola, M.: Bilinear semi-classical moment functionals and their integral representation, J. Approx. Theory 121 (2003), 71Y99. Eynard, B. and Mehta, M. L.: Matrices coupled in a chain: eigenvalue correlations, J. Phys. A, Math. Gen. 31 (1998), 4449, condmat/9710230. Ueno, K. and Takasaki, K.: Toda lattice hierarchy, Adv. Stud. Pure Math. 4 (1984), 1Y95.

4.

5.

6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

19. 20. 21.

Mathematical, Physics, Analysis and Geometry (2006) 9: 53–63 DOI: 10.1007/s11040-005-9002-8

#

Springer 2006

Sharpenings of Li_s Criterion for the Riemann Hypothesis j ANDRE´ VOROS

CEA, Service de Physique The´orique de Saclay, (CNRS URA 2306), F-91191 Gif-sur-Yvette Cedex, France. e-mail: [email protected] (Received: 8 August 2005; in final form: 1 September 2005) Abstract. Exact and asymptotic formulae are displayed for the coefficients
n used in Li_s criterion for the Riemann Hypothesis. For n ! 1 we obtain that if (and only if) the Hypothesis is true,
n
nðA log n þ BÞ (with A > 0 and B explicitly given, also for the case of more general zeta or L-functions); whereas in the opposite case,
n has a non-tempered oscillatory form. Mathematics Subject Classifications (2000): 11M26, 30B40, 41A60 Key words: Riemann zeta function, Riemann hypothesis, Li_s criterion, saddle-point method, Stieltjes constants

Li_s criterion for the Riemann Hypothesis (RH) states that the latter is true if and only if a specific real sequence f
n gn¼1;2;... has all its terms positive [2, 17]. Here we show that it actually suffices to probe the
n for their large-n behavior, which fully encodes the Riemann Hypothesis by way of a clear-cut and explicit asymptotic alternative. To wit, we first represent
n exactly by a finite oscillatory sum (9), then by a derived integral formula (12), which can finally be evaluated by the saddle-point method in the n ! þ1 limit. As a result,
n takes one of two sharply distinct and mutually exclusive asymptotic forms: if RH is true,
n will grow tamely according to (17); if RH is false,
n will oscillate with an exponentially growing amplitude, in both þ and  directions, as described by (18). This dichotomy thus provides a sharp criterion of a new asymptotic type for the Riemann Hypothesis (and for other zeta-type functions as well, replacing (17) by (15)). This work basically reexposes our results of April 2004 announced in [28], but with an uncompressed text; we also update the references and related comments: for instance, we now derive as (24) a large-n expansion surmised by Mas´lanka [20] in the meantime.

j

Institut de Mathe´matiques de Jussieu-Chevaleret (CNRS UMR 7586), Universite´ Paris 7, F-75251 Paris CEDEX 05, France.

54

ANDRE´ VOROS

1. Background and Notations We study the sequence [12, 17] (in the notations of Li, whose
n are n times Keiper_s) X ½1  ð1  1=Þn  ðn ¼ 1; 2; . . .Þ; ð1Þ
n ¼ 

where  are the nontrivial zeros of Riemann_s ðsÞ, grouped by pairs within summations and products as f ¼ 12  ik gk¼1;2;... ;

Re k positive and non-decreasing;

ð2Þ

we also parametrize each such pair by the single number xk ¼ ð1  Þ ¼ 14 þ k 2 . We will use the completed zeta function
ðsÞ (normalized as
ð0Þ ¼
ð1Þ ¼ 1) and a symmetrized form of its Hadamard product formula [7, 26], " # 1 Y sð1  sÞ 1 ; ð3Þ
ðsÞ ¼ sðs  1Þðs=2Þs=2 ðsÞ  xk k¼1 we will also use a Fsecondary_ zeta function built over the Riemann zeros, ZðÞ ¼

1 X

xk  ;

Re  > 12 ;

ð4Þ

k¼1

which extends to a meromorphic function in C having all its poles at the negative half-integers, plus one pole at  ¼ þ 12 [13] of polar part [26] Zð 12 þ "Þ ¼ R2 "2 þ R1 "1 þ Oð1Þ"!0 ;

ð5Þ

with R2 ¼ ð8Þ1 ;

R1 ¼ ð4Þ1 log 2

in this case:

ð6Þ

Our results [28] mainly relate to those of Keiper [12], of which we only learned later (thanks to K. Mas´lanka; they were almost never cited), of BombieriYLagarias [2] on Li_s criterion [17], and of Mas´lanka [20]. Other earlier works considering the
n are [1, 3]. 2. New Exact Forms for
n To reexpress the
n , we start from their generating function [12, 17] f ðzÞ ¼

1
1  X d
n zn1 : log
 dz 1z n¼1

ð7Þ

SHARPENINGS OF LI_S CRITERION FOR THE RIEMANN HYPOTHESIS

55

Now the infinite product formula in (3) implies # " 1 1  X z log 1 þ log
¼ 1z ð1  zÞ2 xk k¼1


¼

1 X ð1Þj

"

ð1  zÞ2

j

j¼1

z

ð8Þ

#j Zð jÞ;

then, expanding ð1  zÞ2j by the generalized binomial formula, reordering in powers of z and substituting the output into (7), we get as first result   n X ð1Þj n þ j  1
n ¼ n Zð jÞ: ð9Þ j 2j  1 j¼1 P j , Other sums related to the ZðkÞ are Z j ¼  j [16, 22, 26] (often Pdenoted n jþ1 ¼ ð1Þ but here we use  as variable). It was already known that

 n j¼1 n Z [12, Equation (27)], and that the Z in turn are complicated polynomials j j j in the Stieltjes constants fk gk< j [22] (for
n and k see also [3, 4, 21] and references therein). Now the latter relations boil down to Z j ¼ 1  ð1  2j ÞðjÞ þð1Þj j1 [26, Equation (46)] simply by promoting logarithmic coefficients j [2, 11] (cf. also [16, Equation (12)]) log ½sð1 þ sÞ  

1 X

n1

n¼1

sn n

ð10Þ

in place of sð1 þ sÞ ¼ 1 

1 X n¼1

n1

ðsÞn : ðn  1Þ!

The
n thus express as affine combinations of the j [2, thm 2]. Remarks:

Y the j are the Stieltjes [constants_] cumulants, up to some relabelings [26, 27];

Y the j admit an arithmetic expression over the primes [2, Equation (4.1)]; see also [10], which cites [25] for the case 0 ¼ ;

Y relation (9) can be inverted also in closed form, by the same technique as for [26, Equation (48)]:   j X 2j nþ1 ð1Þ Zð jÞ ¼
n : jn n¼1

ð11Þ

56

ANDRE´ VOROS

The expression (9) for
n has some distinctive advantages: it involves the functional equation
ðsÞ ¼
ð1  sÞ through (3); and unlike the Z j , the Zð jÞ are positive and gently varying factors: the function ZðÞ is regular and very smooth for real  U 1. Still, (9) is an oscillatory sum, hence difficult to control directly. Now an integral representation, equivalent to (9) simply by residue calculus, will nevertheless prove much more flexible: I ð1Þn n i ð þ nÞð  nÞ ZðÞ; ð12Þ IðÞ d; IðÞ ¼
n ¼  ð2 þ 1Þ C where C is a positive contour encircling just the subset of poles  ¼ þ1; . . . ; þn of the integrand IðÞ. 3. Asymptotic Alternative for
n ; n! !1 The integral formula (12) readily suggests an asymptotic (n ! 1) evaluation by the classic saddle-point method [8], using jIðÞj as height function. First, the integration contour C is to be moved in the direction of decreasing jIðÞj as far down as possible: it will thus pass through some saddle-points * of jIðÞj. Then for large n, IðÞ peaks near each of these points * , where it makes a contribution of the order of magnitude jIð* Þj to the integral: thus the highest saddle-point(s) give(s) the dominant behavior. Consistent asymptotic approximations can also be made inside IðÞ throughout: e.g., here, Stirling formulae used for ðn þ constÞ. This approach, for an integrand not controlled in fully closed form, partly retains an experimental character. We currently advocate it for this problem as a heuristic, rather than rigorous, tool: it predicts the global structure of the results at once, and it treats all the cases readily and correctly, as other techniques confirm. In the present problem, for large n the landscape of the function jIðÞj is dominantly controlled: by its  factors, asymptotically
 ½sin  ð2 þ 1Þ1 n21 for finite ; and by the polar parts of IðÞ near its poles. The induced contour deformation starts as a dilation of C away from the segment ½1; n in all directions, and goes to infinity in the directions j arg j < 2  . The encountered saddle-points can be of two types here (once n is large enough). 1) For  on the segment ð12; 1Þ, jIðÞj
 ½sin  ð2 þ 1Þ1 n21 ZðÞ always has one real minimum r ðnÞ (tending to 12 as n ! 1), which will be reached by the moving contour; other real saddle-points lie below  ¼ 12 and will not get reached here. 2) Complex saddle-points may enter as well, for which we may focus on the upper half-plane alone: the lower half-plane will give complexYconjugate (Fc.c._) contributions. As long as the moving contour stays P inside a half-plane 1 fRe  > 2 þ "g, the integrand can be decomposed as I ¼ k Ik according to (4); then for each individual term and within the Stirling approximation for

SHARPENINGS OF LI_S CRITERION FOR THE RIEMANN HYPOTHESIS

57

d the -ratio, the saddle-point equation is 0 ¼ d log jIk ðÞj
log ð2  n2 Þ  2 log 2  log xk , yielding the saddle-point location

k ðnÞ ¼ n i = 2k :

ð13Þ

Thus any zero on the critical axis (k real) yields a purely imaginary k ðnÞ, not eligible: it lies outside the domain of validity of (4), and its contribution would be subdominant anyway. So in the end, this paragraph excludes the real k . The discussion then fundamentally splits depending on the presence or absence of zeros off the critical axis. 3.1. [RH FALSE] If there is any zero ð12  ik Þ off the critical axis, we select arg k > 0 and assume the case of a simple zero for argument_s sake. Paragraph 2) above fully applies to each such zero: the complex saddle-point k ðnÞ given by (13) lies inside the domain of convergence fRe  > 12g as soon as n > j Im 1=kj1 , and for n ! þ1 n it gives an additive contribution
ðk þ i=2Þ=ðk  i=2Þ (in the usual quadratic approximation of log IðÞ around k ðnÞ), which grows exponentially in modulus and fluctuates in phase; it will indeed exponentially dominate the contribution of the real saddle-point r ðnÞ, to be computed later. This result can also be confirmed rigorously and more directly: by a conformal mapping [2], the function f ðzÞ in (7) has precisely the points zk ¼ ðk  i=2Þðk þ i=2Þ1 and zk* as simple poles of residue 1 in the unit disk; then a general Darboux theorem [6, chap. VII Section 2] applies here to the poles with jzk j < 1, implying that the Taylor coefficients of f (namely, the
n ) indeed have the asymptotic form X zn ðmod oðe"n Þ 8" > 0Þ; n ! 1 ; ð14Þ
n
k þ c:c: fjzk j<1g

which now holds for multiple zeros as well, counted with their multiplicities. Concretely,
n then oscillates between exponentially growing values of both signs. Infinitely many zeros ð12  ik Þ off the critical axis are perfectly admissible P here: their zk satisfy jzk j < 1, zk ! 1, and the corresponding infinite sums k zn k still define valid n ! þ1 asymptotic expansions. On the other hand, the general Darboux formula (14) hopelessly breaks down if the infinitely many zn k (14) have identical and dominant modulus, which is precisely realized in the case [RH true], with all the zk on the unit circle! 3.2. [RH TRUE] Here, Darboux_s theorem only tells that
n ¼ oðe"n Þ 8" > 0, but it fails to give any clue as to an explicit asymptotic equivalent for
n . By contrast, the saddle-

58

ANDRE´ VOROS

point treatment of the integral (12) itself remains thoroughly applicable. Simply now, all the k are real, ZðÞ ¼ OðZðRe Þ jIm j3=2 Þ in fRe  > 12g, and the contour C can be freely moved towards the boundary fRe  ¼ 12g without meeting any of the k ðnÞ (all of which are purely imaginary). Hence the only dominant saddle-point is now r ðnÞ 2 ð12; 1Þ; it is shaped by the double pole of ZðÞ at 12 (itself generated by the totality of Riemann zeros), so that r ðnÞ
12 þ log1 n. This saddle-point is however non-isolated (it tends to the pole), so the standard saddle-point evaluation using the quadratic approximation of log IðÞ around r ðnÞ works very poorly. Here, it is at once simpler and more accurate to keep on deforming a portion of the contour C nearest to  ¼ 12 until it fully encircles this pole (now clockwise), and to note that the ensuing modifications to the integral are asymptotically smaller. Hence for [RH true],
n is given (mod oðnÞ) by [28]
n
ð1Þn 2n Res¼1=2 IðÞ ¼ 2n ½2R2 ð ð 12 þ nÞ  1 þ Þ þ R1  ¼ 2n ½2R2 ð log n  1 þ Þ þ R1  þ Oð1=nÞ ð15Þ (with  0 = estimated by the Stirling formula,  ¼ Euler_s constant, and using the polar structure (5) for ZðÞ). Prior to using (6) to fix the Rj , the argument covers zeros  of a more general (arithmetic) Dirichlet series LðsÞ: as long as the latter has a meromorphic structure and functional equation similar enough to ðsÞ, its secondary zeta function ZðÞ keeps a double pole at  ¼ 12 [27]. A related but more concrete requirement can be put on the function NðTÞ, the number of zeros of LðsÞ with 0 < Im  < T: we ask that for some constants R2 ; R1 and some < 1 (all now depending on the chosen L-series), NðTÞ ¼ 2T ½2R2 ð log T  1Þ þ R1  þ NðTÞ; NðTÞ ¼ OðT Þ for T ! þ1

ð16Þ

(implying R2 U 0). If both conditions (5) and (16) hold, then the polar coefficients of ZðÞ in (5) have to be precisely the Rj from (16). All of that is realized in many cases including, but not limited to, Dedekind zeta functions [13, 15, 27] and some Dirichlet L-functions [5, 18, 27]; see also [14] (discussed at end); in all those instances, NðTÞ ¼ Oðlog TÞ. For the corresponding
n , our saddle-point evaluation then always yields: either (16) ) (15) if all the zeros have Re  ¼ 12 Y or the immediately general result (14) otherwise. Like (14) before, (15) can be derived quite rigorously but by still another method, previously unknown to us, and written for the Riemann zeros by J. Oesterle´ [23] (private communication). We thank him for allowing us to repeat his argument here; we actually word it in the more general present setting (and

59

SHARPENINGS OF LI_S CRITERION FOR THE RIEMANN HYPOTHESIS

slightly streamline it). When all the zeros lie on the critical line, first transform the summation (1) into a Stieltjes integral (where ðTÞ ¼ 2 arctan ð1= 2TÞ): R1 R
n ¼ 2 0 ½1  cos nðTÞ dNðTÞ, then integrate by parts: n1
n ¼ 2 0 sin nNð12 cot 2Þ d. Now replace NðTÞ by its large-T form (16) neglecting NðTÞ and other Oð Þ terms: the error is oð1Þ by the RiemannYLebesgue 1  over the closed interval lemma, mainly because (16) R  makes 1 Nð2 cot 2Þ integrable 1 ½0; ; hence n
n ¼ 0 sin n  ½8R2 ðlog 1  1Þ þ 4R1  d þ oð1Þ. Next, change variable: n ¼ t,R then replace upper t-bound n by þ1 again with an 1 oð1Þ error; thus,
n ¼ n 0 sint t ½8R2 ðlog nt  1Þ þ 4R1  dt ðmod oðnÞÞ, and the last integral evaluates in closed form [9] to yield (15). Unfortunately, we do not see how to extend this purely real-analytic argument to include the [RH false] case as well.

3.3. RECAPITULATION As we ended up with two mutually exclusive large-n behaviors for the
n , (14) and (15), together they provide a sharp equivalence result. For the Riemann zeros, using the explicit values (6): THEOREM (asymptotic criterion for the Riemann Hypothesis). For n ! þ1, the sequence
n built over the Riemann zeros follows one of these asymptotic behaviors: ½RH true

,
n


½RH false

tempered growth to þ 1; as 1 2 nðlog n

 1 þ   log 2Þ

ðmod oðnÞÞ;

ð17Þ

,

non-tempered oscillations; as X
k þ i=2 n þ c:c: ðmod oðe"n Þ 8" > 0Þ:
n
  i=2 k farg  >0g k

ð18Þ This comprehensive asymptotic statement [28] is new on the [RH false] side to our knowledge, and it also completes some earlier results in the [RH true] case. Our main end formula ((17), assuming RH) had actually been displayed by Keiper [12, Equation (37)], but under a somewhat misleading context: in his words, (17) required not just RH but also Fvery evenly distributed_ zeros, and was Fmuch stronger than_ RH; no details or proofs were ever supplied. All that hardly points toward our present conclusion that (17) and RH are strictly equivalent.

60

ANDRE´ VOROS

Oesterle´ had a proof of the statement [RH true] ) (17) (see above), but he neither published nor even posted his typescript [23]. In [2, Cor. 1(c)], rather weak exponential lower bounds
n U  ce"n were shown to imply RH; the backward assertion [RH true] ( (17) is thus also implied by [2] (but cannot be inferred therefrom, as [2] never alludes to asymptotics regarding the
n ). Our saddle-point approach also handles both cases (17)Y(18) at once for the first time. Numerical data [12, 20] agree well with (17) for n < 7000 (and even better in the mean if we add the contribution like (15) but from the next pole of IðÞ,
n ¼ ð1Þn 2n Res¼0 IðÞ ¼ 2Zð0Þ ¼ þ7=4

ð19Þ

[26, Equation (41)], although this correction should not count asymptotically, dominated as it seems by oscillatory terms). Yet the above numerical agreement is inconclusive regarding the Riemann Hypothesis: any currently possible violation of RH would yield a deviation from (17) detectable only at much higher n (see end of Sect. 4). 4. An Even More Sensitive Sequence A slightly stronger difference of behavior follows for the special linear combinations (20) below of the coefficients n themselves (defined by Equation (10) above). Indeed, the definition
ðsÞ ¼ ðs=2Þs=2 sðs  1ÞðsÞ substituted into (7) readily yields a decomposition
n ¼ Sn þ Sn , where [2, thm 2] n   X n ð20Þ j1 Sn ¼  j j¼1 is the contribution of ðs  1ÞðsÞ, and log 4 þ  n þ S^n ; Sn ¼ 1  2

with S^n ¼

n   X n j¼2

j

ð1Þj ð1  2j Þ ð jÞ; ð21Þ

is the contribution of the remaining (more explicit) factor. The large-n behavior of the sum S^n is also computable, by the same route we followed from (9) to (15) through (12). First, ð1Þn n! S^n ¼ 2i

I JðÞ d; C0

JðÞ ¼

ð  nÞ ð1  2 Þ ðÞ; ð þ 1Þ

ð22Þ

61

SHARPENINGS OF LI_S CRITERION FOR THE RIEMANN HYPOTHESIS

integrated around the poles  ¼ 2; . . . ; n of JðÞ; hence for n ! þ1, S^n is asymptotically ð1Þn1 n! Res¼1 JðÞ ¼ 12 n½ ðnÞ þ log 2  1 þ 2;

ð23Þ

now (mod oðnN Þ 8N > 0) because JðÞ has no further singularities; so that finally, using the Stirling expansion for ðnÞ in terms of the Bernoulli numbers B2k , 1 3 X B2k 12k n Sn
12 nðlog n  1 þ   log 2Þ þ  ðn ! 1Þ 4 k¼1 4k ð24Þ

unconditionally:

This at once confirms two empirical conjectures made by Mas´lanka [20, Equations (2.5), (2.8)]: mod oðnÞ, the sequence fSn g expresses the Ftrend_ (17) obeyed by the sequence f
n g under [RH true]; and to all orders in n, fSn g has the asymptotic expansion (24). But whether RH holds or not it makes sense to withdraw the fixed fSn g-contribution from the previous formulae (17) and (18), as we did in [28] to find: Sn ¼ oðnÞ

½RH true

ð25Þ

(a case further discussed in [4, 21, 24]), versus   P k þ i=2 n þ c:c: mod oðe"n Þ 8" > 0 Sn

n
  i=2 k farg  >0g k

½RH false;

ð26Þ

which gives oscillations that blow up exponentially with n. Still, in absolute size, any contribution from (26) will stay considerably smaller than (25) (the background from the set of real j ) up to n  g:ffiffiffiffiffiffiffi i.e., min farg k >0g fjIm 1=k j1p ffi Sn can only reliably signal a zero violating RH up to a height jIm j | n=2; so vice-versa, since such zeros are now known to require jIm j a 109 , they could only be detected by Sn for n a 1018 (see also [1, 23], [14, p. 5]). Our asymptotic criteria may thus not surpass others in practical sensitivity, but their sharpness is theoretically interesting. For instance, Li_s criterion
n > 0 ð8nÞ is now strengthened in that, beyond a finite n-range, it refers to ever larger amplitudes either way, i.e., delicate borderline situations such as
n ! 0 are ruled out. 5. More General Cases: A Summary For the more general zeta-type functions satisfying (5) and (16), the asymptotic alternative for the associated
n is: either (15) if the generalized Riemann

62

ANDRE´ VOROS

Hypothesis (GRH) holds, or (18) otherwise. In the former case, the connection (16) ) (15) often makes the asymptotic form of
n fully explicit without any calculation, and it also ensures R2 U 0 in line with the corresponding generalized Li_s criterion [2, 14, 18, 19]. The
n have also been generalized to L-functions defined by Hecke operators for the congruence subgroup 0 ðNÞ [18] [19, specially remark 5.4]. More recently and in a broader setting (the
n for automorphic L-functions), Lagarias presentedpffiffian ffi alternative approach to estimate the
n with greater accuracy, mod Oð n log nÞ under GRH [14]: in the notations of (20)Y(21), he directly p proves that Sn obeys (15) mod Oð1Þ unconditionally (thm 5.1), then that ffiffiffi Sn ¼ Oð n log nÞ under GRH (thm 6.1). We note that his leading
n -behavior remains tied to the large-T behavior of a counting function according to the rule (16) ) (15) [14, Equations (2.11Y12),(1.18),(5.2)].

References 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15.

Biane, P., Pitman, J. and Yor, M.: Probability laws related to the Jacobi theta and Riemann zeta functions, Bull. Amer. Math. Soc. 38 (2001), 435Y465 [Sect. 2.3]. Bombieri, E. and Lagarias, J. C.: Complements to Li_s criterion for the Riemann hypothesis, J. Number Theory 77 (1999), 274Y287. Coffey, M. W.: Relations and positivity results for the derivatives of the Riemann function, J. Comput. Appl. Math. 166 (2004), 525Y534. Coffey, M. W.: New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants, preprint (Jan. 2005); Toward verification of the Riemann Hypothesis: application of the Li criterion, Math. Phys. Anal. Geom. 8 (2005), 211Y255. Davenport, H.: Multiplicative Number Theory, 3rd edn., revised by H.L. Montgomery, Grad. Texts in Math. 74, Springer, New York, 2000 [chap. 16]. Dingle, R. B.: Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, New York, 1973. Edwards, H. M.: Riemann_s Zeta Function, Academic Press, New York, 1974 [Sect. 1.10]. Erde´lyi, A.: Asymptotic Expansions, Dover, 1956 [Sect. 2.5]. Gradshteyn, I. S. and Ryzhik, I. M.: Table of Integrals, Series and Products, 5th edn, A., Jeffrey (ed.), Academic, 1994, [Equations (3.721(1)) p. 444 and (4.421(1)) p. 626]. Hashimoto, Y.: Euler constants of Euler products, J. Ramanujan Math. Soc. 19 (2004), 1Y14. Israilov, M. I.: On the Laurent expansion of the Riemann zeta-function, Proc. Steklov Inst. Math. 4 (1983), 105Y112 [Russian: Trudy Mat. Inst. i. Steklova 158 (1981), 98Y104]. Keiper, J. B.: Power series expansions of Riemann_s function, Math. Comput. 58 (1992), 765Y773. Kurokawa, N.: Parabolic components of zeta functions, Proc. Japan Acad. Ser. A 64 (1988), 21Y24; Special values of Selberg zeta functions, In: M. R. Stein and R. Keith Dennis (eds), Algebraic K-Theory and Algebraic Number Theory, (Proceedings, Honolulu 1987), Contemp. Math. 83, Amer. Math. Soc. Providence, 1989, pp. 133Y149. Lagarias, J. C.: Li coefficients for automorphic L-functions, Ann. Inst. Fourier, Grenoble (2006, to appear) ½math:NT=0404394 v4. Landau, E.: Einfu¨hrung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, Chelsea, New York, 1949 [Satz 173 p. 89].

SHARPENINGS OF LI_S CRITERION FOR THE RIEMANN HYPOTHESIS

16. 17. 18. 19. 20. 21.

63

Lehmer, D. H.: The sum of like powers of the zeros of the Riemann zeta function, Math. Comput. 50 (1988), 265Y273. Li, X.-J.: The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory 65 (1997), 325Y333. Li, X.-J.: Explicit formulas for Dirichlet and Hecke L-functions, Illinois J. Math. 48 (2004), 491Y503. Li, X.-J.: An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials, J. Number Theory 113 (2005), 175Y200. Mas´lanka, K.: Effective method of computing Li_s coefficients and their properties, Experiment. Math. (to appear) ½math:NT=0402168 v5. Mas´lanka, K.: An explicit formula relating Stieltjes constants and Li_s numbers, preprint

½math:NT=0406312 v2. 22.

23. 24. 25. 26. 27.

28.

Matsuoka, Y.: A note on the relation between generalized Euler constants and the zeros of the Riemann zeta function, J. Fac. Educ. Shinshu Univ. 53 (1985), 81Y82; A sequence associated with the zeros of the Riemann zeta function, Tsukuba J. Math. 10 (1986), 249Y254. Oesterle´, J.: Re´gions sans ze´ros de la fonction zeˆta de Riemann, typescript (2000, revised 2001, uncirculated). Smith, W. D.: A Bgood^ problem equivalent to the Riemann Hypothesis, e-print on http:// www.math.temple.edu/~wds/homepage/works.html (2005 version, unpublished). de la Valle´e Poussin, C.-J.: Recherches analytiques sur la the´orie des nombres premiers I, Ann. Soc. Sci. Brux. 20 (1896), 183Y256 [p. 251]. Voros, A.: Zeta functions for the Riemann zeros, Ann. Inst. Fourier (Grenoble) 53 (2003), 665Y699; erratum: 54 (2004), 1139. Voros, A.: Zeta functions over zeros of general zeta and L-functions, In: T. Aoki, S. Kanemitsu, M. Nakahara and Y. Ohno (eds), Zeta Functions, Topology and Quantum Physics, (Proceedings, Osaka, March 2003), Developments in Math. 14, Springer New York, (2005), pp. 171Y196. Voros, A.: A sharpening of Li_s criterion for the Riemann Hypothesis, preprint (Saclay-T04/ 040 April 2004, unpublished) ½math:NT=0404213 v2.

Mathematical, Physics, Analysis and Geometry (2006) 9: 65–94 DOI: 10.1007/s11040-005-9003-7

#

Springer 2006

Persistence of Eigenvalues and Multiplicity in the Dirichlet Problem for the Laplace Operator on Nonsmooth Domains PIER DOMENICO LAMBERTI and MASSIMO LANZA DE CRISTOFORIS Dipartimento di Matematica Pura ed Applicata, Universita` di Padova, Via Belzoni 7, 35131 Padova, Italy. e-mail: {lamberti, mldc}@math.unipd.it (Received: 12 August 2004; in final form: 28 October 2005) Abstract. We consider the Dirichlet eigenvalue problem for the Laplace operator on a variable nonsmooth domain. We extend a result of Lupo and Micheletti concerning the structure of the set of domain perturbations which leave the multiplicity of an eigenvalue unchanged, and we study the set of perturbations which leave a certain eigenvalue unchanged. Mathematics Subject Classifications (2000): 35P15, 47H30. Key words: Dirichlet eigenvalues and eigenfunctions, domain perturbation, Laplace operator, special nonlinear operators.

1. Introduction This paper concerns the dependence of the eigenvalues of the Laplace operator with Dirichlet boundary conditions under domain perturbation. We are mainly interested in two problems. The first has been considered by Lupo and Micheletti [13Y15], and by Teytel [17], and consists in analyzing the set of domain perturbations for which a given eigenvalue preserves its multiplicity. The second problem consists in analyzing the set of domain perturbations for which a given eigenvalue remains unchanged. In other words, the second problem consists in analyzing a weaker version of the isospectrality problem, which is concerned with the analysis of the domains for which the entire spectrum is the same. For related results, we refer to Colin de Verdie`re [2Y4]. We consider a Freference_ domain
in Rn , and our perturbed domains are taken as images of
by a Lipschitz continuous homeomorphism
of
into a subset of Rn . Thus our problem consists in studying the set of
’s for which a certain eigenvalue preserves its multiplicity, and the set of
’s for which a certain eigenvalue is preserved. We consider our boundary value problems in the perturbed domain
ð
Þ, and by changing the variables in the corresponding equations and boundary conditions by means of the function
, we transform it into a problem in the fixed domain
. As a second step, we transform such

66

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

problem into an eigenvalue problem for an operator in some space of functions defined in
. Such an operator depends on
and turns out to be selfadjoint with respect to a scalar product which also depends on
. Having to deal with
dependent families of operators which are selfadjoint with respect to
dependent scalar products has certain inconveniences, but our approach has the advantage of requiring very little regularity on
and on
. Instead, by employing a different substitution, one could transform the boundary value problem on
ð
Þ into a boundary value problem on
and obtain a family of selfadjoint operators in a space with a fixed scalar product. However, such an approach would require higher regularity on
(see Lupo and Micheletti [13Y15].) We first consider our problem abstractly. We consider a real Hilbert space H, and we introduce the set O of the pairs of the form ðQ; TÞ, with Q in the set of scalar products on H which generate the topology of H, and with T a compact and selfadjoint operator in the Hilbert space H endowed with the scalar product Q, a space which we denote by HQ . Then we show that O is a real analytic Banach manifold of infinite dimension and codimension. Then we turn to analyze the set of pairs ðQ; TÞ in O for which the persistence of the multiplicity of a certain eigenvalue of T in HQ occurs and we show that such set is a real analytic Banach submanifold of O of finite codimension (cf. Theorem 3.2.) In essence, this is a generalization of an idea of Von Neumann and Wigner [19] who observed that the set of the real symmetric matrices with a double eigenvalue has codimension 2 in the space of the real symmetric matrices (see also Arnold [1] and Colin de Verdie`re [3].) Then we consider the case of a family fðQ½x
; T½x
Þgx2X of pairs in O depending on a parameter x in a Banach space X and we extend a result of Lupo and Micheletti [13, Thm. 1, p. 109] concerning the structure of the set of those x in X for which the multiplicity of a certain eigenvalue of T½x
is preserved (cf. Theorem 3.7.) Similar results are obtained also in the study of the persistence of a fixed eigenvalue, for which the FStrong Arnold Hypothesis_, introduced by Colin de Verdie`re [4], plays a crucial role (cf. Theorem 4.4 and Remark 4.7.) Furthermore, our results concerning the persistence of the multiplicity of eigenvalues allow us to provide a Fgeometric_ description of those real analytic families fðQ½
; T½
Þg2R in O for which all the eigenvalues of T½
, which split from a fixed multiple eigenvalue of T½0
, are differentiable functions of the real parameter , whenever such eigenvalues are enumerated according the Courant Min-Max Principle and not necessarily rearranged according to the celebrated Rellich and Nagy Theorem (see Theorems 3.12, 3.13 and Proposition 3.15.) In the second part of the paper, we transfer our conclusions to the set of
’s for which a certain eigenvalue boundary value problem for the Laplace operator has a fixed multiplicity, and to the set of
’s for which a certain eigenvalue is preserved. Concerning the persistence of the multiplicity, our result extends the validity of that of Lupo and Micheletti [13] for the Dirichlet Problem in two senses. Firstly, we can handle some extra degenerate cases (cf. Remark 3.10.) Secondly, we weaken the reguarity

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

67

assumptions for both
and
. Indeed, Lupo and Micheletti require both
and
to be of class C3 . For
, we require Lipschitz continuity. As far as
is concerned, we just require that the imbedding of the Sobolev space W01;2 ð
Þ into the space L2 ð
Þ is compact, a condition which certainly holds in case
is of finite measure (no matter whether
is bounded or not, and with no regularity assumption on the boundary.) See Theorems 5.11, 5.31, and Remark 5.34. The paper is organized as follows. In Section 2, we introduce our preliminaries and we show that the set O is a real analytic manifold. In Section 3, we prove our generalization of the Lupo and Micheletti [13] result concerning the persistence of the multiplicity. At the end of Section 3, we prove some differentiability results for the eigenvalues of a family of operators depending on one real parameter. In Section 4, we prove our result concerning the persistence of an eigenvalue. In Section 5 we present our applications to the Dirichlet eigenvalue problem, and in particular our generalizations of the results of Lupo and Micheletti [13]. 2. Preliminaries and Introduction of the Manifold O We first introduce some technical preliminaries and notation. Let X , Y, Z be real Banach spaces. We denote by LðX ; Y Þ the Banach space of linear and continuous maps of X to Y endowed with its usual norm of the uniform convergence on the unit sphere of X . We denote by BðX  Y; Z Þ the space of the bilinear and continuous maps of X  Y to Z endowed with the norm of the uniform convergence on the cross product of the unit sphere of X and of the unit sphere of Y. We say that the space X is continuously imbedded in the space Y provided that X is a linear subspace of Y, and that the inclusion map is continuous. We denote by Z the set of integer numbers, and by N the set of natural numbers including 0. The inverse function of an invertible function f is denoted f ð1Þ , as opposed to the reciprocal of a complex-valued function g, or the inverse of a matrix A, which are denoted g1 and A1 , respectively. If A is a matrix with real entries, we denote by At the transpose matrix of A. If A is t invertible, we set At  ðA1 Þ . All elements of Rm are thought of as row vectors. We denote by Mm ðRÞ the set of m  m matrices with real entries, and by Sm ðRÞ the set of symmetric elements of Mm ðRÞ. Let ðH; h; iÞ be a real Hilbert space. Let kk denote the norm associated to the scalar product on H. Let dimðHÞ denote the possibly infinite dimension of H. We denote by HQ the linear space H endowed with a scalar product Q defined on H. We denote by kkQ the norm associated to the scalar product Q on H. We denote by I the identity operator in H. We denote by Kð H; H Þ the real Banach subspace of Lð H; H Þ of those elements T which are compact, i.e., which map
bounded  subsets of H to H ; H the real Banach subsets of H with compact closure. We denote by K s Q Q
 ½
T such that Q Tu; v ¼ Q½u; Tv
for all subspace of K HQ ; HQ of those elements
 u; v 2 HQ . As is well known, if T 2 Ks HQ ; HQ , then there exists a subset ½T


68

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

of R, named the spectrum, with ½T
finite or countable, such that R n ½T
is the set of  such that the operator T  I is a linear homeomorphism of HQ . It is also well-known that all the elements  of ½T
n f0g are eigenvalues of T of finite multiplicity, i.e., the null space Ker ðT  I Þ is nontrivial and of finite dimension. Furthermore, 0 is the only possible accumulation point of ½T
. We denote by jþ ½T
the (possibly infinite) number of elements of ½T
\
0; þ1½, each counted with its multiplicity, and we denote by j ½T
the (possibly infinite) number of elements of ½T
\
 1; 0½, each counted with its multiplicity. We also set J þ ½T
 f j 2 Z : 1 r j r jþ ½T
g; J  ½T
 f j 2 Z : j ½T
r j r  1g: Then there exists a uniquely determined function j 7! j ½T
of J ½T
 J  ½T
[ J þ ½T
to R n f0g such that j 7! j ½T
is decreasing on J  ½T
and on J þ ½T
, and such that     ½T
\
0; þ1½ ¼ j ½T
: j 2 J þ ½T
; ½T
\
1; 0½ ¼ j ½T
: j 2 J  ½T
; and such that each eigenvalue is repeated as many times as its multiplicity. We shall require the imbedding of HQ in H to be continuous, and thus that the scalar product Q be coercive on H. Thus we introduce the following Lemma concerning continuous bilinear forms on H, whose verification is straightforward. LEMMA 2.1. Let ðH; h; iÞ be a real Hilbert space. Let ½
be the map of BðH 2 ; RÞ to R defined by ( ) B½u; u
: u 2 H n f0g ; ½B
 inf kuk2 for all B 2 BðH2 ; RÞ. Then we have j½B
j r kBkBðH2 ;RÞ ;

j½B1
 ½B2
j r kB1  B2 kBðH2 ;RÞ ;   for all B, B1 , B2 2 BðH 2 ; RÞ. In particular, the set B 2 BðH 2 ; RÞ : ½B
> 0 is open in BðH 2 ; RÞ. Since scalar products are bilinear and symmetric forms, we introduce the following notation
 
  Bs H2 ; R  B 2 B H 2 ; R : B½u1 ; u2
¼ B½u2 ; u1
8u1 ; u2 2 H : Clearly, Bs ðH 2 ; RÞ is a closed linear subspace of BðH 2 ; RÞ. Then the set of coercive elements of Bs ðH 2 ; RÞ is denoted
 
  Q H 2 ; R  B 2 Bs H 2 ; R : ½B
> 0 : ð2:2Þ Now we observe that Q is a scalar product on H and the imbedding of HQ in H is a homeomorphism if and only if Q 2 QðH 2 ; RÞ. We obviously have 1=2

½Q
1=2 kuk r kukQ r kQkBðH2 ;RÞ kuk;

ð2:3Þ

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

69

for all u 2 H, and
for all Q 2 QðH 2 ; RÞ. We also note that if Q belongs to QðH 2 ; RÞ, then
L HQ ; HQ equals Lð H; HÞ algebraically and topologically. Similarly, K
 HQ ; HQ equals Kð H; H Þ algebraically and topologically. Instead, Ks HQ ; HQ may vary with Q 2 QðH2 ; RÞ, although the topology of HQ does not. We now set 
  M  ðQ; TÞ 2 Bs H 2 ; R  KðH; H Þ : Q½Tu; v
¼ Q½u; Tv
8u; v 2 H : Clearly, M is a closed subset of Bs ðH 2 ; RÞ  Kð H; H Þ. We now turn to study the set

  O  M \ Q H 2 ; R  KðH; H Þ
 
 ¼ ðQ; TÞ 2 Q H 2 ; R  Kð H; H Þ : T 2 Ks HQ ; HQ ; which is obviously open in M. To do so, we first introduce the following technical Lemma. LEMMA 2.4. Let ðH; h; iÞ be a real Hilbert space. Then the following statements hold. (i) For all B 2 BðH2 ; RÞ, there exists a unique S 2 LðH; H Þ such that B½u1 ; u2
¼ h S½u1
; u2 i 8u1 ; u2 2 H :

ð2:5Þ

The map  of BðH 2 ; RÞ to Lð H; H Þ which takes B 2 BðH2 ; RÞ to the operator S defined in (2.5), is a linear homeomorphism. (ii) The map E of BðH 2 ; RÞ  Lð H; H Þ to Lð H; H Þ defined by
 E½Q; T
 ½Q  ðT  IÞ
8ðQ; TÞ 2 B H 2 ; R  Lð H; H Þ ; where T  I½u1 ; u2
 ðT½u1
; u2 Þ

8u1 ; u2 2 H ;

is bilinear and continuous. (iii) The map G of BðH2 ; RÞ  LðH; HÞ to itself defined by
 G½Q; T
 ðQ; E½Q; T
Þ 8ðQ; TÞ 2 B H 2 ; R  LðH; HÞ ; restricts a real analytic diffeomorphism of QðH2 ; RÞ  LðH; H Þ onto itself. Moreover, if ðQ~; T~Þ 2 O, then
  dG½Q~; T~
ðQ_ ; T_ Þ ¼ Q_ ;  Q_  ðT~  IÞ þ Q~  ð T_  IÞ ;

ð2:6Þ

for all ðQ_ ; T_ Þ 2 BðH 2 ; RÞ  Lð H; H Þ. (iv) G maps QðH 2 ; RÞ  Kð H; H Þ onto itself, and O onto QðH 2 ; RÞ Ks ð H; HÞ.

70

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

Proof. Statement (i) is an immediate consequence of the Riesz Representation Theorem for the dual of H. Statement (ii) is an obvious consequence of statement (i). We now prove statement (iii). If G½Q1 ; T1
¼ G½Q2 ; T2
, then Q1 ¼ Q2 , and E½Q1 ; T1
¼ E½Q2 ; T2
. Since  is an isomorphism, we have Q1 ½T1 ½u1
; u2
¼ Q1 ½T2 ½u1
; u2
8u1 ; u2 2 H: If Q1 2 QðH2 ; RÞ, then Q1 is a scalar product and we must have T1 ¼ T2 . Thus G is injective from QðH 2 ; RÞ  Lð H; H Þ to itself. We now prove surjectivity of G. Let ðQ; SÞ 2 QðH 2 ; RÞ  LðH; HÞ. We want to show that there exists T 2 LðH; H Þ such that Q½T½u1
; u2
¼ < S½u1
; u2 >

8u1 ; u2 2 H:

ð2:7Þ

The existence of T follows by the continuity of S, by the coercivity of Q, and by the Lax-Milgram Lemma. Thus G is a bijection of QðH2 ; RÞ  LðH; H Þ onto itself. Moreover, G is obviously real analytic. We now prove that G is a diffeomorphism. It suffices to show that the differential dG½Q~; T~
of G at ðQ~; T~Þ is an isomorphism for all ðQ~; T~Þ 2 QðH2 ; RÞ  LðH; H Þ. By standard Calculus, ~ ~ _ _ _ ~ _ formula (2.6)   holds. If dG½Q; T
ðQ; T Þ ¼ 0, then Q ¼ 0, and Q  ð T  IÞ ¼ 0, i.e., ~ ~ _ Q T ½u1
; u2 ¼ 0, for all u1 ; u2 2 H. Since Q is a scalar product on H, we have T_ ½u1
¼ 0 for all u1 2 H, and thus T_ ¼ 0. Hence, dG½Q~; T~
is injective. We now prove the surjectivity of dG½Q~; T~
. Let ðQ_ ; S_Þ 2 Bs ðH2 ; RÞ Lð H; H Þ. We must show the existence of T_ 2 Lð H; H Þ such that   S_ ¼  Q_  ðT~  IÞ þ Q~  ð T_  IÞ ; i.e., such that     Q~ T_ ½u1
; u2 ¼ < S_½u1
; u2 >  Q_ T~½u1
; u2 8u1 ; u2 2 H: The existence of T_ follows by the continuity of S_, T~, Q_ , by the coercivity of Q~, and by the LaxYMilgram Lemma. Hence, dG½Q~; T~
is surjective. We now turn to the proof of statement (iv). Let ðQ; TÞ 2 QðH 2 ; RÞ  KðH; H Þ. We first prove that T 2 KðH; HÞ if and only if E½Q; T
2 KðH; HÞ. By the Riesz Representation Theorem, and by the definition of E, we have kE½Q; T
ðw1 Þ  E½Q; T
ðw2 Þk j< E½Q; T
ðw1 Þ  E½Q; T
ðw2 Þ; v> j kvk v2Hnf0g

¼ sup

jQ½T½w1
 T½w2
; v
j kvk v2Hnf0g

¼ sup

8w1 ; w2 2 H;

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

71

and thus ½Q
kT½w1
 T½w2
k r kE½Q; T
ðw1 Þ  E½Q; T
ðw2 Þk r kQkBs ðH2 ;RÞ kT½w1
 T½w2
k 8w1 ; w2 2 H: Then if fun gn2N is a bounded sequence in H, the sequence fT½un
gn2N admits a Cauchy subsequence, if and only if the sequence fE½Q; T
ðun Þgn2N admits a Cauchy subsequence. Hence, T is compact if and only if E½Q; T
is compact. We now show that if ðQ; TÞ belongs to QðH 2 ; RÞ  Kð H; HÞ, then T is selfadjoint in HQ if and only if E½Q; T
belongs to Ks ðH; HÞ. It clearly suffices to observe that

< E½Q; T
ðu1Þ; u2 >

¼ Q½T½u1
; u2


8u1 ; u2 2 H ;

and that h; i, Q½; 
are both symmetric forms.

Ì

Then we have the following. For basic definitions and results on manifolds in Banach spaces, we refer to Lang [11]. THEOREM 2.8. Let ð H; h; iÞ be a real Hilbert space. The set O is a real analytic manifold imbedded in QðH 2 ; RÞ  Kð H; H Þ modelled on Bs ðH2 ; RÞ Ks ðH; H Þ. If ðQ~; T~Þ 2 O, then the tangent space to O at ðQ~; T~Þ is given by 
 T ðQ~;T~Þ O  ðQ_ ; T_ Þ 2 Bs H 2 ; R  KðH; H Þ : ð2:9Þ Q_ ½T~½u1
; u2
þ Q~½ T_ ½u1
; u2
¼ Q_ ½u1 ; T~½u2

þ Q~½u1 ; T_ ½u2

 8u1 ; u2 2 H : Proof. The space Ks ðH; H Þ admits a topological supplement in Kð H; H Þ, and a corresponding projection is given by the map 12 ðT þ T*Þ, T 2 KðH; HÞ, where T* denotes the adjoint of T in H. Thus QðH 2 ; RÞ  Ks ð H; H Þ is a Banach submanifold of Bs ðH2 ; RÞ  KðH; H Þ modelled on Bs ðH 2 ; RÞ  Ks ðH; HÞ. By Lemma 2.4, O is a Banach submanifold of Bs ðH 2 ; RÞ  KðH; H Þ modelled on Bs ðH2 ; RÞ  Ks ðH; HÞ. We now compute the tangent space. By Lemma 2.4, G is a diffeomorphism of O onto QðH 2 ; RÞ  Ks ðH; HÞ. Thus we have that  T ðQ~;T~Þ O ¼


 ðQ_ ; T_ Þ 2 Bs H 2 ; R  Kð H; H Þ :


 dG½Q~; T~
ðQ_ ; T_ Þ 2 Bs H2 ; R  Ks ðH; H Þ :

By formula (2.6), we immediately deduce the validity of equality (2.9).

Ì

72

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

Remark 2.10. By Lemma 2.4, we have seen that one can flatten the set O by means of the diffeomorphism G. We now consider an element ðQ; TÞ 2 O, and we observe that the eigenvalue equation for T T½u
¼ u ;

ð2:11Þ

i.e., the equation Q½Tu; v
¼ Q½u; v
8v 2 H ; is equivalent to the Fgeneralized_ eigenvalue equation E½Q; T
ðuÞ ¼ ½Q
ðuÞ :

ð2:12Þ

The advantage of (2.11) with respect to (2.12), is that (2.11) is an eigenvalue equation, while (2.12) is only a Fgeneralized_ eigenvalue equation. The advantage of (2.12) with respect to (2.11) is that E½Q; T
is selfadjoint with respect to the fixed scalar product of H, while T in (2.11) is selfadjoint with respect to the variable scalar product Q. In our exposition, we will stick to the point of view of variable scalar products, i.e., we prefer to handle (2.11), which is more natural than (2.12), and which turns out to be more useful in our applications to domain perturbation problems on nonsmooth domains (see Section 5.) As we have announced in the Introduction, we will consider two problems. The problem of studying the set of ðQ; TÞ’s which preserve the multiplicity of a certain eigenvalue, and the problem of analyzing the set of ðQ; TÞ’s which preserve a certain eigenvalue. (The latter problem is a weaker form of the well known isospectrality problem.) Before doing so, we need to recall a few results concerning the dependence of the eigenvalues and of the eigenfunctions of T upon variation of T. By exploiting the variational formulas for the eigenvalues of a compact selfadjoint operator, one can prove the following (cf. [8, Thm. 2.5].) THEOREM 2.13. Let H be a real Hilbert space. Let j 2 Z n f0g, then the set Aj  fðQ; TÞ 2 O : j 2 J½T
g is open in M. The function j ½
of Aj to R which takes ðQ; TÞ 2 Aj to j ½T
is continuous. Now we consider a certain finite subset F of Z n f0g, and the set of pairs ðQ; TÞ for which F  J½T
and for which the eigenvalues j ½T
with j 2 F do not equal any of the eigenvalues l ½T
of T with l 2 J½T
n F. Thus we introduce the following notation.     = j ½T
: j 2 F 8l 2 J½T
n F : A½F
 ðQ; TÞ 2 O : F  J½T
; l ½T
2 ð2:14Þ

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

73

By Theorem 2.13, the functions j ½
are continuous on A½F
, for all j 2 F, and A½F
is open in M. For each finite subset F of Z n f0g, and ðQ; TÞ 2 A½F
, we define the orthogonal projection PF ½Q; T
of HQ onto the subspace E½T; F
of HQ generated by the set    u 2 HQ : Tu ¼ u; for some  2 j ½T
: j 2 F : Then we have the following Theorem (cf. [10, Thm. 2.18, Props. 2.20, 2.21].) THEOREM 2.15. Let H be a real Hilbert space. Let F be a finite nonempty subset of Z n f0g. Let ðQ~; T~Þ 2 A½F
. Then the following statements hold. ~ of ðQ~; T~Þ in QðH 2 ; RÞ  LðH; HÞ, (i) There exist an open neighborhood W # ~ to LðH; HÞ such that P# ½Q; T
¼ and a real analytic operator PF of W F ~ \ A½F
. T
for all ðQ; TÞ 2 W PF ½Q;   (ii) Let u~j : j ¼ 1; . . . ; jFj be an orthonormal basis of E½T~; F
in HQ~ . Then there exist an open neighborhood W o of ðQ~; T~Þ in QðH 2 ; RÞ  Lð H; H Þ ~ , and jFj real analytic operators uj ½; 
for j ¼ 1; . . . ; jFj of contained in W   W o to H such that uj ½Q; T
: j ¼ 1; . . . ; jFj is an orthonormal basis of # o the  image of PF ½Q; T
in HQ , for all ðQ; TÞ 2 W \ A½F
, and such that uj ½Q; T
: j ¼ 1; . . . ; jFj is an orthonormal subset of the image of P]F ½Q; T
in HQ , for all ðQ; TÞ 2 W o , and such that uj ½Q~; T~
¼ u~j for all j ¼ 1; . . . ; jFj. (iii) Let S be the map of W o to the set MjFj ðRÞ of jFj  jFj-matrices with real entries, defined by S½Q; T
 ðS hk ½Q; T
Þh;k¼1;...;jFj

ð2:16Þ

 ðQ½T ½uk ½Q; T

; uh ½Q; T

Þh;k¼1;...;jFj ; for all ðQ; TÞ 2 W o . Then S½; 
is real analytic, and S½Q; T
is symmetric for all ðQ; TÞ 2 W o \ A½F
. Furthermore, if ðQ; TÞ 2 W o \ A½F
, then the numbers j ½T
for j 2 F are the eigenvalues of S½Q; T
counted with their multiplicity. Finally, if we further assume that j ½T~
assumes a common value ~ for all j 2 F, then the differential of S½; 
at ðQ~; T~Þ is delivered by the formula dS½Q~; T~
ðQ_ ; T_ Þ
 
 ¼ Q~ T_ ½u~k
; u~h h;k¼1;...;jFj 8ðQ_ ; T_ Þ 2 Bs H 2 ; R  LðH; H Þ: ð2:17Þ

74

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

3. On the Set of Scalar Products and Operators Which Preserve the Multiplicity of a Certain Eigenvalue In this section, we consider a finite nonempty subset of Z n f0g, and we analyze the structure of the set   ½F
 ðQ; TÞ 2 A½F
: j ½T
have a common value F ½T
for all j 2 F : ð3:1Þ We do so by means of the following Theorem, which is in the same spirit of some observations of von Neumann and Wigner [19] concerning symmetric matrices. THEOREM 3.2. Let H be a real Hilbert space. Let F be a finite nonempty subset of Z n f0g. If ½F
6¼ ;, then ½F
is a real analytic Banach submanifold of O of codimension ðjFjðjFj þ 1Þ=2Þ  1. Moreover, if ðQ~; T~Þ 2 ½F
, then the tangent space to ½F
at ðQ~; T~Þ is delivered by the formula 
 T ðQ~;T~Þ ½F
 ðQ_ ; T_ Þ 2 Bs H 2 ; R  Kð H; H Þ : Q_ ½T~½u1
; u2
þ Q~½ T_ ½u1
; u2
¼ Q_ ½u1 ; T~½u2

þ Q~½u1 ; T_ ½u2



 ~ ~ _ _ 8u1 ; u2 2 H; Q½ T ½~ uh
; u~k
¼ hk Q½ T ½~ u1
; u~1
; h; k ¼ 1; . . . ; jFj ; ð3:3Þ where hk denotes the Kronecker symbol defined by hk ¼ 1 if h ¼ k, hk ¼ 0 if h 6¼ k and where f~ u1 ; . . . ; u~jFj g is an orthonormal basis of E½T~; F
in HQ~ . ~ ~ Proof. Let ðQ; T Þ 2 ½F
. Let uj ½; 
: j ¼ 1; . . . ; jFj , S be as in Theorem 2.15. It clearly suffices to show that the set G½W o \ ½F

is a submanifold of codimension ðjFjðjFj þ 1Þ=2Þ  1 of QðH 2 ; RÞ  Ks ð H; H Þ. We observe that a pair ðQ; TÞ of W o \ A½F
belongs to ½F
if and only if all the eigenvalues of the matrix S½Q; T
coincide. Let n o 0  A  ðali Þl;i¼1;...;jFj 2 SjFj ðRÞ : a11 ¼ 0 : SjFj Let S^ be the function of W o to MjFj ðRÞ defined by

 ðS li ½Q; T
 li S 11 ½Q; T
Þl;i¼1;...;jFj ; S^½Q; T
 S^li ½Q; T
l;i¼1;...;jFj

ð3:4Þ

for all ðQ; TÞ 2 W o . Obviously, S^ maps W o \ A½F
to S0jFj , and n o ½F
\ W o ¼ ðQ; TÞ 2 W o \ A½F
: S^½Q; T
¼ 0 : Now we denote by F the inverse of the restriction of G to O.
To prove our statement, it suffices to show that the differential of S^  F at Q~; G½Q~; T~
is

75   0 surjective onto SjFj . Since dF Q~; G½Q~; T~
maps Bs ðH 2 ; RÞ  Ks ðH; HÞ onto the 0 , tangent space TðQ~;T~Þ O, it suffices to show that for all A  ðali Þl;i¼1;...;jFj 2 SjFj 2 _ _ there exists a pair ðQ; T Þ 2 Bs ðH ; RÞ  Ks ðH; HÞ such that PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

Q_ ½T~½u1
; u2
þ Q~½ T_ ½u1
; u2


ð3:5Þ

¼ Q_ ½u1 ; T~½u2

þ Q~½u1 ; T_ ½u2

8u1 ; u2 2 H ; i.e., such that ðQ_ ; T_ Þ 2 T ðQ~;T~Þ O, and such that Q~½ T_ ½~ uh
; u~k
 hk Q~½ T_ ½~ u1
; u~1
¼ ahk ; h; k ¼ 1; . . . ; jFj

ð3:6Þ

(see also Theorem 2.8, and Theorem 2.15 (iii).) By taking Q_ ¼ 0, it suffices to _ show the existence
 of T 2 Ks ðHQ~ ; HQ~ Þ such that (3.6) holds. We define G 2 L E½T~; F
; E½T~; F
by setting Q~½G½~ uk
; u~h
¼ ahk

h; k ¼ 1; . . . ; jFj :

Clearly, G is selfadjoint in E½T~; F
endowed with the scalar product Q~. Then we consider the orthogonal projection P~  PF ½Q~; T~
of HQ~ onto E½T~; F
, and we set T_  G  P~. Clearly, T_ 2 Ks ðHQ~ ; HQ~ Þ. Since a11 ¼ 0, condition (3.6) is obviously satisfied. Moreover, T_ is compact because it has finite rank. Finally, the tangent  ~; G½Q~; T~
-image of the null space of the space to ½F
coincides with the dF Q
 differential of S^  F at Q~; G½Q~; T~
, and thus with the right-hand side of (3.3).

Ì

As we shall see in Section 5, we may have to deal with real analytic families of pairs ðQ; TÞ depending on a parameter in Banach space. Then we state the following Theorem, which extends the work of Lupo and Micheletti to the case of Hilbert spaces with variable scalar products (cf. Lupo and Micheletti [13, Thm. 1 p. 109].) THEOREM 3.7. Let H be a real Hilbert space. Let 0 < r 2 N [ f1g. Let F be a finite nonempty subset of Z n f0g. Let X be a real Banach space. Let V be an open subset of X . Let N be a map of class Cr (or real analytic, respectively) of V to O. Let N ½x
 ðQ½x
; T½x
Þ for all x 2 V. Let ½V; F
 f x 2 V : N ½x
2 ½F
g :   x
; F
in HQ~ . Let x~ 2 ½V; F
. Let u~1 ; . . . ; u~jFj be an orthonormal basis of E½T½~ Let  be the dimension of the vector subspace n


o     uh
ð Þ; u~k  hk Q½~ x
djx¼~x T½~ u1
ð Þ; u~1 h;k¼1;...;jFj : 2 X Q½~ x
djx¼~x T½~ ð3:8Þ

76

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

0 of SjFj ðRÞ. Then 0 r  r ðjFjðjFj þ 1Þ=2Þ  1, and there exists an open neighborhood V~ of x~ in V such that ½V; F
\ V~ is contained in a Banach submanifold of X of codimension  and of class Cr (or real analytic, respectively.) Furthermore, if  ¼ ðjFjðjFj þ 1Þ=2Þ  1, then we can choose V~ so that ½V; F
\ V~ is a Banach submanifold of X of codimension  and of class Cr (or real analytic, respectively), whose tangent space at the point x~ equals

 2X : 
    uh
ð Þ; u~k  hk Q½~ x
djx¼~x T½~ u1
ð Þ; u~1 h;k¼1;...;jFj ¼ 0 : Q½~ x
djx¼~x T½~   Proof. Let uj ½; 
: j ¼ 1; . . . ; jFj , S, W o be as in Theorem 2.15 for ðQ; TÞ  ðQ½x
; T½x
Þ and ðQ~; T~Þ  ðQ½~ x
; T½~ x
Þ. Then we consider the map S^ defined in (3.4), and we note that n o ½V; F
\ N ðW o Þ ¼ x 2 N ðW o \ A½F
Þ : S^  N ½x
¼ 0 ; where N denotes the inverse image by the function N . By (2.17) it follows that the image of the differential dðS^  N Þð~ xÞ of S^  N at x~ is the space in (3.8). If  ¼ ðjFjðjFj þ 1Þ=2Þ  1, then the image of dðS^  N Þð~ xÞ coincides with 0 SjFj ðRÞ, and the second part of the statement follows. Otherwise, we consider the 0 ðRÞ onto the image of the space in (3.8), and we observe that projection 0F of SjFj xÞ is surjective, and that accordingly by choosing the differential dð 0F  S^  N Þð~ V~ sufficiently small, the set x 2 V~ : 0F  S^  N ½x
¼ 0g is a manifold of Ì codimension  containing ½V; F
\ V~ . Remark 3.9. Condition  ¼ ðjFjðjFj þ 1Þ=2Þ  1 of Theorem 3.7 can be shown to be equivalent to the transversality of the map N to the manifold ½F
of (3.1) at the point x~. Thus if  ¼ ðjFjðjFj þ 1Þ=2Þ  1, then one could deduce the conclusion on ½V; F
\ V~ of Theorem 3.7 by exploiting the Transversality Theorem for Banach manifolds. The use of the Transversality argument in this kind of matters seems to go back to Arnold [1] (see also Colin de Verdie`re [3], Lupo and Micheletti [13Y15].) However, such an argument seems to have been used to consider maps with values in Ks ðH; HÞ and manifolds in Ks ðH; HÞ. The authors are unaware of similar arguments for maps with values in O and submanifolds of O. The use of the manifold O seems to be necessary in the applications to boundary value problems on nonsmooth domains such as those of Section 5. For such problems the use of maps with values in Ks ðH; HÞ and manifolds in Ks ðH; HÞ would be suitable only under more restrictive regularity assumptions (see Lupo and Micheletti [13Y15].) Remark 3.10. If instead 0 r  < ðjFjðjFj þ 1Þ=2Þ  1, then the Transversality condition is violated, and the lower  is, and the more the Transversality condi-

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

77

tion is violated. At any rate, the information given in Theorem 3.7 seems to be significant also in case 0 <  < ðjFjðjFj þ 1Þ=2Þ  1. Indeed, roughly speaking, Theorem 3.7 says that, it is possible to Fmodify one of  independent parameters_ of a parameter dependent family of operators, in order to make the multiplicity of a certain eigenvalue decrease. Treatment of case 0 r  < ðjFjðjFj þ 1Þ=2Þ  1 seems to be new. By taking Q½
identically equal to the scalar product of H, and X and V equal to Ks ð H; H Þ in the previous statement, we can immediately deduce the validity of the following Corollary. COROLLARY 3.11. Let H be a real Hilbert space. Let F be a finite nonempty subset of Z n f0g. If the set  ½Ks ðH; H Þ; F
¼ T 2 Ks ðH; HÞ :   F  J½T
; l ½T
2 = j ½T
: j 2 F 8l 2 J½T
n F;  j ½T
have a common value F ½T
for all j 2 F ; is not empty, then it is a real analytic submanifold of Ks ð H; H Þ of codimension ðjFjðjFj þ 1Þ=2Þ  1. Corollary 3.11 is a known result (see Colin de Verdie`re [3, Thm. 2, p. 437].) We end this section by considering some applications of the previous results to a different problem, which is however closely related to those considered above. As is well known, given a real analytic family fðQ½
; T½
Þg2I in O depending on  in a certain open neighborhood I of 0 in R, the eigenvalues of T½
are generally not differentiable at  ¼ 0. However, differentiability may occur in certain cases. We now show a link between such a differentiability, and a geometric condition involving the tangent space at ðQ½0
; T½0
Þ to the manifold ½F
. To do so, we first introduce the following consequence of a variant of a result of Rellich and Nagy (see [10, Thm. 2.27, Cor. 2.28].) THEOREM 3.12. Let H be a real Hilbert space. Let F be a finite nonempty subset of Z n f0g. Let ðQ~; T~Þ 2 ½F
. Let W o , S be as
in Theorem 2.15. Let I be an open interval of the real line containing 0. Let Q½
; T ½
2I be  a real analytic family in W o \ A½F
, with ðQ½0
; T½0
Þ ¼ ðQ~; T~Þ. Let u~j : j ¼ 1; . . . ; jFjg be an orthonormal basis for E½T~; F
with respect to the scalar ~ on H. Let S ½Q½
; T½

be the symmetric matrix of (2.16) computed at product Q ðQ½
; T½
Þ, for all  2 I. Then the following statements hold.

78

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

  (i) Possibly shrinking I, there exists a family j ðÞ j2F of real analytic functions of I to R such that for each  2 I there exists a bijection  of F to itself with ð jÞ ðÞ ¼ j ½T½

for all j 2 F. (ii) Let j0 2 F. Then there exist j1 , j2 2 F,  > 0 such that j0 ½T½

¼ j1 ðÞ for 0 r  < , j0 ½T½

¼ j2 ðÞ for  <  r 0. In particular, the function 7!j0 ½T½

has right and left derivatives at 0, and the set of all such right and left derivatives as j0 ranges in F coincides with the set fj0 ð0Þ : j 2 Fg. (iii) Let S_ 0 be the matrix h i

~ ~ ½
; u Q~ dT½
u k h d j¼0

h;k¼1;...;jFj

:

Then S_ 0 is symmetric and
 det I  S_ 0 ¼ j2F ð  j0 ð0ÞÞ ; for all 2 R. Then we have the following. THEOREM 3.13. Let the same assumptions of Theorem 3.12 hold. Then the maps  7! j ½T½

for j 2 F are all differentiable at 0 if and only if dfðQ½
; T½
Þg=dj¼0 belongs to the tangent space TðQ~;T~Þ ½F
of (3.3). Proof. By Theorem 3.12 (ii), the right and left derivatives of j ½T½

at  ¼ 0 exist. Since we have ordered the eigenvalues in decreasing order, we have d d  min F ½T½

U . . . U  max F ½T½

; þ d j¼0 d j¼0þ d d  min F ½T½

r . . . r  max F ½T½

: d j¼0 d j¼0

ð3:14Þ

Thus, a necessary and sufficient condition in order that the right and left derivatives of j ½T½

at  ¼ 0 coincide for all j 2 F, is that all the numbers in (3.14) coincide. Then by Theorem 3.12 (ii), (iii), such condition is equivalent to the condition that the matrix S_ 0 be a multiple of the identity. Since ðQ½
; T½
Þ 2 O for all  2 I, we have dfðQ½
; T½
Þg=dj¼0 2 T ðQ~;T~Þ O. Thus the condition that S_ 0 is a multiple of the identity is equivalent to the condition that Ì dfðQ½
; T½
Þg=dj¼0 belongs to the vector space in (3.3). Theorem 3.13 has a geometric interpretation. It says that the maps  7! j ½T½

are differentiable at the point  ¼ 0 for all j 2 F if and only if the curve fðQ½
; T½
Þg 2 I in O is tangent to the manifold ½F
at the point ðQ½0
; T½0
Þ. In case jFj ¼ 2, we can also prove the following statement, which gives further information on the behaviour of the eigenvalues.

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

79

PROPOSITION 3.15. Let the same assumptions of Theorem 3.12 hold. Let jFj ¼ 2, F  f j1 ; j2 g. If dfðQ½
; T½
Þg=dj¼0 does not belong to the vector space defined in (3.3), then none of the functions fj ½T½

: j 2 Fg is differentiable at  ¼ 0. Moreover, possibly shrinking I, we can assume that the eigenvalues j ½T½

j 2 F are simple for all  2 I n f0g.   Proof. Let j ðÞ j2F be as in Theorem 3.12 (i). If dfðQ½
; T½
Þg=dj¼0 does not belong to the vector space defined in (3.3), then by Theorem 3.13, at least one of the functions of fj ½T½

gj2F is not differentiable at  ¼ 0. Then it follows by Theorem 3.12 (ii), that the derivatives at 0 of the functions j for j 2 F do not coincide. Then by Theorem 3.12 (ii), both the functions of fj ½T½

gj2F are not differentiable at  ¼ 0. Moreover, condition j01 ð0Þ 6¼ j02 ð0Þ, and the real analyticity of j for j 2 F imply that possibly shrinking I, j1 ðÞ 6¼ j2 ðÞ for all Ì  2 I n f0g.

4. On the Set of Scalar Products and Operators which Preserve a Certain Eigenvalue In this section, we analyze the structure of the set of pairs ðQ; TÞ for which a certain eigenvalue is preserved. Then we introduce the following. THEOREM 4.1. Let H be a real Hilbert space. Let F be a finite nonempty subset of Z n f0g. Let ~ 2 R n f0g. Let   ½F; ~
 ðQ; TÞ 2 A½F
: j ½T
¼ ~; 8j 2 F :

ð4:2Þ

If ½F; ~
6¼ ;, then ½F; ~
is a real analytic Banach submanifold of O of codimension equal to ðjFjðjFj þ 1Þ=2Þ. Moreover, if ðQ~; T~Þ 2 ½F; ~
, then the tangent space to ½F; ~
at ðQ~; T~Þ is given by the formula 
 T ðQ~;T~Þ ½F; ~
 ðQ_ ; T_ Þ 2 Bs H2 ; R  KðH; H Þ : Q_ ½T~½u1
; u2
þ Q~½ T_ ½u1
; u2
¼ Q_ ½u1 ; T~½u2

þ Q~½u1 ; T_ ½u2

 uh
; u~k
¼ 0 8h; k ¼ 1; . . . ; jFj : 8u1 ; u2 2 H; Q~½ T_ ½~ ð4:3Þ Proof. Let ðQ~; T~Þ 2 ½F; ~
. Let W o , S be as in Theorem 2.15. We observe that a pair ðQ; TÞ of W o \ A½F
belongs to ½F; ~
if and only if all the eigenvalues of the matrix S½Q; T
coincide with ~. Thus ½F; ~
\ W o ¼ fðQ; TÞ 2 W o \ A½F
: S½Q; T
¼ ~Ig:

80

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

Then by arguing as in the proof of Theorem 3.2, we can show that the differential of S at ðQ~; T~Þ considered as a map of T ðQ~;T~Þ O to SjFj ðRÞ is surjective, and we conclude that ½F; ~
is a submanifold of codimension ðjFjðjFj þ 1Þ=2Þ of O. The formula for the tangent space can be proved by taking the intersection of the Ì null space of the differential of S at ðQ~; T~Þ with T ðQ~;T~Þ O. In order to deal with the case of a real analytic family of pairs ðQ; TÞ, we introduce the following result. THEOREM 4.4. Let H be a real Hilbert space. Let 0 < r 2 N [ f1g. Let F be a finite nonempty subset of Z n f0g. Let ~ 2 R n f0g. Let X be a real Banach space. Let V be an open subset of X . Let N be a map of class Cr (or real analytic, respectively) of V to O. Let N ½x
 ðQ½x
; T½x
Þ for all x 2 V. Let ½V; F; ~
 f x 2 V : N ½x
2 ½F; ~
g :   x
; F
in Let x~ 2 ½V; F; ~
. Let u~1 ; . . . ; u~jFj be an orthonormal basis in E½T½~ HQ~ . Let # be the dimension of the vector subspace n
o   uh
ð Þ; u~k h;k¼1;...;jFj : 2 X ð4:5Þ Q½~ x
djx¼~x T½~ of SjFj ðRÞ. Then 0 r # r ðjFjðjFj þ 1Þ=2Þ, and there exists an open neighborhood V~ of x~ in V such that ½V; F; ~
\ V~ is contained in a Banach submanifold of X of codimension # and of class Cr (or real analytic, respectively.) Furthermore, if # ¼ ðjFjðjFj þ 1Þ=2Þ, then we can choose V~ so that ½V; F; ~
\ V~ is a Banach submanifold of X of codimension # and of class Cr (or real analytic, respectively), whose tangent space at the point x~ equals n o
  uh
ð Þ; u~k h;k¼1;...;jFj ¼ 0 : ð4:6Þ 2 X : Q½~ x
djx¼~x T½~   Proof. Let uj ½; 
: j ¼ 1; . . . ; jFj , S, W o be as in Theorem 2.15 for ðQ; TÞ  ðQ½x
; T½x
Þ and ðQ~; T~Þ  ðQ½~ x
; T½~ x
Þ. Then we note that ½V; F; ~
\ N ðW o Þ ¼ fx 2 N ðW o \ A½F
Þ : S  N ðxÞ ¼ ~Ig : By (2.17), it follows that the image of the differential d ðS  N Þð~ xÞ of S  N at x~ is the space in (4.5). Then we proceed exactly as in the proof of Theorem 3.7.

Ì Also for Theorem 4.4 one could formulate remarks analogous to Remarks 3.9 and 3.10. Remark 4.7. Condition # ¼ ðjFjðjFj þ 1Þ=2Þ corresponds to the so-called FStrong Arnold Hypothesis_ introduced by Colin de Verdie`re [4]. See also Arnold [1].

81

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

By taking Q½
identically equal to the scalar product of H, and X and V equal to Ks ð H; H Þ, we can immediately deduce the validity of the following Corollary. COROLLARY 4.8. Let H be a real Hilbert space. Let F be a finite nonempty subset of Z n f0g. Let ~ 2 R. If the set ½Ks ð H; H Þ; F; ~
  ¼ T 2 Ks ðH; HÞ : F  J½T
; l ½T
6¼ ~ 8l 2 J½T
n F; j ½T
¼ ~ 8j 2 F ; is not empty, then it is a real analytic Banach submanifold of Ks ð H; H Þ of codimension equal to ðjFjðjFj þ 1Þ=2Þ. Corollary 4.8 is a known result (see Colin de Verdie`re [3, Thm. 2, p. 437].)

5. Applications to the Dirichlet Problem for the Laplace Operator In this section, we apply the results of the previous section to the Dirichlet problem for the Laplace operator. Let
be an open subset of Rn . In this paper, we shall consider only case n U 2. We denote by cl
the closure of
and by @
the boundary of
. We denote by L2 ð
Þ the space of square summable real valued measurable functions defined on
, and by W01;2 ð
Þ the Sobolev space obtained by taking the closure of the space Dð
Þ of the C1 functions with compact support in
in the Sobolev space W 1;2 ð
Þ of distributions in
which have weak derivatives up to the first order in L2 ð
Þ, endowed with the norm defined by ( kukW 1;2 ð
Þ 

kuk2L2 ð
Þ þ

n X

)1=2 kuxl k2L2 ð
Þ

8u 2 W 1;2 ð
Þ :

ð5:1Þ

l¼1

Now, we are interested in open connected subsets
of Rn such that W01;2 ð
Þ is compactly imbedded in L2 ð
Þ:

ð5:2Þ

As is well known, if (5.2) holds, then the Poincare´ inequality holds in
(cf., e.g., Evans [6, Proof of Thm. 1, p. 275].) Then we deform
by a Lipschitz continuous homeomorphism of the class A
which we now introduce. We denote by Lipð
Þ the set of Lipschitz continuous functions of
to R, and we set     j
ðxÞ 
ðyÞj n : x; y 2
; x 6¼ y > 0 : A

2 ðLipð
ÞÞ : l
½

 inf jx  yj ð5:3Þ

82

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

We note that l
½

r j det D
ðxÞj1=n ;

ð5:4Þ

for almost all x 2
(cf. [12, Lem. 4.22].) If
2 A
, then
is injective. Then by the Theorem of the Invariance of the Domain (cf., e.g., Deimling [5, Thm. 4.3]),
ð
Þ is open. Condition
2 A
implies that both
and
ð1Þ are Lipschitz continuous, and thus
can be extended to a homeomorphism of cl
onto cl
ð
Þ, which we still denote by
. Clearly
ð@
Þ ¼ @
ð
Þ :

ð5:5Þ

Now it can be verified that if
satisfies (5.2) and if
2 A
, then
ð
Þ also satisfies (5.2) (cf. [8, Prop. 3.7 (vii)].) Accordingly, the Dirichlet eigenvalue problem Z Z t DvDw dy ¼ vw dy 8w 2 W01;2 ð
ð
ÞÞ ð5:6Þ
ð
Þ


ð
Þ

in the unknowns v 2 W01;2 ð
ð
ÞÞ (the Dirichlet eigenfunctions), 2 R (the Dirichlet eigenvalues) has a sequence of eigenvalues 0 < 1 ½

< 2 ½

r . . . ; which we write as many times as their multiplicity. As usual, we introduce the seminorm   j f ðxÞ  f ðyÞj : x; y 2
; x 6¼ y 8f 2 Lipð
Þ ; j f j1  sup jx  yj on Lipð
Þ. It is easily seen that A
is open in ðLipð
ÞÞn (cf. [12, Prop. 4.29], [10, Lemma 3.11].) As is well known, ðLipð
Þ; jj1 Þ is a complete seminormed space. However, we prefer to deal with a normed space, rather than with a seminormed space. Then we will state our results for an arbitrary Banach space X
, continuously imbedded in ðLipð
Þ; jj1 Þ. Alternatively, one could also endow Lipð
Þ with a norm which renders Lipð
Þ a Banach space continuously imbedded in ðLipð
Þ; jj1 Þ, and take X
equal to such Banach space. Then we now wish to analyze the set of
’s in A
\ X n
for which the multiplicity of a certain eigenvalue remains constant, and the set of
’s for which a certain eigenvalue j ½

remains constant in
. Our plan is to change the variables in problem (5.6) by means of
, and to obtain a problem in W01;2 ð
Þ. To do so, we need to introduce some notation, under the assumption that
is a nonempty open connected subset of Rn satisfying (5.2), and that
2 A
. We denote by I the canonical imbedding of W01;2 ð
Þ into L2 ð
Þ. We denote by J


83

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

the operator of L2 ð
Þ to the strong dual W 1;2 ð
Þ of W01;2 ð
Þ which takes u 2 L2 ð
Þ to the element J
½u
of W 1;2 ð
Þ defined by J
½u
½w


Z

uwjdet D
j dx 8w 2 W01;2 ð
Þ ;




ð5:7Þ



2 for all u 2 L2 ð
Þ. We denote by Q
the function of W01;2 ð
Þ to R defined by Z

Q
½u1 ; u2





Du1 ðD
Þ1 ðD
Þt Dut2 jdet D
j dx ;

ð5:8Þ

for all u1 ; u2 2 W01;2 ð
Þ. The form Q
is clearly a scalar product in the space W01;2 ð
Þ, which makes W01;2 ð
Þ a Hilbert space, which we denote by the symbol 1;2 1;2 w1;2 0;
ð
Þ. Then we write simply w0 ð
Þ in case
is the identity. Thus w0 ð
Þ is endowed with its usual energy scalar product

< u1 ; u2 > 

Z


Du1 Dut2 dx

8u1 ; u2 2 W01;2 ð
Þ :

We denote by 
the operator of W01;2 ð
Þ to W 1;2 ð
Þ which takes u 2 W01;2 ð
Þ to the element 
½u
of W 1;2 ð
Þ defined by 
½u
½w
¼ Q
½u; w
for all w 2 W01;2 ð
Þ. Clearly, 
is the operator obtained by pulling back to
the Laplace operator defined on
ð
Þ. Then we introduce the following (see [10, Thm. 3.18].) THEOREM 5.9. Let
be an open connected subset of Rn such that (5.2) holds. Let
2 A
. Then the following statements hold. (i) The operator 
is a linear homeomorphism of W01;2 ð
Þ onto W 1;2 ð
Þ, ð1Þ and the operator T
 
 J
 I is compact and selfadjoint in 1;2 w0;
ð
Þ.

1;2 (ii) The pair ð ; vÞ 2 R  W0 ð
ð
ÞÞ n f0g satisfies (5.6) if and only if > 0 and the pair ð  1 ; u  v 
Þ belongs to the set
0; þ1½W01;2 ð
Þ and satisfies equation u ¼ T
u :





ð5:10Þ

(iii) Equation (5.10) has a decreasing sequence j ½

j2Nnf0g of eigenvalues in
0; þ1½, and j ½

¼ 1 j ½

for all j 2 N n f0g, where f j ½

gj2Nnf0g are the eigenvalues of (5.6) counted with their multiplicity. Each eigenvalue of (5.6) or of (5.10) has finite multiplicity. By applying Theorem 3.7, we can deduce the following generalization of the corresponding result of Lupo and Micheletti [13].

84

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

THEOREM 5.11. Let
be a connected open subset of Rn such that (5.2) holds. Let X
be a Banach space continuously imbedded in Lipð
Þ. Let F be a finite nonempty subset of N n f0g. Let  
½F

2 A
\ X n
:  

l ½

2 = j ½

: j 2 F 8l 2 N n ðF [ f0gÞ;

j ½

have a common value F ½

8j 2 F : ~ Let
~ 2 
½F
. Let v~1 ; . . . ; v~jFj be an orthonormal basis in w1;2 0 ð
ð
ÞÞ of the ~ eigenspace of  corresponding to F ½

. Let Z


 hk ½
 D~ vh D~ vtk  F ½
~
~ vh v~k div 
~ð1Þ dy
~ð
Þ Z h



t i D~ vh D 
~ð1Þ þ D 
~ð1Þ D~ ð5:12Þ  vtk dy;
~ð
Þ

for all 2 ðLipð
ÞÞn , h; k ¼ 1; . . . ; jFj. Let  be the dimension of the vector subspace n o ðhk ½
 hk 11 ½
Þh;k¼1;...;jFj : 2 X n
ð5:13Þ of S0jFj ðRÞ. Then 0 r  r ðjFjðjFj þ 1Þ=2Þ  1, and we can choose an open ~ is contained in a real ~ of
~ in A
\ X n such that 
½F
\ V neighborhood V
n analytic Banach submanifold of X
of codimension . Furthermore, if  ¼ ~ is a real analytic ~ so that 
½F
\ V ðjFjðjFj þ 1Þ=2Þ  1, then we can choose V n Banach submanifold of X
of codimension ðjFjðjFj þ 1Þ=2Þ  1, whose tangent space at
~ equals the space n o ð5:14Þ 2 X n
: ðhk ½
 hk 11 ½
Þh;k¼1;...;jFj ¼ 0 : Proof. By standard calculus in Banach space (see also [10, Proof of Thm. 3.21]), the map
7! ðQ
; T
Þ is real analytic from A
\ X n
to the set  

2

1;2 1;2 O
 ðQ; TÞ 2 Q w0 ð
Þ ; R  K w1;2 0 ð
Þ; w0 ð
Þ :

 1;2 1;2 T 2 Ks ðw0 ð
Þ; QÞ; ðw0 ð
Þ; QÞ : ð5:15Þ By Theorem 2.8, the set O
is a real analytic Banach submanifold of the space 2 1;2 1;2 ~j  v~j 
~, for j ¼ Bs ððw1;2 0 ð
ÞÞ ; RÞ  Kðw0 ð
Þ; w0 ð
ÞÞ. Now we set u 1; . . . ; jFj. By Theorem 5.9 (ii), f~ uj : j ¼ 1; . . . ; jFjg is an orthonormal basis of

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

85

E½T
~; F
in w1;2 ð
Þ. By [10, Lem. 3.26], the ðh; kÞ-entry of the matrix in (5.13) 0;
~ coincides with n h i h io  F ½
~
Q
~ dj
¼
~T
½~ uh
ð Þ; u~k  hk Q
~ dj
¼
~T
½~ u1
ð Þ; u~1 : Also, it is immediate to recognize that   
½F
¼
2 A
\ X n
: ðQ
; T
Þ 2 ½F
; where ½F
has been defined in (3.1) relatively to the Fenvironment_ Hilbert space w1;2 0 ð
Þ. Then we can invoke Theorem 3.7, and conclude that 0 r  r ~ of
~ in A
\ X n ðjFjðjFj þ 1Þ=2Þ  1, and that there exists a neighborhood V
~ such that 
½F
\ V is contained in a Banach submanifold of X n
of codimension Ì , and that the second part of the statement holds. Actually, we can give more information on the codimension  by introducing a proper generalization of a corresponding result of Lupo and Micheletti under very weak assumptions on
(see Theorem 5.31.) Our strategy is as follows. We first show that we can replace 
~ð1Þ in (5.13) by smooth and bounded functions on
~ð
Þ (see Lemma 5.23.) Then we exploit an equality for  (cf. Proposition 5.25), which holds for all bounded ’s, and finally we draw our conclusion on the codimension  (cf. Theorem 5.31.) To do so, we need to introduce some preliminaries. Let
be an open subset of Rn . The space L1 ð
Þ is well known to be canonically isometric to the strong dual of L1 ð
Þ. Accordingly, if f fl gl2N is a sequence in L1 ð
Þ, we shall say that f fl gl2N has a weak* limit f in L1 ð
Þ, if Z lim l!1

fl g dx ¼


Z fg dx

8g 2 L1 ð
Þ :




As is well known, all bounded sequences in L1 ð
Þ have weakly* convergent subsequences. Similarly, we shall speak about weakly* convergent subsequences of ðL1 ð
ÞÞn . In the sequel, we shall exploit the following well known result. THEOREM 5.16. Let D be a subset of Rn . If f 2 LipðDÞ, then there exists f~ 2 LipðRn Þ such that f~jD ¼ f , and j f~j1 ¼ j f j1 . For a proof, we refer to Troianiello [18, Thm. 1.2, p. 12]. We also introduce a family of mollifiers. We take  2 DðRn Þ with support equal to the unit ball centered at 0, and we set l ðxÞ ¼ ðl þ 1Þn ððl þ 1ÞxÞ

8x 2 Rn ;

ð5:17Þ

86

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

for all l 2 N. We denote by Bð0; RÞ the open ball in Rn of radius R > 0 and center 0. Also, if B  Rn , then clB denotes the closure of B. Then we have the following. LEMMA 5.18. The following two statements hold. (i) If 2 LipðRn Þ has compact support, then the sequence f * l gl2N of uniformly in Rn , and weakly * to , and DðRn Þ converges to liml!1 Dð * l Þ ¼ D weakly*, and k jDð * l Þj kL1 ðRn Þ r k jD j kL1 ðRn Þ for all l 2 N. (ii) If 2 LipðRn Þ, then there exists a sequence f l gl2N in DðRn Þ such that liml!1 D l ¼ D weakly*, and such that supl2N k jD l j kL1 ðRn Þ < 1. Proof. Statement (i) holds by standard properties of the convolution in Rn . We now prove statement (ii). For each l 2 N, we consider the function l of clBð0; l þ 1Þ [ ðRn n Bð0; 2ðl þ 1ÞÞÞ to R, which equals on Bð0; l þ 1Þ and 0 on Rn n Bð0; 2ðl þ 1ÞÞ. Clearly, l is Lipschitz continuous. Accordingly, l can be extended to a Lipschitz continuous function ~l on Rn , and j ~l j1 ¼ j l j1 , by Theorem 5.16. We now estimate j l j1 in terms of j j1 . Let x 2 clBð0; l þ 1Þ, y 2 Rn n Bð0; 2ðl þ 1ÞÞ, then we have j l ðxÞ  l ðyÞj ¼ j l ðxÞj r j ðxÞ  ð0Þj þ j ð0Þj r j j1 jxj þ j ð0Þj r j j1 ðl þ 1Þ þ j ð0Þj r ðj j1 þ j ð0ÞjÞðl þ 1Þ r ðj j1 þ j ð0ÞjÞjx  yj: Thus we conclude that supl2N kjD ~l jkL1 ðRn Þ r j j1 þ j ð0Þj. Now we set l  ~l * l . By statement (i), we have supl2N k jD l j kL1 ðRn Þ < 1. Then possibly extracting a subsequence, we can assume that there exists b 2 ðL1 ðRn ÞÞn such that liml!1 D l ¼ b weakly*. Since ~l ¼ in Bð0; l þ 1Þ, a simple computation shows that liml!1 l ¼ uniformly on the compact subsets of Rn , and thus in Ì the sense of distributions in Rn . Then b ¼ D , and the proof is complete. Then we set   D
  2 R
: there exists ’ 2 DðRn Þ such that ’j
¼  : If u is a function of
to R, we denote by u
the function of Rn to R defined by u
ðxÞ  uðxÞ

if x 2
;

u
ðxÞ  0

if x 2 Rn n
:

We note that the symbol u
should not be confused with the symbol uj
, which denotes the restriction of u to
. Then we have the following.

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

87

LEMMA 5.19. Let
be an open subset of Rn . Let
2 A
. Then the following two statements hold. (i) If

2 D
, then there exists a sequence f l gl2N in D
ð
Þ such that 
ð1Þ weakly* in L1 ð
ð
ÞÞ ;

lim l ¼ l!1

lim D l ¼ Dð 
ð1Þ Þ weakly* in ðL1 ð
ð
ÞÞÞn : l!1 (ii) If 2 D
ð
Þ , then there exists a sequence f l gl2N in D
such that lim l!1

l

lim Dð l!1


ð1Þ ¼ weakly* in L1 ð
ð
ÞÞ ; l


ð1Þ Þ ¼ D weakly* in ðL1 ð
ð
ÞÞÞn :

Proof. We first prove statement (i). Clearly, the function
ð1Þ is Lipschitz continuous, and thus 
ð1Þ is Lipschitz continuous. Moreover, the Lipschitz continuity of
ensures that the support of 
ð1Þ is bounded. Then Theorem 5.16 ensures that there exists ~ 2 LipðRn Þ with compact support such that ~j
ð
Þ ¼ 
ð1Þ . Then we set l  ~* l , where l has been introduced in (5.17). We now prove statement (ii). If 2 D
ð
Þ , then 
2 Lipð
Þ. Moreover, the Lipschitz continuity of
ð1Þ ensures that the support of 
is bounded. Then Theorem 5.16 ensures that there exists ~ 2 LipðRn Þ with compact support such that ~j
¼ 
. Then we set l  l * ~. Then we note that if g 2 L1 ð
ð
ÞÞ, then g 
j det D
j 2 L1 ð
Þ, and thus by a change of variables, we easily conclude that liml!1 l 
ð1Þ ¼ weakly* in L1 ð
ð
ÞÞ. Similarly, liml!1 Dð l 
ð1Þ Þ ¼ D weakly* in ðL1 ð
ð
ÞÞÞn . Then we have the following. LEMMA 5.20. Let
be a connected open subset of Rn . Let m 2 N. Let
~ 2 A
. Let X
be a linear subspace of Lipð
Þ containing D
. Let  be a linear map of ðLipð
ÞÞn to Rm . Assume that liml!1 ½ l
¼ ½
in Rm whenever f l gl2N is a sequence in ðLipð
ÞÞn , 2 ðLipð
ÞÞn , and lim D



l!1

l




n2 
~ð1Þ ¼ D 
~ð1Þ weakly* in L1 ð
~ð
ÞÞ :

ð5:21Þ

Then the following equality holds  ½
:



n o  n 2 X n
¼ ½ 
~
: 2 D
~ð
Þ :

ð5:22Þ

88

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

Proof. We denote by W1 and W2 the linear spaces in the left and right-hand sides of (5.22), respectively. Then we set W3  f½
: 2 ðD
Þn g. Since W3 is a finite-dimensional subspace of W1 , then W3 is closed. Now let ½
with is the restriction of an 2 X n
, be an element of W1 . By Theorem 5.16, element of ðLipðRn ÞÞn . Then by Lemma 5.18 (ii) there exists a sequence f l gl2N of ðDðRn ÞÞn such that Z lim

gD

l!1

l

dx ¼




Z

gD dx 8g 2 L1 ð
Þ :


can be verified that ess inf
j det D
~j U ln ½
~
> 0 and that D
~ð1Þ ¼
Now itð1Þ 1 (cf. (5.4), [8, Prop. 3.7 (iv)]). Then by the rule of change of D
~ð
~ Þ variables in integrals (cf. Reshetnyak [16, Thm. 2.2, p. 99]), we obtain Z

fD

lim l!1


~ð
Þ

Z

ð1Þ ~ dy ¼ l


fD
~ð
Þ




~ð1Þ dy 8f 2 L1 ð
~ð
ÞÞ :

Thus, condition (5.21) is satisfied, and our assumptions imply that liml!1 ½ l
¼ ½
. Accordingly, ½
2 W3 , and then W1 ¼ W3 . We now prove that W2 ¼ W3 . Let ½
, with 2 ðD
Þn be an element of W3 . Then by Lemma 5.19 (i), and by our assumptions on , there exists a sequence f l gl2N in ðD
~ð
Þ Þn such that ½
¼ liml!1 ½ l 
~
. Since W2 is closed, we have ½
2 W2 . By the same argument, and by exploiting Lemma 5.19 (ii) instead Ì of Lemma 5.19 (i), we conclude that W2  W3 . Then we have the following. LEMMA 5.23. Let the same assumptions of Theorem 5.11 hold. Let X
be a linear subspace of Lipð
Þ containing D
. Then the space in (5.13) equals the space n




n o  hk ½ 
~
 hk 11 ½ 
~
h;k¼1;...;jFj : 2 D
~ð
Þ :

ð5:24Þ

Proof. By the membership of v~h in W 1;2 ð
~ð
ÞÞ, and by the Ho¨lder inequality, and by the definition of , and by Lemma 5.20, we conclude that the space in Ì (5.13) equals the space in (5.24). Since the functions 
~ for 2 ðD
~ð
Þ Þn are bounded, we can invoke the following technical fact of [10, Lem. 3.26].

89

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

PROPOSITION 5.25. Let the same assumptions of Theorem 5.11 hold. Let v~h 2 W 2;2 ð
~ð
ÞÞ for all h ¼ 1; . . . ; jFj. Then hk ½
¼ 

Z
~ð
Þ

h

i div 
~ð1Þ D~ vtk dy vh D~

for all h; k ¼ 1; . . . ; jFj,

2 ðLipð
Þ \ L1 ð
ÞÞn .

Then the following holds. LEMMA 5.26. Let the same assumptions of Theorem 5.11 hold. Let X
contain D
. Let v~h 2 W 2;2 ð
~ð
ÞÞ for all h ¼ 1; . . . ; jFj. Then the space in (5.13) equals the space 8 < Z :


~ð
Þ

! 
  t t div D~ vh D~ vk  hk D~ v1 D~ v1 dy



: 2 D
~ð
Þ h;k¼1;...;jFj

9

n = ;

:

ð5:27Þ Thus we now must try to understand what is the dimension of the space in (5.27). Then we set o 1;1

W

  ð
Þ  u 2 W 1;1 ð
Þ : u
2 W 1;1 ðRn Þ :

Clearly, o 1;1

W01;1 ð
Þ  W

ð
Þ ;

and equality holds if
is of class C1 . Then we have the following variant of a well known technical fact. LEMMA 5.28. Let
be an open subset of Rn such that @
haso zero n1;1 dimensional Lebesgue measure. If u 2 W 1;1 ð
Þ, then we have that u 2 W ð
Þ if and only if Z

@ u dx ¼ 
@xs

for all s ¼ 1; . . . ; n.

Z

@u dx
@xs

8 2 DðRn Þ ;

ð5:29Þ

90

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS o 1;1

Proof. If u 2 W ð
Þ, then @ðu
Þ=@xs ¼ 0 in Rn n cl
. Thus Z Z Z @ @ @ u dx ¼ u
dx ¼  ðu
Þ dx 8 2 DðRn Þ : n @xs
@xs R cl
@xs Since @ðu
Þ=@xs ¼ @u=@xs in
and @
has zero n-dimensional R Lebesgue measure, then the right-hand side of the above equality equals 
@u=@xs dx and thus (5.29) holds. Conversely, assume that (5.29) holds. Since u 2 W 1;1 ð
Þ, then we have u 2 L1 ð
Þ, and thus u
2 L1 ðRn Þ, and ð@u=@xs Þ
2 L1 ðRn Þ for
Þ=@xs ¼ ð@u=@xs Þ
, i.e., Rs ¼ 1; . . . ; n. Thus Rit suffices to show that @ðu n u @ =@x dx ¼  @u=@x dx for all 2 DðR Þ. Since u
¼ u on
, such s s


Ì fact follows by condition (5.29). LEMMA 5.30. Let
be an open subset of Rn such that @
has zero ndimensional Lebesgue measure. Let r 2 N n f0g. Let f1 , . . . , fr 2 W 1;1 ð
Þ. Let q denote the canonical projection of W 1;1 ð
Þ onto the quotient space o 1;1 1;1 W ð
Þ=Wo ð
Þ. The dimension of the space generated by fqð fl Þgl¼1;...;r in 1;1 W 1;1 ð
Þ=W ð
Þ equals the dimension of the space Z  Z n divð f1 Þ dx; . . . ; divð fr Þ dx : 2 ðD
Þ : V





Proof. Let dq , d denote the dimensions of the space generated by fqð fl Þgl¼1;...;r , and of the space V, respectively. Clearly, r  d equals the dimension of the space ( ) Z r X n r al divð fl Þ dx ¼ 0 8 2 ðD
Þ : V1  ða1 ; . . . ; ar Þ 2 R : l¼1

By Lemma 5.28, ( V1 ¼

ða1 ; . . . ; ar Þ 2 Rr :

r X




o 1;1

al fl 2 W

) ð
Þ :

l¼1 o 1;1

Now we introduce the operator  of Rr to W 1;1 ð
Þ=W ða1 ; . . . ; ar Þ ¼

r X

al qð fl Þ

ð
Þ defined by

8a  ða1 ; . . . ; ar Þ 2 Rr :

l¼1

Then V1 ¼ Ker. Now dim Ker ¼ r  dim Im, where Im denotes the image of . By definition, Im is the space generated by fqð fl Þgl¼1;...;r . Thus Ì dim Ker ¼ r  dq and r  d ¼ r  dq . Then we have the following generalization of a result of Lupo and Micheletti [13, Rem. 8, p. 116].

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

91

THEOREM 5.31. Let the same assumptions of Theorem 5.11 hold. Let @
have zero n-dimensional Lebesgue measure. Let X
contain D
. Let v~h 2 W 2;2 ð
~ð
ÞÞ for all h ¼ 1; . . . ; jFj. Let q denote the canonical projection of W 1;1 ð
~ð
ÞÞ onto o 1;1 1;1 ð
~ ð
ÞÞ=W ð
~ð
ÞÞ. Then  equals the dimension of the the quotient space W o 1;1 ~ subspace of W ð
ð
ÞÞ=W 1;1 ð
~ð
ÞÞ generated by     vtk  hk D~ v1 D~ vt1 : h; k ¼ 1; . . . ; jFj; ðh; kÞ 6¼ ð1; 1Þ : q D~ vh D~ ð5:32Þ Proof. Since the measure of @
is zero, then (5.5) and the Lipschitz convtk 2 tinuity of
imply that the measure of @
~ð
Þ is zero. Since D~ vh D~ 1;1 ~ W ð
ð
ÞÞ, then Lemma 5.30 implies that the dimension of the space in (5.27) equals the dimension of the subspace generated by the set in (5.32). Then Ì we conclude by Lemma 5.26. Remark 5.33. We note that if
~ð
Þ is of class C1;1 , then by standard elliptic regularity theory, we have v~r 2 W 2;2 ð
~ð
ÞÞ for r ¼ 1; . . . ; jFj (cf., e.g., Troianiello [18, Thm. 3.29, p. 195].) We point out however, that the membership of v~r in W 2;2 ð
~ð
ÞÞ may hold also under weaker regularity assumptions on
~ð
Þ. Furthermore, we note that if we assume that
is of class C1;1 , and that ~
2 A
has continuous partial derivatives in
satisfying a Lipschitz condition in
, then
~ð
Þ is of class C1;1 (cf., e.g., [9, Lem. 2.4].) Remark 5.34. Let
be an open connected subset of Rn . Let
~ð
Þ be of class C . Let X
be a linear subspace of Lipð
Þ containing D
. Let F be a finite nonempty subset of N n f0g. Let
~ 2 
½F
. Then
satisfies (5.2) (cf. [8, Prop. 3.7 (vii)]) and by standard elliptic regularity theory, we have v~r 2 W 2;2 ð
~ð
ÞÞ and D~ vr ¼ ð@ ~r =@ Þ for r ¼ 1; . . . ; jFj, where denotes the exterior unit normal to @
~ð
Þ. Then by applying the Divergence Theorem, one can easily see that the dimension of the space generated by the set in (5.27) equals that of the subspace of L1 ð@
~ð
ÞÞ generated by the set   @~ vh @~ vk @~ v1 @~ v1  hk : h; k ¼ 1; . . . ; jFj; ðh; kÞ 6¼ ð1; 1Þ : @ @ @ @ 1;1

By Lipschitz continuity of
~ð1Þ on
~ð
Þ and by (5.5), @
has zero n-dimensional Lebesgue measure. Since @
has zero n-dimensional Lebesgue measure, such dimension coincides with that of the space in (5.32). Thus our Theorems 5.11, and 5.31 can be considered as generalizations to nonsmooth domains of the corresponding result proved by Lupo and Micheletti [13, Rem. 8, p. 116] in case
is of class C3 , and
is of class C3 . We now turn to apply the results concerning the real analytic families of
’s, and which have been deduced with the use of a variant of a Theorem of Rellich and Nagy.

92

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

THEOREM 5.35. Let the assumptions of Theorem 5.11 hold. Let f
 g2I be a ~ of Theorem 5.11 and defined on the real analytic curve in the neighborhood V ~ open neighborhood I of 0. Let
0 ¼
. If  ¼ ðjFjðjFj þ 1Þ=2Þ  1, then the following statements hold. ~ ~ (i) If the curve  f
 g2I  is tangent to the manifold 
½F
\ V at
, then all the functions j ½

2I for j 2 F are differentiable in  at  ¼ 0. ~ at  ¼ 0, (ii) If jFj ¼ 2, and if the  curve f
 g2I is not tangent to 
½F
\ V at  ¼ 0. then both the curves j ½

2I for j 2 F are not differentiable   Furthermore, possibly shrinking I, the eigenvalues j ½

2I are simple for all  2 I n f0g. Proof. By Theorem 5.11, f
 g2I is tangent to 
½F
at  ¼ 0 if and only if d
 =dj¼0 belongs to the set in (5.14). By [10, Lem. 3.26], we have

hk



  d d ~
 ¼  F ½

Q
~ d
¼
~T
½~ uh

 ; u~k : d j¼0 d j¼0

Then by (3.3), d
 =dj¼0 belongs to the set in (5.14) if and only if the vector

 d d ðQ
 ; T
 Þ ¼ d
¼
~ðQ
; T
Þ
 d j¼0 d j¼0 belongs to the tangent space to the manifold ½F
defined in (3.1), relatively to the Fenvironment_ space w1;2 0 ð
Þ. By Theorem 3.13 such condition is equivalent to the differentiability of j ½T

¼ 1 j ½T

at  ¼ 0 for all j 2 F (see also Theorem 5.9 (iii).) In case jFj ¼ 2, the same  argument  shows that f
 g2I is not tangent to 
½F
at  ¼ 0 if and only if ðQ
 ; T
 Þ 2I is not tangent to ½F
at  ¼ 0. Then by Proposition 3.15 such condition implies that the eigenvalues j ½T

¼ 1 j ½T

are not differentiable at  ¼ 0, and that they are simple for Ì  6¼ 0 in a possibly smaller I. Then we conclude the validity of (ii). We now turn to study the structure of the set of
’s which preserve a certain eigenvalue. By applying Theorem 4.4, we obtain the following. THEOREM 5.36. Let
be a connected open subset of Rn such that (5.2) holds. Let X
be a Banach space continuously imbedded in Lipð
Þ. Let F be a finite nonempty subset of N n f0g. Let ~ > 0. Let  
½F; ~

2 A
\ X n
: j ½

¼ ~ 8j 2 F;  ~

l ½

6¼ 8l 2 N n ðF [ f0gÞ :

PERSISTENCE OF EIGENVALUES AND MULTIPLICITY

93

~ Let
~ 2 
½F; ~
. Let v~1 ; . . . ; v~jFj be an orthonormal basis in w1;2 0 ð
ð
ÞÞ of the ~ eigenspace of  corresponding to . Let # be the dimension of the vector subspace n o ð5:37Þ ðhk ½
Þh;k¼1;...;jFj : 2 X n
of SjFj ðRÞ, where  has been introduced in (5.12). Then 1 r # r ~ of
~ in A
\ ðjFjðjFj þ 1Þ=2Þ, and we can choose an open neighborhood V n ~
\ V ~ is contained in a real analytic Banach submanifold of X
such that 
½F; n X
of codimension #. Furthermore, if # ¼ ðjFjðjFj þ 1Þ=2Þ, then we can choose ~ is a real analytic Banach submanifold of X n of ~ so that 
½F; ~
\ V V
codimension ðjFjðjFj þ 1Þ=2Þ, whose tangent space at
~ equals the space n o ð5:38Þ 2 X n
: ðhk ½
Þh;k¼1;...;jFj ¼ 0 : ~
Proof. We proceed as in the proof of Theorem 5.11. We note that 
½F; coincides with the set   ~1
;
2 A
\ X n
: ðQ
; T
Þ 2 ½F; where ½F; ~1
has been defined in (4.2) relatively to the environment space w1;2 0 ð
Þ. Then by [10, Lem. 3.26], the ðh; kÞ-entry of the matrix in (5.37) coinuh
ð Þ; u~k
with u~h  v~h 
, and we conclude by cides with  ~Q
~½dj
¼
~T
½~ exploiting Theorem 4.4. Finally, we note that by setting ¼
~, we obtain hk ½
~
¼ 2hk . Accordingly # U 1. By arguing so as to prove Theorem 5.31, we can prove the following: THEOREM 5.39. Let the same assumptions of Theorem 5.36 hold. Let @
have zero n-dimensional Lebesgue measure. Let X
contain D
. Let v~h 2 W 2;2 ð
~ð
ÞÞ for all h ¼ 1; . . . ; jFj. Let q denote the canonical projection of W 1;1 ð
~ð
ÞÞ onto o 1;1 1;1 ~ the quotient space W ð
ð
ÞÞ=W ð
~ð
ÞÞ. Then # equals the dimension of the o 1;1 subspace of W 1;1 ð
~ð
ÞÞ=W ð
~ð
ÞÞ generated by     q D~ vh D~ vtk : h; k ¼ 1; . . . ; jFj : ð5:40Þ Remark 5.41. As in Remark 5.34, we observe that if
~ð
Þ is a bounded open connected subset of Rn of class C1;1 , and if X
is a linear subspace of Lipð
Þ containing D
, then # equals the dimension of the subspace of L1 ð@
~ð
ÞÞ generated by the set   @~ vh @~ vk : h; k ¼ 1; . . . ; jFj : @ @

94

PIER DOMENICO LAMBERTI AND MASSIMO LANZA DE CRISTOFORIS

Acknowledgements This paper represents an extension of a part of the work performed by P. D. Lamberti in his Doctoral Dissertation at the University of Padova under the guidance of M. Lanza de Cristoforis (see [7].)

References 1. 2. 3. 4. 5. 6. 7. 8.

9.

10.

11. 12. 13. 14. 15.

16. 17. 18. 19.

Arnold, V. I.: Modes and quasimodes, Funktsional. Anal. i Prilozhen. 6 (1972), 12Y20. Colin de Verdie`re, Y.: Construction de Laplaciens dont une partie finie du spectre est donne`e, Ann. Sci. E´cole Norm. Sup. (4) 20 (1987), 599Y615. Colin de Verdie`re, Y.: Multiplicite´s des valeurs propres. Laplaciens discrets et laplaciens continus, Rend. Mat. Appl. (7) 13 (1993), 433Y460. Colin de Verdie`re, Y.: Sur une hypothe`se de transversalite´ d’Arnold, Comment. Math. Helv. 63 (1988), 184Y193. Deimling, K.: Nonlinear Functional Analysis, Springer, Berlin, 1985. Evans, L. C.: Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1998. Lamberti, P. D.: A few spectral perturbation problems, Doctoral Dissertation, University of Padova, Italy, 2002. Lamberti, P. D. and Lanza de Cristoforis, M.: A global Lipschitz continuity result for a domain dependent Dirichlet eigenvalue problem for the Laplace operator, Z. Anal. Anwendungen 24 (2005), 277Y304. Lamberti, P. D. and Lanza de Cristoforis, M.: An analyticity result for the dependence of multiple eigenvalues and eigenspaces of the Laplace operator upon perturbation of the domain, Glasgow Math. J. 44 (2002), 29Y43. Lamberti, P. D. and Lanza de Cristoforis, M.: A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator, J. Nonlinear Convex Anal. 5 (2004), 19Y42. Lang, S.: Introduction to Differentiable Manifolds, Wiley-Interscience, New York, 1962. Lanza de Cristoforis, M.: Properties and pathologies of the composition and inversion operator in Schauder spaces, Rend. Accad. Naz. Sci. XL, Mem. Mat. 15 (1991), 93Y109. Lupo, D. and Micheletti, A. M.: On multiple eigenvalues of selfadjoint compact operators, J. Math. Anal. Appl. 172 (1993), 106Y116. Lupo, D. and Micheletti, A. M.: A remark on the structure of the set of perturbations which keep fixed the multiplicity of two eigenvalues, Rev. Mat. Appl. 16 (1995), 47Y56. Lupo, D. and Micheletti, A. M.: On the persistence of the multiplicity of eigenvalues for some variational elliptic operator depending on the domain, J. Math. Anal. Appl. 193 (1995), 990Y1002. Reshetnyak, Y. G.: Space Mappings with Bounded Distortion, Transl. Math. Monogr. 73, Amer. Math. Soc., Providence, Rhode Island, 1989. Teytel, M.: How rare are multiple eigenvalues? Comm. Pure Appl. Math. 52 (1999), 917Y934. Troianiello, G. M.: Elliptic Differential Equations and Obstacle Problems, Plenum, New York, 1987. ¨ ber das Verhalten von Eigenwerten bei adiabatischen von Neumann, J. and Wigner, E.: U Prozessen, Physikalische Zeitschrift 30 (1929), 467Y470.

Mathematical, Physics, Analysis and Geometry (2006) 9: 95Y108 DOI: 10.1007/s11040-006-9010-3

#

Springer 2006

Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups V. I. ARNOLDj Steklov Mathematical Institute, 8, Gubkina Street, 119991, Moscow, GSP-1, Russia. (Received: 29 March 2006; accepted: 20 April 2006; published online: 20 October 2006) Abstract. Linear combinations x1a1 + . . . + xnan of n given natural numbers as (with nonnegative integral coefficients xs) attain all the integral values, starting from some integer N(a), called the Frobenius number of the vector a; (provided that the integers as have no common divisor, greater than 1). The growth rate of N(a), for large values of s = a1 + . . . + an , depends peculiarly on the direction a 1 of the vector a = sa. The article presents a lower bound of order 1þn1 and an upper bound of order 2 s . Both orders are reached from some directions a. The averaging of N(a) along all directions, p performed for s = 7, 19, 41 and 97, provides the values, confirming the rate s for some p between 3/2 and 2 (for n = 3), excluding neither 3/2 nor 2, for the asymptotic behaviour at large s. One gets p $ 1, 66 for s between 100 and 200. These unexpected results, based on some strange relations of the Frobenius numbers to the higher-dimensional continued fractions geometry, lead to many facts of this arithmetic trubulence theory, discussed in this article both as theorems and as conjectures. Mathematics Subject Classification (2000): 11-XX. Key words: geometry of numbers, diophantine problems, weak asymptotics, continued fractions, convex hulls, Coxeter groups, Weyl chambers, Poincare´ series, Young diagrams, averaging, integer points counting in polyhedral domains.

1. Frobenius Numbers of Additive Semigroups Let ða1 ; . . . ; an Þ be natural integers, having no common divisor greater than 1. DEFINITION 1. The Frobenius number Nða1 ; . . . ; an Þ is the minimal integer, such that every integer l  N is representable in the form l ¼ x1 a1 þ . . . þ xn an

ð1Þ

with integer coefficients xs  0 ðx 2 Znþ Þ. Example. Nð3; 5Þ ¼ 8, since 3 þ 5 ¼ 8;

3 þ 3 þ 3 ¼ 9;

5 þ 5 ¼ 10 :

one may always add three more, representing any l  8, while 7 is not representable. j

Partially supported by RFBR grant 05-01-00104.

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For n ¼ 2 J. Sylvester proved, in [1], that Nða; bÞ ¼ ða  1Þðb  1Þ:

ð2Þ

It follows that for any n the additive semigroup of combinations (1) contains all the integers, starting at some N. But there is no explicit formula for the Frobenius number NðaÞ, a 2 Znþ , for n > 2, and even its asymptotic behaviour for large vectors a is unknown. The present article provides the lower and the upper bounds 1

1þ n1  NðaÞ  2 ;

ð3Þ

where ðaÞ ¼ a1 þ . . . þ an ;

ð4Þ

and a ¼ ðaÞ, the scale independent constants ðÞand ðÞ being described in the proofs below. For n > 2 there exist some examples, where NðaÞ > 2 and some other 1 examples where NðaÞ < "1þ n1 (for some positive constants  and "). The average behaviour of NðaÞ for the majority of the direction vectors  remains unknown. The numeric three-dimensional examples, provided in the present paper (for  ¼ 7, 19, and 41), depend rather peculiarly on the direction vector , and the averaging along all possible directions suggests an intermediate behaviour of N, of order  p, where 1þ

1 3 ¼  p  2: n1 2

The conjecture that p ¼ 1 þ 1=ðn  1Þ for the mean Frobenius number, averaged along a neighbourhood of any fixed direction was announced in [2, 3], but the reasons for this conjecture were some physical arguments rather than the rigorous mathematical proofs. 2. Geometry of Frobenius Numbers Consider formula (1) as a linear mapping l ¼ ðxÞ;  : Rn ! R; of a Euclidean space, depending on the vector a. The additive semigroup generated by this integer vector is ðZnþ Þ  Zþ ¼ fl 2 Z; l  0g: For any positive real number l we define the closed simplex SðlÞ  Rn by the inequalities SðlÞ ¼ fx 2 Rn : xs  0; ðxÞ  lg:

97

GEOMETRY AND GROWTH RATE OF FROBENIUS NUMBERS OF ADDITIVE SEMIGROUPS

The volume of this (rectangular) pyramid in Euclidean space Rn is n   1 Y l ln ; VolSðlÞ ¼ VðlÞ ¼ ¼ n! s¼1 as n!

ð5Þ

where  ¼ a1 . . . an. THEOREM 1. The number MðlÞ of the integer points in the closed simplex SðlÞ is related to the volume of the simplex by the inequalities VðlÞ  MðlÞ  Vðl þ Þ:

ð6Þ

Proof. Associate to any integer point x in our closed simplex SðlÞ the cube fy 2 Rn : xs  ys  xs þ 1; 1  s  ng. These MðlÞ cubes form a polyhedron PðlÞ. We shall prove the inclusions SðlÞ  PðlÞ  Sðl þ Þ:

ð7Þ

For any point z 2 SðlÞ the integer point x, whose coordinates are the integer parts of the corresponding coordinates of the point z, belongs to the simplex SðlÞ. The cube formed at this integer point x of SðlÞ contains the point z, since xs  zs  xs þ 1. Therefore, z 2 PðlÞ, and thus SðlÞ  PðlÞ. For any point y 2 PðlÞ the starting point x of the cube containing y, belongs to the simplex SðlÞ. Thus, ðxÞ  l, and hence ðyÞ  l þ ðy  xÞ  l þ ; since the coordinates ys  xs of the vector y  x are bounded by 1. Therefore ðy  xÞ 

n X

1as ¼ ðaÞ:

s¼1

We have thus proved the inclusion PðlÞ  Sðl þ Þ. The inclusions (7) that we have proved imply the inequalities (6). Remark. Examples show sharper estimates,   2 þ2 Þ  MðlÞ  V l þ : Vðl þ 2 2

ð8Þ

Thus, for a ¼ ð3; 5; 8Þ (n ¼ 3, ðaÞ ¼ 16) we get Mð18Þ ¼ 25, and Vð27Þ  27; 34, while Vð25Þ  21; 70. The arithmetic mean value of the two bounding numbers (8) provides generically an extremely good approximation for the value MðlÞ. In the above example MðlÞ ¼ 25 for l ¼ 18,  ¼ 16, and the mean value of the bounding numbers 24; 52 is a better approximation of MðlÞ ¼ 25 than Vð26Þ  24; 4.

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3. Lower Bounds of Frobenius Numbers We shall now prove the following inequality. THEOREM 2. Suppose that the direction  of the vector a is fixed. Then, for sufficiently long vectors a, the following inequality holds 1

NðaÞ  ðÞ n1 (with some positive constant ). Proof. Consider the r values of the parameter l ¼ N, Nþ1; . . . ; Nþr1. By the definition of the Frobenius number NðaÞ, all these values belong to the additive semigroup ðZnþ Þ. Hence for any of these values l there exists an integer point x in the closed simplex SðlÞ, for which ðxÞ ¼ l, which does not belong to the simplex SðN1Þ: x 2 SðN  1 þ rÞ n SðN  1Þ: We get therefore the inequality MðN þ r  1Þ  MðN  1Þ  r for any r  1. Using inequality (6), we deduce that VðN þ r  1 þ Þ  VðN  1Þ  r: Substituting the expressions (5) for the volumes, we get the inequality ðN þ r  1 þ Þn  ðN  1Þn  rn!: This inequality can be rewritten in the form n1 X ðr þ Þ ðN þ r  1 þ Þs ðN  1Þn1s  rn!: s¼0

Replacing N  1 by the bigger number N þ r  1 þ , we obtain the inequality nðN þ r  1 þ Þn1  n!r=ðr þ Þ: Choosing, say, r ¼ , we deduce that ðN þ 2  1Þn1  ðn  1Þ!=2: Hence, we have proved the inequality rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 ðn  1Þ! : N þ 2  1  2

GEOMETRY AND GROWTH RATE OF FROBENIUS NUMBERS OF ADDITIVE SEMIGROUPS

99

Suppose now that the ratios as =at ¼ s =t are bounded from above by some constant ðÞ > 0 : as  at . Then n1 Y ðn1Þ2 1=n  n  n n1 as   ; as  n1 ; at  ðn1Þ2 ; as ¼ Q  1=n  n : at n t6¼s  n t6¼s These inequalities imply that   n1=n 

ðn1Þ2 n

:

Finally, we prove Theorem 2 in the form  1 2 ðn  1Þ! n1 1 1 ðn1Þ 1 N   n1 2n n  n  ð; nÞ n1 ; 2  1 for the positive constant ¼ ðn  1Þ!=2 n1 =2, provided that  is not too small.

4. Some Euclidean Geometry of Lattices Denote by FðlÞ the ðn  1Þ-dimensional face of the simplex SðlÞ, where hA; xi ¼ l (for the normal vector A of this face with components ðx1 ; . . . ; xn Þ in Euclidean space Rn ). LEMMA 1. The ðn  1Þ -dimensional volume f ðlÞ of the face FðlÞ is provided by the formula f ðlÞ ¼

ln1 jAj ; ðn  1Þ!

ð9Þ

where jAj2 ¼ a21 þ    þ a2n ; and  ¼ a1     an . Proof. The altitude y ¼ hA=jAj of length h ends at a point of the face FðlÞ, where hA; yi ¼ l. Therefore, hðA; AÞ=jAj ¼ l, and thus h ¼ l=jAj. The volume of the simplex SðlÞ of dimension n is equal to 1=n th of the product of the area of the base f ðlÞ with length h of the altitude: VðlÞ ¼ f ðlÞhðlÞ=n: Substituting the value (8) for the volume V and the value l=jAj for the altitude h, we get formula (9) for the area of the base. LEMMA 2. The radius r of the inscribed ball of the Euclidean ðn  1Þ -dimensional simplex FðlÞ equals r¼

l ðÞ; ðaÞ

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V. I. ARNOLD

where the dimensionless coefficient ðÞ has the value ðÞ ¼

n P s¼1

jj  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; s jj2  2s

ð10Þ

ðjj2 being the square of the Euclidean length

n P s¼1

2s Þ.

Proof. The volume f ðlÞ of the ðn  1Þ-dimensional simplex FðlÞ equals 1=ðn  1Þth of the product of the radius r of the inscribed ball with the total Euclidean ðn  2Þ-dimensional area of the boundary of the simplex FðlÞ. Each face Fs ðlÞ of this boundary is defined by the equation ðAs ; xÞ ¼ l in the corresponding coordinate Euclidean subspace xs ¼ 0, the normal vector being As ¼ ða1 ; . . . ; as1 ; asþ1 ; . . . ; an Þ 2 Rn1 s . Calculating the ðn  2Þ-dimensional Euclidean area fs ðlÞ of the face Fs ðlÞ by formula (9) and substituting the expression (9) for the ðn  1Þ-dimensional volume f ðlÞ of the simplex FðlÞ, we calculate the radius r: n 1 r P f ðlÞ; f ðlÞ ¼ n  1 s¼1 s

ðn  1Þf ðlÞ ðn  1Þln1 jAjðn  2Þ! ljAj r¼ X ¼ : ¼ n n n  n2 X  X ðn  1Þ! l fs ðlÞ as jAjs ðas jAjs Þ s¼1

s¼1

s¼1

Replacing the components as of the vector A by the expressions distinguishing the dependence on the length and on the direction  of the vector as ¼ ðaÞs ; we reduce the coefficient of l in the last expression to jAj n P

¼

ðas jAjs Þ

s¼1

proving Lemma 2.

2 ðaÞ

n P

ðaÞjj ðÞ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ; ðaÞ s jj2  2s

s¼1

Ì

LEMMA 3. The fundamental parallelepiped n1 of the lattice of the integer points in the orthogonal hyperplane W : fx 2 Rn : ðA; xÞ ¼ 0g of the vector A in Euclidean space Rn has the ðn  1Þ -dimensional Euclidean volume jj ¼ jAj=d, where d is the greatest common divisor of the components as of the vector A.

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101

Proof. The fundamental parallelepiped of the lattice Zn in the ambient space can be constructed from n1 by the addition of one more vector y, being the closest integer vector to the hyperplane W (not belonging to it). The scalar product hA; yi has the minimal possible positive value for the expression a1 x1 þ . . . þ an xn at the integer points x. This value is divisible by d, and it attains the value d (according to the Euclid algorithm of calculation of the greatest common divisor dða1 ; . . . ; an Þ) at some integer point x. Thus, hA; yi ¼ d. To calculate the distance from the point y to the hyperplane W, it suffices to consider the parallel hyperplane y þ W, and the point z in it closest to the origin. We write it as z ¼ jzjA=jAj and calculate its scalar product with the vector A: hA; zi ¼ jzjðA; AÞ=jAj ¼ hA; yi ¼ dðaÞ: We have calculated the distance from y to the hyperplane W, jzj ¼ d=jAj: The standard integer lattice Zn  Rn has the fundamental parallelepiped of volume 1. Thus, 1 ¼ jn1 jjzj;

jn1 j ¼ 1=jzj ¼ jAj=d;

proving Lemma 3. Consider in Euclidean space Rn , with the integer points lattice Zn, the hyperplane W n1 orthogonal to the integer vector A: W n1 ¼ fX 2 Rn : hA; Xi ¼ 0g: We consider the coordinate flag of Euclidean subspaces R1  R2  . . .  Rn ; generated by the coordinate axis ðx1 Þ, ðx1 and x2 Þ, ðx1 , x2 , and x3 Þ and so on. Their intersections with the hyperplane W n1 define a flag of subspaces in this hyperplane, W n1 W n2 . . . W 1 : Thus, the hyperplane W 1  R2 is defined by the equation a1 x1 þ a2 x2 ¼ 0 on the coordinates x1 and x2 . We shall denote by Xs the points of W s and by As ¼ ða1 ; . . . ; asþ1 Þ the corresponding orthogonal vector of Rsþ1 .

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In each space W s we obtain an s -dimensional lattice of integer points, s ¼ W s \ Zsþ1 . Denote by s the fundamental parallelepiped of this lattice in W s , by js j its s -dimensional volume. We get from Lemma 3 the explicit formula for this volume js j ¼

jAs j ; ds

where jAs j2 ¼ a21 þ    þ a2sþ1 , ds ¼ ða1 ; . . . ; asþ1 Þ being the greatest common divisor of the first components of the vector A. Denote by Rs the minimal posiÌ tive distance W s1 of the point of the lattice s in Euclidean space W s . LEMMA 4. The height Rs of the parallelepiped s over the sub-parallelepiped s1 is Rs ¼

js j jAs jds1 ¼ : js1 j jAs1 jds

Proof. Indeed, the volume of s is the product of the area js1 j of the base with the height Rs . Substituting the preceding formulae for js j and js1 j, we Ì get Lemma 4. COROLLARY. The product of the n1 heights of the parallelepiped n1 is equal to R1 R2  . . . Rn1 ¼ jAj: Proof. Multiplying the ratios of Lemma 4, we get R1  . . . Rn1 ¼

jA1 j jA2 j jAn1 j d0 d1 dn2 jAn1 j d0   ¼ : jA0 j jA1 j jAn2 j d1 d2 jA0 j dn1 dn1

But the components ða1 ; . . . an Þ of the vector An1 ¼ A have no common divisor greater than 1: dn1 ¼ 1. For the number A0 ¼ a1 we get the Fcommon divisor_ d0 ¼ a1 , and finally the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi whole product is reduced to jAn1 j ¼ jAj ¼ a21 þ . . . þ a2n . The geometrical base of the upper bound of the Frobenius number that we shall prove is the following upper bound for the radius of the void ball in Euclidean space, having no common points with a lattice m  Rm. Suppose that this lattice has a flag of sublattices 1  2  . . .  m, such Ì that the sequence of the corresponding heights is R1 ; . . . ; Rm.

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103

THEOREM 3. The radius of a ball in Euclidean space Rm , having no common point with the lattice m , can_t exceed the quantity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 R21 þ . . . þ R2m :

¼ 2 Proof. For m ¼ 1 the statement is obvious, since the complement to the arithmetic progression of step R1 consist of the intervals of this length, and the diameter 2 of the void 1-dimensional ball can_t exceed R1. Suppose Theorem 3 to be true for m ¼ k, and that in the corresponding subspace W k of W kþ1 the radius of the void ball can_t exceed the number pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k ¼ 12 R21 þ . . . þ R2k . Consider a void ball in the space W kþ1. The hyperplanes of W kþ1, parallel to the hyperplane W k and containing points of the lattice kþ1, subdivide the space W kþ1 into slices whose hight has length Rkþ1. Therefore the center of the void ball is at a distance q, not exceeding Rkþ1 =2 , from one of these parallel hyperspaces W k þ y of dimension k. The intersection of the ball with this hypersurface does not intersect its sublattice kþ1 \ ðW k þ yÞ, which is isometric to the sublattice k in W k. Therefore, the radius p of this void ball of dimension k in W k þ y can_t exceed k. By the Pythagoras theorem we have for the radius r of the ðk þ 1Þ-dimensional void ball the formula r 2 ¼ p2 þ q2. Hence r2  2k þ ðRkþ1 =2Þ2 ¼ 2kþ1 ; proving Theorem 3.

Ì

5. Upper Bounds for the Frobenius Numbers THEOREM 4. The Frobenius number Nða1 ; . . . ; an Þ can_t exceed the quantity 2 ðaÞðÞ, where the constant  does not depend on the length of the vector a (its dependence on the direction  of this vector is described below, in the proof of Theorem 4). The idea of the proof is to consider the ðn  1Þ-dimensional simplex FðlÞ for l ¼ N  1. It does not contain any integral point (by the definition of the Frobenius number N). Thus, the inscribed ball of this simplex FðlÞ is void in the Euclidean hyperplane n1 W (having no common point with the sublattice n1 ¼ W n1 \ Zn ). Knowing the radius of the inscribed ball from Lemma 2 and the upper bound for it from Theorem 3, we substitute to this bound the values Rs of the high lengths from Lemma 4, and the resulting inequality provides the desired upper bound for l.

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From Lemma 2 we have the radius of the void ball r¼

l ðÞ; ðaÞ

from Theorem 3 we have 1 2 ðR þ    þ R2n1 Þ; 4 1

r2 

and therefore we get the upper bound l2 

2 ðaÞ ðR21 þ    þ R2n1 Þ : 2 ðÞ 4

ð11Þ

Substituting the values of the heights Rs , provided by Lemma 4, we get R2s ¼

2 ða21 þ    þ a2sþ1 Þds1 : ða21 þ    þ a2s Þds2

ð12Þ

The common divisor ds of the numbers ða1 ; . . . ; asþ1 Þ is a divisor of the numbers ða1 ; . . . ; as Þ, therefore, ds is a divisor of ds1 ¼ ds qs (with an integer qs ). Thus, ds1  ds , jAs j2 > jAs1 j2, and therefore R2s  1. Since the product of the heights Rs is jAj (by the corollary of page 8), we conclude that for every s R2s ¼

jAj2  jAj2 ; R21 . . . R2s1 R2sþ1 . . . R2n1 n1 P

2

and hence R2s  ðn  1ÞjAj . s¼1 We deduce from formula (11) the upper bound 2 ðaÞðn  1ÞjAj2 : l  42 ðÞ 2

Representing jAj as ðaÞjj, we get the inequalities 4 ðaÞðn  1Þjj2 ; l  ðÞ2 ðaÞ; 42 ðÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   where ðÞ ¼ ðn  1Þjj= 2const ðÞ . Finally, since l ¼ NðaÞ  1, we obtain the upper bound for the Frobenius number, l2 

Nða1 ; . . . ; an Þ  1 þ ðÞ2 ðaÞ; proving Theorem 4.

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Remark. In fact, inequality (11) together with the expression (12) for the lengths of the heights, provide more than Theorem 4. Suppose, for instance, that one can order the coefficients ða1 ; . . . an Þ in such a way that the greatest common divisors d1 ¼ ða1 ; a2 Þ; d2 ¼ ða1 ; a2 ; a3 Þ; . . . are all equal to 1. In this case formula (12) provides for all numbers Rs ðs > 1Þ the dimensionless expressions R2s ¼

21 þ    þ 2sþ1 : 21 þ    þ 2s

If the ratios Rs =Rt are all bounded, then this boundedness together with the value R1 . . . Rn1 ¼ jAj provide, for each length Rs , the estimation of the p product ffiffiffiffiffiffi n1 Rs  jAj ðÞ. This estimation leads, with the upper bound (11), to the inequality 1

1

l  1þ n1 !ðÞ   n1 const ðÞ; for which the order of the upper bound is similar to the lower bound order of for the 1 Frobenius number N  const ðÞ n1 , obtained in x 3. 6. Examples of Growth and Mean Rate of Frobenius Numbers We shall provide now some triples a ¼ ða1 ; a2 ; a3 Þ such that NðaÞ  const  2 ðaÞ. Since 2 ðaÞ is higher than 3=2 ðaÞ, it makes impossible the upper bounds of smaller power than 2, NðaÞ  const ðÞ p ðaÞ;

p < 2:

Example. Consider the triple ða; b; a þ bÞ, where ða; bÞ ¼ 1. In this case the Frobenius number is not affected by the presence of c ¼ a þ b, since the summands a þ b might be replaced by a and by b in Equation (1). Thus Nða; b; a þ bÞ ¼ Nða; bÞ ¼ ða  1Þðb  1Þ, by the Sylvester theorem. Supposing that a > 2, b > 2, we have a  1 > a=2;

b  1 > b=2;

Nða; bÞ > ab=4:

Suppose now that a=b < 9, b=a < 9. In this case ða þ bÞ2  ð100=9Þab, and thus in this sector Nða; bÞ >

9 2 ; 400

making any upper bound Nða; b; cÞ < const 3=2 impossible there. The above arguments provide examples of other directions for the vector ða; b; cÞ, for which Nða; b; cÞ < const  3=2 .

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V. I. ARNOLD

Thus, the growth rate of the Frobenius numbers is very different along different directions, and the question on the average behaviour (studying the mean value N^ðÞ of the Frobenius numbers Nða; b; cÞ, averaged along all the simplex a þ b þ c ¼ ), is quite difficult. Now, I shall present the numeric results on the calculation of N^ðÞ, for  ¼ 7; 19, and 41, choosing the prime values of  to avoid the Fcommensurable_ case of the greatest common divisor dða; b; cÞ > 1, where the Frobenius number Nða; b; cÞ does not exist. The following table shows the sum  of the Frobenius numbers along the triangle  ¼ const and the number M of the positive integer points ða > 0; b > 0; c > 0Þ in this triangle:

The triangles of the Frobenius numbers are: for  ¼ 7: 0 0

0 2 0 0 2 2 0 0 0 0 0 0 0

for  ¼ 19:

GEOMETRY AND GROWTH RATE OF FROBENIUS NUMBERS OF ADDITIVE SEMIGROUPS

107

for  ¼ 41:

To evaluate the power p of the conjectural growth rate of the mean values N^ C p ; one should draw their graph on the bilogarithmic paper: ln N^ ln C þ p ln : The slope of this approximately straight-line graph is p

ln N^2  ln N^1 : ln 2  ln 1

The preceding tables, for 2 ¼ 41 and for 1 ¼ 19, provide the slope p

3; 83  2; 20  2; 1; 3; 70  1; 95

(for 1 ¼ 7 the slope p is  2; 5).

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V. I. ARNOLD

However the value  ¼ 41 is too small to decide whether p ¼ 2 would be the asymptotic slope of the graph for  ! 1: this slope might decline (even to the value p ¼ 3=2, suggested by our lower bounds) for larger values of  ¼ a þ b þ c. The calculations at  ¼ 97 and at  ¼ 199, made for me by A. Goder, to whom I am grateful, provide 2 new values of the averaged Frobenius numbers:

Therefore, for ð2 ¼ 97; 1 ¼ 41Þ one finds the slope p  1; 8 and for ð2 ¼ 199; 1 ¼ 97Þ the slope p  1; 66. It makes rather probable my 1999 conjecture that the limit of the slope is p ¼ 1 12 (for  ! 1). pffiffiffiffiffi oscillates from 114  2; 6 to 47  1; 1 at the The unaveraged ratio Nða;b;cÞ 44;3 44;7 abc neighbouring triples ða; b; cÞ ¼ ð7; 14; 20Þ and ð7; 15; 19Þ with  ¼ 41, and it would be interesting to study its local averaging at almost constant directions , suggested in [2] and [3].

References 1. 2. 3.

Sylvester, J. J.: Mathematical questions with their solutions, Education Times 41 (1884), 21. Arnold, V. I.: Weak asymptotics for the numbers of solution of diophantine problems. Funct. Anal. Appl. 33(4) (1999), 292Y293. Arnold, V. I., et al.: Arnold_s Problems. Springer and Phasis, 2005, Problems 1999-8, 1999-9, and 1999-10, pp. 129Y130 and 614Y616.

Mathematical Physics, Analysis and Geometry (2006) 9: 109Y134 DOI: 10.1007/s11040-006-9008-x

#

Springer 2006

Singular Spectrum Near a Singular Point of Friedrichs Model Operators of Absolute Type SERGUEI I. IAKOVLEV Departamento de Matematicas, Universidad Simon Bolivar, Apartado Postal 89000, Caracas 1080-A, Venezuela. e-mail: [email protected]; [email protected] (Received: 21 July 2004; in final form: 29 March 2006; published online: 16 September 2006) Abstract. In L2 ðRÞ we consider a family of selfadjoint operators of the Friedrichs model: Am ¼ jtjm
þ V. Here jtjm
is the operator of multiplication by the corresponding function of the independent variable t 2 R , and V (perturbation) is a trace-class integral operator with a continuous Hermitian kernel vðt; xÞ satisfying some smoothness condition. These absolute type operators have one singular point t ¼ 0 of order m > 0. Conditions on the kernel vðt; xÞ are found guaranteeing the absence of the point spectrum and the singular continuous one of such operators near the origin. These conditions are actually necessary and sufficient. They depend on the finiteness of the rank of a perturbation operator and on the order of singularity m. The sharpness of these conditions is confirmed by counterexamples. Mathematics Subject Classifications (2000): 47B06, 47B25. Key words: analytic functions, eigenvalues, Friedrichs model, linear system, modulus of continuity, selfadjoint operators, singular point, zeros.

1. Introduction The problem of the singular spectrum arising on the continuous one emerges in the analysis of mathematical aspects of quantum scattering theory and quantum solid physics. Note that by the singular spectrum we mean the union of the point spectrum and the singular continuous one. In the study of this problem an important part is played by the selfadjoint Friedrichs model operator S1 : ¼ t
þ V acting in L2 ðRÞ (where t
stands for the operator of multiplication by the independent variable t 2 R, and V is an integral operator with a continuous Hermitian kernel). This operator constitutes quite an apt model of real quantum Hamiltonians. In [1] it was shown how the Friedrichs model can be used for the study of the spectral properties of the Schro¨dinger operator ð
þ qÞ. These operators are related via the integral Fourier transformation. A large body of literature is devoted to this model; we mention the papers [1Y9] by Faddeev and co-workers. For the first time the fact that here the singular spectrum may arise indeed was established in

110

SERGUEI I. IAKOVLEV

the paper [2] by Pavlov and Petras. Radically new conditions have been found in [7, 9] that guarantee the appearance of the singular spectrum. Since, actually, these conditions are necessary and sufficient in the context of the selfadjoint Friedrichs model the problem in question was solved in [9] completely. Further elaboration of this topic seems to be of value. Namely, it is of interest to investigate the singular spectrum of perturbations of the operators of multiplication by a function f ðtÞ of the independent variable (for example, f ðtÞ is equal to cos t or t2 ). Such operators naturally appear when various models of the Schro¨dinger operator are considered in a momentum representation. For example, the operator of multiplication by t2 is obtained if we write the Schro¨dinger operator in a momentum representation. Similarly, the relationship between the Friedrichs model and the one dimensional discrete Schro¨dinger operator S on Z is established with the help of the Fourier series. The operator S is equal to ðU þ U  Þ þ q and is defined on the space l2 ðZÞ of square summable complex sequences u ¼ fun gþ1 n¼1 ; here 1 * U is the operator of right shift, U is its adjoint, and q ¼ fqn g1, so that ðUuÞn ¼ un1 , ðU * uÞn ¼ unþ1 , and ðq uÞn ¼ qn
un [10, 11]. Under the isomorphism between l2 ðZÞ and L2 ð
;
Þ given by the map 1 : u ! u~ðtÞ ¼

þ1 X

un
e{nt ;

ð1:1Þ

n¼1

the operator S turns into S~ acting by the formula Z
~ Su~ðtÞ ¼ 2 cosðtÞ
u~ðtÞ þ q~ðt  xÞ
u~ðxÞ dx ; where q~ðtÞ ¼

P

n

ðS~u~Þ ¼ 1

qn
e{nt , u~ 2 L2 ð
;
Þ: Indeed,

þ1 h
 i X * UþU þq u¼ ðun1 þ unþ1 þ qn un Þe{nt n¼1 {ðn1Þt

X u e þ u e þ q u e{nt n n n n n n n Z
X {nt ¼ 2 cosðtÞ ue þ q~ðt  xÞ
u~ðxÞ dx : n n ¼

X

ð1:2Þ




{ðnþ1Þt

X

ð1:3Þ




Obviously, that the change of variables 2 cos t ¼ x would reduce the study of sing ðS~Þ, the singular spectrum of the operator S~, to that of sing ðS1 Þ. However, since ðcos tÞ0 ¼  sin tj
¼ 0, this substitution is not smooth (that is, not diffeomorphism) near the points 
and, therefore, may lead to a loss of subtle information concerning the structure of sing ðS~Þ. It is clear that, having an idea of the structure of the set sing ðS1 Þ, we can deduce some information about the set sing ðS~Þ (also near the singular points t ¼ 
) with the help of the change of variables, but the results obtained in this way will be expressed in inconvenient terms, and their sharpness near zero will be less than satisfactory.

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

111

Thus, as a model in the theory of continuous spectrum perturbations it seems reasonable to consider the perturbations not of the operator of multiplication by the independent variable t, but of the operator of multiplication by a function of t. In this case the main attention must be paid to the singular spectrum in a neighborhood of so-called singular points next to which it is impossible to introduce a smooth (locally) change of variables reducing our problem to the standard Friedrichs model. It will be shown that in a neighborhood of such points the behavior of the singular spectrum acquires a quite different character. The following two functions f1 ðtÞ ¼ jtjm and f2 ðtÞ ¼ sgn t
jtjm have one zero of order m > 0 at the point t ¼ 0. And near the origin f1 and f2 have different behavior. To these functions there correspond the selfadjoint Friedrichs model operators Am ; m > 0; with one singular point t ¼ 0 Am ¼ jtjm
þ V

ðthe absolute type operatorsÞ ;

ð1:4Þ

and the operators Sm Sm ¼ sgn t
jtjm
þ V

ðthe symmetric type operatorsÞ

ð1:5Þ

also with one singular point t ¼ 0 for m 6¼ 1. The operator S1 ¼ t
þ V is the main operator of the Friedrichs model. It has no singular points. In this paper we study the case of the operators Am . The operators Sm ; m 6¼ 1, were partially considered in [12]. 2. Statement of the Problem and Main Results In L2 ðRÞ we consider a family of selfadjoint operators Am ; m > 0 , given by Am ¼ jtjm


þ

ð2:1Þ

V:

Here jtjm
is the operator of multiplication by the function of the independent variable t 2 R, and V (perturbation) is an integral operator with a continuous Hermitian kernel vðt; xÞ . Thus, the action of the operator Am can be written as follows Z   m ð2:2Þ Am u ðtÞ ¼ jtj
uðtÞ þ vðt; xÞuðxÞ dx : R

We assume that V is nonnegative and belongs to the trace class 1: V  0 ; V 2 1 :

ð2:3Þ

Consequently, the operator Am is defined on the domain of functions uðtÞ 2 L2 ðRÞ such that jtjm uðtÞ 2 L2 ðRÞ. The kernel vðt; xÞ is assumed to satisfy the following smoothness condition vðt þ h; t þ hÞ þ vðt; tÞ  vðt þ h; tÞ  vðt; t þ hÞ  !2 ðjhjÞ ; jhj  1 ;

ð2:4Þ

112

SERGUEI I. IAKOVLEV

with the function !ðtÞ (the modulus of continuity of V) monotone and satisfying a Dini condition: Z 1 !ðtÞ dt < 1 : ð2:5Þ !ðtÞ # 0 as t # 0 ; and t 0 Inequality (2.4) may be regarded as a smoothness condition for the kernel v1=2 ðt; xÞ of the integral operator V 1=2 , because, as shown in R[4], the expression on the left in Inequality (2.4) can be written as the integral R jv1=2 ðt þ h; xÞ  v1=2 ðt; xÞj2 dx (and, therefore, is nonnegative). Together with Inequality (2.4) the fact that V is of class 1 means that the kernel vðt; xÞ satisfies a certain condition of decrease at infinity. The requirement that the operator V be of trace class 1 is sufficient for the absolutely continuous spectrum of Am to coincide with the real semi-axis Rþ ¼ ½0 ; þ1Þ (see [13]). We study the dependence of the behavior of the point and singular continuous spectrum on the smoothness of the kernel vðt; xÞ near the singular point t ¼ 0. As noted above the structure of the singular spectrum sing ðS1 Þ of the operator S1 ¼ t
þ V (the usual Friedrichs model operator) is pretty well studied [1Y9]. In particular, in the papers [7, 9] it was shown that for this operator there exists a sharp pffi  bound of finiteness of the singular spectrum. Namely, if !ðtÞ ¼ O t as t ! 0, the singular spectrum of S1 consists of at most a finite number of eigenvalues of finite multiplicityp(the ffi singular continuous spectrum is absent). On the other hand, if limt!0 !ðtÞ= t ¼ þ1, then examples can be constructed showing that even in the case when V is a rank 1 perturbation the eigenvalues of S1 may have cluster points. By using the simple change of variables jtjm ¼ x, we can show that outside any neighborhood of the origin on the interval ½0 ; þ1Þ the structure of the spectrum sing ðAm Þ is locally identical with that of the operator S1. This result is explained by the smoothness of the above change of variables outside any neighborhood of the origin, and also by the local character of the main results of [1Y9] relating to the structure of sing ðS1 Þ . Here by locality we mean the following. Suppose that conditions (2.4), (2.5) are fulfilled only in some interval ðc; dÞ  R, then the main results of [1Y9] about the structure of sing ðS1 Þ remain true in any closed subinterval
 ðc; dÞ: However, as shown in this paper, in a neighborhood of the origin the behavior of sing ðAm Þ is quite different. Here, near zero, we can still use the change of variables jtjm ¼ x mentioned 0 above, but, since, e.g., ðjtjm Þj0 ¼ 0 for m > 1, this change is not smooth (that is, not a diffeomorphism) near zero. In this sense the zero point is a singular point of the operators Am ; m > 0; so it needs a special inspection. Observe that the origin is also a boundary point of the continuous spectrum of Am , which coincides with the interval ½0 ; þ1Þ. Naturally, there appears a problem of finding sharp, in a sense, conditions on the kernel that guarantee that the singular spectrum is absent near the origin. In this paper it is shown that such sufficient conditions are given in terms of asymptotic behavior of the modulus of continuity !ðtÞ as t tends to zero. It

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

113

appears that for m 2 ð 1 ; 3 these conditions also depend on a rank of the perturbation operator V . Namely, if rank V < 1 , then provided that !ðtÞ ¼ Oðtðm1Þ=2 Þ ; t ! 0; the spectrum near zero is purely absolutely continuous. But if rank V ¼ þ1, then the structure of sing ðAm Þ depends on the value of a constant C in the condition !ðtÞ ¼ Ctðm1Þ=2. The sharpness of these conditions is confirmed by counterexamples. For m  1 the spectrum is always purely absolutely continuous in some neighborhood of the zero point on the interval ½0 ; þ1Þ. At the same time for m > 3 the singular spectrum may appear near zero for any modulus of continuity !ðtÞ. Hence, for m > 3 near zero there is no condition of the singular spectrum absence in terms of !ðtÞ as for m 2 ð 1 ; 3. The content of the paper may be outlined as follows. In Section 3 it is shown how the investigation of the singular spectrum in the selfadjoint Friedrichs model can be reduced to describing the structure of the roots of an analytic operatorfunction with positive imaginary part. Sections 4, 5, and 6 are the main sections of the paper. In Section: 4 we find sufficient conditions on the perturbation V ensuring that the singular spectrum is absent near the origin. For m 2 ð 1 ; 3 these conditions turn out to be different for perturbations of finite and of infinite rank. The counterexamples constructed in Section: 5, 6 show that the results of Section 4 are sharp. Note that some results of this paper (for the case m 2 N ) were announced in [14], and the case of m ¼ 2 has been in detail considered in [15]. 3. An Analytic Operator-Valued Function Tm ðzÞ and the Singular Spectrum For the operator V we write the spectral resolution Xþ1  ð
; ’k Þ ’k ; k  0 ; k ! 0; k ’k k ¼ 1: ð3:1Þ V¼ k¼1 k n o1 1=2 As in [4], we shall use the convenient notation: k ’k ðtÞ ¼: ðtÞ. Since k¼1 V is of trace class, we have ðtÞ 2 L2 ðR ; l2 Þ :

ð3:2Þ

The smoothness condition (2.4) imposed on the kernel vðt; xÞ is equivalent to the following smoothness property of the vector-valued function ðtÞ:   ð3:3Þ kðt þ hÞ  ðtÞk  ! jhj ; jhj  1 : l2

From Equations (3.2), (3.3), and (2.5) we deduce immediately that sup kðtÞk t2R

l2

< 1; lim kðtÞk t!1

l2

¼ 0 :

ð3:4Þ

For z 2 C n ½ 0 ; þ1Þ we define an analytic operator-valued function Tm ðzÞ: E ! E, where E :¼ RðV Þ is the closure of the range of V, as follows: pffiffiffiffi pffiffiffiffi 1 Tm ðzÞ :¼  V ðjtjm  zÞ V : ð3:5Þ

114

SERGUEI I. IAKOVLEV

Here ðjtjm  zÞ1 denotes the operator of multiplication by the corresponding function in L2 ðRÞ. Obviously, that Im Tm ðzÞ  0 if Im z > 0, and Tm ðzÞ 2 1 . The natural Hilbert space isomorphism between E and l2 under the map X 2 f ! f ð f ; ’k Þ g1 ð3:6Þ jð f ; ’k Þj2 ; k¼1 ; k f k ¼ k allows us to write a formula for the operator Tm ðzÞ in the new l2 -representation: Z ð3:7Þ Tm ðzÞ ¼  ðjtjm  zÞ1 ð
; ðtÞÞ ðtÞ dt : R

By making in the integral (3.7) on the interval ð 0 ; þ1Þ the change of variables t m ¼ x and on the interval ð1; 0 Þ the change ðtÞm ¼ x we reshape the expression for Tm ðzÞ as follows: Tm ðzÞ ¼ 

Z

þ1




; ðx1=mÞ



0

  ðx1=m Þ þ
; ðx1=m Þ ðx1=m Þ xz




x1=m1 dx : m ð3:8Þ

Thus, Tm ðzÞ is a sum of terms of the form Z þ1 ð
; gðxÞÞ gðxÞ dx ; xz 0

ð3:9Þ

where gðxÞ 2 L2 ðR ; l2 Þ is a vector-valued function such that kgðx þ hÞ  ~x jhj . Here for every x > 0 the function ! ~x ðtÞ satisfies condition gðxÞkl2  ! (2.5) for small values of t: Now arguing as in the scalar case (see [16]), it is not hard to show that on the interval ð0; þ1Þ the operator-function Tm ðzÞ has boundary values (understood in the strong sense), i.e., for any f 2 l2 we have Tm ðzÞf

! z!þi0

   f ; ðx1=m Þ ðx1=m Þ þ f ; ðx1=m Þ ðx1=m Þ  v:p: x 0     f ; ð1=m Þ ð1=m Þ þ f ; ð1=m Þ ð1=m Þ  i
: m 11=m Z

þ1






x1=m1 dx m

ð3:10Þ Making the inverse change of variables in the above integral, we see that for  > 0 the boundary values Tm ðÞ :¼ Tm ð þ i0Þ ¼ s  limz!þi0 Tm ðzÞ are of the form Z Tm ðÞ ¼ v:p: ðjtjm  Þ1 ð
; ðtÞÞ ðtÞ dt   i


R




; ð1=m Þ

  ð1=m Þ þ
; ð1=m Þ ð1=m Þ m 11=m

:

ð3:11Þ

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

115

In fact, it can be shown (see [4] and also [9, 17]) that the function (3.9) (and, hence, the function (3.8)) is continuous up to the real axis in the trace norm. Thus, the following statement is true. PROPOSITION 3.1. If V satisfies conditions (2.3) Y (2.5), then in the complex plane cut along ½ 0 ; þ1Þ the analytic operator-valued function Tm ðzÞ admits a 1 -norm continuous extension to the upper and the lower parts of the cut on the interval ð 0 ; þ1Þ . The boundary values Tm ðÞ :¼ Tm ð þ i0Þ;  > 0; can be written as in Equation (3.11). The set Nm :¼ f  > 0 : 9g 2 l2 ; g 6¼ 0 ; Tm ðÞg ¼ g g f  > 0 : Kerð I  Tm ðÞÞ 6¼ ; g is called a set of roots of the operator-function Tm . The vector g is called a root vector corresponding to the root  . PROPOSITION 3.2. If V satisfies conditions (2.3) Y (2.5), then sing ðAm Þ, the singular spectrum of the operator Am ¼ jtjm
þ V; m > 0; embeds into the set Nm supplemented by the origin, i.e., sing ðAm Þ ¼ p ðAm Þ [ s:c: ðAm Þ Nm [ f0g ;

ð3:12Þ

where p ðAm Þ is the point spectrum, and s:c: ðAm Þ is the singular continuous spectrum of the selfadjoint operator Am . Proof. The proof of this statement is based on the following result proved in [4]: If 0 2 R is such that for some  > 0 for Mm ðzÞ :¼ I  Tm ðzÞ we have sup

kM1 ðzÞk < 1 ;

ð3:13Þ

0<jz0 j< Imz>0

then the spectrum of Am is absolutely continuous on the interval ð0  ; 0 þ Þ . Now, suppose that 0 > 0 and 0 2 6 Nm , then, since the operator Tm ð0 Þ is compact, the operator Mð0 Þ is continuously invertible. Since MðzÞ is continuous (in the usual operator norm), the same is true for all z with Im z  0 in some neighborhood of 0 . Hence, condition (3.13) is fulfilled at 0 , which proves the Ì proposition. Proposition (3.2) shows that the investigation of sing ðAm Þ reduces to describing the structure of the roots set Nm of the operator function Tm ðÞ with positive imaginary part. It is not hard to deduce that  2 Nm if and only if there is a root vector g ¼ g 6¼ 0 such that

 g ; ð1=m Þ ¼ g ; ð1=m Þ ¼ 0 ; ð3:14Þ

g þ v:p:

Z

ðg ; ðtÞÞ ðtÞ dt ¼ 0 : jtjm   R

ð3:15Þ

116

SERGUEI I. IAKOVLEV

Indeed, according to Equation (3.11), the equality Im ðTm ðÞ g ; g Þ ¼ 0 is equivalent to Equation (3.14). Then the condition Tm ðÞg ¼ g is equivalent to Equation (3.15). Note that, since ðtÞ satisfies Equation (3.3), from Equation (3.14) we see that the integral in Equation (3.15) may be understood in the usual sense. Putting u ¼ 1=m, we write Equations (3.14) and (3.15) in the form ðg ; ðuÞÞ ¼ 0; Z ðg ; ðtÞÞ ðtÞ g þ dt ¼ 0: jtjm  um R

ð3:16Þ ð3:17Þ

In what follows we need the following simple result. LEMMA 3.1. Let u > 0. For any m > 0 there exist constants C1 ; C2 > 0 such that for t 2 ½ 0 ; uÞ the following inequality holds C1 m1 u ðu 

tÞ



um

1 C2  m1 : m t u ðu  tÞ

ð3:18Þ

Proof. Let first 0 < m  1 . Then, obviously, 1  x  1  x m  mð1  xÞ ; x 2 ½ 0 ; 1 : Putting x :¼ t=u we find that for m 2 ð 0 ; 1 and t 2 ½ 0 ; uÞ 1 1 1  :  um1 ðu  tÞ um  tm m um1 ðu  tÞ

ð3:19Þ

ð3:20Þ

Analogously, for m  1 in the inequality 1  x  1  x m  mð1  xÞ ; x 2 ½ 0 ; 1 ; we put x :¼ t=u and obtain that for m  1 and t 2 ½ 0 ; uÞ 1 1 1  m1  m : m1 m m u ðu  tÞ u  t u ðu  tÞ This completes the proof.

ð3:21Þ

ð3:22Þ

Ì

4. Sufficient Conditions Guaranteeing the Absolute Continuity of the Spectrum on the Interval ½0; þ1  Near Zero From the Fredholm analytic alternative (see [18, § 8]) it follows that the set Nm [ f0g  R is a closed set of Lebesgue measure zero. Also Theorem 3 in [15] says that under the condition V  0 the point 0 is not an eigenvalue of the operator Am ¼ jtjm
þ V. Below some conditions on the modulus of continuity !ðtÞ of the perturbation operator V are given guaranteeing the absolute continuity of the spectrum of the operator Am on the interval ½ 0 ; þ1Þ near zero. For m 2 ð 1 ; 3  these conditions depend on a rank of the operator V.

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

117

THEOREM 4.1. Suppose that conditions (2.3) Y (2.5) are fulfilled. Then for m 2 ð0; 1 the roots set Nm is empty in some neighborhood of the origin. And, hence, the spectrum of the operator Am ¼ jtjm
þ V; defined by Equation (2.2), is purely absolutely continuous in some neighborhood of the origin on the interval ½ 0 ; þ1Þ. Proof. If  2 Nm , then conditions (3.16) Y (3.17) are fulfilled at u ¼ 1=m : We take the scalar product of Equation (3.17) by the corresponding root vector g ¼ g ; kgk ¼ 1. Then I:¼ 1 þ

Z

j ðg; ðtÞÞ j2 dt ¼ 0: m m R jtj  u

By virtue of Equation (3.16) the function Therefore,

ð4:1Þ ðtÞ :¼ ðg; ðtÞÞ vanishes at u:

j ðtÞj ¼ j ðg ; ðtÞ  ðuÞÞ j  !ðj t  u jÞ ; jt  uj  1 :

ð4:2Þ

If  is sufficiently close to zero, then the above inequality will be fulfilled on the interval ½u; u. Obviously, we have Zu j ðtÞj2 I 1 dt: ð4:3Þ um  jtjm u

Applying Equations (4.2) and (3.20), we obtain that Zu u

j ðtÞj2 dt  2 um  jtjm

Zu

!2 ðu  tÞ 2 dt  m m u t m um1

0

Zu 0

2 u1m ¼ m

Zu

!2 ðu  tÞ dt ut ð4:4Þ

!2 ðtÞ dt: t

0

Whence 2 u1m I  1  !ðuÞ m

Zu

!ðtÞ dt: t

ð4:5Þ

0

Thus, taking u  1, we have I  1  C !ðuÞ. (Remark that from now on we denote by C various constants which may be different even in a single chain of inequalities.) Since !ðuÞ ! 0 as u ! 0þ, we see that I > 0 for all u > 0 sufficiently small. This contradicts the condition I ¼ 0. The theorem is proved. Ì

118

SERGUEI I. IAKOVLEV

THEOREM 4.2. Suppose that the perturbation V satisfies conditions (2.3)Y(2.5) with the function !ðtÞ ¼ C! t, where  ¼ ðm  1Þ=2, and m 2 ð1; 3: If !1=2 Z 1 ð1  xÞm1 dx ; ð4:6Þ C! < Cm :¼ 2 1  xm 0 then the roots set Nm is empty in some neighborhood of the origin. Consequently, the spectrum of the operator Am ¼ jtjm
þ V , defined by Equation (2.2), is purely absolutely continuous in some neighborhood of the origin on the interval ½ 0 ; þ1Þ . Note: Clearly it follows from Equation (3.3) that for the modulus of continuity !ðtÞ ¼ C! t the greatest possible value of  is 1. The value  ¼ 1 exactly corresponds to m ¼ 3 . Proof. As in the proof of Theorem 4.1 we have that Zu j ðtÞj2 I 1 dt: um  jtjm

ð4:7Þ

u

Then, in view of inequality (4.2), Zu

j ðtÞj2 dt  2 um  jtjm

u

Zu

!2 ðu  tÞ dt  2C2! um  tm

0

Zu

ðu  tÞ2 dt : um  tm

ð4:8Þ

0

By making the change of variables t ¼ ux in the last integral and as 2 ¼ m  1, we find that Z1 ð1  xÞm1 dx ¼ 1  C2! =C2m : ð4:9Þ I  1  2C2! 1  xm 0

Consequently under the condition C! < Cm we obtain that I > 0 , a Ì contradiction. This completes the proof. Observation: It is not difficult to obtain a two-sided estimate for the constant Cm . Indeed, since for m > 1 and x 2 ½ 0; 1 1  x  1  xm  mð1  xÞ ; we have Z 1 Z 1 1 ð1  xÞm1 ð1  xÞm1 dx  dx m 0 1x 1  xm 0 Z 1 ð1  xÞm1  dx : 1x 0

ð4:10Þ

ð4:11Þ

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

119

Whence 

m1 2

1=2

  Cm 

mðm  1Þ 2

1=2 :

ð4:12Þ

R1 At the same time for the values m ¼ 2 and m ¼ 3 the integral 0 ð1  tÞm1 = ð1  tm Þ dt can be evaluated exactly. For m ¼ 2 we obtain C2 ¼ ð1= ln 4Þ1=2 ¼ 0; 849


that coincides with the value of this constant from ffiffiffi the  paper  p [15] (see Theorem 1). Likewise, we find that C3 ¼
= 3  ln 3 1=2 ¼ 1; 182 . . . : Note also that from Equation (4.12) it follows immediately that Ì Cm ! þ0 as m ! 1þ . If the perturbation V is a finite rank operator, the result of Theorem 4.2 can be improved. THEOREM 4.3. Suppose that the perturbation operator conditions  V satisfies  (2.3) Y (2.5) and rank V < 1. If m 2 ð1; 3and !ðtÞ ¼ O tðm1Þ=2 as t ! 0þ , then the origin is not a cluster point of the set of roots Nm of the operator-valued function Tm : Consequently the spectrum of the operator Am ¼ jtjm
þ V , defined by Equation (2.2), is purely absolutely continuous in some neighborhood of the origin on the interval ½ 0 ; þ1Þ . Proof. Arguing by contradiction, suppose that there is a sequence of roots n monotone decreasing to zero. Let un :¼ 1=m , and let gn ¼ gn be the corren sponding root vector. At the points u ¼ un conditions (3.16), (3.17) are fulfilled. Taking the scalar product of Equation (3.17) by gn (we put g ¼ gn and assume that kgn k ¼ 1 ), we get 1 þ

Z

jðgn ; ðtÞÞj2 dt ¼ 0 : m m R jtj  un

ð4:13Þ

By Equation (3.16), the function n ðtÞ :¼ ðgn ; ðtÞÞ vanishes at the points un ; whence, according to Equation (3.3), j

n ðtÞj

¼ j ðgn ; ðtÞ  ðun ÞÞ j  !ðj t  un jÞ ; jt  un j  1 :

ð4:14Þ

If n is sufficiently close to zero, then the above inequality will be fulfilled on the interval ½2un ; 2un . We rewrite Equation (4.13) as follows

1 þ

Zun un

2

0

j n ðtÞj dt þ @ jtjm  um n

Zun 1

1 Zþ1 j n ðtÞj2 A þ dt ¼ 0 : jtjm  um n un

ð4:15Þ

120

SERGUEI I. IAKOVLEV

Here only the second summand is negative, and this summand can be estimated uniformly with respect to n .To establish this we make use of inequalities (3.22), (4.14), and the fact that !ðtÞ  C! tðm1Þ=2 with some constant C! > 0. Then   u   Zn Zun 2 2   j n ðtÞj ! ðun  tÞ   dt   2 dt m  m m1 un ðun  tÞ jtj  un   un

¼

0

Zun

2 um1 n

0

!2 ðtÞ 2 C2 dt  m1! t un

Zun

t m2 dt  C ;

ð4:16Þ

0

uniformly with respect to n 2 N . Consequently, the third term in Equation (4.15) also admits a uniform estimate, so that 0 u 1 0 u 1 Z n Zþ1 Z n Zþ1 2 2 j ðtÞj A mn @ A j n ðtÞj þ dt  þ dt ; Const  @ jtj  um jtjm n 1

1

un

un

ð4:17Þ for all n 2 N . By assumption, we have rank V < 1; hence, dim E ¼ dim RðVÞ < 1, and from the sequence gn 2 E; kgn k ¼ 1, we can select a subsequence converging in l2 to a vector g 6¼ 0 . Not to overload the notation we assume that gn ! g as n ! 1. For the function ðtÞ :¼ ðg; ðtÞÞ we obtain from Equation (4.17) that j ðtÞj2 =jtjm 2 L1 ðRÞ : From

n ðun Þ

ð4:18Þ

¼ 0 it follows that

ð0Þ ¼ ðg ; ð0ÞÞ ¼ lim ðgn ; ðun ÞÞ lim n!1

n!1

n ðun Þ

¼0:

ð4:19Þ

Therefore, taking into account Equation (3.3), we find that j ðtÞj ¼ j ðg ; ðtÞ  ð0ÞÞ j  !ð jtjÞ ; jtj  1 :

ð4:20Þ

Taking the scalar product of relation (3.17) (in which we put u ¼ un and g ¼ gn ) by the vector g, we get Z n ðtÞ ðtÞ dt ¼ 0 : ð4:21Þ ðgn ; gÞ þ m jtj  um R n Now we show that relation (4.21) cannot be valid for all sufficiently large n 2 N: Since ðgn ; gÞ ! 1 as n ! 1, it suffices to show that, starting with some n, the real part of the integral in Equation (4.21) is more than 1=2, that is, Z Re

0

þ 1

þ1 

Z 0

n ðtÞ m

ðtÞ dt > 1=2 : jtj  um n

ð4:22Þ

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

121

We check that the second summand is more than 1=4 for all n large enough (the first one can be treated similarly). Let a > 0, and let  2 ð0; 1Þ. Then 0

Zþ1

B n ðtÞ ðtÞ dt ¼ @ tm  um n

IðnÞ :¼ 0

n ðtÞ tm 



ð1Þu Z n 0

ð1þÞu Z n

þ

ð1Þun

Za

þ

1 Zþ1 C þ A

ð1þÞun

a

ðtÞ dt K I1 þ I2 þ I3 þ I4 : um n ð4:23Þ

For any fixed a we have

I4 ¼

Z

þ1

n ðtÞ tm 

a

ðtÞ dt ! n!1 um n

Z

þ1 a

j ðtÞj2 dt  0 : tm

ð4:24Þ

If n is sufficiently large, according to Equations (4.14) and (4.20), we have for t 2 ½ 0 ; 2un  that j n ðtÞj  !ðj t  un jÞ  C! jt  un j , and j ðtÞj  !ð tÞ  C! t, where  ¼ ðm  1Þ=2 . Together with Equation (3.22) this gives Z

jI2 j 

un

þ

Z

ð1Þun



Z



ð1þÞun

ð1Þun

C2!  m1 un

!

j

un

n ðtÞj
j ðtÞj dt jtm  um nj

j n ðtÞj
j ðtÞj dt þ ðun  tÞ um1 n

un

ð1Þun

Z

ð1þÞun

Z

Z

ð1þÞun

un

j n ðtÞj
j ðtÞj dt ðt  un Þ tm1

!ðjt  un jÞ
!ð tÞ dt jt  un j um1 n ð1þÞun

jt  un j1 ðð1 þ Þun Þ dt

ð1Þun

C2 ð1 þ Þ
un 2 ¼ ! um1 n

Z 0

un

 1 d 

Cun ðun Þ ¼ C : um1 n

Thus, I2 ! 0 as  ! 0þ uniformly with respect to n 2 N .

ð4:25Þ

122

SERGUEI I. IAKOVLEV

Further, using the inclusion (4.18) and the Schwartz inequality, we obtain

jI1 j 

Z

ð1Þun

j

0

Z




n ðtÞj j ðtÞj m um n t

ðun  tÞ j ðtÞj dt un  t 0 Z ð1Þun 1 j ðtÞj tm=2 m=2 dt 1 t ðun Þ 0 ð1Þun



C! um1 n



C! um1 n



C! 1 um1 ðun Þ1 n Z



dt

ð1Þun

!1=2 t m dt




Z

0





C

0

1

1 um n C 1

ð1Þun

Z 0

un

ðð1  Þun Þðmþ1Þ=2

Z

un 0

j ðtÞj2 dt tm

j ðtÞj2 dt tm

!1=2

j ðtÞj2 dt tm

!1=2

!1=2 :

ð4:26Þ

It follows that I1 ! 0 as n ! 1 for any fixed value of  2 ð 0; 1Þ . Next we observe that if t  ð1 þ Þun , then 1 1 un un 1  ¼ ;  ¼ t  un t t ðt  un Þt un
t

ð4:27Þ

so that   1 1 2 1 1þ 
:  t  t

1  t  un

ð4:28Þ

Together with Equation (3.22) this gives Z j n ðtÞj j ðtÞj 2 a j n ðtÞj j ðtÞj dt  dt
m1 m=2
tm=2 t  u t  n ð1þÞun ð1þÞun t !1=2 !1=2 Z a Z a 2 j n ðtÞj2 j ðtÞj2 dt
dt :   tm tm un 0

jI3 j 

Z

a

ð4:29Þ

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

123

In view of the estimate (4.17) uniform with respect to n 2 Nwe finally establish that !1=2 Z a C j ðtÞj2 jI3 j  dt : ð4:30Þ  tm 0 This means that I3 ! 0 as a ! 0þ for any fixed value of  2 ð 0; 1Þ. Clearly, Re IðnÞ  Re I4  jI1 j  jI2 j  jI3 j. Consequently, according to the behavior of the integrals Ii ; i ¼ 1; 4, established above, for any " > 0 we can take  > 0 small enough and then for a fixed value of  we can take a > 0 small enough and sufficiently large n1 2 N such that the following inequality Z þ1 j ðtÞj2 Re IðnÞ  dt  " ð4:31Þ tm 0 will be fulfilled for all n > n1 . Since " > 0 is arbitrary, we conclude that Ì Re IðnÞ > 1=4 for all n 2 N large enough, and the proof is complete. 5. On the Sharpness of the Absence Condition for the Singular Spectrum in the Case of Infinite Rank Perturbation: A Counterexample The main theorem proved in this section tells us that in the case of an infinite rank perturbation (rank V ¼ 1) the absence condition for the singular spectrum of Am ; m 2 ð1; 3 , near the origin is indeed related to the constant C! in the condition !ðtÞ ¼ C!
tðm1Þ=2 ; t ! 0, (see Theorem 4.2). In particular, this means that the result of Theorem 4.3 cannot be extended to the case of a perturbation V of infinite rank. Namely, Theorem 4.2 is sharp in the following sense. THEOREM 5.1. Let m 2 ð 1; 3. For any value of C!  C~m :¼ 2ðm1Þ=2 m there exists an operator V with rank V ¼ 1 , such that V satisfies conditions (2.3) Y (2.5) with !ðtÞ ¼ C! tðm1Þ=2, and the origin is a cluster point of the set of eigenvalues of the operator Am ¼ jtjm
þ V, defined by Equation (2.2). Proof. An explicit construction of the operator V will be given. We fix two that n ! 0 and "n ! 0 monotone sequences of positive numbers n and "n such P1 1=m and assume that as n ! þ1 . Let un :¼ n 1 n < þ1 and "n < ðun  unþ1 Þ=2. We construct the operator V such that the points n will be eigenvalues of the operator Am ¼ jtjm
þ V. Denoting by cn ; n 2 N, positive constants andPputting  ¼ ðm  1Þ=2, we take the eigenfunctions ’n of the operator V ¼ þ1 k¼1 k ð
; ’k Þ ’k defined as follows 8 0 ; t 2 ½0; þ1Þ n ½unþ1 ; un  n  > > > u þ un  n  > < cn ðt  unþ1 Þ ; t 2 ½unþ1 ; nþ1  2 ’n ðtÞ :¼ ð5:1Þ unþ1 þ un  n  > > c ðu    tÞ ; t 2 ½    ; u n n n n n > > 2 : ’n ðtÞ; t < 0 :

124

SERGUEI I. IAKOVLEV

The corresponding eigenvalues n of V will be determined later. Since the functions ’n have mutually disjoint supports, they are orthogonal. The constants cn entering in definition (5.1) are chosen so that the ’n be normalized. Let n :¼ ðun  n  unþ1 Þ=2; then taking into account the evenness of ’n we have 1 ¼

Z

2

j’n ðtÞj dt ¼ 4

R

unþ1 Z þn

c2n ðt  unþ1 Þ2 dt ¼ 4c2n

nm : m

ð5:2Þ

unþ1

Thus, c2n ¼ m=ð4nm Þ :

ð5:3Þ

LEMMA 5.1. Let Am be defined as in Equations (2.1) Y (2.5). A point  is an if eigenvalue of Am and a function v ðtÞ is the corresponding eigenfunction of Am and only if  2 Nm ; v ðtÞ 2 L2 ðRÞ; and for a.e. t 2 R we have v ðtÞ ¼ ðg; ðtÞÞ ðjtjm  Þ, where g 2 l2 ; g 6¼ 0 is a root vector corresponding to the root , that is, Tm ðÞ g ¼ g. Proof. We refer the reader to [9, Lemma 1], where it suffices to replace the Ì operator L ¼ t
þ V by the operator Am ¼ jtjm
þ V. As the root vectors gn corresponding to the points n we take the standard basis in l2 , i.e., the k-th coordinate of gn is ðgn Þk ¼ nk (the Kronecker delta). (Observe that the sequence gn so selected is noncompact, while in the proof of Theorem 4.3 the condition rank V < 1 was used only for establishing the compactness of the set of the root vectors.) By Lemma 5.1, the points n will be eigenvalues of the operator Am provided that n 2 Nm (i.e., conditions (3.16) and (3.17) are fulfilled for u ¼ un and g ¼ gn ; n 2 N;) and the functions vn ðtÞ :¼ ðgn ; ðtÞÞ ðjtjm  n Þ lie in L2 ðRÞ. Since ðgn ; ðtÞÞ ¼ 1=2 n
’n ðtÞ and, by construction, the continuous function ’n has compact support and vanishes identically in some neighborhood of each of the points un ¼ n 1=m , condition (3.16) and the relation vn 2 L2 ðRÞ are fulfilled automatically. It remains to consider condition (3.17): gn þ

Z

ðgn ; ðtÞÞ ðtÞ dt ¼ 0 : jtjm  um R n

ð5:4Þ

Since the supports of the functions ’n ðtÞ are mutually disjoint, we have ðgn ; ðtÞÞ ðtÞ ¼ ¼

n n

1=2 n ’n ðtÞ
i

1=2

n j’n ðtÞj2
ni

’i ðtÞ

o1 i¼1

:

o1 i¼1

ð5:5Þ

125

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

Consequently, in our case relation (5.4) is equivalent to the scalar identity Z

n ’2n ðtÞ dt ¼ 1  2n 0¼1þ m m R jtj  un

Z

un n unþ1

’2n ðtÞ dt ; m um n t

ð5:6Þ

which can be ensured by choosingPappropriate constants n > 0 . Obviously, the operator V ¼ 1 i¼1 n ð
; ’n Þ ’n is nonnegative. We must check that V is of trace class and study the smoothnessP of its kernel. We show that the constants n decrease sufficiently fast, namely, 1 n¼1 n < 1, i.e., V 2 1. To this end we estimate the integral in Equation (5.6) from below. As n :¼ ðun  n  unþ1 Þ=2, we have Z

un n unþ1

’2n ðtÞ dt ¼ m um n t

Z

unþ1 þn

þ

Z

un n



unþ1 þn

unþ1

’2n ðtÞ dt ¼: I1 ðnÞ þ I2 ðnÞ: m um n t

ð5:7Þ

We estimate each of these integrals I1 ðnÞ and I2 ðnÞ separately. By Equation (3.22) 1 1 ;  m m1 ðu  tÞ um m u  t n n n

ð5:8Þ

therefore taking into account the explicit form (5.1) of the function ’n I1 ðnÞ 

c2n

Z

unþ1 þn unþ1

c2n ¼ mum1 n 

Z

c2n m um1 n

n 0

Z

ðt  unþ1 Þ2 dt m um1 ðun  tÞ n x2 dx un  unþ1  x

n 0

x2 dx : un  unþ1

ð5:9Þ

As un  unþ1 ¼ 2n þ "n and 2 þ 1 ¼ m, we find that I1 ðnÞ 

c2n 1 nm :

2n þ "n m m um1 n

ð5:10Þ

Similarly,

I2 ðnÞ ¼ 

c2n

Z

un n un n n

ðun  n  tÞ2 c2n dt ¼ m um1 ðun  tÞ m um1 n n

c2n 1 nm :

"n þ n m m um1 n

Z

n 0

x2 dx "n þ x ð5:11Þ

126

SERGUEI I. IAKOVLEV

If the sequences un and n are chosen in such a way that ðunþ1 =un Þ ! 0 and ðn =un Þ ! 0 as n ! 1, then ðn =un Þ ! 1=2, whence, according to Equations (5.10) and (5.11), lim inf I1 ðnÞ=c2n 

lim

n!þ1

1 ðn =un Þm 1 ¼ 2 m ;
2 m 2ðn =un Þ þ ð"n =un Þ m2

ð5:12Þ

lim inf I2 ðnÞ=c2n 

lim

1 ðn =un Þm 1 ¼ 2 m1 :
2 m ðn =un Þ þ ð"n =un Þ m2

ð5:13Þ

n!þ1

n!þ1

n!þ1

So, passing (if necessary) from un and "n to certain subsequences, we can assume that for any fixed " > 0 the inequalities   1 1 " 2 I1 ðnÞ þ I2 ðnÞ  cn þ  ð5:14Þ m2 2m m2 2m1 2 are fulfilled for all n ¼ 1; 2; . . . . By Equation (5.6) 0 ¼ 1  2n ðI1 ðnÞ þ I2 ðnÞÞ; therefore   1 1 " 2 0  1  2n cn þ  : ð5:15Þ m2 2m m2 2m1 2 That gives n c2n

 

1 1 1 þ " : m2 2m1 m2 2m2

ð5:16Þ

If " > 0 is sufficiently small, then 1=ðm2 2m1 Þ  " > 0, so that n c2n  m2 2m2 :

ð5:17Þ

Recalling Equation (5.3) and the condition 1 X n¼1

P

n < þ1, we finally obtain

n

X X
un  "n  unþ1 m X1 m  C  ¼ C n c2n 2 n n n X X  C um n < þ1 : n ¼C

n  m2 2m2

n

ð5:18Þ

n

Thus, the operator V belongs to the trace class. It remains to check the smoothness condition (2.4) for the kernel of the operator V . For that it is sufficient to establish the following inequality (see Equation (3.3)) kðt þ hÞ  ðtÞk

l2

 C~m jhj ; h 2 R ;

with the constant C~m ¼ 2ðm1Þ=2 m and  ¼ ðm  1Þ=2 .

ð5:19Þ

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

127

~ðtÞ ¼ t ; t  0. Introducing the convenient notations Let ! u1n :¼ un  "n ; u2n :¼ unþ1 ;
n :¼ ½u2n ; u1n  ;

ð5:20Þ


2n :¼ ½ u2n ; ðu1n þ u2n Þ=2  ;
1n :¼ ½ ðu1n þ u2n Þ=2 ; u1n  ; we can write Equation (5.1) as follows 8 t 2 ½ 0; þ1Þ n
n ; > >  0; 2  < ~t  un ; t 2
2n ; cn ! ’n ðtÞ :¼ 1 ~ un  t ; t 2
1n ; c! > > : n t < 0: ’n ðtÞ ;

ð5:21Þ

~ðt1 þ ~ðtÞ ¼ t is semiadditive, that is, ! Note that for  2 ½ 0; 1 the function ! ~ðt1 Þ þ ! ~ðt2 Þ for all t1 ; t2  0 . t2 Þ  ! ~ðtÞ is monotone nondecreasing and semiadditive LEMMA 5.2. Suppose that ! (and thus nonnegative). Then, the function ’n ðtÞ defined by Equation (5.21) satisfies the following smoothness condition ~ðjhjÞ ; j ’n ðt þ hÞ  ’n ðtÞj  cn !

t;h 2 R:

ð5:22Þ

Inequality (5.22) remains true if f
n gþ1 n¼1 is an arbitrary sequence of disjoint finite intervals of the positive semiaxis. =
n, then for some i 2 f1 ; 2g we have Proof. If t1 2
n and t2 2 ~ð jt1  uin jÞ  cn ! ~ðjt1  t2 jÞ : j’n ðt1 Þ  ’n ðt2 Þj ¼ j’n ðt1 Þj ¼ cn !

ð5:23Þ

~ðt þ Þ  ! ~ðtÞ  ! ~ðÞ , according to the semiadditivity of ! ~: For t ;  > 0 , ! Therefore, for t1 ; t2 2
in we obtain   ~ð jt2  uin jÞ ~ð jt1  uin jÞ  ! j ’n ðt1 Þ  ’n ðt2 Þj ¼ cn  ! ~ð jt1  t2 jÞ :  cn !

ð5:24Þ

Now, let t1 2
in ; t2 2
jn , i 6¼ j. Assume that j t1  uin j  j t2  u jn j, then   ~ð jt2  u jn jÞ  ~ð jt1  uin jÞ  ! j ’n ðt1 Þ  ’n ðt2 Þj ¼ cn ! ð5:25Þ ~ð jt2  u jn jÞ ~ð jt1  u jn jÞ  !  cn ! ~ðjt1  t2 jÞ :  cn ! The lemma is proved. Next, it is easy to see that X n j’n ðt þ hÞ  ’n ðtÞj2 kðt þ hÞ  ðtÞk2 ¼ l2

 2 sup ðn c2n Þ
jhj2 : n

Ì

n

ð5:26Þ

128

SERGUEI I. IAKOVLEV

For the proof of this inequality we observe that, since the supports of the functions P ’n are disjoint, for every t the sum n n j’n ðt þ hÞ  ’n ðtÞj2 ¼: S is either zero or consists of at most two terms. If there is only one term, we apply Lemma 5.2. Suppose it consists precisely of two terms; then, if, say, t; h > 0, it is of the form S ¼ i j’i ðtÞj2 þ j j’j ðt þ hÞj2 with i > j . By virtue of Equation (5.1) ’i ðui Þ ¼ 0 and ’j ðujþ1 Þ ¼ 0, therefore, taking into account inequality (5.22), we have S ¼ i j’i ðtÞ  ’i ðui Þj2 þ j j’j ðt þ hÞ  ’j ðujþ1 Þj2  i c2i ðui  tÞ2 þ j c2j ðt þ h  ujþ1 Þ2  i c2i h2 þ j c2j h2 n o  2 max i c2i ; j c2j h2 :

ð5:27Þ

Finally, in view of inequality (5.17), it follows from Equation (5.26) that kðt þ hÞ  ðtÞk  l2

pffiffiffi ðm2Þ=2 22 mjhj  C~m jhj :

Theorem 5.1 is completely proved.

ð5:28Þ

Ì

6. Finite Rank Perturbations: A Counterexample If rank V ¼ 1, then V ¼ ð
; ’Þ’ with ’ 2 L2 ðRÞ. In this case the smoothness condition (2.4) is written in the form   j ’ðt þ hÞ  ’ðtÞj  ! jhj ; j hj  1 ;

ð6:1Þ

with the function !ðtÞ satisfying condition (2.5). eðhÞ :¼ Note that for any function ’ðtÞ its actual modulus of continuity ! sup fj ’ðxÞ  ’ðyÞj : jx  yj < hg always satisfies the additional constraint of eðt1 Þ þ ! eðt2 Þ for all t1 ; t2  0 . eðt1 þ t2 Þ  ! semiadditivity: ! Theorem 4.3 involves the condition !ðtÞ ¼ Oðtðm1Þ=2 Þ as t ! 0 ensuring for the finite rank perturbation operator V the emptiness of the roots set Nm near the origin. This condition appears to be sharp in the class of semiadditive functions !ðtÞ. THEOREM 6.1. Let m > 1. Suppose that !ðtÞ; t  0; is a monotone nondecreasing function satisfying the condition !ð0þ Þ ¼ !ð0Þ ¼ 0 as well as the natural additional condition of semiadditivity: !ðt1 þ t2 Þ  !ðt1 Þ þ !ðt2 Þ for all t1 ; t2  0 . If lim supt!0 !ðtÞ=tðm1Þ=2 ¼ þ1 , then a compactly supported function ’ : R ! R satisfying condition (6.1) is constructed and such that the

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

129

operator Am ¼ jtjm
þ ð
; ’Þ’ has a sequence of positive eigenvalues converging to zero. COROLLARY 6.1. It is not hard to show (see [12, Lemma 2.2]) that if !ðtÞ is a nonnegative semiadditive function and !ðtÞ # 0 as t # 0, then for any a > 0 there exists a constant C > 0 such that Ct  !ðtÞ for t 2 ½ 0; a. Hence, lim supt!0 !ðtÞ=tðm1Þ=2 ¼ þ1 for all m > 3. Therefore, as follows from Theorem 6.1, for every m > 3 and for each monotone and semiadditive function !ðtÞ; t  0; satisfying the condition !ð0þ Þ ¼ !ð0Þ ¼ 0 (and thus nonnegative) a real-valued compactly supported function ’ is constructed satisfying the smoothness condition (6.1) and such that the operator Am ¼ jtjm
þ ð
; ’Þ’ has a sequence of positive eigenvalues converging to zero. This means, in particular, that for m > 3 there is no condition guaranteeing the absence of the singular spectrum of the operator Am ¼ jtjm
þ V near the origin in terms of a modulus of continuity of the perturbation V. COROLLARY 6.2. If m 2 ð 1; 3 , then, according to Theorem 6.1, the sufficient condition !ðtÞ ¼ Oðtðm1Þ=2 Þ as t ! 0 guaranteeing the absence of the singular spectrum of the operator Am near the origin for the finite rank perturbation operator, rank V < 1, (see Theorem 4.3) is sharp. If this condition is not fulfilled, that is, lim supt!0 !ðtÞ=tðm1Þ=2 ¼ þ1, then even in the case when V is a rank 1 perturbation there can exist nontrivial singular spectrum near zero, and, in particular, the operator Am can have a sequence of positive eigenvalues converging to zero. Proof. In the proof of this theorem we combine several lemmas and the result of Theorem 6 from the paper [15] on the solvability of a certain linear equation in the Banach space l1 . LEMMA 6.1. Let ’ 2 L2 ðRÞ. A point  > 0 is an eigenvalue of the operator Am ¼ jtjm
þ ð
; ’Þ’ if and only if Z

j’2 ðtÞj dt ¼ 0 ; m R jtj  

ð6:2Þ

’ðtÞ=ðjtjm  Þ 2 L2 ðRÞ :

ð6:3Þ

1 þ and

If these conditions are fulfilled, then the corresponding eigensubspace is onedimensional and is generated by the function ’ðtÞ=ðjtjm  Þ. Proof. It is sufficient in Lemma 2.1 from the paper [12] to change the Ì operator Sm by Am .

130

SERGUEI I. IAKOVLEV

Let fun g1 1 be a monotone decreasing sequence of real numbers from the interval ð0; 101 Þ that tends to zero and satisfies the condition unþ1 < un =4 . On the real axis we define a sequence of functions ’n ðtÞ as follows:  8  ! jt  u j ; > n <  un  ’n ðtÞ :¼ ! jt  j ; > 2 : 0;

t 2 ½ 34 un ; un ; t 2 ½ 12 un ; 34 un ; t

2 = ½ 12

ð6:4Þ

un ; un :

We shall show that it is possible to choose the un and a bounded sequence of m nonnegative numbers fcn g1 1 in such a way that the points n ¼ ð9un =8Þ will be m eigenvalues of the operator Am ¼ jtj
þ ð
; ’Þ’ with the function þ1 X ’ðtÞ :¼ ðcn Þ1=2 ’n ðtÞ:

ð6:5Þ

n¼1

By Lemma 5.2 we have that j’n ðt þ hÞ  ’n ðtÞj  !ðjhjÞ; h 2 R. The supports of the functions ’n ðtÞ  0 are disjoint, therefore we easily obtain (see the details in [12, Lemma 4.1]) that j’ðt þ hÞ  ’ðtÞj  supðcn Þ1=2
!ðjhjÞ ; h 2 R :

ð6:6Þ

n

By construction, the function ’ðtÞ is compactly supported and bounded. Since un 2 ð0; 101 Þ and un < un1 =4, for m > 1 the value n ¼ ð9un =8Þm < un1 =2: Hence, ’ðtÞ vanishes identically near the points n . Therefore, condition (6.3) is fulfilled for all  ¼ n ; n ¼ 1; 2


. Putting in Equation (6.2)  ¼ n and substituting the expression ’ðtÞ :¼ Pþ1 k 1=2 ’k ðtÞ in Equation (6.2), we obtain a system of linear equations for k¼1 ðc Þ the unknowns ck : n1 þ1 X X ðdnk ck Þ þ dnn cn þ ðdnk ck Þ ¼ 1 ; k¼1

n ¼ 1; 2; . . . ;

ð6:7Þ

k¼nþ1

with the coefficients Zuk dnk :¼ uk =2

’2k ðtÞ dt : j tm  n j

ð6:8Þ

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

131

LEMMA 6.2. The coefficients dnk of the linear system (6.7) satisfy the following inequalities with some constant m  1 1 !2 ðun =4Þ ;
m ðun =4Þm1

dnn 

dnk  m
þ1 X

!2 ðuk =4Þ ðuk =4Þm1

ð6:9Þ

; k ¼ 1; . . . ; n  1 ;

ð6:10Þ

dnk  !2 ðunþ1 Þ=um1 : n

ð6:11Þ

k¼nþ1

Proof. According to Equation (3.22), for arbitrary a ; b > 0 and m  1 the following inequality holds 1 1  m m1 m1 ja  bj
maxfa ; b g m ja  bm j 

1 : ja  bj
maxfam1 ; bm1 g

ð6:12Þ

To prove the lemma, we use inequality (6.12) with elementary estimates and a simple change of variables un  t ¼: x . Indeed, we have Z un !2 ðun  tÞ ð6:13Þ dt dnn  m m 3un =4 ð9un =8Þ  t Z un !2 ðun  tÞ dt  m m 3un =4 ð9un =8Þ  ð3un =4Þ Z un =4 1 !2 ðxÞ ¼ dx : ðun =4Þm 0 ð9=2Þm  3m Now, using Equation (6.12), we obtain that dnn

1 1 
ðun =4Þm mð9=2Þm1 ð9=2  3Þ 

!2 ðun =8Þ
ðun =8Þ ðun =4Þm
mð9=2Þm1 ð3=2Þ

:

Z

un =4

!2 ðxÞ dx

un =8

ð6:14Þ

To complete the proof of Equation (6.9), it remains to recall that !ðtÞ is semiadditive, whence !2 ðun =4Þ  ð2!ðun =8ÞÞ2 so that !2 ðun =8Þ  !2 ðun =4Þ=4:

132

SERGUEI I. IAKOVLEV

Hence dnn 

!2 ðu =4Þ

1

n
m1 mð9=2Þ 12 ðu =4Þm1

ð6:15Þ

:

n

Next, for k ¼ 1; . . . ; n  1 we have dnk ¼

Z

3uk =4

uk =2

 2

!2 ðt  uk =2Þ dt þ tm  ð9un =8Þm

Z

uk

!2 ðuk  tÞ dt tm  ð9un =8Þm

3uk =4

!2 ðuk =4Þ
ðuk =4Þ : ðuk =2Þm  ð9un =8Þm

ð6:16Þ

By assumption we have un  uk =4 for k ¼ 1; . . . ; n  1, so that dnk  2

!2 ðuk =4Þ
ðuk =4Þ !2 ðuk =4Þ 1 ¼ 2 :
m m m1 m ðuk =2Þ  ð9uk =32Þ 2  ð9=8Þm ðuk =4Þ

ð6:17Þ

Finally, using Equation (6.12) we obtain dnk 

!2 ðuk =4Þ

2


ðu =4Þm1 2m1 ð2  9=8Þ

¼

k

!2 ðuk =4Þ


ðu =4Þm1 k

25m : 7

ð6:18Þ

Comparing Equations (6.15) and n (6.18) we see that theoconstant m can be taken from the condition m  max mð9=2Þm1 12 ; 25m =7 : Inequality (6.11) can be established in a similar way: þ1 X

dnk  2

k¼nþ1

Zuk

þ1 X k¼nþ1

 2

Zunþ1

3uk =4

!2 ðunþ1  tÞ dt ð9un =8Þm1 ð9un =8  tÞ

0



!2 ðuk  tÞ dt ð9un =8Þm  tm

!2 ðunþ1 Þ
unþ1
ð9u =8Þm1 9un =8  unþ1 2

:

ð6:19Þ

!2 ðunþ1 Þ
ðun =4Þ !2 ðunþ1 Þ :  9un =8  un =4 um1 n

ð6:20Þ

n

As unþ1 < un =4, we find that þ1 X k¼nþ1

dnk 

2


ð9u =8Þm1 n

Lemma 6.2 is proved.

Ì

SINGULAR SPECTRUM NEAR A SINGULAR POINT OF FRIEDRICHS MODEL OPERATORS

133

Remark. It is easily seen that inequalities (6.9) Y (6.11) allow us to choose a sequence un in such a way that the coefficients dnk of Equation (6.7) constitute an infinite P diagonally dominant matrix, i.e., for all n ¼ 1; 2;


we have dnn > ð1  Þ k: k6¼n dnk ; " 2 ð0; 1Þ. It is well known (see, e.g., [9, ` 2, Lemma 6]) that such a system has a unique solution in the Banach space l1 of bounded sequences. Nevertheless, this does not guarantee that we obtain nonnegative numbers cn ; n ¼ 1; 2;


. As lim supt!0þ !2 ðtÞ=tm1 ¼ þ1 , we assume that the sequence un is chosen in such a way that the sequence !2 ðun =4Þ=ðun =4Þm1 is strictly monotone and tends to þ1 as n ! 1. Passing, if necessary, to a subsequence of un (for which we keep the same notation), we may require that for all n ¼ 1; 2;


the following inequalities be satisfied:  1=2 ; !2 ðunþ1 Þ=um1 n !2 ðun =4Þ m1

ðun =4Þ

 ð22m Þn :

ð6:21Þ ð6:22Þ

LEMMA 6.3. For n ¼ 1; 2;


; the coefficients dnk of the system (6.7) satisfy the inequalities dnk =dkk  2m ;

k ¼ 1;


; n  1 ;

ð22m Þn1 =dnn  1=2 ; þ1 X

dnk  1=2 :

ð6:23Þ ð6:24Þ ð6:25Þ

k¼nþ1

Proof. Relation (6.23) follows from Equations (6.9) and (6.10), inequality (6.25) is a consequence of Equations (6.11) and (6.21), and Equation (6.24) can Ì be deduced from Equations (6.9) and (6.22). Lemma is proved. Having proved this lemma, we see that Theorem 6.1 is an immediate consequence of Theorem 6 from the paper [15] on the solvability of a system of linear equations having a specific form. According to this theorem, in which we put  :¼ 2m , if conditions (6.23) Y (6.25) are fulfilled, then the system (6.7) has a unique solution c ¼ ðc1 ; c2 ;


Þ. The components cn ; n 2 N; are nonnegative, and supn cn  1=2 . Therefore, by Equation (6.6) we obtain that j’ðt þ hÞ  ’ðtÞj  supðcn Þ1=2
!ðjhjÞ < !ðjhjÞ ; h 2 R ;

ð6:26Þ

n

which completes the proof of Theorem 6.1.

Ì

134

SERGUEI I. IAKOVLEV

References 1.

Faddeev, L. D.: On a model of Friedrichs in the theory of perturbations of the continuous spectrum, Trudy Mat. Inst. Steklov 73 (1964), 292Y313. 2. Pavlov, B. S. and Petras, S. V.: The singular spectrum of a weakly perturbed multiplication operator, Funktsional. Anal. i Prilozhen. 4(2) (1970), 54 Y 61. 3. Naboko, S. N.: Uniqueness theorems for operator-functions with positive imaginary part, and the singular spectrum in the selfadjoint Friedrichs model, Dokl. Akad. Nauk SSSR 275(6) (1984), 1310Y1313. 4. Naboko, S. N.: Uniqueness theorems for operator-valued functions with positive imaginary part, and the singular spectrum in the selfadjoint Friedrichs model, Ark. Mat. 25(1) (1987), 115Y140. 5. Mikityuk, Ya. V.: The singular spectrum of selfadjoint operators, Dokl. Akad. Nauk SSSR 303(1) (1988), 33Y36. 6. Yakovlev, S. I.: Perturbations of a singular spectrum in a selfadjoint Friedrichs model, Vestnik Leningrad Univ. Mat. Mekh. Astronom. (1) (1990), 116 Y 117. 7. Naboko, S. N. and Yakovlev, S. I.: Conditions for the finiteness of the singular spectrum in a selfadjoint Friedrichs model, Funktsional Anal. i Prilozhen. 24(4) (1990), 88 Y 89. 8. Yakovlev, S. I.: On the structure of the singular spectrum in selfadjoint Friedrichs model, L., (1991) (Manuscript, Depon. VINITI, no. 2050-B, 17.05.91). 9. Dinkin, E. M., Naboko, S. N. and Yakovlev, S. I.: The boundary of finiteness of the singular spectrum in the selfadjoint Friedrichs model, Algebra i Analiz. 3(2) (1991), 77Y 90. 10. Naboko, S. N. and Yakovlev, S. I.: On point spectrum of the discrete Schro¨dinger operator, Funktsional. Anal. i Prilozhen. 26(2) (1992), 85 Y 88. 11. Naboko, S. N. and Yakovlev, S. I.: Discrete Schro¨dinger operator. The point spectrum on the continuous one, Algebra i Analiz. 4(3) (1992), 183 Y 195. 12. Iakovlev, S. I.: Examples of Friedrichs model operators with a cluster point of eigenvalues, Int. J. Math. Math. Sci. 2003 (2003), 625 Y 638. 13. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics. 4. Analysis of Operators. Academic, New York, 1978. 14. Yakovlev, S. I.: On the singular spectrum of the Friedrichs model operators in a neighborhood of a singular point, Funktsional Anal. i Prilozhen. 32(3) (1998), 91Y 94. 15. Yakovlev, S. I.: The finiteness bound for the singular spectrum of Friedrichs model operators near a singular point, Algebra i Analis, 10(4) (1998), 210 Y 237. 16. Privalov, I. I.: Boundary Properties of Analytic Functions, 2nd edn. GITTL, MoscowLeningrad, 1950; German transl., VEB Deutscher Verlag Wiss., Berlin, 1956. 17. Yakovlev, S. I.: Spectral Analysis of Self-Adjoint Operators of the Friedrichs Model (singular spectrum). PhD thesis, St.Peterburg. (Russian), 1991. 18. Yafaev, D. R.: Mathematical Scattering Theory, Saint-Petersburg University Press, 1994.

Mathematical Physics, Analysis and Geometry (2006) 9: 135Y186 DOI: 10.1007/s11040-005-9004-6

#

Springer 2006

Deformations of Fuchsian Systems of Linear Differential Equations and the Schlesinger System Dedicated to the centenary of the Schlesinger system VICTOR KATSNELSONj and DAN VOLOK Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel. e-mail: {victor.katsnelson, dan.volok}@weizmann.ac.il (Received: 24 June 2005; in final form: 28 October 2005; published online: 30 June 2006) Abstract. We consider holomorphic deformations of Fuchsian systems parameterized by the pole loci. It is well known that, in the case when the residue matrices are non-resonant, such a deformation is isomonodromic if and only if the residue matrices satisfy the Schlesinger system with respect to the parameter. Without the non-resonance condition this result fails: there exist non-Schlesinger isomonodromic deformations. In the present article we introduce the class of the so-called isoprincipal deformations of Fuchsian systems. Every isoprincipal deformation is also an isomonodromic one. In general, the class of the isomonodromic deformations is much richer than the class of the isoprincipal deformations, but in the non-resonant case these classes coincide. We prove that a deformation is isoprincipal if and only if the residue matrices satisfy the Schlesinger system. This theorem holds in the general case, without any assumptions on the spectra of the residue matrices of the deformation. An explicit example illustrating isomonodromic deformations, which are neither isoprincipal nor meromorphic with respect to the parameter, is also given. Mathematics Subject Classifications (2000): 34Mxx, 34M55, 93B15, 47A56. Key words: differential equations in the complex domain, isomonodromic deformation, isoprincipal deformations, Schlesinger system.

NOTATION  C stands for the complex plane.  CP1 stands for the extended complex plane (¼ the Riemann sphere):

CP1 ¼ C [ 1:  Cn stands for the n-dimensional complex space.  In the coordinate notation, a point t 2 Cn will be written as t ¼ ðt1 ; . . . ; tn Þ:

j

The research of Victor Katsnelson was supported by the Minerva Foundation.

136

VICTOR KATSNELSON AND DAN VOLOK

 C*n is the set of points t 2 Cn ; whose coordinates t1 ; . . . ; tn are pairwise

different:

[

Cn* ¼ Cn n

ft : ti ¼ tj g

1 r i;j r n i6¼j

 Mk stands for the set of all k  k matrices with complex entries.  ½ ;  denotes the commutator: for A; B 2 Mk ; ½A; B ¼ AB  BA.  I stands for the identity matrix of an appropriate dimension.



1. Introduction The systematic study of linear differential equations in the complex plane with coefficients dependent on parameters has been started by Lazarus Fuchs in the late eighties of the 19th centuryj [12Y15]. In particular, L. Fuchs investigated the equations whose monodromy does not depend on such parameters. These investigations were continued in the beginning of the twentieth century by L. Schlesinger,jj [35Y37], R. Fuchs,- [16, 17], and R. Garnier, [19]. L. Schlesinger’s research was closely related to the Hilbert 21st problem (a.k.a. the RiemannYHilbert monodromy problem), which requires to construct a Fuchsian system with prescribed monodromy (for the explanation of terminology see Section 1 of the present article). In the paper [35], which appeared exactly 100 years ago Y in 1905, L. Schlesinger proposed the idea that it would be very fruitful to study the deformations of Fuchsian systems X Qj ðtÞ dY ¼ dx 1 r j r n x  tj

 Y;

ð0:1Þ

where the residues Qj depend holomorphically on the pole loci t ¼ ðt1 ; . . . ; tn Þ, and investigate the dependence of the solution Y on t, as well as on x. Emphasizing this idea, L. Schlesinger explained that he was guided by the analogy with the theory of algebraic functions, where he had studied algebraic functions as functions of both the Fmain variable_ and the loci of ramification points considered as parameters (see [34, pp. 287Y288]). Also in the paper [35], the system of PDEs @Qj @tk @Qj @tj j jj

-

½Qj ; Qk  ; 1 r j; k r n; k 6¼ j; tj  tk   P Qj ; Qk ¼ ; 1 r j r n; 1 r k r n tj  tk ¼

k6¼j

L. Fuchs died in 1902. A student of L. Fuchs. The son of L. Fuchs.

ð0:2Þ

DEFORMATIONS OF FUCHSIAN SYSTEMS

137

which is now known as the Schlesinger system, was introduced and the statement that the holomorphic deformation (0.1) is isomonodromic if and only if its coefficients Qj ðtÞ satisfy the system (0.2) was formulated. (See page 294 of [35], four bottom lines of this page.) This formulation was repeated in the book [36], pp. 328Y329, and later in the paper [37], p. 106. (References to the earlier paper [35] are relatively rare. Usually, one refers to the more recent paper [37].) Over the years the Schlesinger system and the isomonodromic deformations of Fuchsian systems were extensively studied; we would like to mention in particular the papers of T. Miwa [32] and of B. Malgrange [29], where it was proved that the Schlesinger system enjoys the Painleve´ property (its solutions are meromorphic functions in the universal covering space over C*n ), and the book by A.R. Its and V. Yu. Novokshenov [22], where the isomonodromic deformations were used in the global asymptotic analysis of the Painleve´ transcendents. However, in the 1990s the famous negative solution to the RiemannYHilbert monodromy problem due to A. A. Bolibrukh (see [4, 5]) gave a strong motivation for the revision of the classical results, concerning the isomonodromic deformations of the Fuchsian systems. For example, it should be noted that in the original works [35Y37] of L. Schlesinger no assumptions concerning the non-resonance of the matrices Qj are made. In such generality the above-cited statement of L. Schlesinger fails. If a holomorphic deformation (0.1) of Fuchsian systems is such that the residues Qj satisfy the Schlesinger system (0.2), then this deformation is isomonodromic, but the converse statement is not true, in general. It was also A. A. Bolibrukh who constructed the first explicit example of the non-Schlesinger isomonodromic deformation. In this example the monodromy is non-trivial and the residues Qj ðtÞ are rational functions of t (see [6] and [7] Y in both papers the same example appears as Example 2; see alsoj Section 3 of the review paper [8], where this example appears as Example 3). At the same time it was shown independently in [23] that almost every isomonodormic deformation of Fuchsian systems with generic rational solutionsjj is non-Schlesinger (for more details see Remark 1 in Section 5 of this article). Thus the isomonodromic property of the deformation (0.1) implies the Schlesinger system for the residues Qj ðtÞ under the non-resonance condition, but not in general. Unfortunately, careless treatment of the non-resonance condition is very common in the history of problems related to the monodromy of Fuchsian systems. It can also be found in some works of V. Volterra and of G. D. Birkhoff (see [18, Chapter XV, Section 9] for details). This tradition continues to certain extent in the above-mentioned paper [32] of T. Miwa on the Painleve´ property of j jj

Unfortunately, to our best knowledge this review has not yet been translated to English. In particular, with trivial monodromy.

138

VICTOR KATSNELSON AND DAN VOLOK

isomonodromic deformations: in this paper the non-resonance condition appears as the equation (2.22), but is omitted both in the introduction and in the formulation of the main result. Without the assumption of non-resonance the main result of [32] does not hold: there exist isomonodromic deformations of the form (0.1), where the residues Qj ðtÞ are not meromorphic in the universal covering space over C*n . (The appropriate example is presented in Section 5 of this article. We note that this phenomenon does not occur in the above-mentioned example of the nonSchlesinger isomonodromic deformation due to A.A. Bolibrukh: in that example the residues Qj ðtÞ are rational functions of t.) The main goal of the present work is to answer the following question: how to describe the class of holomorphic deformations (0.1) with the property that the residues Qj ðtÞ satisfy the Schlesinger system (0.2), when one omits the nonresonance assumption? The presentation of our results is organized as follows. In the first section after this introduction we recall the basic notions concerning the Fuchsian system and introduce a certain canonical multiplicative decomposition of the fundamental solution in a neighborhood of its singular point tj . This is the so-called regular-principal factorization: the fundamental solution is represented as the product of a regular factor, holomorphic and invertible at the point tj ; and a principal factor, holomorphic (multi-valued) and invertible everywhere except at tj : This principal factor is the multiplicative analogue of the principal part in the Laurent decomposition: it contains the information about the nature of the singularity. In Section 2 we introduce the main notion of the present article (Definition 2.9): the so-called isoprincipal j families of Fuchsian systems. These are the holomorphic families (0.1) with the property that all the principal factors of a suitably normalized fundamental solution Yðx; tÞ are, in a certain sense, preserved. We show that every isoprincipal family is also isomonodromic and that the converse is true, when the non-resonance condition is in force. In Section 3 we formulate and prove our main result (Theorem 3.1) that the family (0.1) is isoprincipal if and only if the residues Qj ðtÞ satisfy the Schlesinger system (0.2). This result holds in the general case, without the assumption of nonresonance. In the next section we discuss the isoprincipal deformation of a given Fuchsian system. Using our Theorem 3.1, we also outline how to establish the Painleve´ property of the Schlesinger system and indicate possible generalizations. Finally, in Section 5 we illustrate the general theory with explicit examples of the isoprincipal and isomonodromic deformations. In particular, we give an example of the isomonodromic deformation which is not related to the Schlesinger system and does not possess the Panleve´ property. This example is j

Iso- (from :: o&  equal  in Old Greek) is a combining form.

DEFORMATIONS OF FUCHSIAN SYSTEMS

139

based on the theory of the isoprincipal families of Fuchsian systems with generic rational solutions, developed in [23], [24] and [27]. 2. Fuchsian Differential Systems 2.1. FUCHSIAN DIFFERENTIAL SYSTEMS A Fuchsian differential system is a linear system of ordinary differential equations of the form ! X Qj dY Y; ð1:1Þ ¼ dx x  tj 1rjrn where Qj , 1 r j r n; are square matrices of the same dimension, say Qj 2 Mk , and t1 ; . . . ; tn are pairwise distinct points of the complexplane C. The variable x Flives_ in the punctured Riemann sphere CP1 n t1 ; . . . ; tn , the Funknown_ Y is an Mk -valued matrix function of x. Under the condition X Qj ¼ 0 ð1:2Þ 1rjrn

the point x0 ¼ 1 is a regular point for the system (1.1). If this condition is satisfied (which we always assume in the sequel), then in a neighborhood of the point x0 ¼ 1 there exists a fundamental solution Y ¼ YðxÞ of (1.1) satisfying the initial condition  YðxÞ x¼1 ¼ I: ð1:3Þ This solution Y can be analytically continued into the multi-connected domain CP1 nft1 ; . . . ; tn g. However, for x 2 CP1 nft1 ; . . . ; tn g the value of Y at the point x depends, in general, on the path  from x0 ¼ 1 to x, along which the analytic continuation is performed: Y ¼ Yðx; Þ: More precisely, Y depends not on the path  itself, but on its homotopy class in CP1 nft1 ; . . . ; tn g. Thus Y is a multi-valued holomorphic function in the punctured Riemann sphere CP1 nft1 ; . . . ; tn g or, better to say, Y is a singledvalued holomorphic function on the universal covering surface of the punctured Riemann sphere CP1 nft1 ; . . . ; tn g with the distinguished point 1. 2.2. UNIVERSAL COVERING SPACES Recall (see [10, Chapter 1, Sections 3Y5] if need be) that the universal covering space covðX ; x0 Þ of an arcwise connected topological space X with the dis-

140

VICTOR KATSNELSON AND DAN VOLOK

tinguished point x0 2 X is the set of pairs ðx; Þ; where x is a point in X and  is a homotopy class of continuous mappings  : fs 2 R : 0 r s r 1g 7! X ;

ð0Þ ¼ x0 ; ð1Þ ¼ x:

Such a mapping  is called a path in X from x0 to x. A path in X from x0 to x0 is called a loop with the distinguished point x0 . The product    of two paths ;  in X , where  is a path from a to b and  is a path from b to c, is defined as the path from a to c, obtained by going first along  from a to b and then along  from b to c: 8 1 > < ð2sÞ; 0rsr ; 2 ð  ÞðsÞ ¼ 1 > : ð2s  1Þ; r s r 1: 2 With respect to this product, the homotopy classes of loops in X with the distinguished point x0 form the so-called fundamental group ðX ; x0 Þ of the space X with the distinguished point x0 . The fundamental group ðX ; x0 Þ acts on the universal covering space covðX ; x0 Þ (on the right) as the group of deck transformations ðx; Þ 2 covðX ; x0 Þ;  2 ðX ; x0 Þ:

ðx; Þ 7! ðx; Þ;

ð1:4Þ

2.3. MONODROMY Let Y ¼ Yðx; Þ be the solution of (1.1)Y(1.3) defined on the universal covering surface covðCP1 nft1 ; . . . ; tn g; 1Þ: For each loop  2 ðCP1 nft1 ; . . . ; tn g; 1Þ let us consider the function Y defined on covðCP1 nft1 ; . . . ; tn g; 1Þ by def

Y ðx; Þ ¼ Yðx; Þ:

ð1:5Þ

The expression (1.5) means that the value of Y at the point ðx; Þ of the universal covering surface covðCP1 nft1 ; . . . ; tn g; 1Þ is obtained by the analytic continuation of the solution Y of (1.1)Y(1.3): first along the loop  from the distinguished point x0 ¼ 1 to itself, then along the path  from x0 to x. Thus Y is also a fundamental solution of the linear system (1.1) and, therefore, there exists a unique invertible constant matrix M 2 Mk such that Y ðx; Þ  Yðx; Þ

 M ;

ðx; Þ 2 covðCP1 nft1 ; . . . ; tn g; 1Þ:

ð1:6Þ

DEFINITION 1.1. Let Y ¼ Yðx; Þ be the solution of (1.1)Y(1.3) on the universal covering surface covðCP1 nft1 ; . . . ; tn g; 1Þ and let  2 ðCP1 nft1 ; . . . ; tn g; 1Þ.

141

DEFORMATIONS OF FUCHSIAN SYSTEMS

The constant (with respect to x) matrix M 2 Mk , which appears in the identity (1.6), is said to be the monodromy matrix of the solution Y, corresponding to the loop . Note that for a pair of loops 1 ; 2 2 ðCP1 nft1 ; . . . ; tn gÞ and the corresponding monodromy matrices M1 ; M2 of the solution Y it holds that  Yðx; 1 2 Þ ¼ Y

 M

2

 ðx; 1 Þ ¼ Yðx; 1 Þ

 M

2

¼ Yðx; Þ

 M M : 1

2

Therefore, the monodromy matrices of Y satisfy the following multiplicative identity: M1 2 ¼ M1 M2

81 ; 2 2 ðCP1 nft1 ; . . . ; tn gÞ:

ð1:7Þ

This means that the mapping  7! M is a linear representation of the fundamental group ðCP1 nft1 ; . . . ; tn g; 1Þ: DEFINITION 1.2. Let Y be the solution of (1.1)Y(1.3) on the universal covering surface covðCP1 nft1 ; . . . ; tn g; 1Þ: The linear representation of the fundamental group ðCP1 nft1 ; . . . ; tn g; 1Þ  7! M ;

 2 ðCP1 nft1 ; . . . ; tn g; 1Þ;

ð1:8Þ

where M denotes the monodromy matrix of the solution Y, corresponding to the loop , is called the monodromy representation of the solution Y.

2.4. THE REGULAR-PRINCIPAL FACTORIZATION FOR A FUNDAMENTAL SOLUTION OF A FUCHSIAN SYSTEM: SINGLE-VALUED CASE

Each of the points tj ; 1 r j r n; is a singularity of the solution Y. This means that at least one of the two functions Y and Y 1 is not holomorphic at tj . More information about the nature of the singularity at tj can be obtained from a certain multiplicative decomposition of the solution Y near the point tj , which is called the regular-principal factorization. In order to explain the idea of the regular-principal factorization, let us assume for the moment that the solution Y ¼ YðxÞ is single-valued in the domain CP1 nft1 ; . . . ; tn g Y that is, the monodromy representation of Y is trivial: M ¼ I

8 2 ðCP1 nft1 ; . . . ; tn g; 1Þ:

For 1 r j r n let V j be an open simply connected neighborhood of tj in C, such that tk 62 V j for k 6¼ j. Then it follows, for instance, from G. D. Birkhoff’s results on factorization of matrix functions holomorphic in the annulus (see [1, Section

142

VICTOR KATSNELSON AND DAN VOLOK

7]) that in the punctured neighborhood V j n ftj g the solution YðxÞ admits a factorization of the form YðxÞ ¼ Hj ðxÞ  Pj ðxÞ; x 2 V j nftj g; where the function Hj ðxÞ is holomorphic and invertible in the entire (nonpunctured) neighborhood V j and the function Pj ðxÞ is holomorphic, single-valued and invertible in the punctured plane Cnftj g. The factors Hj ðxÞ and Pj ðxÞ are said to be, respectively, the regular factor and the principal factor of the solu tion Y at its singular point tj . In the general case, when the monodromy representation of Y may be nontrivial, the regular-principal factorization of Y is more involved. Indeed, on the one hand the solution Y is normalized at the distinguished point x0 ¼ 1. In order to consider Y in a neighborhood of tj , we have to choose a homotopy class of paths, connecting the distinguished point x0 ¼ 1 with this neighborhood of tj , and such a choice is not unique. On the other hand, even in the single-valued case x0 ¼ 1 is, in general, a singular point of the principal factor Pj . In the general case Pj will have to be considered as a function on a universal covering surface of the punctured plane Cnftj g. Therefore, Pj needs to be normalized at some distinguished point in Cnftj g (obviously different from x0 ¼ 1) and continued analytically from there. Thus, before we can present the regular-principal factorization of Y in the general case, we need some preparation. 2.5. BRANCHES OF THE SOLUTION OF A FUCHSIAN SYSTEM IN A NEIGHBORHOOD OF THE SINGULAR POINT

We propose the following terminology: DEFINITION 1.3. Let Y be the solution of (1.1)Y(1.3) on the universal covering surface covðCP1 nft1 ; . . . ; tn g; 1Þ: For 1 r j r n assume that: (i) V is a domain in CP1 nft1 ; . . . ; tn g; (ii) p is a point in the domain V; (iii)  is a path in CP1 nft1 ; . . . ; tn g from the distinguished point x0 ¼ 1 to the point p. Define the function Y on the universal covering surface covðV; pÞ by the analytic continuation of the solution Y first along the path , then inside the domain V: def

Y ðx; Þ ¼ Yðx; 

 Þ;

where x 2 V and  is a path in V from p to x.

ð1:9Þ

DEFORMATIONS OF FUCHSIAN SYSTEMS

143

Then the function Y , holomorphic in covðV; pÞ, is said to be the branch of the solution Y in the domain V, corresponding to the path . In what follows we shall be mostly dealing with the branches of the solution Y in domains of the form V j nftj g, where V j is a simply connected neighborhood of the singular point tj , such that tk 62 V j for k 6¼ j. In this case the universal covering surface covðV j nftj g; pj Þ has a simple structure: the fundamental group covðV j nftj g; pj Þ is cyclic, generated by the loop with the distinguished point pj which makes one positive circuit of tj in V j nftj g. DEFINITION 1.4. For 1 r j r n assume that (i) V j  C is an open simply connected neighborhood of the singular point tj , such that tk 62 V j for k 6¼ j; (ii) pj is a point in the punctured neighborhood V j nftj g; (iii) j is a path in CP1 nft1 ; . . . ; tn g from the distinguished point x0 ¼ 1 to the point pj . Let j be the loop in the punctured neighborhood V j n ftj g with the distinguished point pj which makes one positive circuit of tj , and let j be the loop in the punctured sphere CP1 nft1 ; . . . ; tn g with the distinguished point x0 ¼ 1, defined by j ¼ 1 j  j  j def

ð1:10Þ

(that is, the loop j goes from x0 ¼ 1 to pj along j , then makes one positive circuit of tj along the small loop j , then goes again along j , but in the opposite direction: from pj to x0 ¼ 1). Then:  the loop j is said to be the small loop around tj in the punctured neigh-

borhood V j nftj g;  the loop j is said to be the big loop around tj , corresponding to the path j .

Remark 1.5. Note that for a suitable choice of the paths 1 ; . . . ; n the corresponding big loops 1 ; . . . ; n generate the fundamental group ðCP1 n ft1 ; . . . ; tn g; 1Þ. These generators are not free: choosing 1 ; . . . ; n carefully we can ensure, for example, that 1    n ¼ 1: We observe that the surface covðV j nftj g; pj Þ in Definition 1.3 is naturally embedded into the universal covering surface covðCnftj g; pj Þ, which is isomorphic to the Riemann surface of the logarithm ln . Although the basic properties of the function ln  are very well-known, we shall discuss them in some detail, because they are important for our future considerations.

144

VICTOR KATSNELSON AND DAN VOLOK

2.6. THE RIEMANN SURFACE OF THE LOGARITHM For each fixed  2 Cnf0g the equation e ¼  def

has a countable set of solutions ¼ ðÞ ¼ ln . These solutions can be parameterized as ln  ¼ ln jj þ i arg ; where ln jj 2 R

ð1:11Þ

and arg  is an equivalence class of real numbers (the values of arg ) modulo addition by 2. Now we explain how the function ln  can be defined as a single-valued holomorphic function on the universal covering surface covðCnf0g; 1Þ of the punctured plane Cnf0g with the distinguished point 0 ¼ 1. Let us choose a point  2 Cnf0g and a path # in Cnf0g from 0 ¼ 1 to . Then there exists a unique 2 R such that, up to homotopy in Cnf0g, the path # can be parameterized as follows: #ðsÞ ¼ eðln jjþi Þs ;

0rsr1

ð1:12Þ

(here ln jj is the real-valued logarithm). The real number is said to be the value of arg  corresponding to the path #. In this manner we establish a 1-to-1 correspondence between the values of arg  and the homotopy classes of paths in Cnf0g from 0 ¼ 1 to . Thus the function arg  is defined as a single-valued continuous function on covðCnf0g; 1Þ. Accordingly, the function ln  is defined by (1.11) as a singlevalued holomorphic function on the universal covering surface covðCn f0g; 1Þ. In the sequel we shall often refer to the surface covðCnf0g; 1Þ as the Riemann surface of ln . The fundamental group ðCnf0g; 1Þ is cyclic, generated by the loop with the distinguished point 0 ¼ 1 which makes one turn counterclockwise around the origin. The corresponding deck transformation of covðCnf0g; 1Þ is denoted by  7! 

 e2i ;

ð1:13Þ

so that the following monodromy relations hold: argðe2i Þ ¼ arg  þ 2;

lnðe2i Þ ¼ ln  þ 2i:

ð1:14Þ

2.7. TRANSPLANTS OF FUNCTIONS DEFINED ON THE RIEMANN SURFACE LOF THE LOGARITHM

Let us consider the universal covering surface covðCnftj g; pj Þ, where tj is some point in the complex plane C and pj is a distinguished point in the punctured

DEFORMATIONS OF FUCHSIAN SYSTEMS

145

plane Cnftj g. Let us choose some value j of argðpj  tj Þ and let #j denote the corresponding path (see (1.12)) in Cnf0g from 0 ¼ 1 to  ¼ pj  tj . Let us define a mapping from the universal covering surface covðCnftj g; pj Þ into the Riemann surface of ln  as follows. To each point ðx; Þ 2 covðCnftj g; pj Þ; where  is a path in Cnftj g from pj to x, we associate the point ðx  tj ; tj #j Þ 2 covðC n f0g; 1Þ, where the path tj , _ which leads in Cnf0g from pj  tj to x  tj , is obtained by the parallel translation of the path : tj ðsÞ ¼ ðsÞ  tj ;

0 r s r 1:

This mapping is an isomorphism between covðC n ftj g; pj Þ and the Riemann surface of ln . It will be denoted by ! x  tj ; x argðpj tj Þ¼ j

x 2 covðCnftj g; pj Þ; x  tj 2 covðCnf0g; 1Þ;

ð1:15Þ

and the inverse mapping will be denoted by ! x þ tj ; x argðpj tj Þ¼ j

x 2 covðCnf0g; 1Þ; x þ tj 2 covðCnftj g; pj Þ:

ð1:16Þ

Accordingly, we shall denote the deck transformation of covðC n ftj g; pj Þ, corresponding to the loop with the distinguished point pj which makes one positive circuit around tj in C n ftj g, by x 7! tj þ ðx  tj Þ

 e2i ;

x 2 covðCnftj g; pj Þ:

ð1:17Þ

With this notation we can consider the function lnðx  tj Þ as a function of x, holomorphic on the universal covering surface covðC n ftj g; pj Þ and such that (see (1.14))  ln ðx  tj Þ

 e2i



¼ lnðx  tj Þ þ 2i:

ð1:18Þ

More generally, we propose the following DEFINITION 1.6. Let a function EðÞ be defined on the Riemann surface of ln . Let tj 2 C, let pj 2 Cnftj g and let us choose a value j of argð pj  tj Þ. For each x 2 covðCnftj g; pj Þ let x  tj denote the image of x in the Riemann surface of ln  under the isomorphism (1.15). The function Eðx  tj Þ, defined as a function of x on the universal covering surface covðCnftj g; pj Þ, is said to be the transplant of the function EðÞ into covðCnftj g; pj Þ, corresponding to the value j of argð pj  tj Þ.

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VICTOR KATSNELSON AND DAN VOLOK

2.8. THE REGULAR-PRINCIPAL FACTORIZATION FOR A FUNDAMENTAL SOLUTION OF A FUCHSIAN SYSTEM: GENERAL CASE

THEOREM 1.7. Let Y be the solution of the Fuchsian system (1.1)Y(1.2), satisfying the initial condition (1.3). For j ¼ 1; . . . ; n assume that: (i) V j  C is an open simply connected neighborhood of the singular point tj , such that tk 62 V j for k 6¼ j; (ii) pj is a point in the punctured neighborhood V j nftj g; (iii) j is a path in CP1 nft1 ; . . . ; tn g from the distinguished point x0 ¼ 1 to the point pj . Then for each j, 1 r j r n; the branch Yj of the solution Y in the punctured neighborhood V j n ftj g, corresponding to the path j , admits the factorization Yj ðxÞ ¼ Hj ðxÞ

 P ðxÞ; j

x 2 covðV j nftj g; pj Þ;

ð1:19Þ

where the factors possess the following properties: (R) the matrix function Hj ðxÞ is holomorphic and invertible in the entirej (non-punctured) neighborhood V j ; (P) the matrix function Pj ðxÞ is holomorphic and invertible on the universal covering surface covðCnftj g; pj Þ. DEFINITION 1.8. The factorization (1.19), where the factors Hj and Pj possess the properties (R) and (P), is said to be the regular-principal factorization of the branch Yj of the solution Y in a punctured neighborhood of the singular point tj : The factors Hj and Pj are said to be, respectively, the regular factor and the principal factor of the branch Yj . Proof of Theorem 1.7. For j ¼ 1; . . . ; n let j be the big loop around tj , corresponding to the path j (see Definition 1.4), and let Mj be the corresponding monodromy matrix of Y. Then, according to Definitions 1.1 and 1.3, the monodromy matrix Mj is given by Mj ¼ Y1j ðxÞYj ðtj þ ðx  tj Þ

 e2i Þ;

8x 2 covðV j nftj g; pj Þ;

ð1:20Þ

where Yj is the branch of Y in V j n ftj g, corresponding to the path j . Since the matrix Mj is invertible, there exists a matrix denoted by ln Mj , such thatjj eln Mj ¼ Mj :

j

In particular, the function Hj ðxÞ is single-valued in V j . Here we refer to [18, Chapter VIII, Section 8]. Such a matrix ln Mj is not unique, of course, but for our purposes any choice of ln Mj will do. jj

147

DEFORMATIONS OF FUCHSIAN SYSTEMS

Let us choose a transplant lnðx  tj Þ of the function ln  into covðCnftj g; pj Þ. Then, according to (1.18), the matrix function 1

1

def

ðx  tj Þ 2i ln Mj ¼ elnðxtj Þ 2i ln Mj ; which is holomorphic and invertible on covðC n ftj g; pj Þ, satisfies the relation ððx  tj Þ

 e2iÞ

1 2i

ln Mj

1

¼ ðx  tj Þ 2i ln Mj

 M ; j

x 2 covðC n ftj g; pj Þ:

Hence, in view of (1.20), the branch Yj of the solution Y in V j nftj g has the form Yj ðxÞ ¼ ðxÞ

 ðx  tjÞ

1 2i

ln Mj

;

x 2 covðCnftj g; pj Þ;

where ðxÞ is a matrix function, holomorphic, invertible and single-valued in the punctured neighborhood V j n ftj g. Now, according to [1, Section 7], we can factorize the function ðxÞ as ðxÞ ¼ þ ðxÞ

  ðxÞ;

x 2 V j nftj g

where þ ðxÞ and  ðxÞ are matrix functions, single-valued, holomorphic and invertible in, respectively, the entire (non-punctured) neighborhood V j and the punctured plane C n ftj g. We set Hj ðxÞ

def

Pj ðxÞ

def

¼ þ ðxÞ;

¼  ðxÞ

 ðx  tjÞ

1 2i

ln Mj

and obtain the desired factorization (1.19) with the properties (R) and (P). This Ì completes the proof. Remark 1.9. Of course, the principal and regular factors of the branch Yj of the solution Y in a punctured neighborhood of the singularity tj are determined only up to the transformation Pj ðxÞ ! TðxÞ

 P ðxÞ;

Hj ðxÞ ! TðxÞ1

j

 H ðxÞ; j

ð1:21Þ

where TðxÞ is an invertible entire matrix function. However, once the choice of, say, the regular factor Hj is fixed, the principal factor Pj is uniquely determined. Moreover, if we choose a different path from x0 to pj , say 0j , then, in view of Definitions 1.1 and 1.3, the branches Yj and Y0 of the solution Y are related by j

Y0 ðxÞ ¼ Yj ðxÞ j

 M

1 0 j j

;

x 2 covðC n ftj g; pj Þ;

0 is the monodromy matrix of Y; corresponding to the loop where M1 j j 1 0 j j 2 ðCP1 nft1 ; . . . ; tn g; 1Þ. Hence the branch Y0j admits the regular-

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VICTOR KATSNELSON AND DAN VOLOK

principal factorization Y0j ðxÞ ¼ H0j ðxÞ where Hj ðxÞ P0j ðxÞ

¼ ¼

 P ðxÞ; 0 j

x 2 covðCnftj g; pj Þ;

H0j ðxÞ; 0; Pj ðxÞ  M1 j j

x 2 V j; x 2 covðCnftj g; pj Þ:

Thus the regular factor at the singular point tj can be chosen independently of the choice of the path j and will be denoted simply by Hj ðxÞ. Remark 1.10. Up to now, we have made no use of the fact that the system (1.1) is Fuchsian (that is, each singularity is a simple pole for the coefficients of the system). In particular, the regular-principal factorization (1.19) of the fundamental solution of a linear differential system also takes place in a neighborhood of the isolated singularity, where the coefficients of the system have a higher order pole or even an essential singular point. However, in the special case when the system is Fuchsian, the general form of the fundamental solution in a neighborhood of its singular point is quite well known (see, for instance, [18, Chapter XV, Section 10]) and thus much more precise statements concerning the principal factors of the solution of the system (1.1) can be made. If the matrix Qj is non-resonant,j then the principal factor Pj can be chosen in the form Pj ðxÞ ¼ ðx  tj ÞAj ; where Aj is a matrix, similar to the matrix Qj : Aj ¼ C1 j Q j C j ;

ð1:22Þ ð1:23Þ

where Cj is an invertible matrix. In the general case (without the assumption that the matrix Qj is non-resonant) the principal factor Pj can be chosen in the form Pj ðxÞ ¼ ðx  tj ÞZj

 ðx  tj ÞA

j

;

ð1:24Þ

where Zj is a diagonalizable matrix with integer eigenvalues l1 ; . . . ; lk and Aj is ^ p are related to the eigenvalues p of a non-resonant matrix, whose eigenvalues the matrix Qj by the equations ^ p ¼ p  lp ; 1 r p r k: ð1:25Þ The matrices Zj ; Aj also possess certain additional properties, but we shall not go into further details, because in the sequel our considerations will be mostly based on the existence of the regular-principal factorization (1.19) described in Theorem 1.7 rather than on the specific form of the factors. j A square matrix Q is said to be non-resonant if distinct eigenvalues of Q do not differ by integers or, in other words, if the spectra of the matrices Q þ nI and Q are disjoint for every n 2 Z n 0.

DEFORMATIONS OF FUCHSIAN SYSTEMS

149

3. Holomorphic Families of Fuchsian Differential Systems 3.1. FAMILIES OF FUCHSIAN SYSTEMS, PARAMETERIZED BY THE POLE LOCI In the present paper we consider a family of linear differential systems of the form

X Qj ðtÞ dY Y: ð2:1Þ ¼ dx x  tj 1r jrn The variable x Flives_ in the punctured Riemann sphere CP1 nft1 ; . . . ; tn g, where t1 ; . . . ; tn are pairwise distinct points of the complex plane C. However, now t1 ; . . . ; tn are not fixed but serve as the parameters of the family. The string t ¼ ðt1 ; . . . ; tn Þ is considered as a point of Cn* and the residue matrices Qj are assumed to depend on the the parameter t. The Funknown_ Y is a square matrix function depending both on the Fmain variable_ x and on the parameter t. DEFINITION 2.1. Assume that the matrix functions Qj ðtÞ; 1 r j r n; are defined and holomorphic for t 2 D , where D is a domain in Cn* : Then the family (2.1) is said to be a holomorphic family of Fuchsian systems, parameterized by the pole loci. In what follows we assume that the condition X Qj ðtÞ  0; t 2 D;

ð2:2Þ

1rjrn

holds. Thus for every fixed t 2 D the point x ¼ 1 is a regular point for the system (2.1), considered as a differential system with respect to x. For each fixed t 2 D , the fundamental solution Y ¼ Yðx; tÞ of the differential system (2.1) with the initial conditionj  Yðx; tÞ  ¼I ð2:3Þ x¼1

is defined and holomorphic as a function of x on the universal covering surface covðCP1 nft1 ; . . . ; tn g; 1Þ. In the present section our goal is to compare the properties of Yðx; tÞ, such as the monodromy representation or the principal factors, for different t. More precisely, we would like to understand what does it mean that Bthe monodromy representation or the principal factors of Yðx; tÞ are the same for different t?^ In the previous section these notions were defined in terms of homotopy classes of paths on the punctured Riemann sphere CP1 nft1 ; . . . ; tn g. However, for different t ¼ ðt1 ; . . . ; tn Þ the domains CP1 nft1 ; . . . ; tn g are different. Thus one ought to explain how to consider Fthe same paths for different t.` This can be done because we can confine ourselves to local considerations. j

In view of (2.2), for each fixed t 2 D the point x ¼ 1 is regular for the system (2.1) and hence the initial condition (2.3) can be posed.

150

VICTOR KATSNELSON AND DAN VOLOK

3.2. CYLINDRICAL NEIGHBORHOODS OF POINTS IN Cn DEFINITION 2.2. Let W j ; 1 r j r n, be subsets of the complex plane C. The Cartesian product W ¼ W 1      W n  Cn is said to be the cylindrical set with the bases W j ; 1 r j r n. DEFINITION 2.3. Let t ¼ ðt1 ; . . . ; tn Þ be a point in Cn . For j ¼ 1; . . . ; n let W j be an open neighborhood of tj in C, such that: (i) the set W j is simply connected; (ii) the set C n W j is connected. Denote by W the cylindrical set with the bases W j , 1 r j r n. The set W is said to be an open cylindrical neighborhood of the point t in Cn . DEFINITION 2.4. Let subsets S ; O of Cn be such that: (i) the set O is open; (ii) the closure S is compact; (iii) S  O . Then we say that the set S is compactly included in O and denote this relation by S Ð O: 3.3. ISOMONODROMIC FAMILIES OF FUCHSIAN SYSTEMS Let t 0 be a point in the domainj D and let W be a cylindrical neighborhood of t 0 , such that W Ð D . Then W p \ W q ¼ ;: 1 r p; q r n; p 6¼ q; ð2:4Þ S hencejj the set C n k W k is connected. For a fixed t 2 W each homotopy class of loops in the punctured sphere CP1 nft1 ; ; tn g with the distinguished point S x0 ¼ 1 has a representative which is a loop in the perforated sphere CP1 n k W k . j

Recall that D is the domain where the residue matrices Qj ðtÞ from (2.1) are defined and holomorphic. Since D Cn* , the coordinates t01 ; t02 ; . . . ; t0n of every point t 0 2 D are pairwise distinct. jj Here we refer to a relatively delicate result from general topology: if K1 ; K2 are two disjoint compact subsets of Rm and each of two sets Rm nK1 , Rm nK2 is connected, then the set Rm nfK1 [ K2 g is connected, as well. (See for example [20], Corollary of Theorem VI.10.) Actually we do not need this result in full generality. For our goal it is enough to consider only the case of m ¼ 2 and some very special compact sets K  R2 , such as finite unions of disks, etc. In this particular case, the above stated result is elementary.

151

DEFORMATIONS OF FUCHSIAN SYSTEMS

S

On the other hand, if  is a loop in the perforated sphere CP1 n k W k with the distinguished point x0 ¼ 1, then for each fixed t 2 W the loop  serves as a path of the analytic continuation with respect to x for the solution Yðx; tÞ of (2.1)Y(2.3) and one can consider the corresponding monodromy matrixj M ðtÞ: Although the loop  does not depend on t, the corresponding monodromy matrix M ðtÞ does, in general. We distinguish the following special case: DEFINITION 2.5. Let (2.1) be a holomorphic family of Fuchsian systems, where the residue matrices Qj ðtÞ are holomorphic in a domain D Cn* and satisfy (2.2). For each t 2 D , let Y ðx; tÞ be the solution of (2.1)Y(2.3). The family (2.1) is said to be an isomonodromic family of Fuchsian systems with the distinguished point x0 ¼ 1 if for every t0 2 D there exists a cylindrical open neighborhood W Ð D of t 0 , such that the following holds. S For every loop  in the perforated sphere CP1 n k W k with the distinguished point x0 ¼ 1 and every pair of points t 0 ; t00 2 W the monodromy matrices M ðt0 Þ; M ðt00 Þ of the solutions Yðx; t0 Þ, Yðx; t00 Þ, which correspond to this loop , are equal: M ðt 0 Þ ¼ M ðt 00 Þ

ð2:5Þ

Remark 2.6. Note that if the family (2.1) is isomondromic with the distinguished point x0 ¼ 1 and t0 2 D , then the monodromy matrices of Y are constant with respect to t in every cylindrical open neighborhood W of t0 , such that W Ð D . 3.4. ISOPRINCIPAL FAMILIES: THE INFORMAL DEFINITION Now we introduce the notion of the isoprincipal family of Fuchsian systems, which is the central notion in the present article. Again, we assume that (2.1) is a holomorphic family of Fuchsian systems, where the residue matrices Qj ðtÞ are holomorphic in a domain D Cn* and satisfy (2.2), and consider the solution Yðx; tÞ of (2.1)Y(2.3). According to Proposition 1.7, for each fixed t 2 D a branch Yj of the solution Y in a neighborhood of tj admits the regular-principal factorization, but now both the regular factor Hj and the principal factor Pj may depend on t: Yj ðx; tÞ ¼ Hj ðx; tÞ  Pj ðx; tÞ: ð2:6Þ For example, if the principal factor is chosen in the form (1.24), then Zj ¼ Zj ðtÞ; Aj ¼ Aj ðtÞ and Pj ðx; tÞ ¼ ðx  tj ÞZj ðtÞ j

See Definition 1.1.

 ðx  tj ÞA

j ðtÞ

:

ð2:7Þ

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VICTOR KATSNELSON AND DAN VOLOK

Roughly speaking, the family (2.1) is isoprincipal if for every j; 1 r j r n, the matrices Zj and Aj in (2.7) do not depend on t : Zj ðtÞ  Zj ; Aj ðtÞ  Aj , and Pj ðx; tÞ ¼ ðx  tj ÞZj

 ðx  tj ÞA

j

:

ð2:8Þ

If the matrices Zj and Aj in (2.7) do not depend on t, then the principal factor Pj ðx; tÞ of the form (2.8) possesses the following property: it depends only on the difference x  tj . For our goals, the specific form (2.8) of the principal factors is of no importance. We just need each principal factor Pj ðx; tÞ to depend only on the difference x  tj : This means that there exist functions Ej ; such that Pj ðx; tÞ ¼ Ej ðx  tj Þ; or, in the language of differential equations, @Pj ¼ 0; ‘ 6¼ j; @t‘  @Pj @Pj dEj  ¼ : ¼ @x @tj d ¼xtj

ð2:9Þ

ð2:10aÞ ð2:10bÞ

The formal definition of the isoprincipal family of Fuchsian systems (see Definition 2.9 below) is more involved, since for each t the branch Yj ðx; tÞ and the principal factor Pj ðx; tÞ depend on the choice of a path j in the punctured sphere CP1 n ft1 ; . . . ; tn g, which connects the distinguished point x0 ¼ 1 with a neighborhood of x ¼ tj . Moreover, for each t the principal factor Pj ðx; tÞ should be a transplantj of a function Ej ðÞ, holomorphic on the Riemann surface of ln , into a universal covering surface over C n ftj g and these transplants should be defined coherently with respect to t. 3.5. ISOPRINCIPAL FAMILIES: THE FORMAL DEFINITION DEFINITION 2.7. Let a function EðÞ be defined on the Riemann surface of ln . Let W j be a simply connected domain, compactly included in C. Let pj 2 C n W j and let us choose a branch of argðpj  tj Þ, continuous with respect to tj in W j (since W j is simply connected and W j Ð C n fpj g, such a choice can be made). For every tj 2 W j let us specify the value of argðpj  tj Þ in this manner and consider the corresponding transplant Eðx  tj Þ of the function EðÞ into covðC n ftj g; pj Þ: j

See Definition 1.6.

153

DEFORMATIONS OF FUCHSIAN SYSTEMS

Then the family of transplants fEðx  tj Þgtj 2W j is said to be coherent with respect to tj in W j . DEFINITION 2.8. Let t 0 be a point in a domain D Cn* and let W ; V be a pair of open cylindrical neighborhoods of the point t 0 , such that V Ð Cn*

and W Ð fV \ D g:

Then the pair W ; V is said to be a nested pair of open cylindrical neighborhoods of the point t0 in D : DEFINITION 2.9 (Formal definition of the isoprincipal family). Let (2.1) be a holomorphic family of Fuchsian systems, where the residue matrices Qj ðtÞ are holomorphic in a domain D Cn* and satisfy (2.2). For t 2 D let Yðx; tÞ be the solution of (2.1)Y(2.3). The family (2.1) is said to be an isoprincipal family of Fuchsian systems with the distinguished point x0 ¼ 1 if for every t 0 2 D there exists a nested pair of open cylindrical neighborhoods of t0 : V ¼ V 1      V n;

W ¼ W1      Wn;

W Ð V;

such that the following holds. S For every path j in the perforated sphere CP1 n k W k from the distinguished point x0 ¼ 1 to a point pj 2 V j n W j ; 1 r j r n, there exists a coherent family of transplants fEj ðx  tj Þgtj 2W j of a function Ej ðÞ, holomorphic and invertible on the Riemann surface of ln , such that for each t 2 W the branch Yj ðx; tÞ of the solution Yðx; tÞ in the punctured domain V j n ftj g admits the representation Yj ðx; tÞ ¼ Hj ðx; tÞ

 E ðx  tj Þ; j

x 2 covðV j n ftj g; pj Þ;

ð2:11Þ

where Hj ðx; tÞ is a function, holomorphic (with respect to x) and invertible in the entire domain V j : Remark 2.10. In view of Definition 1.8, Definition (2.9) means that the family (2.1) is isoprincipal with the distinguished point x0 ¼ 1 if every branch of the solution Yðx; tÞ in a neighborhood of each singular point x ¼ tj admits the regular-principal factorization (2.11), where the principal factor is the appropriately shifted copy of a function Ej ðÞ, which is holomorphic and invertible on the Riemann surface of ln  and does not depend on t. The meaning of the words Fappropriately shifted copy_ is made precise in Definition 2.7: this is what we call a coherent family of transplants of Ej ðÞ.

154

VICTOR KATSNELSON AND DAN VOLOK

Thus Definition 2.9 is a formal interpretation of the informal definition in Section 2.4. We would also like to note that it suffices to consider only the branches corresponding to a certain choice of the paths 1 ; . . . ; n Y the same choice as the one mentioned in Remark 2.13 below. 3.6. EVERY ISOPRINCIPAL FAMILY IS AN ISOMONODROMIC ONE THEOREM 2.11. Let (2.1) be a holomorphic family of Fuchsian systems, where the residue matrices Qj ðtÞ are holomorphic in a domain D Cn* and satisfy (2.2). Assume that the family (2.1) is isoprincipal with the distinguished point x0 ¼ 1. Then this family is isomonodromic with the distinguished point x0 ¼ 1. Before we turn to the proof of Theorem 2.11, let us introduce the following Ftdependent_ counterpart of Definition 1.4: DEFINITION 2.12. Let t0 be a point in the domain D and let W Ð V be a nested pair of open cylindrical neighborhoods of t0 in D . For 1 r j r n let j be a 1 S path in the perforated sphere CP n k W k from the distinguished point x0 ¼ 1 to a point pj 2 V j n W j : Furthermore, assume that j is the loop in the annulus V j n W j with the distinguished point pj which makes one positive S circuit of the set W j , and let j be the loop in the perforated sphere CP1 n k W k with the distinguished point x0 ¼ 1, defined by def

j ¼ 1 j

 j  j :

ð2:12Þ

Then:  the loop j is said to be the small loop around the set W j in the annulus

V j n W j;  the loop j is said to be the big loop around the set W j , corresponding to the

path j . Remark 2.13. Similarly to the case of a fixed t (see Remark 1.5), for a suitable choice of the paths 1 ; . . . ; n the S corresponding big loops 1 ; . . . ; n generate the fundamental group ðCP1 n k W k ; 1Þ. Proof of Theorem 2.11. Let t0 be a point in D and let W Ð V be a nested pair of open cylindrical neighborhoods of t0 as in Definition 2.9. 1 S In view of Remark 2.13, it suffices to prove that if j is a path in CP n k W k from the distinguished point x0 ¼ 1 to a point pj 2 V j n W j and j is the

155

DEFORMATIONS OF FUCHSIAN SYSTEMS

corresponding big loop around W j ; then the monodromy matrix Mj ðtÞ of Yðx; tÞ does not depend on t: Mj ðtÞ ¼ const;

t 2 W:

ð2:13Þ

In view of (1.20), for each fixed t 2 W the monodromy matrix Mj ðtÞ is given by Mj ðtÞ ¼ Y1j ðx; tÞ

 Y ðtj þ ðx  tj Þe2i; tÞ; j

8x 2 covðV j n ftj g; pj Þ;

where Yj ðx; tÞ is the branch of Yðx; tÞ in V j n ftj g, corresponding to the path j . Substituting the expression (2.6) for Yj into the above identity and taking into account that the factor Hj ðx; tÞ is a single-valued function of x, we obtain Mj ðtÞ

2i ¼ E1 j ðx  tj Þ  Ej ððx  tj Þe Þ   1  ; ¼ Ej ðÞ  Ej ðe2i Þ  ¼xtj

8x 2 covðV j n ftj g; pj Þ:

2i Thus the function E1 j ðÞ  Ej ðe Þ, holomorphic on the Riemann surface of ln , is constant with respect to  on a certain non-empty open subset of this surface. Therefore, this function is identically constant on the Riemann surface of ln  and we write

Mj ðtÞ ¼ E1 j ðÞ

 E ðe2iÞ j

8t 2 W ;  2 covðC n f0g; 1Þ:

ð2:14Þ

But the right-hand side of the last identity does not depend on t, hence we obtain Ì (2.13). 3.7. EVERY NON-RESONANT ISOMONODROMIC FAMILY IS AN ISOPRINCIPAL ONE The converse to Theorem 2.11 is only conditionally true: it holds under the assumption that all the matrices Qj are non-resonant (see footnote 9). In general, however, an isomonodromic family can be non-isoprincipal. The appropriate counterexample will be presented in Section 5 of this paper. LEMMA 2.14. Let (2.1) be a holomorphic family of Fuchsian systems, where the residue matrices Qj ðtÞ are holomorphic in a domain D Cn* and satisfy (2.2). Assume that this family is isomonodromic with the distinguished point x0 ¼ 1. Then the family (2.1) is isospectral in the following sense: for every pair of points t 0 ; t 00 2 D and each j, 1 r j r n, the spectra spec Qj ðt0 Þ and spec Qj ðt 00 Þ are equal:j spec Qj ðt0 Þ ¼ spec Qj ðt00 Þ; j

8t0 ; t00 2 D ; 1 r j r n:

As usual, the spectra are considered Fwith multliplicities_.

ð2:15Þ

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THEOREM 2.15. Let (2.1) be a holomorphic family of Fuchsian systems, where the residue matrices Qj ðtÞ are holomorphic in a domain D Cn* and satisfy (2.2). Assume that this family satisfies the following conditions: (i) It is isomonodromic with the distinguished point x0 ¼ 1; (ii) At least for one point t 2 D , each of the matrices Qj ðtÞ; j ¼ 1; . . . ; n; is nonresonant. Then the family (2.1) is isoprincipal with the distinguished point x0 ¼ 1. Remark 2.16. It should be noted that, unlike the rest of our considerations in the present article, the proofs of Lemma 2.14 and Theorem 2.15 utilize the explicit form of the principal factors mentioned in Remark 1.10. Proof of Lemma 2.14. Let t0 be a point in D and let W Ð V be a nested pair of open cylindrical neighborhoods of t0 in D as in Definition S 2.5. For j ¼ 1; . . . ; n let us choose a path j in the perforated sphere CP1 n k W k from the distinguished point x0 ¼ 1 to a point pj 2 V j n W j . As usual, for each t 2 W we denote by Yj ðx; tÞ the branch of the solution Yðx; tÞ of (2.1)Y(2.3) in the punctured domain V j n ftj g, corresponding to this path j . Then, in view of Remark 1.10 (see (1.24) and (1.25)), the branch Yj ðx; tÞ admits the regular-principal factorization Yj ðx; tÞ ¼ Hj ðx; tÞ t 2 W;

 ðx  tj ÞZ

j ðtÞ

 ðx  tj ÞA

j ðtÞ

;

x 2 covðV j nftj g; pj Þ;

where

  spec e2iAj ðtÞ ¼ spec e2iQj ðtÞ :

Let j be the big loop around W j , corresponding to the path j . Then the monodromy matrix Mj ðtÞ of Yðx; tÞ, corresponding to the loop j is given by Mj ðtÞ ¼ Y1j ðx; tÞYj ðtj þ ðx  tj Þe2i ; tÞ ¼ e2iAj ðtÞ : Therefore,

 specðMj ðtÞÞ ¼ spec e2iQj ðtÞ ;

t 2 W:

Since the family (2.1) is isomonodromic, we have Mj ðtÞ ¼ Mj ðt0 Þ;

t 2 W;

hence   0 spec e2iQj ðtÞ ¼ spec e2iQj ðt Þ ;

t 2 W:

ð2:16Þ

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DEFORMATIONS OF FUCHSIAN SYSTEMS

This means that the spectra spec Qj ðtÞ and spec Qj ðt 0 Þ coincide modulo integers. But the function Qj ðtÞ is continuous with respect to t in W ; hence spec Qj ðtÞ ¼ spec Qj ðt0 Þ;

t 2 W:

Since the above identity holds for all t in a neighborhood of every point t0 2 D , Ì we obtain the identity (2.15). Proof of Theorem 2.15. Let t 0 be a point in D and let W ÐV V be a nested pair 0 of open cylindrical neighborhoods of t in D as in Definition 2.5. As in the proof of Lemma S 2.14, for j ¼ 1; . . . ; n let us choose a path j in the perforated sphere CP1 n k W k from the distinguished point x0 ¼ 1 to a point pj 2 V j n W j and consider for each fixed t 2 W the branch Yj ðx; tÞ of the solution Yðx; tÞ of (2.1)Y(2.3) in the punctured domain V j n ftj g, corresponding to this path j . Since by Lemma 2.14 the family (2.1) is isospectral, the matrix Qj ðtÞ is nonresonant for every t 2 D . Hence, in view of Remark 1.10 (see the expressions (1.22) and (1.23)), the branch Yj admits the regular-principal factorization Yðx; tÞ ¼ Hj ðx; tÞ  ðx  tj ÞAj ðtÞ ;

x 2 covðV j n ftj g; pj Þ;

where the matrix Aj ðtÞ is similar to the matrix Qj ðtÞ. Therefore, specðAj ðtÞÞ ¼ specðAj ðt0 ÞÞ;

t 2 W:

ð2:17Þ

In view of Definition 2.9, it remains to prove that the matrix Aj ðtÞ does not actually depend on t. Let j be the big loop around W j , corresponding to the path j . Then the monodromy matrix Mj ðtÞ of Yðx; tÞ, corresponding to the loop j , is given by (see (2.16)) Mj ðtÞ ¼ e2iAj ðtÞ ;

t 2 W:

Since the family (2.1) is isomonodromic, we have Mj ðtÞ ¼ Mj ðt0 Þ;

t 2 W;

hence 0

e2iAj ðtÞ ¼ e2iAj ðt Þ ;

t 2 W:

But, in view of (2.17), the last identity implies Aj ðtÞ ¼ Aj ðt0 Þ ¼ const;

t 2 W:

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4. Isoprincipal Families of Fuchsian Systems and the Schlesinger System 4.1. THE SCHLESINGER SYSTEM A natural question arises: how to express the property of a family of Fuchsian systems

X Qj ðtÞ dY Y; ð3:1Þ ¼ dx x  tj 1rjrn to be isoprincipal in terms of the residues matrix functions Qj ðtÞ? Here, as always, we assume that the residue matrices Qj ðtÞ are holomorphic in a domain D Cn* and satisfy X Qj ðtÞ  0; t 2 D : ð3:2Þ 1rjrn

It turns out that an answer to this question is given by the so-called Schlesinger system of PDEs: 8 @Qi > > > > @tj < @Qi > > > > : @ti

½Qi ; Qj  ; ti  tj   X Qi ; Qj ¼ ; t  tj 1rjrn i ¼

1 r i; j r n; i 6¼ j; 1 r i r n:

ð3:3Þ

j6¼i

The following theorem is the main result of the present article. THEOREM 3.1 (The main result). Let (3.1) be a holomorphic family of Fuchsian systems, where the residue matrices Qj ðtÞ are holomorphic in a domain D Cn* and satisfy (3.2). Then the family (3.1) is isoprincipal with the distinguished point x0 ¼ 1 if and only if the residue matrices Qj ðtÞ satisfy with respect to t the Schlesinger system (3.3) in the domain D . Remark 3.2. Note that (3.3) implies @ X Qi ¼ 0; @tj 1 r i r n

1 r j r n;

P that is, i Qi ðtÞ is a first integral of the Schlesinger system. In particular, if functions Qj ðtÞ, 1 r j r n; satisfy the Schlesinger system (3.3) in the domain D and at some point t0 2 D it holds that X Qj ðt 0 Þ ¼ 0; 1rjrn

then these functions Qj ðtÞ satisfy the relation (3.2).

DEFORMATIONS OF FUCHSIAN SYSTEMS

159

Remark 3.3. We would like to stress that in Theorem 3.1 no assumptions on the spectra of the residue matrices Qj ðtÞ are made. Thus Theorem 3.1 for the isoprincipal families of Fuchsian systems can be viewed as an amended version of L. Schlesinger’s statement, concerning the isomonodromic deformations (see [35] and the introduction of the present article). In the case of the isomonodromic families of Fuchsian systems Theorem 3.1 implies the following: (i) If the residue matrices Qj ðtÞ satisfy the Schlesinger system (3.3), then the family (3.1) is isoprincipal and hence by Theorem 2.11 also isomonodromic. (ii) If the family (3.1) is isomonodromic and, in addition, all the residue matrices Qj ðtÞ are non-resonant (at least at some point), then by Theorem 2.15 the family (3.1) is isoprincipal and hence the residue matrices Qj ðtÞ satisfy the Schlesinger system (3.3). We remark that in the statement (ii) the assumption of non-resonance for the residues Qj ðtÞ cannot be omitted: in Section 5 we shall present an example of the isomonodromic family (3.1), where the residue matrices Qj ðtÞ are resonant and do not satisfy the Schlesinger system (3.3) (thus contradicting the statement of L. Schlesinger). Nevertheless, our proof of the Fonly if_ part of the Theorem 3.1 largely follows the original proof of L. Schlesinger for the isomonodromic case (see also [21, Section 3.5], where the modern adaptation of Schlesinger’s proof is presented). In particular, the overdetermined linear system (3.6), which appears in Proposition 3.6 below and is crucial in the derivation of the Schlesinger system, can be found in [35, Section II]. The proof of Theorem 3.1 will be split into parts and presented as a series of propositions, culminating with Propositions 3.15 and 3.16. 4.2. THE AUXILIARY SYSTEM RELATED TO THE ISOMONODROMIC FAMILY OF FUCHSIAN SYSTEMS

In order to prove Theorem 3.1, we have to study the partial derivatives of the solution Yðx; tÞ, satisfying the initial condition  ð3:4Þ Yðx; tÞ x¼1 ¼ I; with respect to the parameters t1 ; . . . ; tn . First of all, let us choose and fix a point t0 2 D and let W Ð D be a cylindrical open neighborhood of t 0 . Since the coefficients of the system (3.1) and the initial condition (3.4) depend holomorphically on t, the solution Yðx; tÞ is holomorphic 1 S jointly in x and t in the Cartesian product covðCP n k W k ; 1Þ  W : @Y 0 In particular, the partial derivatives @tj ðx; t Þ are defined and holomorphic 1 S 0 with respect to x in covðCP n k W k ; 1Þ: Since these definitions of @Y @tj ðx; t Þ

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VICTOR KATSNELSON AND DAN VOLOK

agree for various choices of W as long as W is sufficiently small, we conclude 0 that for each fixed t 0 2 D the partial derivatives @Y @tj ðx; t Þ are defined and holomorphic as functions of x on the the same surface as the function Yðx; t0 Þ itself Y the universal covering surface covðCP1 n ft01 ; t0n g; 1Þ. It turns out that in terms of these partial derivatives of Y the property of the family (3.1) to be isomonodromic can be expressed as follows: PROPOSITION 3.4. Let (3.1) be a holomorphic family of Fuchsian systems, where the residue matrices Qj ðtÞ are holomorphic in a domain D Cn and satisfy (3.2). Then the family (3.1) is isomonodromic with the distinguished point x0 ¼ 1 if and only if the solution Yðx; tÞ of (3.1), (3.4) satisfies a linear system of the form 8 @Y > > > < @x > @Y > > : @tj

¼

X Qj ðtÞ x  tj 1rjrn

¼ Tj ðx; tÞ

 Y;

 Y; ð3:5Þ 1 r j r n;

where for each t 2 D the functions  Tj ðx; tÞ, 1 r j r n; are single-valued holomorphic with respect to x in CP1 n t1 ; . . . ; tn . DEFINITION 3.5. Let the family (3.1) of Fuchsian systems be isomonodromic with the distinguished point x0 ¼ 1. The system (3.5) with the single-valued coefficients Tj ðx; tÞ, 1 r j r n; which appears in Theorem 3.4, is said to be the auxiliary linear system related to the isomonodromic family (3.1) of Fuchsian systems. Proof of Proposition 3.4. The first equation of the system (3.5) is just the Fuchsian system (3.1) itself, hence we only need to prove that for each fixed t 2 D the logarithmic derivatives def

Tj ðx; tÞ ¼

@Y ðx; tÞ @tj

 Y 1 ðx; tÞ;

which a priori are defined as holomorphic functions of x on the universal  1 covering surface covðCP n t ; . . . ; t ; 1Þ; are single-valued in the punctured 1 n   1 sphere CP n t1 ; . . . ; tn : Let us choose a point t 0 2 D and a cylindrical open neighborhood W Ð D of 1 S 0 t . Let  2 ðCP n k W k ; 1Þ and let us denote by [ x 7! x  ; x 2 covðCP1 n W k ; 1Þ k

161

DEFORMATIONS OF FUCHSIAN SYSTEMS

S

the deck transformation of the universal covering surface covðCP1 n k W k ; 1Þ, correspondingj to this loop . According to Definition 1.1, the monodromy matrix M ðtÞ of the solution Yðx; tÞ, which corresponds to the loop , is given by [ x 2 covðCP1 n W k ; 1Þ; t 2 W : M ðtÞ ¼ Y 1 ðx; tÞ  Yðx; tÞ; 1rkrn

The monodromy matrix M ðtÞ does not depend on x and hence is a holomorphic single-valuedjj function of t in W . Differentiating the equality Yðx; tÞ ¼ Yðx; tÞ

 M ðtÞ

with respect to tj , we obtain @Y @Y ðx; tÞ ¼ ðx; tÞ @tj @tj

 M ðtÞ þ Yðx; tÞ 

@M ðtÞ: @tj

Therefore, the logarithmic derivative Tj ðx; tÞ ¼

@Y ðx; tÞ @tj

 Y 1 ðx; tÞ

satisfies the monodromy relation Tj ðx; tÞ ¼ Tj ðx; tÞ þ Yðx; tÞ



@M ðtÞ @tj

 Y 1ðx; tÞ:

The last equality implies that the following two statements are equivalent:(1) The monodromy matrix M ðtÞ does not depend on tj : @M ðtÞ  0; @tj

t 2 W:

(2) It holds that Tj ðx; tÞ  Tj ðx; tÞ;

x 2 covðCP1 n

[

W k ; 1Þ;

t 2 W:

1rkrn

S However, the statement 2) holds for every  2 ðCP1 n k W k ; 1Þ if and only if for each t 2 W the function Tj ðx; tÞ is a single-valued function of x in CP1 n ft1 ; . . . ; tn g. In view of Definition 2.5, this completes the proof. Ì j

See (1.4). Recall (see Definition 2.2) that all the bases W k of the cylindrical neighborhood W are simply connected. This equivalence is stronger than the statement of Proposition 3.6 in the sense that it holds for each individual loop  and each individual index j. jj

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4.3. THE AUXILIARY SYSTEM RELATED TO THE ISOPRINCIPAL FAMILY OF FUCHSIAN SYSTEMS

Now we turn to the case, when the family (3.1) is not only isomonodromic but, moreover, isoprincipal. We claim that in this special case the auxiliary linear system (3.5) can be written explicitly in terms of the residues Qj ðtÞ. PROPOSITION 3.6. Let (3.1) be a holomorphic family of Fuchsian systems, n where the residue matrices Qj ðtÞ are holomorphic in a domain D C and satisfy (3.2). Assume that the family (3.1) is isoprincipal with the distinguished point x0 ¼ 1. Then the solution Yðx; tÞ of (3.1) and (3.4) satisfies the following auxiliary system: 8 X Qj ðtÞ @Y > >  Y; ¼ > < @x x  tj 1rjrn ð3:6Þ > Qj ðtÞ @Y > > ¼  Y; 1 r j r n: : @tj x  tj In the proof of Proposition 3.6 we shall use the following LEMMA 3.7. Let U; V be simply connected domains in the complex plane C, such that: (i) U Ð V; (ii) the set V n U is connected. Let W be a domain in Cn and let Hðx; tÞ be a function of x 2 V and t 2 W , possessing the following properties: (a) the function Hðx; tÞ is holomorphic (jointly in x and t) in fV n Ug  W ; (b) for each fixed t 2 W the function Hðx; tÞ is holomorphic with respect to x in the entire domain V. Then the function Hðx; tÞ is holomorphic (jointly in x and t) in the domain V  W. Remark 3.8 Lemma 3.7 is a special case of the well-known Hartogs lemma. However, in this simple case the conclusion follows immediately from the Cauchy integral formula. Indeed, let  be a smooth loop in the annulus V n W which makes one positive circuit of the set W and let ÐW W be a polydisk:  ¼ 1     n :

163

DEFORMATIONS OF FUCHSIAN SYSTEMS

Then for each t 2  and x 2 W it holds that I Hj ð; tÞ 1 Hj ðx; tÞ ¼ d 2i x j 0 1 I I I Hj ð; 1 ; . . . n Þ 1 B C d ; d1    dA  ¼ @ n nþ1 x  t Þ    ð  t Þ ð ð2iÞ 1 1 n n j

@n

@1

where @k denotes the boundary of the disk k . Since the contours of integration lie in the domain fV n Wg  W , where Hj ðx; tÞ is jointly holomorphic in x and t (in particular, continuous), the integral represents a function, jointly holomorphic in x and t for x in a neighborhood of W j and t 2 . Proof of Proposition 3.6. Let t0 2 D and let W Ð V be a nested pair of cylindrical open neighborhoods of t0 . For ‘ ¼ 1; . . . ; n let us consider the logarithmic derivative T‘ ðx; tÞ ¼

@Y ðx; tÞ @t‘

 Y 1ðx; tÞ:

ð3:7Þ

Since by Theorem 2.11 the isoprincipal family (3.1) is also isomonodromic, Proposition 3.4 implies that for each fixed t 2 W the function T‘ ðx; tÞ is singlevalued holomorphic with respect to x in the punctured sphere CP1 n ft1 ; . . . ; tn g. We have to prove that Q‘ ðtÞ T‘ ðx; tÞ ¼  : ð3:8Þ x  t‘ To this end let us introduce the auxiliary function Q‘ ðtÞ def ; ð3:9Þ F‘ ðx; tÞ ¼ T‘ ðx; tÞ þ x  t‘ which is single-valued holomorphic with respect to x in the punctured sphere CP1 n ft1 ; . . . ; tn g: We are going to show that for j ¼ 1; . . . ; n the following statement holds: (j) The point x ¼ tj is a removable singularity of the function F‘ ðx; tÞ. Then, according to Liouville theorem, the function F‘ ðx; tÞ is constant with respect to x. Since, as follows from (3.4) and (3.9)   ð3:10Þ T‘ ðx; tÞ x ¼ 1 ¼ F‘ ðx; tÞ x ¼ 1 ¼ 0; we can then conclude that F‘ ðx; tÞ  0; which leads to the desired result (3.8).

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Now we turn to the proof of the statement ðjÞ.STo begin with, let us choose and fix a path j in the perforated sphere CP1 n k W k from the distinguished point x0 ¼ 1 to a point pj 2 V j n W j : Then, according to Definition 2.9, for each t 2 W the branch Yj ðx; tÞ of the solution Yðx; tÞ in V j n ftj g admits the regularprincipal factorization Yj ðx; tÞ ¼ Hj ðx; tÞ

 E ðx  tj Þ;

x 2 covðV j n ftj g; pj Þ;

j

ð3:11Þ

where the family fEj ðx  tj Þgtj 2W j is a coherent family of transplants of a function Ej ðÞ, holomorphic and invertible on the Riemann surface of ln , and the function Hj ðx; tÞ is holomorphic with respect to x and invertible in the entire (non-punctured) domain V j . Then the principal factor Ej ðx  tj Þ is jointly holomorphic in x and tj in covðC n W j ; pj Þg  W j ; and hence the regular factor Hj ðx; tÞ ¼ Y1j ðx; tÞ

 E ðx  tj Þ j

is jointly holomorphic (single-valued) inx and t in fV j n W j g  W . Since the function Hj ðx; tÞ is also holomorphic with respect to x in the entire domain V j , Lemma 3.7 implies that it is jointly holomorphic in x and t in V j  W . Thus we can differentiate the equality 3.11 with respect to t‘ , 1 r ‘ r n. First we consider the case ‘ 6¼ j. Then, since @Ej ðx  tj Þ ¼ 0; @t‘

‘ 6¼ j;

ð3:12Þ

we obtain for x 2 V j n ftj g T‘ ðx; tÞ ¼

@Yj @Hj ðx; tÞ ¼ ðx; tÞ  Ej ðx  tj Þ @t‘ @t‘

¼

@Hj ðx; tÞ @t‘

 Hj1 ðx; tÞ  Y ðx; tÞ:

F‘ ðx; tÞ ¼

@Hj ðx; tÞ @t‘

 Hj1 ðx; tÞ þ xQ‘ðtÞt ;

j

Hence x 2 V j n ftj g; ‘ 6¼ j;

‘

which proves the statement ðjÞ in the case ‘ 6¼ j. Next we differentiate the equality (3.11) with respect to tj . Since  @Ej ðx  tj Þ @Ej ðx  tj Þ dEj ðÞ  ¼ ¼ ; @x @tj d  ¼ xtj

ð3:13Þ

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DEFORMATIONS OF FUCHSIAN SYSTEMS

we obtain Tj ðx; tÞ ¼

@Hj ðx; tÞ @tj

Hj ðx; tÞ 

 Hj1ðx; tÞ 

@Ej ðx  tj Þ @x

1  E1  ðx  tj Þ  Hj ðx; tÞ; j

x 2 covðV j n ftj g; pj Þ:

On the other hand, differentiating the equality 3.11 with respect to x, we get @Yj @Hj ðx; tÞ  Y1j ðx; tÞ ¼ ðx; tÞ  Hj1 ðx; tÞ þ @x @x @Ej ðx  tj Þ 1 þ Hj ðx; tÞ   E1 x 2 covðV j n ftj g; pj Þ; j ðx  tj Þ  Hj ðx; tÞ; @x hence



Fj ðx; tÞ ¼

@Hj @Hj ðx; tÞ þ ðx; tÞ Hj1 ðx; tÞ þ @x @tj

þ

Qj ðtÞ @Yj  ðx; tÞ @x x  tj

 Y1ðx; tÞ; j

x 2 covðV j n ftj g; pj Þ:

Taking into account that Yj ðx; tÞ is a branch of the solution Yðx; tÞ of the Fuchsian system (3.1), we obtain

X Qk ðtÞ @Hj @Hj ðx; tÞ Hj1 ðx; tÞ  ; ðx; tÞ þ Fj ðx; tÞ ¼ x  tk @x @tj 1rkrn k6¼j

x 2 V j nftj g: Thus the function Fj ðx; tÞ has at x ¼ tj a removable singularity, which proves the Ì statement ðjÞ in the case j ¼ ‘. Remark 3.9. Note that the relations (3.12), (3.13), which are instrumental in the proof of Proposition 3.6, are precisely the relations (2.10) in the informal definition of the isoprincipal family of Fuchsian systems in Section 2.4. Remark 3.10. We observe that the auxiliary linear system (3.6) leads to a linear system for the regular factor Hj ðx; tÞ of the regular-principal factorization (3.11). Indeed, in view of (3.12) and (3.13), we can differentiate the equality (3.11) to obtain @Yj @Hj ðx; tÞ  Hj1 ðx; tÞ ¼ ðx; tÞ  Y1j ðx; tÞ; j 6¼ ‘; @t‘ @t‘



@Yj @Yj @Hj @Hj 1 ðx; tÞ  Hj ðx; tÞ ¼ ðx; tÞ  Y1j ðx; tÞ; ðx; tÞ þ ðx; tÞ þ @tj @x @tj @x

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VICTOR KATSNELSON AND DAN VOLOK

and therefore @Hj @t‘ @Hj @Hj þ @tj @x

¼ ¼

Q‘ x  t‘

P 1r‘rn

 Hj ;

‘ 6¼ j;

 Hj :

Q‘ xt‘

ð3:14Þ

‘6¼j

Using the change of variables  ¼ x  tj ;

ð3:15Þ

def

ð3:16Þ

Lj ð; tÞ ¼ Hj ð þ tj ; tÞ; one can rewrite the system (3.14) in the following form: @Lj @t‘

¼

@Lj @tj

¼

Q‘  þ tj  t‘ X Q‘

1r‘rn

 Lj ;

 þ tj  t‘

‘ 6¼ j;

 Lj :

ð3:17Þ

‘6¼j

The system (3.17) is nothing more than the system (3.6) with the constraint x  tj ¼  ¼ const: Note that, although x ¼ tj is a singularity of the Fuchsian system (3.1) and the auxiliary system (3.6), the right-hand side of the system (3.17) is holomorphic with respect to  at  ¼ 0 (compare with Remark 1.2 in [29]). This occurs because the function Lj ð; tÞ, defined in (3.16), is holomorphic with respect to  and invertible at  ¼ 0. 4.4. THE FROBENIUS THEOREM The auxiliary system (3.6), related to the isoprincipal family (3.1) is an overdetermined system of PDEs. The criterion for the existence of solution of such a system is known (see [33, Section 2.11]) as the Frobenius theorem: THEOREM 3.11 (The Frobenius theorem for Pfaffianj systems). Let p and q be domains in, respectively, Cp and Cq . Consider the following system of PDEs: @ ¼ j ð ; Þ; @ j j

1 r j r q;

A Pfaffian system is a system of the form (3.18).

ð3:18Þ

DEFORMATIONS OF FUCHSIAN SYSTEMS

167

where ð Þ is an unknown Cp -valued function of the variable ¼ ð 1 ; . . . ; q ; Þ ¼ ð 1; j ð ; Þ; . . . ; p; j ð ; ÞÞ, 1 r j r q; are given Cp q Þ 2 C and j ð valued functions, holomorphic with respect to ; in the domain p  q . Then the following statements (i) and (ii) are equivalent: Þ of (i) For every pair of points 0 2 p ; 0 2 q there exists a solution ð the system (3.18), holomorphic in a neighborhood of 0 and satisfying the initial conditionj ð 0 Þ ¼ 0 : (ii) The C-valued functions i; j ð ; Þ satisfy the equations X @ i;j ð @ i;j ð ; Þ ; Þ þ  ‘;k ð ; Þ ¼ @ k @ ‘ 1r‘rp X @ i;k ð @ i;k ð ; Þ ; Þ þ  ‘; j ð ; Þ; ¼ @ j @ ‘ 1r‘rp

ð3:19Þ

1 r i r p; 1 r j; k r q; in the domain p  q . DEFINITION 3.12. The condititon (3.19), formulated in the Frobenius theorem 3.11, is said to be the compatibility condition for the overdetermined system of PDEs (3.18). The overdetermined system of PDEs (3.18) which satisfies the compatibility condition (3.19) is said to be compatible. Remark 3.13. Note that the compatibility condition (3.19) can be obtained by substituting the (3.18) into the identity @ 2 i @ 2 i ¼ ; @ j @ k @ k @ j

1 r i r p; 1 r j; k r q:

Thus the statement (ii) of the Frobenius theorem (3.11) follows from the statement (i) immediately. In what follows, we shall often deal with linear overdetermined systems of PDEs, depending on a parameter. Because of the global existence theorem for such systems, the following stronger version of the Frobenius theorem (3.11) holds in this case: THEOREM 3.14. (The Frobenius theorem for linear systems with a parameter). Let q and r be domains in, respectively, Cq and Cr . Consider the following linear systems of PDEs: @ ¼ j ð ;  Þ  ; 1 r j r q; ð3:20Þ @ j j

Note that, according to the uniqueness theorem for ordinary differential equations, such a solution ð Þ is necessarily unique.

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where 2 q is the Fmain_ variable,  2 r is a parameter and j ð ;  Þ, 1 r j r q; are Mp -valued functions, holomorphic with respect to in the domain q for each fixed  2 r . Then: (i) For each fixed value of the parameter  , say  ¼  0 2 r , the linear system @ ¼ j ð ;  0 Þ @ j

 ;

ð3:21Þ

1 r j r q;

;  0 Þ if and only if the functions  j ð ;  0Þ has a fundamental solutionj ð satisfy the equations @j ð ;  0 Þ @k ð ;  0Þ  ¼ ½k ð ;  0 Þ; j ð ;  0 Þ; @ j @ k 2 q ;

ð3:22Þ 1 r j; k r q:

(ii) If the functions j ð ; Þ are jointly holomorphic with respect to ;  in the domain q  r and satisfy the equations @j ð ;  Þ @k ð ;  Þ  ¼ ½ k ð ;  Þ;  j ð ;  Þ; @ j @ k 2 q ;  2 r ;

ð3:23Þ 1 r j; k r q;

then for every point 0 2 q and every Mp -valued function 0 ð Þ; holomorphic and invertible in r ; there exists a unique fundamental solution ð ;  Þ of the system (3.20), jointly holomorphic with respect to ;  in the domain q  r and satisfying the initial condition  ð ;  Þ  ¼ 0 ¼ 0 ð Þ:

ð3:24Þ

4.5. PROOF OF THEOREM 3.1 PROPOSITION 3.15. Let (3.1) be a holomorphic family of Fuchsian systems, where the residue matrices Qj ðtÞ are holomorphic in a domain D Cn and satisfy (3.2). Assume that the family (3.1) is isoprincipal with the distinguished point x0 ¼ 1. Then the residues Q1 ðtÞ; . . . ; Qn ðtÞ satisfy the Schlesinger system (3.3) in the domain D . j

In other words, an Mp -valued solution ð ;  0 Þ, such that det ð ;  0 Þ 6¼ 0 for 2 q .

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DEFORMATIONS OF FUCHSIAN SYSTEMS

Proof. According to Proposition 3.6, the solution Yðx; tÞ of (3.1) and (3.4) satisfies the auxiliary linear system (3.6). Therefore, Yðx; tÞ is the fundamental solution of the overdetermined linear system (3.6) with the initial condition Yð1; t 0 Þ ¼ I; where t 0 is an arbitrary fixed point in the domain D . Hence, in view of Theorem 3.14, the linear system (3.6) (which is a special case of the system (3.20) with q ¼ n þ 1, r ¼ 0, ð Þ ¼ Yðx; tÞ) is compatible. The compatibility condition (3.23) takes in this case the form of the following two equations:

X @ Qi

X Qj ½Qi ; Qj  @ þ ¼ ; @x x  tj @tj x  ti ðx  ti Þðx  tj Þ 1rirn 1rirn ð3:25Þ 1 r j r n;



Qj ½Qj ; Qi  @ @ Qi ;  ¼ @ti x  tj @tj x  ti ðx  ti Þðx  tj Þ

1 r i; j r n:

ð3:26Þ

Since the residues Qj do not depend on x, from Equation (3.25) we obtain X @Qi @tj 1rirn



X ½Qi ; Qj  1 ¼ ; x  ti 1 r i r n ðx  ti Þðx  tj Þ

1 r j r n:

Both sides of the last equality are rational functions of x, which are holomorphic in the punctured sphere CP1 n ft1 ; . . . ; tn g and have simple poles at the points x ¼ ti , 1 r i r n. Equating the residues at each pole x ¼ ti , 1 r i r n, we obtain Equation (3.3). Equation (3.26) leads in a similar way to the first of the Equation (3.3) and, Ì therefore, provides no additional information. PROPOSITION 3.16. Let (3.1) be a holomorphic family of Fuchsian systems, where the residue matrices Qj ðtÞ are holomorphic in a domain D Cn* and satisfy (3.2). Assume that the residue matrices Qj ðtÞ satisfy the Schlesinger system (3.3). Then the family (3.1) is isoprincipal with the distinguished point x0 ¼ 1. Proof. Let us choose t 0 2 D and a nested pair of open cylindrical neighborhoods W ÐV V of t0 in D : Let Yðx; tÞ be the solution of (3.1) and (3.4) and for 1 r j r n let j be a path from x0 ¼ 1 to a point pj 2 V j n W j . Then, according to Theorem 1.7 the branch Yj ðx; t 0 Þ of the solution Yðx; t0 Þ in V j n ft0 g, corresponding to the path j , admits the regular-principal factorization Yj ðx; t 0 Þ ¼ Hj ðx; t 0 Þ

 P ðx; t0Þ; j

x 2 covðV j n ft 0 g; pj Þ;

ð3:27Þ

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VICTOR KATSNELSON AND DAN VOLOK

where the factors Hj ðx; t0 Þ and Pj ðx; t0 Þ are holomorphic with respect to x and invertible in, respectively, V j and covðC n ft0j g; pj Þ: From here we proceed in four steps: Step 1. Firstly, we construct a function Ej ðÞ, holomorphic and invertible on the Riemann surace of ln  and such that the principal factor Pj ðx; t0 Þ is a transplant of Ej ðÞ into covðC n ft0j g; pj Þ: Pj ðx; t0 Þ ¼ Ej ðx  t0j Þ: To this end we choose some value 0j of argðpj  t0j Þ and consider the corresponding isomorphism from the Riemann surace of ln  onto covðC n ft0j g; pj Þ (see (1.16)): !  þ t0j ;  argðpj t0j Þ¼ 0j

ð3:28Þ

 2 covðCnf0g; 1Þ;

þ

t0j

2

covðCnft0j g; pj Þ:

We set def

Ej ðÞ ¼ Pj ð þ t0j ; t0 Þ;

ð3:29Þ

where  þ t0j denotes the image in covðC n ft0j g; pj Þ of the point  2 covðC n f0g; 1Þ under the isomorphism (3.28). Step 2. Secondly, we construct the regular factor Hj ðx; tÞ for the solution Yðx; tÞ of (3.1) and (3.4). In view of Remark 3.10, we consider the overdetermined linear system 8 @Lj ð; tÞ > > > @t > < i @Lj ð; tÞ > > > > : @tj

Qi ðtÞ  Lj ð; tÞ;  þ tj  ti X Qi ðtÞ ¼  Lj ð; tÞ;  þ tj  ti 1rirn ¼ 

1 r i r n; i 6¼ j;

i 6¼ j

ð3:30Þ with the initial condition  Lj ð; tÞ t ¼ t0 ¼ Hj ð þ t0j ; t 0 Þ;

ð3:31Þ

where Hj ðx; t0 Þ is the regular factor in the factorization (3.27). Here  2 C is a small parameter: jj < ; where  > 0 is such that x1 2 W j ) fx : jx  x1 j < g  V j ;

1 r j r n:

ð3:32Þ

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DEFORMATIONS OF FUCHSIAN SYSTEMS

We claim that the overdetermined linear system (3.30), depending on the parameter , is compatible (see Calculation 1 below). Therefore, the system (3.30) has a solution Lj ð; tÞ, satisfying the initial condition (3.31). This solution Lj ð; tÞ is jointly holomorphic in ; t for jj < ; t 2 W and invertible. We define the function Hj ðx; tÞ by def

Hj ðx; tÞ ¼ Lj ðx  tj ; tÞ;

t 2 W ; x 2 Vj

ð3:33Þ

Step 3. Thirdly, we consider the product def

Zj ð; tÞ ¼ Lj ð; tÞ

 E ðÞ:

ð3:34Þ

j

Note that, in view of (3.31), for t ¼ t0 we have Zj ð; t0 Þ ¼ Yj ð þ t0j ; t0 Þ; hence the function Zj ð; t 0 Þ satisfies the linear system X

dZj ð; t0 Þ ¼ d

1rirn

Qi ðt 0 Þ  þ t0j  t0i

 Z ð; t0Þ: j

ð3:35Þ

Also, as follows from (3.30), we have @Zj ð; tÞ Qi ðtÞ ¼  þ tj  ti @ti

 Z ð; tÞ; j

X @Zj ð; tÞ Qi ðtÞ ¼  Zj ð; tÞ:  þ tj  ti @tj 1rirn

1 r i r n; i 6¼ j;

ð3:36Þ

ð3:37Þ

i6 ¼ j

Furthermore, the function Zj ð; tÞ can be shown (see Calculation 2 below) to satisfy with respect to  the linear system X @Zj ð; tÞ Qi ðtÞ ¼ @  þ tj  ti 1rirn

 Z ð; tÞ: j

ð3:38Þ

We note that for each fixed t 2 W the linear differential system (3.38) with respect to  has no singularities in the punctured domain def

V tj ¼ f :  þ tj 2 V j g n f0g; hence its fundamental solution Zj ð; tÞ is holomorphic with respect to  on a universal covering surface over the domain V tj . Therefore, the function Lj ð; tÞ ¼ Zj ð; tÞ

 E1  ðÞ j

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VICTOR KATSNELSON AND DAN VOLOK

is holomorphic with respect to  and invertible on this universal covering surface. On the other hand, the function Lj ð; tÞ is also single-valued holomorphic with respect to  and invertible in the open disk f : jj (see (3.32)). Hence the function Lj ð; tÞ is single-valued holomorphic with respect to  and invertible in the non-punctured domain V tj [ f0g: It follows that the function Hj ðx; tÞ, defined in (3.33), is holomorphic (singlevalued) with respect to x and invertible in the entire domain V j . Step 4. Finally, we consider the coherent family of transplants fEj ðx  tj Þgtj 2W j (see Definition 2.7), corresponding to the unique branch of argðpj  tj Þ which is continuous with respect to tj in W j and takes the value 0 , chosen at Step 1, at the point tj ¼ t0j . For each t 2 W we define the function Yj ðx; tÞ by def

Yj ðx; tÞ ¼ Hj ðx; tÞ

 E ðx  tj Þ ¼ Z ðx  tj ; tÞ; j

j

ð3:39Þ

x 2 covðV j n ftj g; pj Þ: Note that in view of (3.29), (3.31) this definition agrees for t ¼ t 0 with (3.27) and the notation Yj ðx; t0 Þ for the branch of the solution Yðx; t 0 Þ in V j n ft0j g, corresponding to the path j . Now we show that for every t 2 W the function Yj ðx; tÞ, defined in (3.39), is the branch of the solution Yðx; tÞ of (3.1) and (3.4) in the punctured domain V j n ftj g, corresponding to the path j . First of all, in view of (3.38), the function Yj ðx; tÞ satisfies with respect to x the system (3.1): X Qi ðtÞ @Yj ðx; tÞ ¼ x  ti @x 1rirn

 Y ðx; tÞ: j

Hence for each t 2 W the function Yj ðx; tÞ can be analytically continued along the path j in the opposite direction: from pj to 1. The value of this continuation at x ¼ 1 will be denoted by Y^ð1; tÞ; in particular it holds that Y^ð1; t0 Þ ¼ Yð1; t 0 Þ ¼ I:

ð3:40Þ

Furthermore, the value Y^ð1; tÞ can be considered as the initial value at the distinguished point x0 ¼ 1 for a fundamental solution Y^ðx; tÞ of the Fuchsian system (3.1), defined on the universal covering surface covðCP1 n ft1 ; . . . ; tn g; 1Þ. The function Yj ðx; tÞ is the branch of this solution Y^ðx; tÞ in the punctured domain V j n ftj g, corresponding to the path j . Now we note that, in view of (3.36)Y(3.37), @ Y^ Qi ðtÞ ðx; tÞ ¼  @ti x  ti

 Y^ðx; tÞ;

1 r i r n;

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DEFORMATIONS OF FUCHSIAN SYSTEMS

in particular,   @ Y^ ðx; tÞ ¼ 0; @ti x¼1

ð3:41Þ

1 r i r n:

Combining (3.41) with (3.40), we observe that the solution Y^ðx; tÞ satisfies the initial condition  Y^ðx; tÞ x¼1 ¼ I;

t 2 W;

and by the uniqueness theorem for linear differential systems coincides with the fundamental solution Yðx; tÞ of (3.1) and (3.4). Thus the function Yj ðx; tÞ, defined in (3.39), is the branch of the solution Yðx; tÞ of (3.1) and (3.4) in the punctured domain V j n ftj g, corresponding to the path j . The equality (3.39) itself can now be considered as the regular-principal factorization of the branch Yj ðx; tÞ. In view of Definition 2.9, we conclude that the family (3.1) is isoprincipal with the distinguished point x0 ¼ 1. In order to complete the proof, it remains to present the calculations, omitted in the above reasonings. Calculation 1. We show that the overdetermined linear system (3.30), depending on the parameter , is compatible. In this case the compatibility condition (3.23) of Theorem (3.14) is represented by the following two equalities: @ @tj



Qi  þ tj  ti



X

@ þ @ti 1rkrn



Qk  þ tj  tk

¼

k6¼j

ð3:42aÞ

X

½Qk ; Qi  ¼ ; ð þ tj  tk Þð þ tj  ti Þ 1rkrn

1 r i r n; i 6¼ j;

k6¼j

and @ @tk



Qi  þ tj  ti



@  @ti



Qk  þ tj  tk

¼ ð3:42bÞ

¼

½Qk ; Qi  ; ð þ tj  tk Þð þ tj  ti Þ

1 r i; k r n; i; k 6¼ j:

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VICTOR KATSNELSON AND DAN VOLOK

The equality (3.42a) can be simplified as follows: @Qi @tj



X @Qk 1 þ  þ tj  ti 1 r k r n @ti



1  þ tj  tk

k6¼j

X ½Qk ; Qi  1 1 : ¼  t  ti  þ tj  tk  þ tj  ti 1rkrn k k6¼i; j

In view of (3.3), the right-hand side of the last equality can be rewritten as

X @Qk 1 1  @ti  þ tj  tk  þ tj  ti 1rkrn k6¼i; j

¼

X @Qk @Qj 1 1  þ  ; @ti  þ tj  ti 1 r k r n @ti  þ tj  tk k6¼j

where we have used the fact that, in view of Remark 3.2, X @Qk ¼ 0; @ti 1rkrn

1 r i r n:

Since, as follows from (3.3), @Qi @Qj ¼ ; @tj @ti

1 r i; j r n; i 6¼ j;

we conclude that the equality (3.42a), indeed, holds. The equality (3.42b) can be verified analogously. Calculation 2. We show that the function Zj ð; tÞ, defined in (3.34), satisfies Equation (3.38) with respect to . This can be done as follows. We consider the auxiliary function def

Xj ð; tÞ ¼

X @Zj Qi ðtÞ ð; tÞ   Zj ð; tÞ:  þ tj  ti @ 1rirn

ð3:43Þ

It will be shown below that Xj ð; tÞ satisfies the linear system @Xj @ti @Xj @tj

Qi  X j ;  þ tj  ti X Qi ¼  Xj :  þ tj  ti 1rirn ¼

1 r i r n; i 6¼ j; ð3:44Þ

i6¼j

(Note that this system is the same as the system (3.30) for the function Lj .)

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DEFORMATIONS OF FUCHSIAN SYSTEMS

Since, in view of (3.35), the solution Xj ð; tÞ of the linear system (3.44) satisfies the initial condition  X ð; tÞ  0 ¼ 0; t¼t

j

the uniqueness theorem for linear differential systems implies Xð; tÞ  0: Therefore, the function Zj ð; tÞ satisfies (3.38). Now let us prove that Xj ð; tÞ satisfies, indeed, the linear system (3.44). In view of (3.37), we have X @Xj ¼ Qi @tj 1rirn



i6 ¼ j

 Z

j



@ @



Zj  þ tj  ti



X @ Qi  @tj  þ tj  ti 1rirn

X

Qi Qk ð þ tj  ti Þð þ tj  tk Þ 1 r i;k r n

 Z

j

k6 ¼ j

¼

X

X @Qi @Zj Qi    þ tj  ti @ @tj 1rirn 1rirn



1  þ tj  ti

 Z

j



i6 ¼ j



X

Qk Qi ð þ t  ti Þð þ tj  tk Þ j 1 r i;k r n

 Z : j

i6 ¼ j

Now we substitute (3.3) to obtain @Xj @tj

¼

X

Qi  þ tj  ti 1rirn i6 ¼ j

X @Zj Qi Qj  @ ð þ tj  ti Þ 1rirn

Qi Qk ð þ tj  ti Þð þ tj  tk Þ 1 r i;k r n

0

i; k6 ¼ j

B X @

1

Qi C A  þ t  t j i 1rirn



i6 ¼ j

¼

X

Qi  þ tj  ti 1rirn

 Z

j



i6¼j

X



¼



 Z

j

X @Zj Qk  @  þ tj  tk 1rkrn

 Z

! j

 X : j

i6 ¼ j

The first equation of the system (3.44) is obtained analogously.

Ì

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VICTOR KATSNELSON AND DAN VOLOK

5. Isomondromic and Isoprincipal Deformations of Fuchsian Systems DEFINITION 4.1. Let a Fuchsian system dY ¼ dx

X

Q0j

1rjrn

x  t0j

! Y;

ð4:1Þ

where t 0 ¼ ðt01 ; . . . ; t0n Þ 2 C*n , Q01 ; . . . ; Q0n 2 Mk and X

Q0j ¼ 0;

ð4:2Þ

1rjrn

be given. Let a holomorphic family of Fuchsian systems dY ¼ dx

X Qj ðtÞ Y; x  tj 1rjrn

ð4:3Þ

where the residue matrices Qj ðtÞ are holomorphic in a neighborhood D  C*n of t0 , be such that: X

Qj ðtÞ  0;

t 2 D;

ð4:4Þ

1rjrn

Qj ðt0 Þ ¼ Q0j ;

1 r j r n:

ð4:5Þ

If the holomorphic family of Fuchsian systems (4.3) is isoprincipal (respectively, isomonodromic) with the distinguished point x0 ¼ 1, then it is said to be an isoprincipal (respectively, isomonodromic) deformation with the distinguished point x0 ¼ 1 of the Fuchsian system (4.1). Remark 4.2. Note that, according to Theorem 2.11, an isoprincipal deformation (4.3) of a given Fuchsian system (4.1) is also an isomonodromic one. As Theorem 2.15 implies, the converse is true under the condition that all the matrices Q01 ; . . . ; Q0n are non-resonant. Now we are going to address the question of the existence of an isoprincipal deformation of a given Fuchsian system. It follows from Theorem 3.1 and Remark 3.2 that the holomorphic family (4.3) is an isoprincipal deformation with

177

DEFORMATIONS OF FUCHSIAN SYSTEMS

the distinguished point x0 ¼ 1 of the Fuchsian system (4.1) if and only if the residues Qj ðtÞ satisfy the Schlesinger system 8 ½Qi ; Qj  @Qi > > ¼ ; 1 r i; j r n; i 6¼ j; > > > ti  tj < @tj   X Qi ; Qj ð4:6Þ @Q i > ¼ ; 1 r i r n: > > > t  tj > 1rjrn i : @ti j6 ¼ i

Thus the question is, whether the Cauchy problem for the Schlesinger system with the initial condition (4.5) is solvable. An answer to this question follows from the Frobenius theorem (Theorem 3.11): PROPOSITION 4.3. Let t 0 ¼ ðt01 ; . . . ; t0n Þ 2 C*n , Q01 ; . . . ; Q0n 2 Mk be given. Then there exist a neighborhood D  C*n of t0 and unique matrix functions Q1 ðtÞ; . . . ; Qn ðtÞ, holomorphic in D , which satisfy the Schlesinger system (4.6) and the initial condition (4.5). Proof. According to Theorem 3.11 and Remark 3.13, we have to to verify the identity @ 2 Qi @ 2 Qi ¼ ; @tj @tk @tk @tj

1 r i; j; k r n;

substituting the expressions (4.6) for the partial derivatives of Qi : In the case i ¼ k 6¼ j we have 0 1   @ 2 Qi @ B X Qj ; Qi C ¼ @ A @tj 1 r k r n ti  tj @ti @tj k6¼i

¼

½Qi ; ½Qi ; Qj   ½Qi ; Qj  ðti  tj Þ

2

þ

X ½Qi ; ½Qj ; Qk  þ ½Qk ; ½Qi ; Qj  ðti  tj Þðti  tk Þ 1rkrn k6¼i

¼

½Qi ; ½Qi ; Qj   ½Qi ; Qj  2



ðti  tj Þ 

 Qi ; Qj @ @ 2 Qi ¼ ; ¼ @ti ti  tj @tj @ti

X

½Qj ; ½Qk ; Qi  ðt  tj Þðti  tk Þ 1rkrn i k6¼i

where we have used the Jacobi identity ½A; ½B; C þ ½B; ½C; A þ ½C; ½A; B ¼ 0;

8A; B; C 2 Mk :

In the case k 6¼ i 6¼ j the computations are analogous and will be omitted. Ì

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VICTOR KATSNELSON AND DAN VOLOK

Remark 4.4. Proposition 4.3 was originally proved by L. Schlesinger in [35]. Here we would also like to mention the paper [3] by P. Boalch, where some general considerations concerning the compatibility of systems from a class, containing the Schlesinger system, are presented. As an immediate consequence of Theorem 3.1 and Proposition 4.3, we obtain THEOREM 4.5. Let a Fuchsian system (4.1)Y(4.2) be given. Then there exists a neighborhood D  C*n of t0 and a unique isoprincipal deformation (4.3) of the Fuchsian system (4.1) with the distinguished point x0 ¼ 1, where the residue matrices Qj ðtÞ are holomorphic in the neighborhood D . Remark 4.6. Theorem 4.5 can also be proved in another way. Let W Ð V be a nested pair of open cylindrical neighborhoods of t0 in C*n and for 1 r j r n let j be a path from x0 ¼ 1 to a point pj 2 V j n W j . Let Yðx; t0 Þ be the solution of (4.1) with the initial condition  Yðx; t0 Þ x ¼ 1 ¼ I and let Yj ðx; t 0 Þ ¼ Hj ðx; t 0 Þ

 E ðx  t0j Þ; j

x 2 covðV j n ft0j g; pj Þ

be the regular-principal factorization of the appropriate branch Yj ðx; t0 Þ in the punctured domain V j n ft0j g (here the function Ej ðÞ, holomorphic and invertible in the Riemann surface of ln , is defined in the same way as in the proof of Proposition 3.16 Y see (3.29)). First we assume that the family (4.3), where the residue matrices Qj ðtÞ are holomorphic in W , is an isoprincipal deformation of the Fuchsian system (4.1) with the distinguished point x0 ¼ 1 and consider the solution Yðx; tÞ of (4.3) with the initial condition  Yðx; tÞ x ¼ 1 ¼ I:

ð4:7Þ

Since Yðx; S tÞ is jointly holomorphic in x and t and invertible in the domain covðCP1 n k W k ; 1Þ  W , so is the Fratio` Rðx; tÞ ¼ Yðx; tÞ  Y 1 ðx; t0 Þ: def

179

DEFORMATIONS OF FUCHSIAN SYSTEMS

Moreover, taking into account the initial condition (4.7) and the fact that the family (4.3) is by Theorem 2.11 isomonodromic, we reach the following conclusion: (R) The function Rðx; tÞ is holomorphic with respect to x; t and   S (single-valued) invertible in the domain CP1 n k W k  W and it holds that  t 2 W: Rðx; tÞ x ¼ 1  I; Similarly, considering the coherent family of transplants fEj ðx  tj Þgtj 2W j ; we observe that the Fratio` def

Fj ðx; tÞ ¼ Ej ðx  tj Þ

0  E1  ðx  tj Þ; j

x 2 covðC n W j ; pj Þ; t 2 W

  is holomorphic (single-valued) with respect to x; t and invertible in C n W j  W: Since the family (4.3) is assumed to be isoprincipal, there exist functions Hj ðx; tÞ (the regular factors of Yðx; tÞ), such that: (H) Each function Hj ðx; tÞ, 1 r j r n, is holomorphic (single-valued) with respect to x; t and invertible in V j  W ; (Pb) Fj ðx; tÞ  Hj1 ðx; t0 Þ ¼ Hj1 ðx; tÞ  Rðx; tÞ; t 2 W ; x 2 V j n W j;

1 r j r n:

Note that the function Fj ðx; tÞ  Hj1 ðx; t0 Þ on the left-hand side of the equality (Pb) is holomorphic (single-valued) with respect to x; t and invertible in fCn W j g  W . It is determined entirely in terms of the initial data t0 ; Q01 ; . . . ; Q0n . The equality (Pb) itself can be viewed as a factorization problem for this function, where one looks for the factors Hj ðx; tÞ, Rðx; tÞ, possessing the analyticity properties (H) and (R). If such factors can be found, then, reversing the reasonings above, one arrives at the isoprincipal deformation of the Fuchsian system (4.1). The uniqueness of the solution, possessing the properties (H) and (R), for the factorization problem (Pb) follows immediately from the Liouville theorem. The existence of this solution for t in a sufficiently small neighborhood W of t 0 can be established by elementary means, since for t ¼ t0 the solution exists (it is trivial: R ¼ I). However, analyzing the factorization problem (Pb) more carefully and systematically (see, for instance, [9, Section 5]) and taking into account Theorem 3.1, one can reach the stronger conclusion that the functions Qj ðtÞ; which satisfy the Schlesinger system (4.6) with the initial condition (4.5), are meromorphic in the universal covering space} covðC*n ; ; t0 Þ: This result was obtained by B. Malgrange in [29] and (in the non-resonant case) by T. Miwa in [32]. Our proof,

180

VICTOR KATSNELSON AND DAN VOLOK

which involves the isoprincipal deformations and is outlined above, will be presented in more detail elsewhere. Remark 4.7. As was already mentioned (see Remarks 1.10 and 2.16), most of the considerations of the present article need not be restricted to the case of linear differential systems with only Fuchsian singularities. For instance, one can consider linear systems of the form ! pj n X X Qj;k dY Y; ¼ kþ1 dx j¼1 k¼0 ðx  tj Þ where n X

Qj;0 ¼ 0:

j¼1

In this case the regular-principal factorization of the solution and the notion of the isoprincipal family ! pj n X X Qj;k ðtÞ dY ¼ Y kþ1 dx j¼1 k¼0 ðx  tj Þ can be introduced as in Definitions 1.8, 2.9. Furthermore, the auxiliary linear system related to the isoprincipal family takes the form (compare with (3.5)) ! 8 pj n P > P Q ðtÞ @Y j;k > > Y; ¼ > kþ1 > < @x j¼1 k¼0 ðx  tj Þ ! > pj > P Qj;k ðtÞ @Y > > > Y; 1 r j r n: : @t ¼  kþ1 j k¼0 ðx  tj Þ The compatibility condition for this overdetermined system is given by the system 8 @Qi;k > > > > > @tj > > < @Qi;k > > > > > @ti > > :

X

¼

l



ð1Þ

0 r ‘ r pi k 0 r q r pj

¼

X

X

lþq l

ð1Þ

1 r j r n 0 r ‘ r pi k j6¼i



½Qi;kþl ; Qj;q  ðti  tj Þqþlþ1

l



lþq l



; 0 r k r pi ; 1 r i; j r n; i 6¼ j;

½Qi;kþl ; Qj;q  ðti  tj Þqþlþ1

; 0 r k r pi ; 1 r i r n;

0 r q r pj

which itself is compatible and contains the Schlesinger system 4.6 as a special case (corresponding to p1 ¼ . . . ¼ pn ¼ 0).

181

DEFORMATIONS OF FUCHSIAN SYSTEMS

6. An Example In order to illustrate the distinction between the isoprincipal and the isomonodromic deformations of Fuchsian systems in the case when the nonresonance condition is violated, we present the following explicit example. Let us consider the linear differential system 0 1 1 0 dY @ xðx1Þ AY: ð5:1Þ ¼ 1 dx 0 ðx2Þðx3Þ Note that the system (5.1) is of the form dY ¼ dx



Q00 Q01 Q02 Q03 þ þ þ Y; x x1 x2 x3

where Q00

¼

Q01

¼

1 0

0 ; 0

Q02

¼

Q03

¼

0 0

0 ; 1

and hence it is Fuchsian and resonant. Moreover, x ¼ 1 is a regular point of the system (5.1), since Q00 þ Q01 þ Q02 þ Q03 ¼ 0: The solution YðxÞ satisfying the initial condition Yj1 ¼ I is a rational matrix functions (in particular, single-valued); it has the following form: 0 x 1 0 B C def YðxÞ ¼ @ x  1 x  2 A ¼ Y 0 ðxÞ: 0 x3 The principal factors x P0 ðxÞ ¼ 0 1 P2 ðxÞ ¼ 0

of Y 0 can be chosenj in the form 1



0 0 ; ; P1 ðxÞ ¼ x1 1 0 1



1 0 0 ; P3 ðxÞ ¼ : 1 0 x3 x2

Now we are going to construct the isoprincipal deformation of the system (5.1). For simplicity, we assume that the singular points x ¼ 1; 2; 3 are fixed j

See Remark 1.9.

182

VICTOR KATSNELSON AND DAN VOLOK

while the position of one singularity (x = 0) varies: x ¼ t. Then we need to determine a holomorphic family of matrix functions Yðx; tÞ such that   Yðx; tÞ t¼0 ¼ Y 0 ðxÞ; ð5:2Þ Yðx; tÞ x ¼ 1 ¼ I; P0 ðx  tÞ is the principal factor of Yðx; tÞ at x ¼ t and for k ¼ 1; 2; 3 Pk ðxÞ is the principal factor of Yðx; tÞ at x ¼ k. Such a family is unique and consists of rational matrix functions of the form: 1 0 xt 0 C B Yðx; tÞ ¼ @ x  1 x  2 A: 0 x3 The function Yðx; tÞ satisfies with respect to x the Fuchsian system 0 1 t1 0 C dY B ðx  tÞðx  1Þ CY ¼B @ A 1 dx 0 ðx  2Þðx  3Þ 0

Q0 Q01 Q02 Q03 þ þ þ ¼ Y; xt x1 x2 x3

ð5:3Þ

and the constant functions Qj ðtÞ  Q0j ;

j ¼ 0; 1; 2; 3;

satisfy, of course, the Schlesinger system 8 dQj ½Q0 ; Qj  > > ¼ ; j ¼ 1; 2; 3; > > < dt tj > dQ0 > > > : dt

¼

3 X ½Qj ; Q0  j¼1

tj

;

with the initial condition Qj ð0Þ ¼ Q0j ;

j ¼ 0; 1; 2; 3:

However, the deformation (5.3) is not the only possible isomonodromic deformation of the system (5.1). Indeed, let us consider a family of rational functions 0 1 xt 2tðx  tÞhðtÞ B C Yðx; tÞ ¼ @ x  1 ðx  1Þðx  3Þðt  3Þ A; x2 0 x3

183

DEFORMATIONS OF FUCHSIAN SYSTEMS

where hðtÞ is a function, holomorphic in a neighborhood of t ¼ 0 (outside this neighborhood hðtÞ may have arbitrary singularities). Such a family also satisfies the normalizing conditions (5.2); moreover, we have 0

x1 B xt ¼@

Yðx; tÞ1

0

0

@Yðx; tÞ Yðx; tÞ1 @x

1 2thðtÞ ðx  2Þðt  3Þ C A; x3 x2

1 2tðx  tÞhðtÞ ðx  1Þðx  2Þðx  3Þðt  3Þ C C: A 1 ðx  2Þðx  3Þ

t1 B ðx  tÞðx  1Þ ¼B @ 0

Thus we obtain the following deformation of the system (5.1): 0

t1 B dY B ðx  tÞðx  1Þ ¼ dx @ 0 ¼

1 2tðx  tÞhðtÞ ðx  1Þðx  2Þðx  3Þðt  3Þ C CY A 1 ðx  2Þðx  3Þ

Q0 ðtÞ Q1 ðtÞ Q2 ðtÞ Q3 ðtÞ þ þ þ Y; xt x1 x2 x3

where Q0 ðtÞ  Q00 ;

Q2 ðtÞ ¼

0 0

Q1 ðtÞ ¼

tðt  1ÞhðtÞ t3 0

1 0

2tðt  2ÞhðtÞ t3 1

!

;

Q3 ðtÞ ¼

0 0

!

thðtÞ : 1

Since the monodromy of Yðx; tÞ for every t is trivial, the deformation (5.4) is isomonodromic, but if hðtÞ 6 0, then it is not isoprincipal and the coefficients Qj ðtÞ do not satisfy the Schlesinger system. Moreover, since we only require hðtÞ to be holomorphic in a neighborhood of t ¼ 0, the behavior of the functions Qj ðtÞ outside this neighborhood may be arbitrary. In particular, these functions Qj ðtÞ need not be meromorphic with respect to t in C n f1; 2; 3g. Remark 5.1. The example presented above is based on the theory of holomorphic families (4.3) of Fuchsian systems, whose solutions Yðx; tÞ are generic rational matrix functions of x for every fixed t. This theory was developed

184

VICTOR KATSNELSON AND DAN VOLOK

in [23], [24] and [27] (see also the electornic version of the latter work [28].) For such Fuchsian systems the number of poles n is even, so we write 2n instead of n. All such Fuchsian systems with this property can be parameterized as follows. If k is the dimension of the square residue matrices Qj , then to each pole tj a k  1-vector is related as a Ffree` parameter. Therefore, to each 2n-tuple t ¼ ðt1 ; . . . ; t2 nÞ the total of ðk  1Þ  2n complex parameters is related. To every choice of these parameters (satisfying a certain non-degeneracy condition) corresponds a different system of the form (4.3), whose solution Yðx; tÞ is a generic rational matrix function. Considering t as variable, we assign ðk  1Þ  2n complex parameters to each t 2 C2n *. Thus the families (4.3), possessing the property that Yðx; tÞ is a generic rational matrix function of x for each fixed t, can be parameterized by ðk  1Þ  2n complex valued functions of t, and these functional parameters are free. To obtain holomorphic families, we have to require that these complex valued functions are holomorphic. Of course, the monodromy of any rational matrix function of x is trivial. Hence we can parameterize the class of the isomonodromic deformations (4.3) of Fuchsian systems with generic rational solutions by ðk  1Þ  2n functional parameters, and these parameters can be arbitrary (up to the mentioned non-degeneracy condition) holomorphic functions of t. It turns out that the deformation corresponding to a given choice of the functional parameters is isoprincipal if and only if all these parameters are constant functions. In particular, the class of the isomonodromic deformations (considered without the non-resonance condition) is much richer than its subclass of the isoprincipal deformations.

References 1.

Birkhoff, G. D.: The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proc. Amer. Acad. Arts Sci. 49 (1913), 521Y568. Reprinted in: [2], 259Y306. 2. Birkhoff, G. D.: Collected Mathematical Papers, Vol. I, American Mathematical Society, New York, 1950, pp. iYxi, 754 pp. 3. Boalch, P.: Symplectic manifolds and isomonodromic deformations, Adv. Math. 163 (2001), 137Y205. 4. BmjhBrt, A. A.: Oomajeka PukalaYDhjmaeoqa. Ecgtb vfntv. yfer, 45(2) (1990), 3Y47. Engl. transl.: Bolibruch, A. A.: The RiemannYHilbert problem, Russian Math. Surveys 45(2) (1990), 1Y47. 5. BmjhBorta, A.: 21-z ghjajeka Dhjmaeoqa lkz khleqyzt sricjdst chcqek. Hferf, 1994 (Th. Man. bycn. CntrJIjdf, Pjcc. ArfL. yfer, 206). (In

Russian). English transl.: Bolibrukh, A.: The 21st Hilbert Problem for Linear Fuchsian Systems, Providence: Amer. Math. Soc. 1995 (Proc. Steklov Inst. Math. 6.

Bolibruch, A.: On isomonodromic deformations of Fuchsian systems, J. Dynam. Control Systems 3(4) (1997), 589Y604.

DEFORMATIONS OF FUCHSIAN SYSTEMS

7.

185

BmjhBort, A.: Ja bpjvjyjlhjvymlt pjbzybzt syicmbzt mcmaellmcqevq. yfer, 221 (1998), 127Y142 (In Russian). English transl.: Bolibrukh, A.: On isomonodromic confluences of fuchsian singularities, Proc. Steklov Inst. Math. 221 (1998), 117Y132.

B m j h B o r t , A . : L b sseoel w b f k m y m lt yh f d y t y b z c v th j v jh s y m lv b rjpssbwbtynfvb. Cnh. 29Y82 d: Cjdhtvtyyst Ghjaktvs Vfnyfnbrb, Dmiecr 1, Sovremennye Problemy Matematiki, Vypusk 1, MATtM. IHCT. CTtrkjdf, Pocc. Frfl. yfer, (In Russian). DjLIBRUKH, A.: Differential equations with meromorphic coefficients, pp. 29Y82 in: Modern Problems of Mathematics 1, Steklov Inst. Math., Russian Acad. Sci., Moskow, 2003. 9. Bolibruch, A. A., Its, A. R. and Kapaev, A. A.: On the RiemannYHilbertYBirkhoff inverse monodromy problem and the Painleve´ equations, Fkut,hf b fyfkbp, 16(1) (2004), 121Y162, and also St. Petersburg Math. J. 16(1) (2005), 105Y142. 10. Forster, O.: Riemannschen Fla¨chen, Springer, Berlin Heidelberg New York, 1977, x+223 pp. (in German). English Transl.: Forster, O.: Lectures on Riemann Surfaces, (Graduate Texts in Mathematics, 81) Springer, Berlin Heidelberg New York, 1981. viii+254 pp. Russian Transl.: AJHCNTH, O.: Hbvfyjds Gjdthtlmpqh Vbh, Vjcrdf, 1980. 8.

11. 12.

13.

14.

15.

16. 17.

Fuchs, L.: Gesammelte Mathematische Werke. Band 3, (Herausgegeben by R. Fuchs und L Schlesinger). Mayer & Mu¨ller, Berlin, 1909. Fuchs, L.: Zur Theorie der linearen Differentialgleichungen, Sitzungsberichte der K. preuss. Akademie der Wissenschaften zu Berlin. Einleitung und No. 1Y7, 1888, S. 1115Y1126; No. 8Y15, 1888, S.1273Y1290; No. 16Y21, 1889, S. 713Y726; No. 22Y31, 1890, S. 21Y38. Reprinted in: [11], S. 1Y68. Fuchs, L.: U€ ber lineare Differentialgleichungen, welche von Parametern unabh€ angige substitutionsgruppen besitzen, Sitzungsberichte der K. preuss. Akademie der Wissenschaften zu Berlin. 1892, S. 157Y176. Reprinted in: [11], S. 117Y139. Fuchs, L.: U€ ber lineare Differentialgleichungen, welche von Parametern unabh€ angige Substitutionsgruppen besitzen, Sitzungsberichte der K. preuss. Akademie der Wissenschaften zu Berlin. Einleitung und No. 1Y4, 1893, S. 975Y988; No. 5Y8, 1894, S. 1117Y1127. Reprinted in: [11], S. 169Y195. Fuchs, L.: U€ ber die Abh€ angigkeit der L€ osungen einer linearen differentialgleichung von den in den Coefficienten auftretenden Parametren, Sitzungsberichte der K. preuss. Akademie der Wissenschaften zu Berlin. 1895, S. 905Y920. Reprinted in: [11], S. 201Y217. Fuchs, R.: Sur quelquea e´quations diffe´rentielles line´ares du second ordre, C. R. Acad. Sci., Paris 141 (1905), 555Y558. ¨ ber lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Fuchs, R.: U Endlichen gegebenen wesentlich singula¨ren Stellen, Math. Ann. 63 (1907), 301Y321.

18.

UAHTMAXEP, a. P. Remohz vfnhbw. 2-t bplfybt. Yferf, 1966, 575 c. (In Russian). English transl.: Gantmacher, F. R.: The Theory of Matrices, Chelsea, New York, 1959, 1960.

19.

Garnier, R.: Sur une classe d’e´quations diffe´rentielles dont les inte´grales ge´ne´rales ont leurs points critiques fixes, C. R. Acad. Sci., Paris 151 (1910), 205Y208. Hurewich, W. and Wallman, H.: Dimension Theory, Princeton University Press, Princeton, 1941. Iwasaki, K., Kimura, H., Shimomura, S. and Yoshida, M.: From Gauß to Painleve´, (A Modern Theory of Special Functions) Aspects of Mathematics: E; vol. 16. Friedr. Vieweg & Sohn, Braunschweig, 1991, xii+347 pp.

20. 21.

186 22. 23.

24.

25.

26.

27.

28.

29. 30. 31.

32. 33.

34. 35. 36. 37.

VICTOR KATSNELSON AND DAN VOLOK

Its, A. R. and Novokshenov, V. Yu.: The Isomonodromic Deformation Method in the Theory of Painleve´ Equations, Lect. Notes in Math, 1191, Springer, Berlin-New York, 1986. Katsnelson, V.:, Fuchsian differential systems related to rational matrix fuctions in general position and the joint system realization, in: L. Zalcman (ed.), Israel Mathematical Conference Proceedings, Vol. 11 (1997), Proceedings of the Ashkelon Workshop on Complex Function Theory (May 1996), pp. 117Y143. Katsnelson, V.: Right and left joint system representation of a rational matrix function in general position (System representation theory for dummies), in: D. Alpay and V. Vinnikov (eds.), Operator Theory, System Theory and Related Topics (The Moshe Livsˇic anniversary volume. Proceedings of the Conference on Operator Theory held at Ben-Gurion University of the Negev, Beer-Sheva and in Rehovot, June 29YJuly 4, 1997), (Operator Theory: Advances and Applications, 123), Birkha¨user Verlag, Basel, 2001, pp. 337Y400. Katsnelson, V. and Volok, D.: Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions I, in: J. A. Ball, J. W. Helton, M. Klaus and L. Rodman (eds.), Current Trends in Operator Theory and its Applications, (Operator Theory: Advances and Applications, 149), Birkha¨user Verlag, Basel, 2004, pp. 291Y348. Katsnelson, V. and Volok, D.: Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions I, arXiv.org e-print archive: http://arxiv.org, math. CA/0304108. Katsnelson, V. and Volok, D.: Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions II, in: D. Alpay, and V. Vinnikov (eds.), Operator Theory, System Theory And Scattering Theory: Multidimensional Generalizations, (Operator Theory: Advances and Applications, 157), Birkha¨user Verlag, Basel, 2005, pp. 165Y203. Katsnelson, V. and Volok, D.: Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions II, arXiv.org e-print archive: http://arxiv.org, math. CA/0406489. Malgrange, B.: Sur les de´formations isomonodromiques. I. Singularites regulieres, (In French) [On isomonodromic deformations. I. Regular singularities.] pp. 401Y426 in [31]. Malgrange, B.: Sur les deformations isomonodromiques. II. Singularites irregulieres, (In French) [On isomonodromic de´formations. II. Irregulare singularities.] pp. 427Y438 in [31]. Mathe´matique et Physique. (Se´minaire de l’Ecole Normale Supe´rieure 1979Y1982), (In French). [Mathematics and Physics. (Seminar of Ecole Normale Supe´rieure 1979Y1982.)] in: L. Boutet de Monvel, A. Douady and J.-L. Verdier (eds.), Progr. in Math. 37, Birkha¨user, Boston Basel Stuttgart, 1983. Miwa, T.: Painleve’ property of monodromy preserving deformation equations and the analyticity of  functions, Publ. Res. Inst. Math. Sci. 17(2) (1981), 703Y721. Narasimhan, R.: Analysis on real and complex manifolds, (Advanced Studies in Pure Mathematics, Vol. 1). Masson & Cie, E´diteur, Paris; North-Holland Publishing Co., Amsterdam; 1973. x+246 pp. Russian Transl.: YAPACbvXAH, P. Ayfkbp yf ltqcntbntkmymlt rjvgktrcymlt vyjcjjaofpbzt. VBH, Vjcrdf 1971, 232 c. Schlesinger, L.: Sur la de´termination des fonctions alge´briques uniformes sur une surface de  Riemann donne´e, (French.) Ann. Sci. Ecole Norm. Sup. Se´r. 3 20 (1903), 331Y347. ¨ Schlesinger, L.: Uber die Lo¨sungen gewisser linearer Differentialgleichungen als Funktionen der singula¨ren Punkte, J. Reine Angew. Math. 129 (1905), 287Y294. Schlesinger, L.: Vorlesungen u¨ber lineare Differentialgleichungen, Leipzig und Berlin, 1908. ¨ ber eine Klasse von Differentialsystemen beliebiger Ordnung mit festen Schlesinger, L.: U kritischen Punkten, J. Reine Angew. Math. 141 (1912), 96Y145.

Mathematical Physics, Analysis and Geometry (2006) 9: 187Y201 DOI: 10.1007/s11040-006-9009-9

#

Springer 2006

Gaussian Beam Construction for Adiabatic Perturbations M. DIMASSI1, J.-C. GUILLOT2 and J. RALSTON3 1

De´partement de Mathe´matiques, Universite´ Paris 13, Villetaneuse, France. e-mail: dimassi@ math.univ-paris13. fr 2 Centre de Mathe´matiques Aplique´s, CNRS-UMR 7641 Ecole Polytechnique, 91128 Palaiseau Cedex, France. e-mail: [email protected] 3 University of California, Los Angeles, CA 90095, USA. e-mail: [email protected] (Received: 6 September 2005; accepted: 29 March 2006; published online: 15 June 2006) Abstract. We construct wave packets concentrated near a single space-time trajectory of the semiclassical Hamiltonian for a Bloch electron in a crystal lattice subject to slowly varying external electric and magnetic fields. The use of an analog of the Gaussian beam Ansatz make it possible to construct packets for arbitrarily long finite times. Key words: Bloch electron, Gaussian beam construction, Floquet eigenvalues. Mathematics Subject Classifications (2000): 35P20, 58G99, 81Q15.

1. Introduction The quantum dynamics of a Bloch electron in a crystal subject to external electric and magnetic fields, ry VðyÞ and ry  AðyÞ, which vary slowly on the scale of the crystal lattice, is governed by the Schro¨dinger equation Dt u ¼ P u; ujt¼0 ¼ u ðyÞ; D ¼

1 @ i

ð1Þ

 2 y P ¼ Dy þ AðyÞ þ V0 ð Þ þ VðyÞ: 

Here V0 is periodic with respect to the crystal lattice   R3, and it models the electric potential generated by the lattice of atoms in the crystal. There has been a growing interest in the rigorous study of dynamics of Bloch electrons in the presence of slowly varying external perturbations (see [1, 3Y7, 11, 12, 14, 16, 18, 20, 22, 27]). Since the work of Peierls [21] and Slater [26], it is well known that, if  is sufficiently small, then solutions of (1) for suitable initial data u are in some sense governed by the Fsemi-classical_ Hamiltonian Hðy; Þ ¼ ð þ AðyÞÞ þ VðyÞ:

188

MOUEZ DIMASSI ET AL.

Here ðÞ is one of the Fband functions_ describing the Floquet spectrum of the unperturbed Hamiltonian P0 ¼ x þ V0 ðxÞ; x ¼ y=. The purpose of this paper is to construct nontrivial asymptotic solutions of (1) which are concentrated near  ¼ fðyðsÞ; sÞ; 1 < s < þ1g, the projection into space-time of a trajectory of the Hamiltonian H. We can allow A and V to have a slow time dependence as well, i.e., A ¼ Aðx; tÞ and V ¼ Vðx; tÞ, which changes H to Hðy; ; sÞ ¼ ð þ Aðy; sÞÞ þ Vðy; sÞ:

This construction was presented in [7] with only a sketch of the FGaussian beam_ method which gives asymptotic solutions over arbitrarily long time intervals. We give the details of that method here. 2. Preliminaries Let  ¼ 3i¼1 Zai be the lattice generated by the basis a1 ; a2 ; a3 ,ai 2 R3: The reciprocal lattice * is defined as the lattice generated by the dual basis fa*1 ; a*2 ; a*3 g determined by ai a*j ¼ 2ij ; i; j ¼ 1; 2; 3: Let E be a fundamental domain for , and let E* be a fundamental domain for *: If we identify opposite edges of E (resp. E*) then it becomes a flat torus denoted by T ¼ R3 = (resp. T* ¼ R3 =*). Let V0 be a real-valued potential, C1 and -periodic. For k in R3 ; we define H0 ðkÞ ¼ ðDx þ kÞ2 þ V0 ðxÞ:

as an unbounded operator on L2 ðTÞ with domain H 2 ðTÞ. The Hamiltonian H0 ðkÞ is a semi-bounded and self-adjoint. Since the resolvent of ðDx þ kÞ2 is compact, the resolvent of H0 ðkÞ is also compact, and therefore H0 ðkÞ has a complete set of (normalized) eigenfunctions n ð; kÞ 2 H 2 ðT*Þ, n 2 N; called Bloch functions. The corresponding eigenvalues accumulate at infinity and we enumerate them according to their multiplicities, 1 ðkÞ  2 ðkÞ  ::::

Since eix* H0 ðkÞeix* ¼ H0 ð* þ kÞ; n ðkÞ is periodic with respect to *: The function n ðkÞ is called a band function and the closed intervals n :¼ n ðT*Þ are called bands. Standard perturbation theory shows that n ðkÞ is a continuous function of k and is real analytic in a neighborhood of any k such that n1 ðkÞ < n ðkÞ < nþ1 ðkÞ:

ð2Þ

We make the following hypothesis on the spectrum of the unperturbed Schro¨dinger operator (H1) For a given n0 we will assume that n0 ðkÞ satisfies (2) for all k

GAUSSIAN BEAM CONSTRUCTION FOR ADIABATIC PERTURBATIONS

189

Under this assumption we can choose the eigenfunction n0 ðx; kÞ associated to n0 ðkÞ to be a real-analytic function of k with values in H 2 ðTÞ: Turning now to Equation (1), we assume that Aðy; sÞ and Vðy; sÞ are smooth

and that the derivatives @y;s A and @y;s V are bounded on R3  R for j j  1 and j j  0. To construct asymptotic solutions of (1) we use, as in [4, 7, 8, 11, 12], the two-scale expansion method in which the electron coordinate x and the slowly varying space variable y ¼ x are regarded as independent variables, and we introduce the slow time variable s ¼ t. Thus, we consider the following equation in the independent variables x; y and s : Ds v ¼ Pv; where Pv ¼

h

i 2 Dy þ Dx þ Aðy; sÞ þV0 ðxÞ þ Vðy; sÞ v:

ð3Þ

Note that, if in the solution vðx; y; s; Þ of (3) we let x ¼ y= and t ¼ s= then it becomes a solution of (1). In the variable x; vðx; y; s; Þ is required to be periodic. We look for approximate solution to (3), which have the form: i h vðx; y; s; Þ ¼ eiðy;sÞ= m0 ðx; y; sÞ þ m1 ðx; y; sÞ þ    þ N mN ðx; y; sÞ :

ð4Þ

Now substituting (4) into (3) and collecting terms which are the same order in ; we get i h ðDs  PÞv ¼ eiðy;sÞ= c0 ðx; y; sÞ þ c1 ðx; y; sÞ þ    þ Nþ2 cNþ2 ðx; y; sÞ

ð5Þ

where 

 @ c0 ðx; y; sÞ ¼  H0 ðkðy; sÞÞ  Vðy; sÞ m0 ; @s

ð6Þ

  @ ðy; sÞ m1 ; c1 ðx; y; sÞ ¼ Km0  H0 ðkðy; sÞÞ þ Vðy; sÞ  @s

ð7Þ

and for j ¼ 2; 3; :::; N þ 2   @ cj ðx; y; sÞ ¼ Kmj1 þ y mj2  H0 ðkðy; sÞÞ þ Vðy; sÞ  ðy; sÞ mj : @s

Here K¼i

h @H

0

@k

ðkðy; sÞÞ

@ @ @ i þ  kðy; sÞ  @y @y @s

190

MOUEZ DIMASSI ET AL.

and kðy; sÞ ¼ Aðy; sÞ þ

@ ðy; sÞ: @y

Notice that, when  is real-valued, (4) is the standard ansatz of geometric optics. In the construction of geometric optics solutions one requires that c0 ðx; y; sÞ ¼ 0; ðeikonal equationÞ

ðEÞ

cj ðx; y; sÞ ¼ 0; ðtransport equationsÞ:

ðTjÞ

One can solve these equations locally by applying the classical Fmethod of characteristics._ However, as is well known, if we try to construct WKB-solutions globally (that is in some large given region)  may develop singularities at Fcaustics,_ and the transport equations then become undefined. The consideration of these difficulties, beginning with Keller [17] and Maslov [19], lead to the development of the theory of Fourier integral operators, e.g., as given by Ho¨rmander [15]. Instead of Fourier integral operators we are going to use FGaussian Beams,_ as in [2, 22, 23]. Let us explain briefly the main idea of this construction. Given T > 0; we want to find ; m0 ; ::::; mN such that kðDs  PÞvkL2 ðR3 =R3 ½T;T Þ ¼ OðN Þ:

ð8Þ

Since we are going to choose  so that =  cdððy; sÞ; Þ2 ;

in view of (4) it suffices, assuming for instance that the intersection of the support of ci ðx; y; sÞ with jsj < T is compact, to have ðy; sÞ ! ci ðx; y; sÞ vanish to high order on  uniformly with respect to x 2 R3 =: In fact, (8) will follow from the obvious equality dððy; sÞ; Þm jei= j ¼ Oðm=2 Þ

which implies that, when cj ðx; y; sÞ vanishes to order m on  uniformly in x; Z

jcj ðx; y; sÞei= j2 dy ¼ Oðmþ3=2 Þ;

jsj
uniformly on x:

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191

3. Construction of the Phase Function From now on we assume that (H1) holds, and we let n0 be the normalized eigenfunction corresponding to n0 ðkÞ : H0 ðkÞn0 ¼ n0 ðkÞ; hn0 ; n0 i ¼ 1:

In view of (H1) we can assume that both n0 ðkÞ and n0 ðx; kÞ are real-analytic in k: From here on we will suppress the index n0, since it remains fixed throughout the construction. Equation (E) tells us that for all ðy; sÞ; m0 ðx; y; sÞ is an eigenfunction of H0 ðkðy; sÞÞ with eigenvalue @ @s ðy; sÞ  Vðy; sÞ: Hence, under assumption (H1), we can satisfy (E) by choosing   @ @ ðy; sÞ   Aðy; sÞ þ ðy; sÞ þ Vðy; sÞ Gðy; sÞ :¼ @s @y ¼ 0; (eikonal equation);

ð9Þ

and setting m0 ðx; y; sÞ ¼ f0 ðy; sÞðx; kðy; sÞÞ:

Set hðy; s; ; Þ ¼  ð Aðy; sÞ þ Þ  Vðy; sÞ;

and let ðyðsÞ; s; ðsÞ; ðsÞÞ be a trajectory of the Hamiltonian flow generated by hðy; s; ; Þ; i.e., y_ ¼ 

@ ðAðy; sÞ þ Þ; s_ ¼ 1 @k

ð10Þ

and _ ¼ 

3 X

y_i

i¼1

@Ai @V @A @V ðy; sÞ þ ðy; sÞ; _ ¼ y_ ðy; sÞ þ ðy; sÞ: @y @y @s @s

ð11Þ

The physical significance of these equations becomes clearer when they are rewritten in the coordinates ðy; s; k; Þ ¼ ðy; s;  þ AðyÞ; Þ: It follows from (9) that y_ ¼ 

@ ðkÞ; s_ ¼ 1 @k

k_ ¼ _ þ

3 X @A i¼1

@yi

ðy; sÞ y_ i þ

@A ðy; sÞ s_ @s

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¼

3  X @A i¼1

@yi

ðy; sÞ 

¼ y_  Bðy; sÞ þ

@Ai @A @V ðy; sÞ y_i þ ðy; sÞ þ ðy; sÞ @y @s @y

@A @V ðy; sÞ þ ðy; sÞ: @s @y

Here B ¼ ry  Aðy; sÞ is the magnetic field corresponding to Aðy; sÞ: REMARK. Notice that, the classical flow exists for all s: Indeed, the Hamiltonian vector field ð@; h; @y;s hÞ is uniformly bounded with respect to ðy; s; ; Þ; and hence no classical trajectory can blow up in a finite time. From now on, we fix a trajectory fðyðsÞ; s; ðsÞ; ðsÞÞ; 1 < s < a þ 1g with initial data ðy0 ; 0; 0 ; 0 Þ satisfying hðy0 ; 0; 0 ; 0 Þ ¼ 0: We denote by  its projection into the ðy; sÞ-space. As was indicated above, we are not going to attempt to solve (9) exactly. We only need to have Gðy; sÞ vanish to high order on : More precisely, we are going to prove: THEOREM 1. Let 0 ðyÞ be a smooth function such that the Hessian, =@y2 0 ðy0 Þ; is positive definite. Then, given N and T > 0; there is a smooth function ðy; sÞ such that ðy; 0Þ  0 ðyÞ vanishes to order N at y0 ; and ðy; sÞ has the properties: (i) the gradient of  satisfies ð@y ðyðsÞ; sÞ; @s ðyðsÞ; sÞÞ ¼ ððsÞ; ðsÞÞ (ii) in a small neighborhood  of  we have Gðy; sÞ ¼ Oðdððy; sÞ; ÞN Þ;

(iii)

and

=ðy; sÞ > Cdððy; sÞ; Þ2

ð12Þ ð13Þ

uniformly on ðy; sÞ 2  \ fjsj < Tg. Proof. In this section and the next we will use the following coordinates Y ¼ ðy1 ; y2 ; y3 ; sÞ and Z ¼ ð1 ; 2 ; 3 ; Þ: With this notation, the condition that (12) holds to third order on ðyðsÞ; sÞ is 4 @G @h X @h @ 2  ¼ þ ¼ 0; @Yi @Yi l¼1 @Zl @Yi @Yl

ð14Þ

4 h X @2G @2h @2h @2 @2h @2 i ¼ þ þ þ @Yi @Yj @Yi @Yj l¼1 @Yi @Zl @Yi @Yl @Yi @Zl @Yj @Yl

ð15Þ

GAUSSIAN BEAM CONSTRUCTION FOR ADIABATIC PERTURBATIONS 4 X 4 h X l¼1 m¼1

193

4 X @2h @2 @2 @h @3 þ ¼ 0; @Zl @Zm @Ym @Yj @Yi @Yl l¼1 @Zl @Yi @Yj @Yl

Remembering the Equations (10)Y(11), (14) is just the compatibility condition ð _ ðsÞ; _ ðsÞÞ ¼ ðð@y ðyðsÞ; sÞÞs ; ð@s ðyðsÞ; sÞÞs Þ:

To study (15), we introduce the matrices MðsÞ; AðsÞ; BðsÞ and CðsÞ defined by ðMðsÞÞij ¼

@2 @2h ðyðsÞ; sÞ; ðAðsÞÞij ¼ ðyðsÞ; s; ðsÞ; ðsÞÞ; @Yi @Yj @Yi @Yj

ðBðsÞÞij ¼

@2h @2h ðyðsÞ; s; ðsÞ; ðsÞÞ; ðCðsÞÞij ¼ ðyðsÞ; s; ðsÞ; ðsÞÞ: @Yi @Zj @Zi @Zj

By observing that

P4

@3  @h l¼1 @Zl @Yi @Yj @Yl

¼

d ds

h

@2  @Yi @Yj

i ðyðsÞ; sÞ ; (15) becomes

MðsÞ _ þ MðsÞCðsÞMðsÞ þ MðsÞBt ðsÞ þ BðsÞMðsÞ þ AðsÞ ¼ 0;

ð16Þ

2 which is a nonlinear Ricatti equation. The initial value Mð0Þ ¼ @y;s ðy0 ; 0Þ is 2 2 determined by the requirement @y ðy0 ; 0Þ ¼ @y 0 ðy0 Þ and conclusion i) of the Theorem which implies Mð0Þy_ð0Þ ¼ _ ð0Þ: Note that this makes =Mð0Þ positive semi-definite with null space spanned by ðy_ð0Þ; 1Þ: Since (16) is a nonlinear ODE, one has no assurance that solutions will exist for all s: However, with the choice of Mð0Þ in the preceding paragraph we will have that here. To see this, we proceed as in [23].

LEMMA 1. Let A; B and C be as above and assume that the initial value Mð0Þ is a symmetric matrix, Mð0Þ ¼ Mð0Þt ; such that =Mð0Þ is positive semi-definite with null space spanned by ðy_ð0Þ; 1Þ: Then (16) has a solution defined for all s: Moreover, MðsÞ ¼ MðsÞt ; and =MðsÞ is positive semi-definite with null space spanned by ðy_ðsÞ; 1Þ: Proof. Consider the system of equations Y_ ¼ BðsÞY þ AðsÞZ Z_ ¼ CðsÞY  Bt ðsÞZ:

Let ðY 1 ðsÞ; Z 1 ðsÞÞ and ðY 2 ðsÞ; Z2 ðsÞÞ be solutions of (17) and set GðsÞ ¼ ððY 1 ðsÞ; Z1 ðsÞÞ; ðY 2 ðsÞ; Z 2 ðsÞÞÞ ¼def Y 2 ðsÞZ 1 ðsÞ  Y 1 ðsÞZ2 ðsÞ:

ð17Þ

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Since AðsÞ ¼ At ðsÞ and CðsÞ ¼ Ct ðsÞ; we have G_ ðsÞ ¼ 0; i ¼ 1; 2:

Hence, the flow defined by (17) preserves the canonical two-form : Now let ðY i ðsÞ; Zi ðsÞÞ be the solution of (17) with initial data ð^ ei ; Mð0Þ^ ei Þ; where e^i ; i ¼ 1; ::; 4; is the standard basis for R4 : We denote by KðsÞ and HðsÞ the 4  4 matrices with columns Y 1 ðsÞ; Y 2 ðsÞ; Y 3 ðsÞ; Y 4 ðsÞ and Z 1 ðsÞ; Z2 ðsÞ; Z 3 ðsÞ; Z 4 ðsÞ

respectively. We claim that KðsÞ is for all s: To see this suppose Pinvertible P4 4 i i a Y ðs Þ ¼ 0; and let wðsÞ ¼ a ðY ðsÞ; Zi ðsÞÞ: Since the coefficient 0 i¼1 i i¼1 i matrices in (17) have real entries, the complex conjugate wðsÞ is also a solution of (17). Since is constant on pairs of solutions of (17), we obtain 0 ¼ ðwðs0 Þ; wðs0 ÞÞ ¼ ðwð0Þ; wð0ÞÞ ¼ 2ih=Mð0Þv0 v0 i;

P where v0 ¼ 4i¼1 ai Y i ð0Þ: Since =Mð0Þ is positive semi-definite with null space spanned by ðy_ð0Þ; 1Þ; it follows that v0 ¼ cðy_ð0Þ; 1Þ for some scalar c: Differentiating (10)Y(11) with respect to s shows that ððy_ðsÞ; 1Þ; ð_ ðsÞ; _ ðsÞÞ isa solution of (17), and uniqueness for the initial value problem implies 4 X

ai Y i ðs0 Þ ¼ ðcy_ðs0 Þ; cÞ:

i¼1

Thus c ¼ 0 which implies v0 ¼ 0 and hence ai ¼ 0; i ¼ 1; ::; 4; since fY i ; i ¼ 1; ::; 4g is the standard basis for R4 : Thus KðsÞ is invertible for all s; and we define MðsÞ ¼ HðsÞKðsÞ1 : We have M_ ¼ N_ Y 1  NY 1 Y_ Y 1 ¼ ðCY  Bt NÞY 1  NY 1 ðBY þ ANÞY 1 ¼ C  Bt M  MB  MAM:

On the other hand, by definition of KðsÞ and HðsÞ; MðsÞY i ðsÞ ¼ Zi ðsÞ: Combining this with the fact that and C are constant in s; we deduce that the matrix MðsÞ has the properties that Mð0Þ has for s ¼ 0: In particular, MðsÞ ¼ MðsÞt and =MðsÞ is positive definite on the orthogonal complement of ðy_ðsÞ; 1Þ: This completes the proof of this lemma. In view of the preceding lemma, if we require @Y ðYðsÞÞ ¼ ððsÞ; ðsÞÞ and 2 @Y ðYðsÞÞ ¼ MðsÞ; (12) holds for N ¼ 3: Next we determine the higher order derivatives of  on  by setting higher order derivatives of G equal to zero on :

GAUSSIAN BEAM CONSTRUCTION FOR ADIABATIC PERTURBATIONS

195

However, the differential equations along  that we obtain for @ ðyðsÞ; sÞ for j j > 2 will be linear. From @y G ¼ 0 on  with j j  3 we obtain @Z h@Y ð@y Þ þ

X

c ; @y  þ d ¼ 0;

j j¼j j

where c ; and d depend on @Y ðyðsÞ; sÞ for jj < j j: Taking into account the Hamiltonian Equations (10)Y(11), we get X d c ; @y ðyðsÞ; sÞ þ d ¼ 0: ½@y ðyðsÞ; sÞ þ ds j j¼j j

ð18Þ

on ðyðsÞ; sÞ: This gives us a linear system for the partial derivatives of a fixed order f@y ðyðsÞ; sÞ; j j ¼ mg: The remaining partial derivatives of  on  are determined by the relations d

½@ ðyðsÞ; sÞ ¼ @y ð@ ÞðyðsÞ; sÞy_ðsÞ þ ð@s @ ÞðyðsÞ; sÞ: ds

We solve the system (18) starting with m ¼ 3; then m ¼ 4 and so on. Since the equations are linear, we have solutions defined for 1 < s < 1: 2 Now, given the formal Taylor series of  on ; and using that =MðsÞ ¼ =@y;s ðyðsÞ; sÞ is positive definite on the orthogonal complement of ðy_ðsÞ; 1Þ; we construct  satisfying all the properties of Theorem 1 from its Taylor series on :

4. Propagation of the Amplitude In the following, we assume that  has been chosen so that (12) holds for given N: We will denote H0 ðkÞ; ðx; kÞ and ðkÞ simply by H0 ;  and ; respectively. We will use the Einstein summation convention (like indices are summed from 1 to 3 ). Recall that m0 ðx; y; sÞ ¼ f0 ðy; sÞðx; kðy; sÞÞ:

ð19Þ

On the other hand, since x ! m1 ðx; ; Þ is required to be -periodic, it is natural to write m1 ðx; y; sÞ ¼ f1 ðy; sÞðx; kðy; sÞÞ þ m? 1 ðx; y; sÞ;

with hð; kðy; sÞÞ; m? 1 ðx; y; sÞi ¼ 0:

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MOUEZ DIMASSI ET AL.

Substituting (19) and the definition of m? 1 into (7) and using (9), we obtain h i c1 ðx; y; sÞ ¼ Km0  Gðy; sÞm1 ðx; y; sÞ  H0 ðkðy; sÞÞ  ðkðy; sÞÞ m? 1:

ð20Þ

Since we have already made Gðy; sÞm1 ðx; y; sÞ vanish to arbitrary order on ; we can omit it, and consider the equation h i 0 ¼ c1 ðx; y; sÞ ¼ Km0  H0 ðkðy; sÞÞ  ðkðy; sÞÞ m? 1:

ð21Þ

By the Fredholm alternative, the (21) has a solution m1 if and only if the term involving m0 in (21) is orthogonal to ker½H0 ðkðy; sÞÞ  ðkðy; sÞÞ :

In view of the definition of K and (H1) this is equivalent to 0 ¼ hKm0 ; i ¼ h;

@m0 @H0 @m0 @   @y  @y kðy; sÞm0 i: @s @k

ð22Þ

Since ðkÞ is simple eigenvalue, we may assume that ð; kÞ is analytic in k: Thus, taking the gradient with respect to k in i @ h @ @H i h 0 ¼  H0   ; @k @k @k

and taking the inner product with  we get @ @H0 i: ¼ h; @k @k

ð23Þ

Here we have used the fact that hð; kÞ; ð; kÞi ¼ 1: Substituting (19) into the right hand side of (22) and using (23), we get the transport equation for f0 0¼

@f0 @ @f0   þ nðy; sÞf0 ; @s @k @y

where n is the function given by nðy; sÞ ¼ h;

@ @H0 @ @  i  @y kðy; sÞ: i  h; @k @y @s

The real and imaginary parts of n were computed in [7, x5]
1 @ @  ðkðy; sÞÞ; 2 @y @k

i @ @ @ @ @ =n ¼  hðH0  Þ Þ i þ ih i; ; ðM ; i  ih; 2 @kj @k j @kj @yj @s

ð24Þ

GAUSSIAN BEAM CONSTRUCTION FOR ADIABATIC PERTURBATIONS

197

where M is the anti-symmetric matrix corresponding to the magnetic field B ¼ ry  Aðy; sÞ: 0

0 @ B3 B2

B3 0 B1

1 B2 B1 A : 0

Introducing n @ @ @ @ @ @ o L ¼ = ðhðH0  Þ ; i; hðH0  Þ ; i; hðH0  Þ ; iÞ @k2 @k3 @k3 @k1 @k1 @k2

one can write =n ¼ LB  ih; _ i

L is an angular momentum which satisfies the usual commutation relations. As in Section 1, we want to choose f0 so that the right hand side of (24) vanishes on  to order N: Differentiating the coefficient of the right hand side of (24), and setting the derivatives equal to zero on ; we obtain linear differential equation for @Y f0 with an inhomogeneous term involving the derivatives of f0 up to order j j  1: @ @ @ ð@Y f0 Þ   ð@ f0 Þ þ nðy; sÞ@Y f0 þ C ð f0 ; @Y f0 ; :::; @Yj j1 f0 Þ ¼ 0; @s @k @y Y

with C ¼ 0 for ¼ 0: We recall that Y ¼ ðy1 ; y2 ; y3 ; sÞ: Along YðsÞ ¼ ðyðsÞ; sÞ the above differential equation takes the form d j j1 ½@ f0 ðYðsÞÞ þ nðYðsÞÞ@Y f0 ðYðsÞÞ þ C ð f0 ; @Y f0 ; :::; @Y f0 ÞðYðsÞÞ ¼ 0; ds Y j j1

Since the coefficient C ð f0 ; @Y f0 ; :::; @Y f0 Þ depends on all the partials up to order j j  1 and this equation is linear, we can solve it recursively for all s: Thus, we may assume that f0 is chosen so that for all M 2 N; hKm0 ; i ¼ OM ðdððy; sÞ; ÞM Þ;

ð25Þ

uniformly on  \ jsj < T: Next we set i h 1 m? ¼ ðH  Þ  hKm ; i ; Km 0 0 1

ð26Þ

where ðH  Þ1 denotes the inverse which maps the orthogonal complement of  in L2 ðEÞ into itself. In view of hypothesis (H1), the right-hand side of (26) is well defined and depends smoothly on ðx; y; sÞ: To complete the construction of m1 , we have to compute f1 : As in the case of the equation c1 ¼ 0; modulo a negligible term the equation c2 ¼ 0 has a solution m2 if and only if hKm1 þ m0 ; i ¼ 0;

ð27Þ

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MOUEZ DIMASSI ET AL.

which yields h; Kð f1 Þi ¼ h; Km? 1 þ y m0 i:

ð28Þ

Clearly, the left-hand side of (28) equals that of (23) with f1 instead of f0 : Thus, (23) yields @f1 @E @f1 1 @ @E    ð  Þ f1 þ ðiLB þ h; _ iÞ f1 ¼ ih; Km?1 þ y m0 i: @s @k @y 2 @y @k

Thus the problem of making hKm1 þ m0 ; i vanish to high order on  is also reduced to solving linear ordinary differential equation along YðsÞ: Hence, we may assume that f1 ðy; sÞ is constructed such that hKm1 þ m0 ; i ¼ Oðdððy; sÞ; ÞM Þ;

ð29Þ

uniformly on ðy; sÞ 2  \ fjsj < Tg: Summing up (20), (21), (22), (25), (26), (27) and (29) we have constructed m0 ; m1 such that c0 ðx; y; sÞ ¼ Gðy; sÞf0 ðy; sÞðx; kðy; sÞÞ ¼ Oðdððy; sÞ; ÞN Þ;

ð30Þ

c1 ðx; y; sÞ ¼ Gðy; sÞm1 ðx; y; sÞ þ hKm0 ; iðx; kðy; sÞÞ ¼ Oðdððy; sÞ; ÞN Þ;

ð31Þ

uniformly on x 2 T and ðy; sÞ 2  \ fjsj < Tg . The situation for the equations cj ðx; y; sÞ ¼ 0; j  2 is quite similar: we set mj ðx; y; sÞ ¼ fj ðy; sÞðx; kðy; sÞÞ þ m? j ;

with

i h 1 m? Kmj1 þ mj2  hKmj1 þ mj2 ; i : j ¼ ðH0  Þ

This shows that m? j only depends on mj1 and mj2 : Next, the function fj is determined from the equation hKmj þ mj1 ; i ¼ 0;

which leads to a linear differential equation for fj with an inhomogeneous term involving mj1 and mj2 :

5. Main Results Before stating the main results of this paper let us specify the form of the initial u in (1). Recall that 0 was an arbitrary smooth function such that =@y2 0 ðy0 Þ

GAUSSIAN BEAM CONSTRUCTION FOR ADIABATIC PERTURBATIONS

199

3 was positive definite. For any f 2 C1 0 ðR Þ we can carry out the preceding construction with

u ðyÞ ¼ f ðyÞe

i0 ðyÞ 

y @0 ðyÞÞ: ð ; Aðy; 0Þ þ @y 

THEOREM 2. Given T > 0 and N 2 N; there exists 0 > 0 and an approximate solution  ðsÞ concentrated near the projection of the null-bicharacteristc of the Hamiltonian hðy; s; ; Þ ¼  ð Aðy; sÞ þ Þ  Vðy; sÞ with initial data ðy0 ; 0; @ 0 ðy0 Þ; 0 Þ such that for all  2 0; 0 ½ k  ðy; 0Þ  u ðyÞk ¼ OðÞ;

ð32Þ

kðDt  P Þ  k ¼ ON ðN Þ;

ð33Þ

uniformly on t 2 ½T; T : Proof. Let  ¼ fðyðsÞ; s; ðsÞ; ðsÞÞ; 1 < s < 1g be the null bicharacteristic of hðy; s; ; Þ with initial data yð0Þ ¼ y0 ; ð0Þ ¼ @y 0 ðy0 Þ; and let ðy; sÞ be the corresponding function given by Theorem 1 with @y ðy0 ; 0Þ ¼ @y 0 ðy0 Þ: The results of Section 3 yield  ðy; sÞ

y @ ðy; sÞÞ þ OðÞ: ¼ f ðy; sÞeiðy;sÞ= ð ; Aðy; sÞ þ  @y

On the other hand, from the estimates (13) and (30) we have k f ð; 0Þei

ð;0Þ 

ðx; Að; 0Þ þ

0 ðÞ @ @0 ðÞÞkL2 ðR3y Þ ð; 0ÞÞ  f ðÞei  ðx; Aðy; 0Þ þ @y @y

¼ OðM Þ;

uniformly on x 2 T: Clearly (32) follows from the two above equalities. In Sections 1, 2 and 3, we have constructed functions ðx; y; s; Þ for each M of the form ðx; y; s; Þ ¼ ei

ðy;sÞ 

i h m0 ðx; y; sÞ þ m1 ðx; y; sÞ þ ::: þ M mM ðx; y; sÞ

so that i h ðDs  PÞ ¼ eiðy;sÞ= c0 ðx; y; sÞ þ ::: þ Mþ2 cM1 þ2 ðx; y; sÞ ;

with cj ðx; y; sÞ ¼ Oðdððy; sÞ; ÞM Þ; =ðy; sÞ  Cðdððy; sÞ; Þ2 Þ;

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MOUEZ DIMASSI ET AL.

uniformly on x 2 T. Consequently, kðDs  PÞ ðx; ; s; Þk2L2 ðR3y Þ ¼

Z

e2=ðy;sÞ= jc0 ðx; y; sÞ þ ::: þ Mþ2 cMþ2 ðx; y; sÞj2 dy ¼ OðMþ3=2 Þ

ð34Þ

uniformly on x 2 T: Here we have used the remarks in the end of Section 1. Set  ðy; sÞ ¼ ðy ; y; s; Þ: Since ðDs  P Þ



h i y ¼ ðDs  PÞ ð ; y; s; Þ; 

Equation (33) follows from (34).

Acknowledgement The authors thank Nicolay Tanushev for suggesting an improvement in the proof of Lemma 1 that we have used here.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Allaire, G. and Piatniski, A.: Homogenization of the Schro¨dinger equation and effective mass theorems, Commun. Math. Phys. 258 (2005), 1 Y 22. Arnaud, J. A.: FHamiltonian theory of beam mode propagation,_ in E. Wolf (ed), Progress in Optics XI, North Holland, 1973, pp. 249 Y 304. Bellissard, J. and Rammal, R.: An algebric semi-classical approach to Bloch electrons in a magnetic field, J. Pysique France 51 (1990), 1803. Buslaev, V. S.: Semi-classical approximation for equations with periodic coefficients, Russ. Math. Surv. 42 (1987), 97 Y 125. Chang, M. C. and Niu, Q.: Berry phase, hyperorbits, and the Hofstadter spectrum, Phys. Rev. Lett. 75 (1996), 1348Y1351. Chang, M. C. and Niu, Q.: Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical in magnetic Bloch bands, Phys. Rev., B 53 (1996), 7010 Y 7022. Dimassi, M., Guillot, J.-C., and Ralston, J.: Semi-classical asymptotics in magnetic Bloch bands, J. Phys. A: Math. G. 35 (2002), 7597 Y 7605. Dimassi, M., Guillot, J.-C., and Ralston, J.: On effective Hamiltonians for adiabatic perturbations of magnetic Schro¨dinger operators, J. Asymptot. Anal. 40 (2004), 137 Y 146. Dimassi, M. and Sjo¨strand, J.: Spectral Asymptotics in the Semi-Classical Limit. London Math. Soc. Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999. Duistermaat, J. J.: Oscillatory integrals, Lagrange immersions and unfolding of singularities. Commun. Pure Appl. Math. 27 (1974), 207 Y 281. Ge´rard, C., Martinez, A., and Sjo¨strand, J.: A mathematical approach to the effective Hamiltonian in perturbed periodic problems, Commun. Math. Phys. 142 (1991), 217 Y 244. Guillot, J. C., Ralston, J., and Trubowitz, E.: Semi-classical methods in solid state physics. Commun. Math. Phys. 116 (1988), 401 Y 415. Helffer, B. and Sjo¨strand, J.: On diamagnetism and the de HaasYvan Alphen effect, Annales I.H.P. (Physique the´orique) 52 (1990), 303 Y 375.

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14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

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Hoˆvermann, F., Spohn, H., and Teufel, S.: Semi-classical limit for the Schro¨dinger equation with a short scale periodic potential, Commun. Math. Phys. 215(3) (2001), 609 Y 629. Ho¨rmander, L.: Fourier integral operator I, Acta Math. 127 (1971), 79 Y 183. Horn, W.: Semi-classical construction in solid state physics, Commun. P.D.E. 16 (1993), 255 Y 290. Keller, J. B.: Corrected BohrYSommerfeld quantum conditions for non-separable systems, Ann. Phys. 4 (1958), 180 Y 188. Kohomoto, M.: Berry’s phase of Bloch electrons in electromagnetic fields, J. Phys. Soc. Jpn. 62 (1993), 659 Y 663. Maslov, V. P. and Fedoriuk, M. V.: Semiclassical Approximation in Quantum Mechanics, D. Reidel, Dordrecht, 1981. Panati, G., Spohn, H., and Teufel, S.: Effective dynamics for Bloch electrons: Peierls substitution and beyond, Commun. Math. Phys. 222 (2003), 547 Y 578. Peierls, R.: Zur Theorie des diamagnetimus von leitungselektronen, Z. Phys. 80 (1933), 763 Y 791. Ralston, J.: Magnetic breakdown, Aste´risque 210 (1992), 263 Y 2282. Ralston, J.: On the construction of quasimodes associated with stable periodic orbits, Commun. Math. Phys. 51 (1976), 219 Y 242. Ralston, J.: Approximate eigenfunctions of the Laplacian, J. Differ. Geom. 12 (1977), 87 Y 100. Simon, B.: Holonomy, the quantum adiabatic theorem, and Berry’s phase, Phys. Rev. Lett. 51 (1983), 2167 Y 2170. Slater, J. C.: Electrons in perturbed periodic lattices, Phys. Rev. 76 (1949), 1592 Y 1600. Sundaram, G. and Niu, Q.: Wave packet dynamics in slowly perturbed crystals: Gradient corrections and Berry phase effects, Phys. Rev. B 59 (1999), 14915 Y 14925.

Math Phys Anal Geom (2006) 9:203–223 DOI 10.1007/s11040-006-9011-2

A Geometrical Interpretation of ‘Supergauge’ Transformations Using D-Differentiation D. J. Hurley · M. A. Vandyck

Received: 15 December 2004 / Accepted: 4 August 2006 / Published online: 2 December 2006 © Springer Science + Business Media B.V. 2006

Abstract D-transport is employed to construct, within the limited setting of a non-graded manifold, a geometrical framework that yields a generalisation of the ‘supergauge’ transformations of Supergravity. Killing’s equation is shown to be at the origin of the ‘gauged’ supersymmetry transformations. The presence of a fielddependent Lorentz transformation is traced to the fact that, for every given X, the difference of two D-differentiation operators 1D X and 2D X is a linear transformation that necessarily depends on X. Key words supergravity · gauge transformation · D-differentiation Mathematics Subject Classifications (2000) 83E50 · 58C20 · 58Z05

1 Introduction In a previous article [5], we introduced the concept of D-differentiation, which we employed [6] to provide new geometrical insight into the problem of the semiclassical movement of electrons in crystals. Later, in a monograph [7], we further applied D-differentiation to the classical dynamics of rigid bodies and to geometrical optics in General Relativity, thereby providing an alternative geometrical setting for these fields. More recent investigations have now led us to the conclusion that certain aspects of Supergravity also benefit from being recast into the language of D-differentiation. In order to see this, let us return to the foundations of Supergravity.

D. J. Hurley (B) Department of Mathematics, National University of Ireland, Cork, Ireland e-mail: [email protected] M. A. Vandyck Department of Physics, National University of Ireland, Cork, Ireland

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At present, two very different approaches to the field equations of Supergravity are widely known: one based on an analogy with General Relativity [10–12], and the other exploiting torsion constraints [4, 16–18] on a suitable graded manifold. (Comprehensive references to the literature may be found, e.g., in [2, 14, 15].) Whatever approach is preferred, one always encounters the so-called ‘supergauge transformations’ which express [19, 20] the fundamental invariances of the theory under, for instance, general coordinate transformations or supersymmetry transformations. It is, therefore, of great importance to acquire a deeper understanding of them by studying them from different points of view. The most common presentation of supergauge transformations consists in relating them [19] to gauge theories. Consequently, owing to the fact that a gauge theory involves a connection form defined on a principal fibre bundle P over a manifold M, the central role [9] in that approach is played by covariant differentiation on a bundle E associated with P. In this work, we are going to consider the problem from a more general perspective, which will be based on D-differentiation. This operation [5–7] reduces to covariant differentiation ∇ in a special case (to be specified later), but does not possess all the properties of ∇. For instance, the crucial difference between D and ∇ is that, in general, D f X = f D X ,

(1.1)

for all functions f and all vector fields X defined on the manifold M. The property (1.1) is reminiscent of Lie differentiation, which, indeed, is also contained in the class of D-differentiation operators. The observation (1.1) that an arbitrary D X fails to be linear (with respect to functions) in its differentiating slot X has the consequence that the torsion and the curvature constructed from D are, in general, non-linear operators, in contrast with what happens for covariant differentiation. It may thus seem unlikely that there would be any relationship between general D-differentiation and Supergravity, which possesses a linear torsion and curvature. Surprisingly, such a connection exists. Hereafter, we are going to construct a framework based on D-differentiation of a very general kind, which will lead to transformations identical in form to the supergauge transformations of Supergravity. For the two sets of transformations to be also identical in content, it will suffice to restrict the operator D to metric-compatible covariant differentiation. In full generality, the proposed framework will have a clear geometrical interpretation. Moreover, it will enable us to relate some of the observations frequently made about supergauge transformations in Supergravity to the mathematical structure of D-differentiation. This will provide an explanation for these observations and, at the same time, prove that they have a more general validity than what is necessary for Supergravity. It is important to emphasise that, in principle, a complete discussion of Supergravity requires the use of graded manifolds, also called ‘supermanifolds.’(Numerous references to the literature are available, e.g., in [1] or [3].) However, in order to avoid the complications due to the fact that several inequivalent definitions of such manifolds exist, we shall adopt the more modest setting [8] of an ordinary (C ∞ ) manifold modelled on Rn .

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This will compel us to compare our results, not directly with those of Supergravity itself, but with those that would hold in Supergravity if the underlying manifold were non-graded. We impose this limitation in scope solely for the sake of mathematical clarity. Moreover, just as covariant differentiation and Lie differentiation can be generalised to graded manifolds [1, 3], so can D-differentiation. (This statement is not immediately obvious, but a simple argument will be provided in Appendix 3, and a full investigation will be pursued elsewhere.) At a later stage, it should therefore be possible to extend the treatment presented hereafter to the situation of a graded manifold. Beginning with the next section, we shall present a brief summary of D-differentiation. Then, in Section 3, we shall introduce compound differentiation, which results from the combination of two D-differentiation operators. The particular case where one of them is Lie differentiation will be given special attention, with a view to relating it to supergauge transformations later. Section 4 will be concerned with the transport, induced by compound differentiation, of quantities along integral curves of vector fields. We shall first transport points in the bundle PL∗M of cotangent bases in M. The expression obtained for the so-called ‘compound transport’ of these bases will be interpretable as the product of two infinitesimally commuting transformations, one of which will be determined by Lie differentiation. Furthermore, if the transport is employed in the reduced bundle PO∗M of (oriented) orthonormal dual bases, a generalised Killing equation will have to be satisfied by the vector field along the direction of differentiation. (This equation will be seen as the geometrical origin of gauged supersymmetry transformations, for instance.) In Section 5, we shall investigate the compound transport of D-differentiation matrices. The transport will involve two stages: Lie differentiation will be applied first, and then the result will be modified by a transformation generated by the operator D involved in the compound differentiation. For compound transport to be self-consistent, it will be necessary to restrict D to a particular class, namely that for which the curvature operator is tensorial. This class will be shown to be wide enough to include the operators commonly encountered in Physics. In Section 7, we shall combine compound differentiation with a certain kind of linear transformation, as required for supergauge transformations later. The corresponding expressions for the compound transport will also be obtained. Finally, in Section 8, we shall relate our results to Supergravity, within the abovementioned limitation of the non-graded manifold employed here. We shall see how the well-known supergauge transformations follow from a special case of our geometrical framework, and it will be clear that the construction is valid in a broader context. 2 Summary of D-Differentiation Given that D-differentiation, albeit relatively new, is now well described in the literature [5–7], it is unnecessary to repeat the details of its construction. However, a brief outline is useful, since it will enable us to fix the notation that will be employed hereafter.

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Let M be a C∞ manifold of dimension n. Let also the set of smooth functions and vector fields defined on M be denoted by F and X , respectively. A field of local bases of X will be written as {e(i) : 1  i  n}, and {e(i) : 1  i  n} will be its dual basis. The Einstein summation convention will systematically apply. For every X and Y in X , the expression for D X Y in the basis {e(i) }, reads D X Y = {X(Y i ) + i j(X) Y j}e(i) ,

(2.1)

with i a j b

i j(X) ≡ λi jk X k − A

e(a) (X b ).

(2.2)

i a

In Eq. 2.2, A j b denotes the components of a tensor A, which expresses [5, 7] the fact that, in general, D f X Z fails to agree with f D X Z , as mentioned in Eq. 1.1. Together with the symbols λi jk , the tensor A provides a complete description [5] of any particular D. Important examples are Lie and covariant differentiation. For instance, covariant differentiation is uniquely characterised by A = 0, and λi jk reduces then to the connection coefficients. Note that, for Eq. 2.1 to be independent of the choice of the basis, the symbols λi jk must satisfy a very special transformation rule under a change of basis [5, 7], which generalises the well-known transformation law of connection coefficients. A formula similar to Eq. 2.1 can be developed [7] when Y is replaced by a oneform α of the cotangent space to M. Also, if f is a function of F , one assumes D X f = X( f ) = df (X).

(2.3)

Furthermore, the curvature R is expressible in terms of λi jk and A as displayed in [5, 7]. An equivalent form, which is elementary to establish from [5, 7], will be more convenient for our purposes: R(X, Y, Z ) ≡ (D X DY − DY D X − D[X,Y] ) Z ≡ i j(X, Y) Z j e(i) i a j b (X)

i j(X, Y) = P i jk (X) Y k − Q

e(a)(Y b ),

(2.4) (2.5)

where i n

P i jk (X) ≡ Ri jmk X m − S jk i a

i

a

Q j b (X) ≡ −S jm

b

e(n)(X m )

m

Xm − W

i

na jmb

(2.6)

e(n)(X m )

(2.7)

Ri jab ≡ e(a)(λi jb ) − e(b )(λi ja ) + λi ra λr jb − λi rb λr ja − ir s j s e(r)(D ab )

−λi jr Dr ab + A i u ja b

≡ A

uv

≡ A

S W

i jab

ir jb

Du ra + Airub λr ja − A

i u j r

i u j b

δ v a + Airua A

r v j b

(2.8)

i v u j aδ b

−A

r u j b

Dr b a − A

i u j b)

λi ra − e(a)(A

− Airvb A

r u j a.

(2.9) (2.10)

The construction of D-differentiation does not require the presence of any metric g on the manifold M. However, if a metric happens to be available, its Dderivative is readily evaluated [5, 7]. An operator D with the property that D X g vanishes for all X is called ‘metric-compatible’.

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207

In the context of Supergravity, orthonormal bases, denoted here by a caret over the corresponding indices, take a prominent place. It is easy to prove from [5, 7] that, in such a basis, metric-compatibility is equivalent to the requirement (ˆıjˆ) (X) = 0 for all X. ˆı aˆ , jˆ bˆ

In terms of λˆı jˆkˆ and A

(2.11)

the constraint (2.11) reads λ(ˆıjˆ)kˆ = A(ˆıjˆ) aˆ bˆ = 0.

(2.12)

Finally, a simple calculation, exploiting Eqs. 2.5, 2.6, 2.7, 2.8, 2.9, and 2.10, shows that the curvature matrix of a metric-compatible operator D satisfies (ˆıjˆ) = 0.

(2.13)

This information, which is not directly available in our previous publications, will be of great importance in Section 5.

3 Compound Differentiation When an operator of D-differentiation has been selected on the manifold M, it becomes possible to ‘D-transport’ a vector field Y along any integral curve of a vector field X as follows. If t  → Ft (P) denotes the integral curve of X that passes through the point P at t = 0, one defines the D-transport D YFt (P) of Y at the point Ft (P) as being the vector D

YFt (P) ≡ Y Ft (P) − t D X Y + o(t2 ).

(3.1)

At the first order in t, it is not necessary to specify, in Eq. 3.1, at which point along the curve one must evaluate D X Y. In Supergravity, one needs an additional differential operator, δ X , which we shall call ‘compound differentiation.’ More precisely, if 1D X and 2D X denote two operations of D-differentiation on the manifold M, let δ X be defined by δ X ≡ 1D X + k (2D X ),

(3.2)

in which k is a constant to be specified hereafter. In general, δ X is not a D-operator, because δ does not satisfy Eq. 2.3, unless k vanishes. Furthermore, if k takes the value −1, then δ X is a linear transformation, for every X fixed. The special case k = −1 of Eq. 3.2 will come to prominence in Section 5, where we shall employ some of its properties, which we shall now investigate. Let the transformation  be defined by −(X, Y) ≡ (1D X − 2D X )(Y).

(3.3)

Owing to the fact that  is linear in Y (but not necessarily in X), one may write (X, •) ≡ i j(X) e(i) ⊗ e( j) .

(3.4)

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D. J. Hurley and M. A. Vandyck

Moreover, one easily proves, by introducing Eqs. 2.1 and 2.2 in Eqs. 3.3 and 3.4, that i j(X) reads i j(X) = P ijk X k − Qijab e(a)(X b ),

(3.5)

where P ijk and Qijab are related to the characteristics λi jk and A of 1D and 2D by λ

2 i

jk

= 1λi jk + P ijk

,

2

i a j b

A

i a j b

= 1A

+ Qijab .

(3.6)

This observation shows that the linear transformation defined by Eq. 3.3 is very special because of its dependence on X, in the manner specified by Eqs. 3.4 and 3.5. In other words, if one wished to reverse the process and define an operator 2D of D-differentiation from 1D by combining it with a linear transformation as in Eq. 3.3, this would require that  be of the form that we just determined. A simple example of an  that does not allow such a construction is one where (X, Y) would be assumed independent of X, because Eq. 3.5 would constrain P and Q to vanish. In the context of Supergravity, one says that  is necessarily a field-dependent linear transformation. In order to make closer contact with Supergravity, let us rewrite Eq. 3.2, for an arbitrary k, in terms of the torsion 2 T of the operator 2D. One obtains δ X Y = (1D X Y + k[X, Y]) + k {2DY X + 2 T(X, Y)}.

(3.7)

If, in Eq. 3.7, the operator 1D is chosen as −kL, the simplified form of Eq. 3.7 reads δ X Y = k {2DY X + 2 T(X, Y)}.

(3.8)

In particular, if Y happens to be a basic vector field e(•) on M, or a dual basic vector e(•) , one has δ X e(•) = k {2De(•) X + 2 T(X, e(•) )} ,

δ X e(•) = −k {e(•) [2De(i) X + 2 T(X, e(i) )]} e(i) , (3.9)

where, for simplicity, we have omitted the index specifying the basic vector under consideration. At this stage, it is important to realise that k must be equal to −1, otherwise the operator 1D cannot satisfy Eq. 2.3. To proceed further, one now observes that the right-hand side of Eq. 3.9, for k = −1, coincides with what Supergravity theories call [19, 20] a ‘supergauge transformation’ of the basis {e(•) }, provided 2D is selected as being (super)-covariant differentiation. Therefore, we shall henceforth identify 1D with Lie differentiation. However, we shall refrain, for the moment, from restricting 2 D to covariant differentiation, because, as it stands, the statement (3.9) is stronger than its Supergravity analogue. Indeed, as mentioned earlier, the torsion is a nonlinear operator for a general D-operator, although it becomes linear for covariant differentiation. It is thus instructive to investigate how far it is possible to go, in the non-linear case. Note that, given that the operator 1D is now taken as being Lie differentiation, we shall henceforth omit the upper-left index 2 in all expressions of the kind 2D, 2 T(X, Y) etc.

Geometrical Interpretation of ‘Supergauge’ Transformations Using D-Differentiation

209

4 Compound Transport of Vectors and Forms An operator δ X of compound differentiation enables one to define the ‘compound transport’ T Y (or Tα) of a vector field Y (or a one-form α) along the integral curves of a vector field X. By analogy with Eq. 3.1, which applies to D-transport, one writes then (at the first order in t): αFt (P) ≡ α Ft (P) − tδ X α

T

(4.1)

= α Ft (P) − t(L X α − D X α).

(4.2)

To gain a more geometrical understanding of Eq. 4.2, and, at the same time, to prepare the investigation of the compound transport of a matrix • • (X) of D-differentiation in Section 5, it is convenient to introduce the principal fibre bundle PL∗ M of cotangent bases above M. The fibre of PL∗ M above a point P of M is constituted by all the possible bases of T P∗ M. More precisely, let {e(•) } denote again a field of dual bases over M. Select a (local) cross-section σ of PL∗ M. This means that, with every point P of an open subset (•) of M, one associates a particular point e P of the fibre above P, namely a particular (•) (•) basis of T P∗ M. (We have underlined the index in e P to distinguish e P from the field (•) (•) (•) of bases {e } evaluated at P.) Because e P and {e P } are bases of the same tangent space T P∗ M, one may write (•)

e P ≡ E• • e(•) P ,

(4.3) (m)

where the scalars Em i represent the components of e P in the basis {e(i) P }, and form the regular matrix E• • . It is thus clear, by virtue of Eqs. 4.1, 4.2, and 4.3, that compound transport takes place in PL∗ M as T (•) e Ft (P)

(•)

≡ e Ft (P) − tδ X e(•) ,

(4.4)

with δ X e(•) = {[X(E• • ) − E• • L• • (X)] − [X(E• • ) − E• • • • (X)]} e(•) = {[X(E• • ) − E• • L• • (X)] − [X(E• • ) − E• • • • (X)]} E• • e(•) .

(4.5) (4.6)

In Eq. 4.6, the symbol E• • denotes the inverse matrix of E• • , and L• • (X) stands for the matrix L • • (X) describing Lie differentiation in the language of Ddifferentiation. The explicit form of L• • (X) reads [5, 7]

L• • (X) = −D• •k X k − e(•) (X • ),

(4.7)

where D• •• denotes the commutation coefficients [5, 7] of the basis {e(•) }. Similarly, in Eq. 4.6, the matrices L• • (X) and • • (X) are expressed in the basis {e(•) }. It is enlightening to reformulate them in the basis e(•) , as we shall see. To do this, one must recall the transformation law [5, 7] of a D-differentiation matrix  in the cases at hand:

L• • (X) = E• • L• • (X) E• • − X(E• • ) E• • •

 • (X) = E

•

•

•

 • (X)E

•

•

•

•

− X(E • ) E • .

(4.8) (4.9)

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D. J. Hurley and M. A. Vandyck

After an elementary treatment, Eq. 4.6 simplifies then as δ X e(•) = [−L• • (X) + • • (X)] e(•) ,

(4.10)

from which it follows that the prescription (4.4) for the compound transport in PL∗ M becomes, at the first order in t, T (•) e Ft (P)

(•)

(•)

= e Ft (P) − t[−L• • (X) + • • (X)] e Ft (P)

(4.11)

(•)

= {I + t[L• • (X) − • • (X)]} e Ft (P)

(4.12)

(•)

= [I − t• • (X)] [I + tL• • (X)] e Ft (P) . •

(4.13) •

At the first order in t, the matrices [I − t • (X)] and [I + tL • (X)] commute in Eq. 4.13. The result (4.13) provides an intuitive interpretation of the compound transport (•) (•) in PL∗ M. It means that the transported basis Te Ft (P) is obtained from the basis e Ft (P) by the successive application (in either order) of the two linear transformations [I − t• • (X)] and [I + tL• • (X)]. For future reference, let us observe that it is permissible to perform, in Eq. 4.6, exclusively the substitution (4.9), and ignore Eq. 4.8. The corresponding expression for δ X e(•) , which is equivalent to Eq. 4.10, reads δ X e(•) = {[X(E• • ) − E• • L• • (X)] + • • (X) E• • } e(•) ,

(4.14)

which will be analysed in detail in Section 8, when we shall compare our results with the methods of Supergravity. So far, we have considered the transport (4.11), (4.12) and (4.13) in the principal bundle PL∗ M of dual bases above M. Let us now investigate whether Eqs. 4.11, 4.12, and 4.13 may be used in the reduced principal bundle PO∗ M of (oriented) orthonormal dual bases above M. The structure group of PO∗ M is SO( p, q), the group of orthogonal matrices with unit determinant, which preserve the metric of signature p − q. In PO∗ M, the candidate-transport would be T (ˆ•) e Ft (P)

(ˆ•)

= {I + t[L•ˆ •ˆ (X) − •ˆ •ˆ (X)]} e Ft (P) ,

(4.15)

where the carets emphasise that orthonormal bases are understood. Moreover, the analogue of Eq. 4.14 would read δ X e(ˆ•) = {[X(E•ˆ • ) − E•ˆ • L• • (X)] + •ˆ •ˆ (X) E•ˆ • } e(•) .

(4.16)

It is clear that Eq. 4.15 is self-consistent if and only if the transformation leading (ˆ•) (ˆ•) from e Ft (P) to Te Ft (P) belongs to SO( p, q) at the first order in t. Equivalently, the criterion of self-consistency is that the matrix L•ˆ •ˆ (X) − •ˆ •ˆ (X) be antisymmetric. Such will be the case if and only if

L X g − D X g = 0.

(4.17)

The Eq. 4.17 will be called the ‘generalised Killing equation’, and it is closely related to supersymmetry transformations in Supergravity, as we shall show in Section 8. At present, we already see that, if one attempts to enforce the transport (4.15) in the bundle PO∗ M, it is necessary and sufficient to restrict attention to those metrics g and those directions X of transport related together by Eq. 4.17. Moreover, if D is metric-compatible, Eq. 4.17 reduces to the Killing equation.

Geometrical Interpretation of ‘Supergauge’ Transformations Using D-Differentiation

211

5 Compound Transport of D-Differentiation Matrices The operations of compound differentiation and transport apply to vectors and one-forms, and may be extended to tensors. However, it does not seem obvious how to give a meaning to the concept of the transport of a matrix • • (X) of D-differentiation, because such a matrix is not made up of tensor components. An identical difficulty would arise, for instance, if one tried to parallel-transport a connection or to differentiate it covariantly. On the other hand, the following remarks give an indication on how to proceed: 1. In contrast with the object D X Y, a matrix • • (X) is basis-dependent. Indeed, • • (X) transforms in a very special manner [5, 7] under a change of basis. One should thus expect the law of compound transport of such a matrix to be related to the law of compound transport of the basis, namely Eq. 4.13. 2. Lie differentiation is always uniquely defined in any smooth manifold M, whereas, in general, there may not be any other operator of D-differentiation on M. Moreover, even if another such operator does exist, it need not be unique. (For instance, many connections may be put on M, and each gives rise to its own operation of covariant differentiation.) In that sense, Lie differentiation is the most fundamental of D-operators. In the same spirit, one may re-interpret the (•) law (4.13) of transport of the basis by saying that, from e Ft (P) , one first constructs (•)

the intermediate basis Ie Ft (P) as I (•) e Ft (P)

(•)

≡ [I + tL• • (X)] e Ft (P) ,

(5.1) (•)

which is (uniquely) generated by Lie differentiation. Then, Ie Ft (P) is further (•) modified to obtain Te Ft (P) as T (•) e Ft (P)

(•)

≡ [I − t• • (X)] Ie Ft (P) ,

(5.2)

which depends on the particular (i.e. non-unique) choice made for • • (X). In keeping with the two-step procedure that we have just sketched, let us begin by defining the Lie transport of a matrix of D-differentiation. To avoid confusion with the matrix • • (X), we shall denote the matrix to be transported by • • (X). Let X and Y be two vector fields, and let α be a one-form. An elementary calculation based on the properties of Lie differentiation yields the identity (L X α)(Y) = X{α(Y)} − α(L X Y).

(5.3)

This equation enables one to define the Lie derivative of • • as (L X • • )(Y) ≡ X{ • • (Y)} − • • (L X Y).

(5.4)

The right-hand side of Eq. 5.4 is meaningful because the vector field X can act on the matrix-valued function • • (Y) just as well as it could act, in Eq. 5.3, on the scalarvalued function α(Y). The question of the basis-dependence of this definition will be addressed later.

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With an expression for L X • • at one’s disposal, it becomes possible to define the  •• of • • along an integral curve of X as Lie transport

 ••F (P) ≡ • •Ft (P) − tL X • • .

t

(5.5)

Our first task is thus completed. • The second step consists in obtaining the compound transport T • of • • by altering the Lie transport (5.5). To this end, we put

T

•

•Ft (P)

≡ • •Ft (P) − tδ X • • ,

(5.6)

in which the compound derivative δ X • • is the following modification of the Lie derivative (δ X • • )(Y) ≡ (L X • • )(Y) + • • ( • • , X, Y) •

•

•

(5.7) •

= X{ • (Y)} − • (L X Y) +  • ( • , X, Y),

(5.8)

by virtue of Eq. 5.4. Of course, the definition (5.8) is incomplete, so long as  has not been specified. In order to find an expression for , let us return to Eqs. 5.1 and 5.2. The former, being the intermediate stage in the transport, is the analogue of Eq. 5.5, whereas the latter, being the final stage, is the analogue of Eq. 5.6. According to Eq. 5.2, the final stage follows from the intermediate one after multiplication by the matrix [I − t• • (X)]. It is therefore reasonable to construct the quantity • • ( • • , X, Y) of Eq. 5.7 from that matrix. The candidate for • • ( • • , X, Y) is obvious, because, to construct a matrix from • • , one may apply to • • (Y) the transformation law [5, 7] for a matrix of D-differentiation under the change of basis e



(•)

= [I − t• • (X)] e(•) .

(5.9)

At the first order in t, the transformation law implies



• • (Y) = • • (Y) + t{Y[• • (X)] + • • (Y) • • (X) − • • (X) • • (Y)}.

(5.10)

For this reason, we define  by • • ( • • , X, Y) ≡ c {Y[• • (X)] + • • (Y) • • (X) − • • (X) • • (Y)},

(5.11)

where c denotes a constant to be fixed later. We have now completed the second, and final, step required to define the transport of • • , and this is as far as one can go for the transport of an arbitrary • • . In Supergravity, however, one only needs to transport • • itself. In this special case, Eqs. 5.6, 5.8, and 5.11 become •



T

•

•Ft (P)

≡ • •Ft (P) − tδ X • • •

•

(5.12)

(δ X  • )(Y) = X{ • (Y)} −  • (L X Y) + c U,

(5.13)

U ≡ Y[• • (X)] + • • (Y) • • (X) − • • (X) • • (Y).

(5.14)

with

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213

After a lengthy calculation, summarised in the Appendix 1, the result (5.13) simplifies as (δ X • • )(Y) = • • (X, Y) + (c + 1) U,

(5.15)

in which • • (X, Y) stands for the curvature matrix. It is important to realise that the statements (5.12), (5.13), (5.14), and (5.15) refer to the basis e(•) employed in Section 4, as emphasised by the underlined indices. The question thus arises of the dependence of Eqs. 5.12, 5.13, 5.14, and 5.15 on the choice of the basis. In other words, Eq. 5.12 is a prescription to construct a • • matrix T • (Y) from the original • • (Y). However, it is not obvious that T • (Y) • satisfies the transformation law that must hold if T • (Y) is to be a legitimate matrix of D-differentiation. This matter, which is purely technical, is studied in Appendix 2. The result is • that T • (Y) transforms correctly when c takes the value −1 in Eq. 5.15. With all the indices omitted, the transport of a D-differentiation matrix along the integral curves of a vector field X, and its compound derivative thus read (T)(Y) = (Y) − t(X, Y)

(5.16)

(δ X )(Y) = (X, Y).

(5.17)

Caution, however, must be exercised when one interprets Eq. 5.16, because the fact that a -matrix transforms correctly under a change of basis is no guarantee that  is of the form that we have always considered, namely Eq. 2.2. One needs to verify that the transformed -matrix is the matrix of a D-operator. To clarify this issue, we rewrite Eq. 5.16 as i



T

j Ft (P) (Y)

= i j Ft (P) (Y) − ti j(X, Y)

(5.18)

= i j Ft (P) (Y) + i j(−tX, Y),

(5.19)

which is reminiscent of the relationship between the -matrices of two D-operators differing from each other by a linear transformation with matrix i j(Y):  j(Y) = 1i j(Y) + i j(Y).

2 i

(5.20)

(See Eq. 3.6.) Let us now compare the explicit expressions (2.5) and (3.5) for i j(−tX, Y) and i j(Y): i a j b (−tX)

i j(−tX, Y) = P i jk (−tX) Y k − Q i a j b

i j(Y) = P ji k Y k − Q

e(a)(Y b )

(5.21)

e(a)(Y b ).

(5.22)

i a j b

have to transform like ten-

Recall that, by virtue of Eq. 3.6, the quantities Q

i a

sor components. Consequently, the same has be true of Q j b (−tX) for all t in a neighborhood of zero. As a result of Eq. 2.7, all the D-operators characterised by i

S

a jm b

=0=W

i

na jm b

i a j b (−tX)

are (trivially) of this kind, because their Q

vanishes for all t.

(5.23)

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D. J. Hurley and M. A. Vandyck

It then follows from Eq. 3.6 that the quantities Pi jk (and thus P i jk (−tX)) have to behave like tensor components. By virtue of Eqs. 2.5 and 2.6, this condition is satisfied by all the operators of the class (5.23). The conclusion of this investigation is that, for the D-operators of the class (5.23), the compound transport (5.16) of a -matrix induces a mapping in the space of D-operators. In terms of the characteristics of such operators, the mapping is given by λ

T i

jk

T i a Ajb

= λi jk + P i jk (−tX) = λi jk − t Ri j m k X m i a j b.

= A

(5.24) (5.25)

The class (5.23) seems sufficiently large for many applications, because it contains covariant differentiation, Lie differentiation, and other operators employed in Physics [7]. Finally, let us recall that, in many contexts, one imposes a constraint on λi jk i a

and A j b , the typical example being metric-compatibility. It is then necessary to check that the transport (5.24), (5.25) preserves the chosen constraint. The case of metric-compatibility is most simply studied in an orthonormal frame. By virtue of Eq. 2.12, the constraint is preserved iff λ(ˆıjˆ)kˆ = TA(ˆıjˆ)

T

aˆ bˆ

= 0.

(5.26)

As a result of the property (2.13) of antisymmetry of the curvature matrix ˆıjˆ , it is clear that Eq. 5.26 holds, which shows that metric-compatibility is preserved by the transport (5.24), (5.25).

6 Analysis Recall that, apart from taking 1D = L and k = −1 in Eq. 3.2, we have imposed no constraint other than Eq. 5.23 on the operator D underpinning our whole construction. Much insight, however, can be gained from investigating special cases. For instance, let us assume (temporarily) that the operator D is Lie differentiation, which is equivalent to putting, for all X, • • (X) = L• • (X).

(6.1)

Thus, it follows from Eqs. 4.4 and 4.12 that δ X e(•) = 0

,

T (•) e Ft (P)

(•)

= e Ft (P) .

(6.2)

for all X and all t. The first equality of Eq. 6.2 is algebraically obvious. Indeed, if one returns to Eq. 3.2 with k = −1, one sees that δ X vanishes when 1D X = 2D X = L X . From the geometrical point of view, Eqs. 5.1 and 5.2 show that the intermediate (•) (•) basis Ie between e(•) and Te is obtained from the original one by Lie transport. Then, one retraces one’s steps when applying Eq. 5.2 to generate the transported (•) basis Te , because, in the special case (6.1), the matrices [I − t• • (X)] and [I + • tL • (X)] are the inverses of each other (at the first order in t). In other words, the

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215

second equality of Eq. 6.2 states no more than the fact that, if one goes forwards by Lie transport, and backwards by Lie transport as well, the compound motion is equivalent to staying at rest. As a result of the above geometrical reasoning, one understands that the special case under consideration leads to trivial compound transport. One realizes then that it is important, for self-consistency, that the matrix  be also transported trivially in this case. Owing to the fact that the curvature of Lie differentiation vanishes, Eqs. 5.16 and 5.17 imply δX  = 0

,

 = ,

T

(6.3)

as anticipated. Note that Eq. 6.3 would not have held if, in Eq. 5.15, the value −1 had not been selected for the constant c. Let us emphasise that it would not have been possible to study the enlightening special case of Lie differentiation, had we restricted attention, in Section 3, to the operator D being, for instance, covariant differentiation. It is thus genuinely useful to have at one’s disposal an operation, such as D-differentiation, which contains, as special cases, Lie differentiation, covariant differentiation, and other operators. If one wishes to analyse the framework beyond the case of identically-vanishing curvature, one may notice that, by virtue of Eq. 2.5, all the operators of D-differentiation of the kind Eq. 5.23 have a curvature operator that happens to be a tensor. It does not follow, however, that their torsion is always tensorial. A counter-example is Lie differentiation, which does belong to the class (5.23), but which possesses a non-tensorial torsion. An important special case is here covariant differentiation. Then, the coefficients λi jk are the usual connection coefficients, and their transport according to Eq. 5.24 is an illustration of the classical theorem [8, 13] guaranteeing that the difference between the coefficients of two connections is a tensor.

7 Generalised Derivative and Transport We saw, in Section 3, how it is possible to obtain an operator 2D of D-differentiation from an operator 1D by combination with a linear transformation . The same considerations enable one to introduce a linear transformation in the compound derivative and the compound transport. The corresponding treatment is quite simple, from the mathematical point of view, but the result plays such an important role in Supergravity that we are going to elaborate a little; details will, however, be left to the reader. Let us return to Eq. 3.2, and put D X ≡ D X + (X, •).

2

(7.1)

Then, Eq. 3.2 implies δ X Y = 1D X Y + k {D X Y + (X, Y)} = ( D X Y + k[X, Y]) + k {DY X + T(X, Y) + (X, Y)}. 1

(7.2) (7.3)

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D. J. Hurley and M. A. Vandyck

If, in Eq. 7.3, the operator 1D is again chosen as Lie differentiation, and the constant k takes the value −1, one finds δ X Y = −{DY X + T(X, Y) + (X, Y)},

(7.4)

which generalizes Eq. 3.8 to incorporate the transformation , linear in Y. Moreover, by virtue of Eq. 7.2, the generalisation (7.4) follows from the original expression (3.8) after the replacement i j(X) −→ i j(X) + i j(X),

(7.5)

which is allowed as a consequence of Eq. 3.6. That replacement yields at once the compound derivative and the transport in the bundle PL∗ M, in whichever of the equivalent forms we may prefer. For our purposes, the most convenient ones will be δ X e(•) = {[X(E• • ) − E• • L• • (X)] + • • (X) E• • + • • (X) E• • } e(•) T (•) e Ft (P)

(7.6)

= {e(•) [De(i) X + T(X, e(i) )] + • • (X) E• i } e(i)

(7.7)

(•) • • (X)]} e Ft (P) ,

(7.8)

= {I + t[L• • (X) − • • (X) −

which should be compared with Eqs. 4.14, 3.9 and 4.12. Similarly, in the bundle PO∗ M, the transport reads T (ˆ•) e Ft (P)

(ˆ•)

= {I + t[L•ˆ •ˆ (X) − •ˆ •ˆ (X) − •ˆ •ˆ (X)]} e Ft (P) (ˆ•)

= {I − t•ˆ •ˆ (X)} {I + t[L•ˆ •ˆ (X) − •ˆ •ˆ (X)]} e Ft (P) .

(7.9) (7.10)

Given that Eq. 7.10 refers to the bundle of orthonormal frames, it is natural to require that the matrix {I − t•ˆ •ˆ (X)} belong to the group SO( p, q), at the first order in t. Therefore, we shall assume that •ˆ •ˆ (X) is antisymmetric. Consequently, the criterion (4.17) of self-consistency of the transport remains valid:

LX g − DX g = 0 ,

when (ˆ• •ˆ ) (X) = 0.

(7.11)

The generalisation of the transport of a -matrix proceeds along analogous lines. One begins by performing the replacement (7.5) in Eq. 5.11, so that Eq. 5.15 becomes (δ X • • )(Y) = • • (X, Y) + (c + 1) U + c V • • (X, Y) •

•

•

•

V • (X, Y) ≡ Y{ • (X)} + [ • (Y) ,  • (X)],

(7.12) (7.13)

where the square brackets denote the commutator of matrices. Then, for c = −1, the transport may be written T •

λ

•k

T • a A•b

= λ• • k − t {R• • m k X m − e(k) {• • (X)} − [λ• • k , • • (X)]} =

• a A•b

+

• a t [A • b

, • • (X)].

(7.14) (7.15)

Note that the new terms in Eqs. 7.14 and 7.15 are consistent with the constraint of metric-compatibility, if one wishes to impose it. The complete framework is now at our disposal, and the time has come to compare the present construction with that employed in Supergravity. This will be done in the next section.

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217

8 Comparison with Supergravity Let us begin by re-emphasising that a complete discussion of Supergravity requires graded manifolds, in which scalar and tensor fields have a parity (even, odd, or mixed). In our present setting of ordinary manifolds, however, the concept of parity is not defined. Therefore, a comparison between our results and those of Supergravity amounts to one between our results and those that would hold in Supergravity if all the objects involved were even. Supergravity considers, at each point P of the manifold M, a set of special bases analogous to the fibre of PO∗ M above P, introduced in Section 4. The field of bases {e(•) } in the cotangent spaces to M is usually taken as holonomic: i e(i) P = dx P ,

(8.1)

where xiP is the i-th coordinate of P in a certain chart. Moreover, the matrix E•ˆ • of the decomposition of e(ˆ•) in e(•) , as displayed in Eq. 4.3, is called [19, 20] the ‘vielbein’. The transport in PO∗ M is postulated [19] to be given by Eqs. 4.4 and 4.16. Note that, in Supergravity, it is more common to express the transport in terms of δ X E•ˆ • than through δ X e(ˆ•) . The relationship between the two is (δ X E•ˆ • ) dx• ≡ δ X e(ˆ•) .

(8.2)

A heuristic argument is then invoked to identify the matrix •ˆ •ˆ (X) of Eq. 4.16 as the ‘field-dependent Lorentz transformation’ ˆ

•ˆ •ˆ (X) ≡ λ•ˆ •ˆ kˆ X k ,

(8.3)

in which the coefficients λ •ˆ •ˆ kˆ are interpreted as the connection of an operator of covariant differentiation. Finally, Eqs. 4.16 and 8.3 are manipulated into the form (3.9), which is the one employed for practical calculations [19, 20] under the name of a ‘supergauge transformation’ of the vielbein. With the present framework, there is no need for any heuristic argument to lead from Eq. 4.16 to 3.9, because Eqs. 4.16 and 3.9 are equivalent, for any operator D, not necessarily covariant differentiation. This is why we did not impose any restriction on 2D in Section 3, so that the results obtained here are stronger than those of Supergravity. Furthermore, to recover conventional Supergravity, for which an arbitrary operator D would be too general, it suffices to demand that D be metric-compatible i k covariant differentiation. Then, Eq. 8.3 is a consequence of Eq. 2.2 with A j l = 0. In addition, by virtue of Eq. 2.12, metric-compatibility guarantees that the matrix [1 − t •ˆ •ˆ (X)] belongs to SO( p, q), at the first order in t, and generates (in Supergravity language) a field-dependent Lorentz transformation. With the operator D now specialised as covariant differentiation, the transport (5.24) of λi jk is interpreted as that of the connection on M. It coincides with what is known [19, 20] in Supergravity as a ‘supergauge transformation’ of the connection. The generalised transports developed in Section 7, which we shall call ‘generalised supergauge transformations’, may be analysed in a similar fashion. This is not the place for a detailed study of generalised supergauge transformations in Supergravity. However, a few remarks will be enlightening.

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Supergravity considers two configurations {E•ˆ • , λ•ˆ •ˆ •ˆ } and {E •ˆ • , λ •ˆ •ˆ •ˆ } as equivalent iff they are related by a generalised supergauge transformation. Such a transformation involves the position-dependent parameters Xx and •ˆ •ˆ (Xx ). Some of the degrees of freedom of X and  are employed to cast {E•ˆ • , λ•ˆ •ˆ •ˆ } in a canonical form, which reduces to the flat-space one when gravity is ignored. The remaining degrees of freedom express the transformation of the components of the canonical form, inter alia, under supersymmetry transformations. With our present construction, the geometrical origin of supersymmetry transformations is quite clear, because, owing to the metric-compatibility of D, the consistency requirement (7.11) reduces to the Killing equation

L X g = 0.

(8.4)

Moreover, given that the canonical form of the vielbein reduces to its flat-space value when gravity is absent, the Killing vector fields X also reduce to their flat-space limit in the same circumstances. This is why, in the terminology of Supergravity, one manages to ‘gauge’ supersymmetry. Finally, in Supergravity, one does not normally recognise, a priori, that the matrix •ˆ •ˆ depends on X. It is only a posteriori, in the course of the establishment of supersymmetry transformations, that one discovers that one is compelled to allow •ˆ •ˆ to depend on X. This is emphasised by saying [19] that ‘a gauged supersymmetry transformation must be accompanied by a field-dependent Lorentz transformation’. In contrast with Supergravity, the present construction shows why, a priori, •ˆ •ˆ depends on X. It is a consequence of the fact that the difference of two operators 1D X and 2D X of D-differentiation is a linear transformation that necessarily depends on X, as emphasised in Section 3. This feature applies to the difference of any two Doperators, and not only of two covariant derivatives. We thus see that the observation made in the context of Supergravity has a much wider validity.

9 Conclusion The purpose of this work was to present a general geometrical construction that would shed light, in a special case, on the supergauge transformations employed in Supergravity. By not restricting attention at once to Supergravity and its particularities, one gains a deeper understanding of the mathematical structure of the framework. The main tool in this investigation was the operation of D-differentiation, which we briefly recalled in Section 2. D-differentiation unifies, inter alia, Lie differentiation and covariant differentiation. It is entirely determined by a matrix • • , called a D-differentiation matrix. It also leads to a concept of ‘transport’ of a vector field Y (or a one-form α) along the integral curves of a vector field X. By combining together two D-operators, we defined ‘compound differentiation’ δ in Section 3. One of the D-operators was then selected as Lie differentiation (the other remaining general), and we investigated, in Sections 4 and 5, how the corresponding δ enables one to transport points of the fibre bundles PL∗ M and PO∗ M, as well as D-differentiation matrices • • . That kind of transport was called ‘compound transport’.

Geometrical Interpretation of ‘Supergauge’ Transformations Using D-Differentiation

219

For compound transport to be self-consistent in PO∗ M, the constraint (4.17) had to be satisfied, which reduces to the Killing equation (8.4) if the D-operator chosen is metric-compatible. Moreover, the transport of a D-differentiation matrix was shown to be self-consistent for all operators D of the class (5.23), which is characterised by tensorial curvature. In Section 7, we generalised compound transport to incorporate an SO( p, q) transformation generated by a matrix •ˆ •ˆ . Generalised compound transport was then compared to the (generalised) supergauge transformations of Supergravity in Section 8. It was established that these transformations are identical to generalised compound transport if the operator D at the basis of the framework is taken as metric-compatible covariant differentiation. Moreover, in this case, the Killing equation (8.4) is the geometrical origin of the gauged supersymmetry transformations of Supergravity. Finally, the matrix •ˆ •ˆ was identified with the ‘field-dependent Lorentz transformation’ encountered in Supergravity. The field-dependence of  was traced to a property of general D-differentiation, not restricted to the special case of Supergravity. Finally, it is important to repeat the remark made earlier that a complete comparison of the present framework with Supergravity would require the setting of a graded manifold. In particular, it would be necessary to have at one’s disposal the graded extension of D-differentiation, which one may call ‘super D-differentiation’. This new operator can be defined, but its construction is too lengthy to be presented here. However, a simple reasoning is provided in Appendix 3 to show its existence.

Appendix 1 Establishing that Eq. 5.15 follows from Eqs. 5.13 and 5.14 is not particularly instructive, but it is necessary for the completeness of the framework. The strategic point worth mentioning about the proof is that, in Eq. 5.13, the constant c may be written as [−1 + (c + 1)]. There then only remains to be shown that X{• • (Y)} − • • (L X Y) − U = • • (X, Y).

(9.1)

Let us begin by simplifying the notation. We shall, hereafter, systematically omit free indices, and retain exclusively summation ones. For instance, λk , , and Aa b • a will stand for λ• • k , • • , and A • b . Matrix multiplication will be understood over the suppressed indices. Finally, we shall cease to distinguish between underlined indices and ordinary ones. With these conventions, the object U of Eq. 5.14 reads U = Y[(X)] + (Y) (X) − (X) (Y).

(9.2)

The following three lemmas play an important role in the treatment:– 1. Lemma 1: 2Aa b e(a) (X m ) e(m) (Y b ) − 2Aa b e(m) (X b ) e(a) (Y m ) = = [e(s) (X a ) e(t) (Y b ) − e(s) (Y a ) e(t) (X b )](−At a δ s b + As b δ t a ). (9.3)

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D. J. Hurley and M. A. Vandyck

2. Lemma 2: −λk An l Y k e(n) (X l ) − Aa b λm e(a) (Y b ) X m + +λk An l X k e(n) (Y l ) + Aa b λm e(a) (X b ) Y m = = (λk An l − An l λk )[X k e(n) (Y l ) − Y k e(n) (X l )].

(9.4)

3. Lemma 3: 2Aa b An l e(a) (Y b ) e(n) (X l ) − 2Aa b An l e(a) (X b ) e(n) (Y l ) = = [e(s) (X a ) e(t) (Y b ) − e(s) (Y a ) e(t) (X b )](−As a At b + At b As a ). (9.5) They are easily obtained by ordinary tensor manipulations. Due attention must be paid, however, to the order of the factors in products, because matrix multiplication is understood. The main proof proceeds in three stages. In the first step, one evaluates the quantity Z ≡ X{• • (Y)} − • • (L X Y)

(9.6)

by inserting in the definition (9.6) of Z the value of • • (L X Y) as well as Eq. 2.2. After Lemma 1 is employed, one finds: Z = [X(λm ) + λr e(m) (X r ) − λr Dr km X k + Aa b e(a) (Db km ) X k ] Y m − −X(Aa b ) e(a) (Y b ) − Aa l Dn ka X k e(n) (Y l ) + 1 + [e(s) (X a ) e(t) (Y b ) − e(s) (Y a ) e(t) (X b )](−At a δ s b + As b δ t a ) − 2 −Aa b e(a) [e(m) (X b )] Y m + [X k e(n) (Y l ) − Y k e(n) (X l )] An b Db kl .

(9.7)

The same method is used, in the second step, to evaluate the quantity U of Eq. 9.2. This time, Lemmas 2 and 3 are invoked, and yield U = Y(λk ) X k + λk Y(X k ) − Y(Aa b ) e(a) (X b ) − Aa b Y[e(a) (X b )] + +(λm λk − λk λm ) X k Y m + +[X k e(n) (Y l ) − Y k e(n) (X l )] (λk An l − An l λk ) + 1 + [e(s) (X a ) e(t) (Y b ) − e(s) (Y a ) e(t) (X b )](−As a At b + At b As a ). 2

(9.8)

In the last step, Eqs. 9.7 and 9.8 are combined together to form the difference Z − U, which constitutes the left-hand side of Eq. 9.1. Simple algebraic manipulations enable one to recognise in Z − U the coordinate-expression 2.5, 2.6, 2.7, 2.8, 2.9, and 2.10 of (X, Y), which establishes Eq. 9.1.

Appendix 2 The purpose of the present appendix is to prove that a -matrix transported in accordance with Eqs. 5.12, 5.14 and 5.15 satisfies the correct transformation law

Geometrical Interpretation of ‘Supergauge’ Transformations Using D-Differentiation

221

under a basis change, when c = −1. In keeping with the simplified notation adopted in Appendix 1, we shall rewrite the relevant transformation law [5, 7] as  (Y) = M (Y) N − Y(M) N,

(9.9)

where M is the matrix relating the two sets of basic vectors, and N is its inverse. We also recall [5, 7] that, under the same basis change, the curvature matrix transforms as  (X, Y) = M (X, Y) N.

(9.10)

Moreover, for later convenience, let us reformulate the law of transport of a -matrix as (Y) = (Y) − t[(X, Y) + (c + 1)U]

T

˜ = (Y) − t (c + 1) U,

(9.11) (9.12)

˜ by which amounts to defining the matrix  ˜ (Y) ≡ (Y) − t(X, Y).

(9.13)

Two lemmas will be required:– ˜ transforms as 1. Lemma 1: Under a basis change, the matrix  ˜ ˜ (Y) = M (Y) N − Y(M) N. 

(9.14)

2. Lemma 2: Under a basis change, the quantity U of Eq. 5.14 transforms as U = M U N + W,

(9.15)

with W ≡ X(M) (Y) N + M (Y) X(N) − Y[X(M)] N − Y(M) X(N). (9.16) These lemmas are elementary: For Lemma 1, it suffices to insert Eqs. 9.9 and 9.10 ˜ (Y), which is Eq. 9.13 with  replaced by  , and  replaced in the definition of 

by  . For Lemma 2, one inserts Eq. 9.9 in the definition of U , which is Eq. 5.14 with  replaced by  . The result is then simplified by invoking the identity N X(M) = −X(N) M for all X.

(9.17)

The main proof is now straightforward: ˜ (Y) − t (c + 1) U

 (Y) ≡ 

T

(9.18)

˜ = [M (Y) N − Y(M) N] − t (c + 1) [M U N + W]

(9.19)

= [M (Y) N − Y(M) N] − t (c + 1) W.

(9.20)

T

The first two terms on the right-hand side of Eq. 9.20 constitute the correct transformation law for the matrix T(Y). Therefore, to eliminate the contribution involving W, one must put c = −1 in Eq. 9.20.

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Appendix 3 As emphasised in the introduction, the calculations presented above are performed exclusively within the framework of an ordinary (smooth) manifold modelled on Rn , so that the question of extending D-differentiation to a graded manifold never arises, at least from the mathematical point of view. On the other hand, if the results are to be interpretable physically in terms of Supergravity theory, it is necessary that D-differentiation generalise to graded manifolds. We do not wish to enter any detailed discussion of that problem, because a complete construction of ‘super D-differentiation’ deserves a separate publication. Be that as it may, a simple reasoning indicates that D-differentiation generalises to a graded manifold, although it does not show how to achieve this goal. For X fixed, the operator D X is a derivation [5, 7] of the algebra of tensors, which preserves tensor rank. All such derivations may be obtained [8] as the sum of a Lie derivative and a tensor. Both Lie differentiation and tensors have been generalised [1, 3] to the graded case, so that a graded version of D-differentiation can be defined. It is, of course, beyond the power of this limited reasoning to determine the detailed properties of this super D-differentiation. As an illustration, let us investigate the generalisation of the ordinary covariant derivative ∇. It is clear that there exists a tensor such that, for X fixed, the operator ∇ may be written as ∇ X = L X + (X, •),

(9.21)

in which a dot indicates a suppressed argument. For instance, on vector fields, the object is given in terms of the torsion by (X, •) ≡ T(X, •) + ∇• X,

(9.22)

from which it is obvious that is a tensor. According to our simple reasoning, ‘supercovariant’ differentiation S ∇ can be defined as S

∇ X ≡ SL X + S (X, •),

(9.23)

in which the superscript S refers to the graded (or ‘super’) version of the corresponding quantities. Indeed, owing to the fact that the concepts of the super Lie derivative and a super tensor, which appear on the right-hand side of Eq. 9.23, are well defined, so must be the supercovariant derivative on the left-hand side, although our simple reasoning does not provide the explicit form of S (X, •). The conclusion of the existence of supercovariant differentiation is known to be correct, and the construction of S ∇ has been carried out completely. (See, for instance, [1, 3] and the references therein.) A similar reasoning can be invoked to generalise rank-preserving derivations of the algebra of tensors other than covariant differentiation, and, in particular, D-differentiation. This is as far as one can go without entering the specificities of the axiomatics that one may wish to adopt for a graded manifold. It is well known that several axiomatic frameworks have successively been tried [1, 3], so that a complete discussion of super D-differentiation is impossible before a particular choice of axiomatics is made for the underlying graded manifold.

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References 1. Bartocci, C., Bruzzo, U. and Hernández-Ruipérez, D.: The Geometry of Supermanifolds, Kluwer, Dordrecht, The Netherlands, 1991. 2. Castellani, L., D’Auria, R. and Fré, P.: Supergravity and Superstrings, World Scientific, Singapore, 1991. 3. DeWitt, B.: Supermanifolds, Cambridge University Press, Cambridge, UK, 1992. 4. Grimm, R., Wess, J. and Zumino, B.: A complete solution of the Bianchi identities in superspace with supergravity constraints, Nuclear Phys., B 152 (1979), 255–265. 5. Hurley, D. and Vandyck, M.: A unified framework for Lie and covariant differentiation (with application to tensor fields), J. Math. Phys. 42 (2001), 1869–1886. 6. Hurley, D. and Vandyck, M.: An application of D-differentiation to solid-state physics, J. Phys. A 33 (2000), 6981–6991. 7. Hurley, D. and Vandyck, M.: Topics in Differential Geometry; A New Approach Using DDifferentiation, Springer-Praxis, London, UK, 2002. 8. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Interscience, New York, NY, 1963. 9. Nakahara, M.: Geometry, Topology and Physics, Adam Hilger, Bristol, UK, 1990. 10. Nath, P. and Arnowith, R.: Generalized supergauge symmetry as a new framework for unified gauge theories, Phys. Lett. B 56 (1975), 177–180. 11. Nath, P. and Arnowith, R.: Supergravity and gauge symmetry, Phys. Lett. B 65 (1976), 73–77. 12. Nath, P. and Arnowith, R.: Supergravity geometry in superspace, Nuclear Phys., B 165 (1980), 462–482. 13. Synge, J. L. and Schild, A.: Tensor Calculus, Dover, New York, NY, 1942. 14. van Nieuwenhuizen, P.: Supergravity, Phys. Rep. 68 (1981), 189–398. 15. Weinberg, S.: The Quantum Theory of Fields, Vol. 3, Cambridge University Press, Cambridge, UK, 2000. 16. Wess, J. and Zumino, B.: Superspace formulation of supergravity, Phys. Lett. B 66 (1977), 361–364. 17. Wess, J. and Zumino, B.: Superfield Lagrangian for supergravity, Phys. Lett. B 74 (1978), 51–53. 18. Wess, J. and Zumino, B.: The component formalism follows from the superspace formulation of supergravity, Phys. Lett. B 79 (1978), 394–398. 19. Wess, J. and Bagger, J.: Supersymmetry and Supergravity, Princeton University Press, Princeton, NJ, 1992. 20. West, P.: Introduction to Supersymmetry and Supergravity, World Scientific, Singapore, 1990.

Math Phys Anal Geom (2007) 9:225–231 DOI 10.1007/s11040-007-9013-8

Reflection in a Translation Invariant Surface Brendan Guilfoyle · Wilhelm Klingenberg

Received: 1 July 2005 / Accepted: 1 July 2006 / Published online: 27 January 2007 © Springer Science + Business Media B.V. 2007

Abstract We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. In addition, we show that the focal curve is not physically visible. This is done by constructing explicitly the focal set of the reflected line congruence (2-parameter family of oriented lines in R3 ) with the aid of the natural complex structure on the space of all oriented affine lines. Key words line congruence · focal set · reflection · caustic Mathematics Subject Classifications (2000) 53A25 · 78A05 · 53C80 PACS 42.15.-i The purpose of this paper is to prove the following Theorem: Main Theorem: The focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source and is not physically visible, while the focal surface is translation invariant. B. Guilfoyle (B) Department of Mathematics and Computing, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland e-mail: [email protected] W. Klingenberg Department of Mathematical Sciences, University of Durham, DH1 3LE Durham, United Kingdom e-mail: [email protected]

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In contrast to the focal surface, the reflected wavefront is not translation invariant, in general. This generalises a similar result for reflection in a cylinder [5]. There have been many investigations of generic focal sets of line congruences [1, 2, 6]. Rather than work in the generic setting, we compute the focal set explicitly in this special case. This we do by applying recent work on immersed surfaces in the space L of oriented affine lines in R3 [3, 4]. The next section contains a summary of the background material on the complex geometry of L and the focal sets of arbitrary line congruences. It also details the reflection of a line congruence in an oriented surface in R3 . In Section 2 we solve the problem of reflection of a point source off an arbitrary translation invariant surface (Proposition 2). We then compute the focal set and thus prove the Main Theorem.

1 The Parametric Approach to Geometric Optics Let L be the set of oriented affine lines in Euclidean R3 , which, by parallel translation, can be identified with the tangent bundle to the 2-sphere, TP1 . We summarise now the main features of this identification (further details can be found in [3, 4]). The canonical projection π : L → P1 assigns an oriented line, or ray, to its direction, and we also have the double fibration:

In the diagram, the left-hand map is projection onto the first factor. The mapping on the right, denoted by , takes (γ , r) ∈ L × R to the point on the oriented line γ in R3 that lies an affine parameter distance r from the point on γ closest to the origin (as shown). Let ξ be the local holomorphic coordinate on P1 obtained by stereographic projection from the south pole. This can be extended to coordinates (ξ, η) on L minus the fibre over the south pole. The map (ξ, η, r) → (ξ, η, r) = (z(ξ, η, r), t(ξ, η, r)) has the following cooordinate expression [3]: z=

2(η − ηξ 2 ) + 2ξ(1 + ξ ξ )r (1 + ξ ξ )2

2

t=

−2(ηξ + ηξ ) + (1 − ξ 2 ξ )r (1 + ξ ξ )2

,

(1.1)

where z = x1 + ix2 , t = x3 and (x1 , x2 , x3 ) are Euclidean coordinates in R3 .

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Definition 1 A line congruence is an immersed surface f :  → L, i.e. a 2parameter family of oriented lines in R3 . A smoothly parameterised line congruence is a smoothly immersed surface f :  → L together with an open cover {U α } of  and diffeomorphisms C → U α : μα → γ . For short we denote a parameterisation simply by μ, and assume that all maps are at least C1 -smooth. The first order properties of such a family can described by two complex functions, the optical scalars: ρ, σ :  × R → C, which are defined relative to an orthonormal frame in R3 adapted to the congruence. The real part θ and the imaginary part λ of ρ are the divergence and twist of the congruence, while σ is the shear [7]. For a parameterised line congruence we compose with the coordinates above to get μ → (ξ(μ, μ), ¯ η(μ, μ)). ¯ The optical scalars then, with a natural choice of orthonormal frame, have the following expressions [3]: ρ = θ + λi =

∂ + η∂ ξ − ∂ − η∂ξ ∂ − η∂ − η − ∂ + η∂ + η

σ =

∂ + η∂ξ − ∂ − η ∂ ξ ∂ − η∂ − η − ∂ + η∂ + η

,

(1.2)

where ∂ + η ≡ ∂η + r∂ξ −

2ηξ ∂ξ 1 + ξξ

∂ − η ≡ ∂η + r∂ξ −

2ηξ ∂ξ 1 + ξξ

,

and ∂ and ∂¯ are differentiation with respect to μ and μ, ¯ respectively. Definition 2 The curvature of a line congruence is defined to be κ = ρ ρ¯ − σ σ¯ . A line congruence is flat if κ = 0. A line congruence  ⊂ L is flat iff the rank of the projection π : L → P1 restricted to  is non-maximal. Definition 3 A point p on a line γ in a line congruence is a focal point if ρ and σ blow-up at p. The set of focal points of a line congruence  generically form surfaces in R3 , which will be referred to as the focal surfaces of . Theorem 1 The focal set of a parametric line congruence  is 

  (γ , r) ∈ R3  γ ∈  and 1 − 2θ0r + (ρ0 ρ 0 − σ0 σ 0 )r2 = 0 ,

where the coefficients of the quadratic equation are given by (1.2) at r = 0. Proof In terms of the affine parameter r along a given line, the Sachs equations, which σ and ρ must satisfy, are [7]: ∂ρ = ρ2 + σ σ ∂r

∂σ = (ρ + ρ)σ. ∂r

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These are equivalent to the vanishing of certain components of the Ricci tensor of the Euclidean metric. They have solution: ρ=

ρ0 − (ρ0 ρ 0 − σ0 σ 0 )r 1 − 2θ0r + (ρ0 ρ 0 − σ0 σ 0 )r2

σ =

σ0 , 1 − 2θ0r + (ρ0 ρ 0 − σ0 σ 0 )r2

where σ0 , θ0 and ρ0 are the values of the optical scalars at r = 0. The theorem follows.   This has the following corollary: Corollary 1 Let  be a line congruence, ρ = θ + λi, σ the associated optical scalars and ρ0 , θ0 , λ0 , σ0 their values at r = 0. If  is flat with non-zero divergence, then there exists a unique focal surface S given by r = (2θ0 )−1 . If it is flat with zero divergence, then the focal set is empty. If  is non-flat, then there exists a unique focal point on each line iff |σ0 |2 = 2 λ0 , there exist two focal points on each line iff |σ0 |2 < λ20 and there are no focal points on each line iff |σ0 |2 > λ20 . The focal set is given by 1

r=

θ0 ± (|σ0 |2 − λ20 ) 2 . ρ0 ρ¯0 − σ0 σ¯ 0

Proof The focal set of a parameterised line congruence are given by r = r(μ, μ) ¯ satisfying the quadratic equation in Theorem 1. If κ = 0, then there is none or one solution depending on whether θ0 = 0 or not. If κ = 0 then there are two, one or no solutions iff |σ0 |2 − λ20 is greater than, equal to or less than zero (respectively). The solution of the quadratic equation in each case is as stated.   Given a line congruence f :  → L, a map r :  → R determines a map  → R3 by γ → ( f (γ ), r(γ )) for γ ∈ . With a local parameterisation μ of , we get a map C → R3 which comes from substituting r = r(μ, μ) ¯ in (1.1). Of particular interest are the surfaces in R3 orthogonal to the line congruence – when the line congruence is normal. These exist iff the twist of the congruence vanishes, and the surfaces are obtained from the solutions of the following equation [3]: ¯¯ ¯ ¯ = 2η∂ ξ + 2η¯ ∂ξ . ∂r (1 + ξ ξ¯ )2

(1.3)

We turn now to the reflection of an oriented line in a surface in R3 . This is equivalent to the action of a certain group on the space of oriented lines, as described by: Theorem 2 [4] Consider a parametric line congruence ξ = ξ1 (μ1 , μ¯ 1 ), η = η1 (μ1 , μ¯ 1 ) reflected off an oriented surface with parameterised normal line

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congruence ξ = ξ0 (μ0 , μ¯ 0 ), η = η0 (μ0 , μ¯ 0 ) and r = r0 (μ0 , μ¯ 0 ) satisfying (1.3) with ξ = ξ0 and η = η0 . Then the reflected line congruence is 2ξ0 ξ¯1 + 1 − ξ0 ξ¯0 , (1 − ξ0 ξ¯0 )ξ¯1 − 2ξ¯0

ξ= η=

(ξ¯0 −ξ¯1 )2 η ((1−ξ0 ξ¯0 )ξ¯1 −2ξ¯0 )2 0

−

(1+ξ0 ξ¯1 )2 η¯ ((1−ξ0 ξ¯0 )ξ¯1 −2ξ¯0 )2 0

+

(1.4)

(ξ¯0 −ξ¯1 )(1+ξ0 ξ¯1 )(1+ξ0 ξ¯0 ) r0 , ((1−ξ0 ξ¯0 )ξ¯1 −2ξ¯0 )2

(1.5)

where the incoming rays are only reflected if they satisfy the intersection equation η1 =

(1+ξ¯0 ξ1 )2 η (1+ξ0 ξ¯0 )2 0

−

(ξ0 −ξ1 )2 η¯ (1+ξ0 ξ¯0 )2 0

+

(ξ0 −ξ1 )(1+ξ¯0 ξ1 ) r0 . 1+ξ0 ξ¯0

(1.6)

By virtue of the intersection equation, an alternative way of writing (1.5) is η=

−(1+ξ0 ξ¯0 )2 η¯ ((1−ξ0 ξ¯0 )ξ¯1 −2ξ¯0 )2 1

+

2(ξ¯0 −ξ¯1 )(1+ξ0 ξ¯1 )(1+ξ0 ξ¯0 ) r0 . ((1−ξ0 ξ¯0 )ξ¯1 −2ξ¯0 )2

(1.7)

The geometric content of this is: reflection of an oriented line can be decomposed into a sum of rotation about the origin (the derived action of PSL(2,C) on TP1 ) and translation (a fibre-mapping on TP1 ).

2 Reflection Off a Translation Invariant Surface Consider a translation invariant surface with axis lying along the x3 −axis in R3 . Such a surface can be parametrised by (u, v) → (z = z0 (u), t = v) for (u, v) ∈ R2 , where z = x1 + ix2 , t = x3 and (x1 , x2 , x3 ) are Euclidean coordinates in R3 . Proposition 1 The normal congruence of a translation invariant surface is: 

z˙ 0 ξ0 = ± − z˙¯ 0 η0 = 12 (z0 − 2vξ0 − z¯ 0 ξ02 )

 12

r0 =

(2.1) 1 2



 ξ¯0 z0 + ξ0 z¯ 0 ,

(2.2)

where a dot represents differentiation with respect to u and the choice of sign of ξ0 is one of orientation of the normal. Proof Let ξ0 ∈ S2 be the direction of the normal. This corresponds to the unit vector

0 = V

2ξ0 2ξ¯0 1 − ξ0 ξ¯0 ∂ ∂ ∂ + + 1 + ξ0 ξ¯0 ∂z 1 + ξ0 ξ¯0 ∂ z¯ 1 + ξ0 ξ¯0 ∂t

230

∂ ∂u

B. Guilfoyle, W. Klingenberg

The vanishing of the inner product of this vector with the push forward of ∂ and ∂v yields 1 − ξ0 ξ¯0 = 0

z˙ 0 ξ¯0 + z˙¯ 0 ξ0 = 0

and (2.1) follows. Equations (2.2) come from inverting (1.1) for η and r.

 

By a suitable choice of parameterisation u we can remove the sign ambiguity of (2.1) and make it positive, which we do from here on. We consider the reflection of a point source off a surface that is translation invariant along the x3 -axis. By a translation of surface and source we move the point source to the origin. The line congruence consisting of all oriented lines through the origin is given by η1 = 0 and ξ1 ∈ S2 . Proposition 2 The reflection of a point  the origin off a translation  source at invariant surface is ξ = −ξ02 ξ¯1 and η = ξ¯0 − ξ0 ξ¯12 ξ02r0 where 1

v + (v 2 + z0 z¯ 0 ) 2 ξ1 = z¯ 0

(2.3)

and ξ0 is given by (2.1). Proof The reflection equations contained in Theorem 2, with η1 = 0, ξ1 ∈ S2 yield the stated reflected line congruence and the intersection equation has solution 1

ξ1 =

v ± (v 2 + z0 z¯ 0 ) 2 . z¯ 0

Here the ± refers to the two oriented lines from the source that intersect any given point in R3 , and is chosen so that the ray goes from the source to the point of reflection – which gives a plus sign and, hence, (2.3).   Main Theorem: The focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source and is not physically visible, while the focal surface is translation invariant. Proof The local parameter we choose is μ = u + iv and the optical scalars are computed by inserting the reflected line congruence in Proposition 2 into (1.2). The focal set is then determined by inserting the reflected line congruence and the solutions r of the quadratic equation in Theorem 1 into (1.1). The reflected line congruence is found not to be flat, so the quadratic has two solutions and we obtain the following focal set: z=

z0 z˙¯ 0 − z¯ 0 z˙ 0 z˙¯ 0

t = 0,

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and z=

2¨z0 z˙¯ 0 z20 z¯ 0 − 2z¨¯ 0 z˙ 0 z20 z¯ 0 + z˙ 30 z¯ 20 − 2z˙ 20 z¯˙ 0 z0 z¯ 0 + z˙ 0 z˙¯ 20 z20 , 2¨z0 z˙¯ 0 z0 z¯ 0 − 2z¨¯ 0 z˙ 0 z0 z¯ 0 − z˙ 20 z˙¯ 0 z¯ 0 + z˙ 0 z˙¯ 20 z0

t=

2v z0 z¯ 0 (z¨¯ 0 z˙ 0 − z¨ 0 z˙¯ 0 ) . 2¨z0 z˙¯ 0 z0 z¯ 0 − 2z¨¯ 0 z˙ 0 z0 z¯ 0 − z˙ 20 z˙¯ 0 z¯ 0 + z˙ 0 z˙¯ 20 z0

The first of these is a curve in the x1 x2 -plane parameterised by u, and the second is a surface that is invariant in the x3 -direction. To show that the curve is not physically visible, we note that the vector

from the focal point to the point of reflection is a positive multiple of the W

reflected direction V: ¯

= (1 + ξ2 ξ2 )z0 z¯ 0 V

W 1 2 v + (v 2 + z0 z¯ 0 ) 2 Thus the focal curve lies on the opposite side of the surface, and is not physically visible.  

References 1. Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps, vol. 1, Birkhaeuser, Basel (1986) 2. Bruce, J., Giblin, P., Gibson, C.: On caustics by reflection. Topology 21, 179–199 (1982) 3. Guilfoyle, B., Klingenberg, W.: Generalised surfaces in R3 . Math. Proc. R. Ir. Acad. 104A(2), 199–209 (2004) 4. Guilfoyle, B., Klingenberg, W.: Reflection of a wave off a surface. J. Geom. 84, 55–72 (2006) 5. Guilfoyle, B., Klingenberg, W.: A Kähler surface with applications in geometric optics. In: Proceedings of the Programme on the geometry of pseudo-Riemannian manifolds with application in physics, Erwin Schrödinger Institute, Vienna (in press) 6. Izumiya, S., Saji, K., Takeuchi, N.: Singularities of line congruences. Proceedings of the Royal Society of Edinburgh 133A, 1341–1359 (2003) 7. Penrose, R., Rindler, W.: Spinors and spacetime, vol. 1 and 2, Cambridge University Press, Cambridge (1986)

Math Phys Anal Geom (2007) 9:233–262 DOI 10.1007/s11040-006-9012-1

On Separation of Variables for Homogeneous SL(r) Gaudin Systems Gregorio Falqui · Fabio Musso

Received: 18 October 2006 / Accepted: 4 December 2006 / Published online: 25 January 2007 © Springer Science + Business Media B.V. 2007

Abstract By means of a recently introduced bihamiltonian structure for the homogeneous Gaudin models, we find a new set of Separation Coordinates for the sl(r) case. Key words Gaudin systems · separation of variables · bihamiltonian geometry Mathematics Subject Classifications (2000) 70H06 · 70H20 · 37K10

1 Introduction In this paper we will discuss the Gaudin system with sl(r)-valued spins defined by the Hamiltonian HG =

n 

Tr(Ai · Aj),

Ak ∈ sl(r)

(1.1)

i< j

on the manifold M = sl(r)n equipped with the standard product Lie-Poisson structure. We will refer to it, with a slight abuse of notation, as the homogeneous sl(r) XXX Gaudin system (as in [11]).

G. Falqui (B) Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, via R. Cozzi 53, 20125 Milano, Italy e-mail: [email protected] F. Musso Dipartimento di Fisica, Università di Roma TRE, and Instituto Nazionale di Fisica Nucleare, Sezione di Roma TRE, Via Vasca Navale 84, 00146 Roma, Italy e-mail: [email protected]

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The ‘conventional’ approach to the integrability of this quite well studied problem is based on the Lax representation and the r-matrix theory (see, e.g., [21, 30]). Fixing n distinct parameters ai , i = 1, . . . , n one introduces the matrix L(λ) =

n  i=1

Ai , λ − ai

(1.2)

to be considered as an element of the Loop algebra sl(r)((λ)). Along the Hamiltonian flow defined by (1.1), the Lax matrix L(λ) evolves according to a Lax equation dL(λ) = [L(λ), M]. dt Thanks to the existence of an r-matrix for the Lax matrix (1.2) the spectral invariants Ii(α) = Res

Tr(L(λ)α ),

i = 1, . . . , n,

α = 2, . . . , r,

(1.3)

λ=ai

are in involution. These integrals, together with the integrals of the motion associated with the invariance of the system under the global SL(r) action given by Ai → GAi G−1 ,

(1.4)

to be referred to as global gauge invariance, provide a complete set of constants of the motion for HG . The separability of the Hamilton-Jacobi equations associated with the Gaudin Hamiltonian (1.1) was first studied [30, 31], for the low r cases, as a kind of byproduct of the solution of the Bethe Ansatz equations associated with the quantum Gaudin system. Separability was then proved for the general case in [16, 29]. and (implicitly) framed within the theory of Algebraically Complete Integrable Systems in [1, 2, 6, 7]. In this scheme, it turns out that one can find a set of ‘algebro-geometrical’ Darboux coordinates (ζi , λi ) as coordinates of a set of dn = r(r − 1)(n − 1)/2 distinguished points on spectral curve (ζ, λ) = Det(ζ − L(λ)),

(1.5)

 (r − 1)  (n − 2)r + (r − 2) . 2 In the recent paper [11] we have reconsidered the (homogeneous XXX) Gaudin model, and generalized to the case of an arbitrary Lie algebra g an alternative set of integrals of the motion for HG (see, e.g. [3]), introduced in the Hopf-algebraic approach to the integrability of the system. The distinguished feature of such integrals, which in the case of g = sl(2) are given by the very simple expressions 2  l  Kl = Tr Al , (1.6) whose genus is g =

i=1

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235

is that they are independent of the (fake) parameters entering the definition of the Lax matrix (1.2). These integrals were also introduced, in a different context [22], as Hamiltonians of a notable class of Hamiltonian flows on the moduli space of n + 3-sided polygons in R3 , and later generalized in [14] to the Rd case.1 This moduli space turns out to be a suitable Marsden-Weinstein Hamiltonian quotient of the Poisson manifold su(2)n+3 associated with the corresponding Gaudin model. The Hamiltonian flows associated with (1.6) were termed ‘bending flows’ due to the following fact: if one draws, from a chosen vertex, the n possible diagonals of an n + 3-sided polygon, the flow associated with the Hamiltonian Kk geometrically represents the bending of one side of the polygon along the kth diagonal (the other side being kept fixed). The key point for the analysis performed in [11] was the introduction, along with the standard Lie-Poisson structure P, of a particular second Poisson structure, hereinafter called R. In the n = 3 case, this structure is defined by its Hamiltonian vector fields as follows:

⎧ ∂F ∂F ⎪ ˙ ⎪ A1 = A1 , + ⎪ ⎪ ∂ A2 ∂ A3 ⎪ ⎪ ⎪



⎨ ∂F ∂F ∂F ∂F ˙ + A2 , − + (1.7) A2 = A1 , ⎪ ∂ A1 ∂ A2 ∂ A2 ∂ A3 ⎪ ⎪





⎪ ⎪ ∂F ∂F ∂F ∂F ∂F ⎪ ⎪ ⎩ A˙ 3 = A1 , − + A2 , − + 2 A3 , , ∂ A1 ∂ A3 ∂ A2 ∂ A3 ∂ A3 ∂F are elements of sl(r) to be properly defined in Section 2. ∂ Ai The Poisson pencil R − λP and the integrals (1.6) fulfill standard LenardMagri relations, namely one can check that     i = 1, . . . , n, Rd Tr A21 = 0, Pd Tr Ai2 = 0,  a   2 Ai = PdKa , a = 2, . . . , n. (1.8) Rd Tr

where

i=1

For the general sl(r) case one can show that it is possible to find a sufficient number of polynomial functions in involution that provide a set of integrals of the motion Kl(α) alternative to the set defined by the Lax matrix (1.2). They share with the integrals (1.6) the property of being defined independently of the parameters entering the Lax matrix (1.2). In the last Section of [11] we addressed the problem of separability of such flows, in the framework of the so-called bihamiltonian approach to the SoV problem (see, e.g., [4, 13, 23, 26], and the references quoted therein). In particular we solved it for the sl(2) case by means of explicit computations, showing that the separation coordinates associated with the pencil R − λP

1 We

thank J. Harnad for drawing our attention to these references.

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are rational functions of the natural coordinates (hi , ei , fi ) in sl(2)n , and the separation relations are quadratic equations in these coordinates. In this paper we will solve the corresponding problem for the sl(r) case, with r arbitrary. This task will be accomplished by means of a careful mixing of techniques of the theory of Lax equations with r-matrix structure, and the theory of bihamiltonian systems such as those exposed in a series of papers by Gel’fand and Zakharevich [17, 18] and Magri and collaborators [9, 10, 23]. In particular, we will make extensive use of (refinements of) the results presented in [12, 13] concerning the Separation of Variables of systems with an arbitrary number of Lenard-Magri chains. The key points for the analysis we are going to develop in this paper are: 1. It is possible to deform the Poisson tensor R into R˜ in such a way that R˜ is still compatible with P and restricts to the (generic) symplectic leaf S of P. b) The integrals Kl(α) defined by the pencil R − λP are in involution also ˜ although the recurrence relation they satisfy in relation with w.r.t. R, the new pencil R˜ − λP are more complicated than the usual LenardMagri relations. a)

2. It is possible to define n − 1 Lax matrices La linearly depending on a ‘spectral parameter’ λ such that any formal vector field X(λ) which is ‘Hamiltonian w.r.t. the pencil R − λP induces a Lax equation on each of the matrices La ’. Thanks to the first property it will be possible to endow the generic symplectic leaf S of P with a special geometric structure, that is, a (1, 1) tensor with vanishing Nijenhuis torsion, whose ‘spectral’ data will provide us with a set of separation coordinates for the H–J equations associated with HG . Thanks to the second property, as well as other specific features of the deformation R˜ of R, (to be fully discussed in the core of the paper) we will be able to show that the separation relations are provided by the spectral curves of the matrices La . The distinguished feature of such a SoV scheme is that the separation coordinates are defined iteratively in subsets of dr = r(r − 1) coordinates, which are coordinates of a set of r(r − 1)/2 points on a genus g = (r − 1)(r − 2)/2 curve, irrespectively of the number n of ‘sites’ of the Gaudin model. A word of warning: the set of coordinates defined in this way on S must be completed by a set of r(r − 1) coordinates associated with the global SL(r) invariance of the model, just like the set of integrals coming either from the Lax matrix (1.2) or from the construction discussed in [11] must be supplemented by the set of integrals associated with the global gauge invariance of the model. However, since these integrals are associated with a sort of ‘cyclic’ coordinates, they will trivially enter the H-J equations and the problem of separability. So in the core of the paper, we will often ‘forget’ about them.

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The scheme of the paper is as follows: In Section 2 we will fix some conventions and notations to be used throughout the paper, review the results of [11] to be used in the sequel, and introduce the Lax matrices La . In Section 3 we will briefly recall the main points of the bihamiltonian scheme for SoV, and, in Section 3.1 we will discuss how to apply such a picture to the sl(r) Gaudin models. Finally, in Section 4 we will give examples of our constructions in the sl(2) and sl(3) case. In the last section we briefly summarize the content of the paper and add a few comments. In order to simplify the presentation, we collected the proofs of some important but somewhat technical points in three Appendices.

2 The Bihamiltonian Structures and the Lax Matrices Let g be the Lie algebra sl(r). It is known that it (as well as any simple Lie algebra) can be identified with its dual, e.g., via the dual pairing given by the trace in the fundamental representation. In this paper we will constantly use such an identification. The Lie Poisson structure on M = gn is the one defined, in the natural coordinates {A1 , . . . , An } by its Hamiltonian vector fields:

∂F , (2.1) A˙ i = Ai , ∂ Ai where, if X = (Xi , . . . , Xn ) represents a tangent vector to M, the elements ∂F ∈ sl(r), i = 1, . . . , n are those matrices defined by means of the expression ∂ Ai of the Lie derivative of F w.r.t. X as   n  ∂F Lie X (F) = . (2.2) Tr Xi · ∂ Ai l=1

We will hereinafter denote the Poisson tensor associated with the Lie-Poisson natural bracket by P. From, e.g., [24, 28]) we know that we can endow M with a multi-parameter family of Poisson structures which are compatible with the natural one (2.1). In [11] a further linear Poisson structure, to be denoted by R, has been introduced. It can be described as follows. We notice that relation (2.1) can be written as:

n  ∂F ˙ pijk Ak , , with pijk = δik δij. (2.3) Ai = ∂ Aj j,k=1

The new Poisson tensor R is analogously defined by the expression:

n  ∂F ˙ Ai = rijk Ak , , ∂ Aj

(2.4)

j,k=1

with ‘structure constants’ given by rijk = (k − 1)δijδ jk − θ(i−k) δij + θ( j−i) δik + θ(i− j) δ jk

(2.5)

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where δ is the usual Kronecker symbol and θ(i) equals 1 if i > 0, and vanishes for i  0. Explicitly, the Hamiltonian vector field associated by R with a function F is given by:  i−1 

N   ∂F ∂F ∂F ∂F ˙ + Ai = Ai , (i − 1) + Ak , − . (2.6) ∂ Ai ∂ Ak ∂ Ak ∂ Ai k=i+1

k=1

The following facts can be proven [11]: Proposition 2.1 1. The pencil of bivectors R − λP is a bihamiltonian structure on M, that is, R is a Poisson structure compatible with P. (1) 1 2. The functions Hα,1 = α+1 Tr(Aα+1 are common 1 ), α = 1, , . . . , r − 1 Casimirs for R and P. The Lenard-Magri chains starting at 1 a = 2, . . . , n Tr(Aaα+1 ), α+1 provide us with further d = (n − 1)r(r − 1)/2 functionally independent integrals (a) = Hα,1

(a) Hα, p,

a = 2, . . . , n,

p = 2, . . . , α + 1,

α = 1, . . . ,

r − 1.

(2.7) 3. Taking into account the integrals associated with the global SL(r) invariance n Ai ), of the model, that is the ring of functions generated by Fξ = Tr(ξ · i=1 those integrals insure complete Liouville integrability of the model. Remarks 1. The Gaudin Hamiltonian (1.1) is expressed in terms of the integrals (2.7) as   a−1  n n    a HG = H2,2 = Tr Aa · Ab . a=2

b =1

a=2

2. A convenient choice of the integrals associated with the global SL(r) invariance can be done as follows. We pick the r − 1 independent elements Fh1 , . . . , Fhr−1 associated with, say, the standard Cartan subalgebra of sl(r), and the Gel’fand-Cetlyn invariants, that is, the Casimirs of the nested subalgebras sl(2) ⊂ sl(3) ⊂ · · · ⊂ sl(r),

(2.8)

under the map sl(r) n → sl(r) sending the n-tuple {A1 , . . . , An } into the total sum, Atot = i=1 Ai . n

For the sequel of the paper the following construction is crucial. Let us introduce n − 1 Lax matrices: La = (λ − (a − 2))Aa +

a−1  k=1

Ak

a = 2, . . . , n,

(2.9)

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239

It holds: Proposition 2.2 Let F be a smooth function on M and let us consider the pencil of vector fields X Fλ = Pλ dF := (R − λP)dF (we say that X Fλ is Hamiltonian w.r.t. the pencil Pλ ). Then, along X Fλ , the matrices Li of (2.9) evolve according to a Lax equation, Lie X Fλ (La ) = [La (λ), Ma (λ)]

(2.10)

with n  ∂F ∂F Ma (λ) = (a − 1 − λ) + ∂ Aa ∂ Ab b =a+1

Proof Let us denote αi = given by: Lie X λ (Ai ) = (Pλ dF)i =

∂F , ∂ Ai

i = 1, . . . , n. The vector field X Fλ is explicitly



  (rijk − λpijk ) Ak , αk =

F

j,k

=



(k − λ − 1)δijδ jk − θ (i − k)δij + θ ( j − i)δik + θ (i − j)δ jk

j,k

=

i−1  



⎡

Ak , αk − αi + ⎣ Ai , (i − λ − 1)αi +

k=1

N 



 Ak , αk =

⎤ αk ⎦

k=i+1

Substituting in La (λ) we get: ⎛ Lie X λ (La ) = (λ − a + 2) ⎝

a−1  

F



⎡

Ak , αk − αa + ⎣ Aa , (a − 1 − λ)αa +

k=1

N 

⎤⎞ αk ⎦⎠ +

k=a+1

⎛ ⎡ ⎤⎞ a−1  a−2 N     ⎝ Ak , αk − α j + ⎣ A j, ( j − λ − 1)α j + + αk ⎦⎠ = j=1

k=1

⎡⎛ = ⎣⎝(λ − a + 2)Aa +

k= j+1 a−1  j=1

⎞ ⎛ A j⎠ , ⎝(a − 1 − λ)αa +

N 

⎞⎤ αk ⎠⎦

 

k=a+1

We can interpret this result by saying that we can associate with the homogeneous n-particle Gaudin system a set of n − 1 matrices depending on a parameter λ, satisfying a Lax equation along the ‘formal’ (i.e., depending on the parameter λ) flows of vector fields that are Hamiltonian with respect to the pencil Pλ .

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Proposition 2.3 The coefficients Kα(a) (λ) of the expansion in powers of μ of the characteristic polynomial det(μ − La (λ)) = μr +

r−1 

Kα(a) (λ)μr−α−1

α=1

of every Lax matrix La (λ) are polynomial Casimirs of the pencil Pλ = R − λP. Moreover, along any vector field X associated with any of the non-trivial coefficient of such polynomial Casimir, all matrices La (λ) evolve according to Lax equations Lie X (La (λ)) = [La (λ), Ma (X)] for suitable matrices Ma (X). Proof These assertion follow from the general theory of bihamiltonian pencils on loop algebras (see, e.g., [27] and [28]). We sketch the proof for com(α) = 1/(α + pleteness, considering the equivalent set of spectral invariants Hm α+1 1)tr(Lm ) . To prove the first statement, we must show that, for any one-form v, that we can assume to be exact, v = dF we have   (l) v, Pλ dHm = 0. Now, switching the action of the Poisson pencil on v = dF the LHS of this equation reads L X Fλ (Ha(l) ) = L X Fλ (1/(α + 1)Tr(La (λ))α+1 ) 1   = Tr La (λ) p · L X Fλ (La (λ)) · La (λ)α− p α + 1 p=0    = Tr Laα · L X Fλ (La ) = tr Laα (λ) · [La (λ), MaF (λ)] = 0 α−1

(2.11) This proves the first assertion of the proposition, and, in particular, shows that (a) all the vector fields Xα, p associated (say, via P) with the coefficients of the expansion  (a) p Kα(a) = Kα, (2.12) pλ p 0

are indeed bihamiltonian vector fields. To prove the second statement we notice, using a very simple trick well known to experts in the bihamiltonian theory of integrable systems, that (a) (a) Xα, vector field w.r.t. the pencil, p = PKα, p−1 can be written as a Hamiltonian   considering the ‘truncated’ polynomial λ− p Kα(a) (λ) + , where (·)+ denotes the nonnegative part of the expansion in λ. So we see that the bihamiltonian vector fields of the hierarchy are as well Hamiltonian vector fields w.r.t. the bihamiltonian pencil Pλ . The assertion then follows from Proposition 2.2.  

On separation of variables for homogeneous SL(r) Gaudin systems

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Let us now focus our attention on a single Lax matrix, say La¯ ; calling, for simplicity, Ba¯ = ab¯ −1 =1 Ab , we have that the matrix La¯ = (λ − a¯ + 2)Aa¯ + Ba¯ is a Lax matrix with spectral parameter that evolve according to Lax equations along the vector fields of the hierarchy. Clearly, the Poisson brackets induced on M(2) = sl(r) × sl(r) by the map M → M(2) defined by {Aa¯ , Ba¯ } are nothing but the Lie Poisson brackets on M(2) . So, applying the formalism of [1, 2, 6, 7, 30], i.e., according to the Sklyanin ‘magic recipe’ [32], we can get, for every fixed a¯ a set of canonical coordinates {ξαa¯ , λaβ¯ }. Actually, we shall do this in Section 3.1. The point is that, to get a set of canonical coordinates for the whole systems, we have to compare the different sets of coordinates coming from the different Lax matrices La , a = 2, . . . , n (and those coming from the global gauge invariance of the model). To solve this problem, we shall make use of the bihamiltonian structure of the problem, and, namely, frame the Gaudin systems within the so-called bihamiltonian scheme for SoV. For the case of sl(2), we were able to solve the problem by means of straightforward computations. For the general case, we have to use some slightly more sophisticated ideas and techniques of the bihamiltonian theory, to be discussed in the next Section.

3 Bihamiltonian Geometry and Separation of Variables As we already remarked in the Introduction, a theory of Separation of variables based on the notions of bihamiltonian geometry has been quite recently introduced in the literature. The basic property of such a theoretical scheme which will enable us to solve the SoV problem of this paper can very simply stated as follows: Proposition 3.1 Let (M, P1 − λP0 ) be a bihamiltonian manifold and suppose that there exist functions f, g, λ f , λg , with λ f = λg , (eventually, λ f and/or λg might be constant) satisfying P1 df = λ f P0 df,

P1 dg = λg P0 dg.

(3.1)

Then { f, g}0 = { f, g}1 = 0. Proof The assertion easily follows from the equations { f, g}1 = df, P1 dg = λg { f, g}0 {g, f }1 = dg, P1 df = −λ f { f, g}0 ,   In words, calling (with a slight abuse of language) a function f satisfying (3.1) an ‘eigenvector’ of the pair P1 , P0 relative to the ‘eigenvalue’ λ f as

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in [20], this proposition simply says that eigenvectors belonging to different eigenspaces mutually commute. If the Poisson tensors P1 and P0 do not share the same image and kernel, then a complete set of eigenvectors cannot be found. This is a typical instance in the Gel’fand-Zakharevich theory of bihamiltonian integrable systems, (and happens for the Poisson tensors R and P that we have considered so far). In general, the bihamiltonian theory of SoV suggests to consider a suitable !1 of P1 , such that it restricts to the (generic) symplectic leaves deformation P of P and it is still compatible with P. Upon restriction, the generic symplectic leaf S of P will be endowed with a regular bihamiltonian structure (that is, a bihamiltonian structure in which one element of the pencil is invertible). So, in the terminology of [13] the generic symplectic leaves of P are ωN manifolds, that is are symplectic manifolds (with symplectic form naturally induced by P), endowed with a compatible Nijenhuis (or hereditary) tensor N. In terms of the Poisson structures, the Nijenhuis tensor on the symplectic leaf S is defined by !1 |S · P−1 |S . N=P 0 !1 , one can adopt the To concoct out of P1 the suitable deformation P following strategy: First one fixes a complete set C1 , . . . , Ck of Casimirs of P0 , considers the first vector fields of the Lenard chains associated with Ca , i.e., Xa = P1 dCa ,

a = 1, . . . , k,

and a distribution Z , transversal to the symplectic leaves of P0 . For any basis W1 , . . . , Wk in Z , the matrix [G0 ]a,b = LieWb (Ca ) is nonsingular (say on an open set U ⊂ M). So, the tensor defined by  !1 = P1 − Xa ∧ [G−1 P 0 ]a,b Wb

(3.2)

(3.3)

a,b

is well defined and restricts to the generic symplectic leaf S of P, since, by !1 dC j = 0, j = 1, . . . , k. Notice that, if we define a new basis in construction, P Z by  a Za = G−1 so that Lie Z a (Cb ) = δa,b , (3.4) 0 b Wb , b

!1 simplifies to the expression of the deformed tensor P  !1 = P1 − P Xa ∧ Z a .

(3.5)

a

We will call a basis of Z satisfying (3.4) a normalized basis for the transversal distribution. The proof of the following Proposition can be found in [5, 12].

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Proposition 3.2 Let (M, P1 − λP0 ) be a 2n + k dimensional bihamiltonian manifold with corank(P0 ) = k, and suppose that there exists an integrable distribution Z ⊂ T M of dimension k, s.t.: 1. Z intersect transversally the symplectic foliation of P0 . 2. The space of functions invariant under Z is a Poisson subalgebra for the whole pencil (M, P1 − λP0 ). !1 is the deformation of P1 defined by (3.3), P !1 − λP0 is still a Poisson Then, if P pencil on M, and its restriction endows the generic symplectic leaves of P0 with the structure of a ωN manifold. Furthermore, if Z 1 , . . . , Z k are a set of generators of Z , normalized w.r.t. a given complete set C1 , . . . , Ck of Casimir functions of P0 , condition 2 above translates into the equations: Lie Z a P0 = 0,

Lie Z a P1 =

k  [Z a , Xb ] ∧ Z b ,

where

b =1

Xa = P1 dCa ,

a = 1, . . . , k.

(3.6)

Definition 3.3 We say that a bihamiltonian manifold (M, P1 − λP0 ), endowed with a transversal distribution Z satisfying the assumptions of Proposition 3.2 admits an affine structure if it is possible to choose a complete set of Casimir of P0 , and a corresponding basis of normalized flat generators Z b , b = 1, . . . , k in Z such that, for every Casimir of the Poisson pencil H a (λ) and every b , c = 1, . . . k one has, in addition to (3.6) Lie Z b Lie Z c (H a (λ)) = 0.

(3.7)

The notion of affine structure for a bihamiltonian manifold was studied in [13] in connection with the problem of the Stäckel separability of a bihamiltonian system. For the purposes of the present paper, we remark that an affine Poisson pencil satisfies special properties, to be illustrated in the following. Let (M, P1 − λP0 ) be a corank k affine bihamiltonian manifold, and let Z a , a = 1, . . . , k be a set of normalized flat generators for the transversal distribution Z . Let us consider the polynomial Casimirs H (a) (λ) = λna H0a + · · · + Hnaa , and their deformations along the flat generators, that is, the k2 polynomials Dab (λ) = Lie Z b H (a) (λ) = λna δba − Dab, 1 λna −1 − · · · − Dab, na .

(3.8)

The polynomials Dab (λ) are invariant along Z , so that they can be considered as functions on the generic symplectic leaves of P0 . They are the building blocks of the bihamiltonian set-up for SoV for GZ systems. Indeed it holds: Proposition 3.4 Let (λ) be the determinant of the matrix Dab of (3.8). Then 1. The roots λi of (λ) satisfy !1 dλi = λi P0 dλi , P

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! ! 2. Let D(λ) denote the classical adjoint matrix of D(λ), and let [ D(λ)] a,c be ! ! non-identically vanishing. Then any ratio ρ(λ) := [ D(λ)]a,b /[ D(λ)]a,c of ! elements belonging to the a-th row of D(λ), evaluated at the roots λi of (λ) satisfy the equation !1 dρ(λi ) = λi P0 dρ(λi ). P

(3.9)

The proof of this Proposition is contained in Appendix A. 3.1 Separation of Variables for the sl(r) Gaudin Systems In this subsection we will specialize the results of Section 3 and show how the bihamiltonian structure Pλ = R − λP associated in Section 2 with the parameter independent integrals of the Gaudin model provides a set of separation coordinates and relations for the H–J equations associated with HG . The first step is to show that Pλ induces a ωN manifold structure on the generic symplectic leaf S of P, that is, that the tensor R can be suitably deformed. We consider in M = sl(r)n the n(r − 1) vector fields Wiα :=

∂ , ∂[Ai ]r,α

i = 1, . . . , n,

α = 1, . . . , r − 1,

(3.10)

that is, the vector fields defined by their action on the n-tuple of matrices (A1 , . . . , An ) by LieWiα (A1 , . . . , An ) = (0, 0, · · · , eα,r , · · · , 0), "#

%$(3.11)

i-th place

where eα,r is the elementary matrix (eα,r )ij = δi,α δ j,r . Proposition 3.5 The distribution Z spanned by the vector fields Wiα satisfies the hypotheses of Proposition 3.2. The proof of this Proposition is contained in Appendix B. To construct a set of flat generators Z iα for Z , we can argue as follows. In the case of a single copy of sl(r), we normalize the W α with respect to the coefficients C1 , . . . , Cr−1 of the characteristic polynomial of A. The normalization for the n site case is done site by site. Since the determinant of a matrix is a linear function of each of its entry, it is not difficult to realize that such normalized generators Z iα provide the GZ manifold M, R − λP with the structure of an affine GZ manifold, according to Definition 3.3. Let us now consider the Lax matrices La = (λ − a + 2)Aa +

a−1  b =1

Ab ,

a = 2, . . . , n,

as well as L1 = A1 .

On separation of variables for homogeneous SL(r) Gaudin systems

245

!i (λ, ξ ). Define Mi (λ, ξ ) = ξ I − Li (λ), and denote their classical adjoint with M The determinants of the matrices Mi define, thanks to Proposition 2.3, polynomial Casimirs Kα(i) (λ) for R − λP, via: Det(Mi (λ, ξ )) = ξ r +

r−1 

Kα(i) (λ)ξ r−α−1 .

(3.12)

α=1

In particular, Kα(1) are the common Casimirs of P and R, while K(a) (λ) are, for a = 2, . . . , n, the non-trivial polynomial Casimirs of the pencil R − λP. Let us consider the n2 matrices Dij defined by  α Dij β = Lie Z αj (Kβ(i) (λ)). (3.13) The proof of the following Proposition, which is based on a few elementary  properties of the matrices Dij is contained in Appendix C. Proposition 3.6 The determinant Da (λ) of the matrices factors as



 Daa , a = 2, . . . , n,

Da (λ) = (λ − a + 2)r−1 a (λ),

(3.14)

where a (λ) is a monic polynomial of degree r(r − 1)/2. Let λas , a = 2, . . . , n, s = 1, . . . , r(r − 1)/2, be the roots of a (λ), and let us consider a row (say, the s & & first) δα (λ) = D aa (λ) 1,α of the adjoint matrix Daa (λ) of Daa (λ). Finally, let ξa be s the functions obtained by evaluating in λ = λa the ratios δr−2 (λ)/δr−1 (λ). Then, these (n − 1)r(r − 1)/2 pairs of functions {ξas , λas } satisfy 1) The Jacobi separation relations Det(Ma (ξas , λas )) = 0. 2) The differential relations ! as = λas Pdλas , Rdλ

! as = λas Pdξas . Rdξ

In particular, their brackets, (say, with respect to the Lie Poisson structure P), are of the separate form: {λsb , ξat } P = δ st δab ϕas (ξas , λas ),

(3.15)

where ϕas (ξas , λas ) are functions of the two variables (ξas , λas ). The meaning of this Proposition can be rephrased as follows. For every integer a = 2, . . . , n we can construct a Lax matrix, whose characteristic polynomial gives us a family of Casimirs of the bihamiltonian pencil R − λP defined on the manifold M = sl(r)n . Separated coordinates are constructed, according to the bihamiltonian scheme, by deforming such Casimirs along normalized generators of a suitable distribution Z defined in M. In particular, for each a = 2, . . . , n, we can algebraically construct a ‘cluster’ of (r(r − 1)) variables {λas , ξas }s=1,...r(r−1)/2 that are, in the terminology of [20], algebro-geometrical Nijenhuis coordinates, that is, satisfy properties 1 and 2 of Proposition 3.6.

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To finish our job we have to: (a) Discuss about the coordinates associated with the global gauge invariance of the Gaudin Systems (b) Explicitly construct, out of the coordinates found so far, a set of canonical separated coordinates (that is, a set of Darboux-Nijenhuis coordinates). Point (a) can be solved as follows. One notices thatany function ϕ n depending only on the ‘global’ matrix variable AT = i=1 Ai , which is invariant along the distribution Z satisfies the differential relation ! = (n − 1)Pdϕ. Rdϕ

(3.16)

In particular, this family includes the mutually commuting Hamiltonians of Gel’fand-Cetlyn type discussed in the Remark after Proposition 2.1. The property (3.16) follows from the fact that they trivially satisfy the relation Rdϕ = (n − 1)Pdϕ w.r.t. the undeformed pencil, and from the property that ϕ commutes with all the Hamiltonians of the hierarchy. Inside this ring of functions one can find a set of r(r − 1)/2 canonical coordinates that complement the Gel’fand-Cetlyn Hamiltonians. Thanks to (3.16) they will have vanishing Poisson brackets with the Nijenhuis coordinates of Proposition 3.6. The solution to point (b) above can be simply done by means of a direct computation of the Poisson brackets between ξas and λas . In particular, this computation will implicitly prove that these quantities are functionally independent. Proposition 3.7 The Poisson brackets, w.r.t. the Lie Poisson pencil P of the coordinates λas , ξas defined above are given by {λas , ξas } = (λas − a + 2)(λas − a + 1)

(3.17)

Proof The proof of this proposition follows verbatim that of Theorem 1.3 in [1, 2], to which we refer for full details. Indeed, the coordinates λas , ξas can be !a (λ, ξ ). So we can seen as common zeroes of the first row of the matrix M apply all the considerations of [1, 2], the only difference being that the Poisson brackets of the entries of the matrix Ma (λ, ξ ) are given by: ( ' ij Ma (λ, ξ ), Makl (σ, η) =    a−1    = tr (λ − a + 2)(σ − a + 2)Aa + Ar ekjδli − eli δkj =

=

r=1

1 (λ − a + 1)(σ − a + 2)(Majk (λ, ξ )δil − Mail (λ, ξ )δ jk ) + λ−σ

+ (λ − a + 2)(σ − a + 1)(Mail (σ, η)δ jk − Majk (σ, η)δil ) . (3.18)

On separation of variables for homogeneous SL(r) Gaudin systems

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The presence of the factors (λ − a + 2)(σ − a + 1) and (λ − a + 2)(σ − a + 1) is responsible for the factor (λas − a + 2)(λas − a + 1) in (3.17).  

4 Examples In this Section we will specialize the constructions presented in the paper for the cases of sl(2) and sl(3). 4.1 The sl(2) Case Here we briefly reframe the explicit computations of the last Section of [11] within the formalism exposed in this paper. We consider the manifold M = sl(2)n , endowed with the Poisson pencil R − λP of Section 2. It is explicitly parametrized in terms of the n matrices   hi ei . (4.1) Ai = fi − hi The generic symplectic leaf S of P is a 2n dimensional symplectic manifold, defined by the equations ) i = 1, . . . , n, Ci = 1 2 TrAi2 = hi2 + ei fi , and can be (generically) endowed with the 2n coordinates (hi , fi ), i = 1, . . . , n. A set of normalized transverse vector fields are given in this case by Zi =

1 ∂ , fi ∂ei

(4.2)

The matrices La are explicitly given by    a−1 h (λ − a + 2)e + e (λ − a + 2)ha + a−1 b a b b =1 b =1 La (λ) = .  a−1 (λ − a + 2) fa + a−1 f −((λ − a + 2)h + b a b =1 b =1 hb )

(4.3)

As canonical coordinates associated with the global SL(2) invariance one can choose the two functions n n  hi fi , φ1 = i=1 . λ1 = n i=1 fi i=1 The non-trivial separation coordinates are gotten simply considering the zeroes za of the elements [La ]2,1 , and the values μa on these zeroes of the elements [La ]2,2 , normalized as in the previous Section. One sees that a−1 za = −

k=1

fa

fk

+ (a − 2),

a = 2, . . . , n

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Shifting these values by the unessential term a − 2, we find that the separation coordinates are given, for a = 2, . . . , n, by  a−1 fk λa ha + a−1 k=1 hk (4.4) λa = − k=1 , μa = − fa λa (λa − 1) They fulfill the separation relations ⎛ ⎛  2 ⎞⎞   a−1 a−1   1 ⎝Ca2 λa2 + Tr Aa μa2 = λa + Tr ⎝ Ab A b ⎠⎠ . 2(λa (λa − 1))2 b =1

b =1

(4.5) In other words, the separation coordinates are coordinates of suitable points on the rational curves (4.5). The corresponding Hamilton-Jacobi equations can be explicitly solved by means of algebraic functions. 4.2 The sl(3) Case We consider the Poisson manifold M = sl(3)n , endowed with the Poisson pencil R − λP and parametrized by the n matrices ⎞ ⎛ e3,i h1,i e1,i ⎟ ⎜ ⎟ (4.6) Ai = ⎜ ⎝ f1,i h2,i − h1,i e2,i ⎠ i = 1, . . . , n. f3,i f2,i

−h2,i

On this manifold the Poisson tensor P has 2n Casimirs: )   )   Ci2 = 1 2 Tr (Ai2 ) , Ci3 = 1 3 Tr (Ai )3 i = 1, . . . , n.

(4.7)

The a−1characteristic polynomials of the Lax matrices La = (λ − a + 2)Aa + b =1 Ab are expressed as  a (μ, λ) = μ3 − μH2(a) (λ) − H3(a) (λ).

(4.8)

The transversal distribution Z is generated by the 2n flat generators:

) ∂ ∂ 2 + ( f2,i h1,i − f3,i e1,i ) Z i = 1 d ( f3,i (h1,i − h2,i ) + f2,i f1,i ) ∂e2,i ∂e3,i

) ∂ ∂ Z i3 = 1 d f2,i − f3,i ∂e3,i ∂e2,i 2 2 f1,i − f3,i e1,i d = f2,i f3,i (2h1,i − h2,i ) + f2,i

The symplectic leaves of P are generically parametrized by matrices Ai of the form: ⎛ ⎞ h1,i e1,i 3,i ⎜ ⎟ ⎜ f1,i h2,i − h1,i 2,i ⎟ ⎝ ⎠ f3,i f2,i

− h2,i

On separation of variables for homogeneous SL(r) Gaudin systems

249

where 2,i and 3,i are suitable functions of the coordinates h1,i , h2,i , f1,i , f2,i , f3,i , e1,i , parametrically depending on the Casimirs (4.7). The coordinates {λas , ξas } can quite explicitly be found by means of the following steps: !a (λ, ξ ). We consider the matrix Ma (λ, ξ ) = ξ − La (λ) and its adjoint M ! We have to look for the common zeroes of the elements Ma (λ, ξ )3,1 and !a (λ, ξ )3,2 , that is, for the common zeroes of M     −La (λ)2,1 ξ − La (λ)2,2 ξ − La (λ)1,1 −La (λ)1,2 Det , Det . (4.9) −La (λ)3,1 − La (λ)3,2 −La (λ)3,1 −La (λ)3,2 Taking into account the form of the vector fields Z aα and of the characteristic polynomial (4.8), we can identify the system (4.9) with , ξ Lie Z a2 H2(a) + Lie Z a2 H3(a) = 0 , (4.10) ξ Lie Z a3 H2(a) + Lie Z a3 H3(a) = 0 where Det(Ma (ξ, λ)) = ξ 3 − H2(a) ξ − H3(a) .

(4.11)

As we have noticed in Section 3.1 , we can factor out (λ − a + 2) from each line of this system, and consider, in matrix form the equivalent system:   a a ξ, 1| G2,2 G2,3 , with Gaα,β = Lie Z aα Hβ(a) /(λ − a + 2). (4.12) Ga3,2 Ga3,2 We notice that Gaα,β are polynomials in λ of degree α − 1, so that the three zeroes λa1 , λa2 , λa3 of the equation   a G2,2 Ga2,3 =0 (4.13) a (λ) = Det Ga3,2 Ga3,2 are the compatibility condition for the system (4.10); the corresponding coordinates ξa1 , ξa2 , ξa3 are thus given by, e.g.,  ξas = −Ga3,3 /Ga2,3 λ=λs (4.14) a

We remark that our procedure for finding separation coordinates exactly matches the one introduced, in the framework of r-matrix theory, in [6, 7]. Finally, defining ζas = λas − a + 2, a = n, . . . , n and considering the pairs {ζas , ρas } = ξas /ζas (ζas − 1),

s = 1, 2, 3,

a = 2, . . . , n,

we see that the solution W of the (stationary) Hamilton-Jacobi equations of the sl(3) Gaudin model can be expressed as:   3 - s n 3   Pa ξ dζ  + HTs qsT , (4.15) W= ζ (ζ − 1) α=s a=2 s=1 where Paα denotes the point (ξaα , ζaα ) on the genus g = 1 algebraic curve defined by (4.11), and HTs and qTS denote, respectively, a suitable complete family of

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Gel’fand-Cetlyn Hamiltonians associated with the global SL(3) invariance of the model, and their canonically conjugated variables. Standard arguments show that the linearization of the flows associated with the mutually commuting Hamiltonians we have considered in this paper, and hence also the flow associated with the ‘physical’ Hamiltonian of the sl(3) Gaudin system can be achieved by means of the Abel maps associated with the differentials (the last four being of the third kind) dζ , ξa

ζ dζ , (ζ − 1)ξa

where ξa =

dζ , (ζ − 1)ξa

dζ , ζ (ζ − 1)ξa

ξ dζ , (4.16) ζ (ζ − 1)ξa

∂Det(Ma (ξ, λ)) . The case of sl(r), r > 3 can be treated analogously. ∂ξ

5 Conclusion and Discussion In this paper we have reconsidered the SoV problem for the (homogeneous XXX) Gaudin systems based on the Lie algebras sl(r), from a particular standpoint. Namely, we considered the n-site system as a Gel’fand-Zakharevich system defined on the manifold M = sl(r)n , with respect to the Poisson pencil Pλ = R − λP, where P is the usual Lie Poisson bracket on M, while R, given by (2.4) and (2.5) is a further linear Poisson structure on M that was introduced in [11]. We showed that the system admits a set of n − 1 ‘Lax’ matrices, linear in the spectral parameter λ that evolve according to Lax equations along any vector field that is Hamiltonian with respect to the Poisson pencil Pλ . Thanks to this property, by using the bihamiltonian set-up for SoV, we managed to define a set of separating coordinates, quite explicitly given in (4.4) for the sl(2) case and by (4.13) and (4.14) for the sl(3) case. We notice that this set of coordinates provide an alternative set of separation coordinates for the Hamilton-Jacobi equations associated with the Gaudin Hamiltonian w.r.t. the coordinates that can be found by means of the conventional approach based on the rational Lax matrix (1.2), i.e., L(λ) =

n  i=1

Ai . λ − ai

(5.1)

In this respect, a few remarks are in order: First we notice that the existence of different sets of separation coordinates is to be ascribed to the super-integrability of the system. This is particularly clear in the bihamiltonian setting, where the separation coordinates are obtained, by means of the procedure outlined in Section 3, from the polynomial Casimirs of a Gel’fand-Zakharevich Poisson pencil. The general problem of the connections between super-integrability and ‘multi-separability’ is, to the best of our knowledge, still an open problem (see, [25] and the references quoted therein). In particular, the classification problem for these systems have been solved only for systems with a small number of degrees of freedom.

On separation of variables for homogeneous SL(r) Gaudin systems

251

The homogeneous XXX Gaudin models are systems with an arbitrarily high number of degrees of freedom where the connection between super integrability and multi-separability happens, and their study might shed light on the structural properties of this phenomena. The second remark is the following. The SoV scheme based on the Lax matrix (5.1) leads to the definition of a divisor of degree d R = r(r − 1)(n − 1)/2 on the spectral curve R(λ, μ) = Det(μ − L(λ)) (see, e.g., [19]). It is not difficult to ascertain that the genus of this spectral curve is, for g = sl(r) gL =

 (r − 1)  (n − 2)r + (r − 2) , 2

that is, it grows linearly with the number n of sites of the model. As we have shown in Section 3.1 and exemplified in Section 4 for r = 2, 3, the SoV scheme herewith outlined leads to consider the set of n − 1 Lax matrices La (λ) given by La (λ) = (λ − a + 2)Aa + Ba ,

Ba =

a−1 

Ab .

(5.2)

b =1

The separation coordinates parametrize sets of degree r(r − 1)/2 divisors on the spectral curves  a (λ, μ) = Det(μ − La (λ)). We see that the genus of such curves is ga = (r − 1)(r − 2)/2, that is depends only on the rank r of the algebra, and not on the number n of sites, showing that the equations of motions can be explicitly solved by means of θ functions of genus ga for all n’s. This also implies the existence of canonical transformations between (suitable open sets of) the degree dL Jacobian of the spectral curve R(μ, λ) associated with L(λ) and the (corresponding open subsets) of the Cartesian product of the degree r(r − 1)/2 Jacobians associated with the curves  a (λ, μ).2 This simply follows from the fact that both the algebro-geometrical coordinates found from L(λ) and those we discovered in this paper are canonical coordinates for the standard Lie-Poisson bracket P0 on sl(r)n , and, in particular, (together with those coordinates associated with the global SL(r) invariance) are Darboux coordinates for the restriction of P0 to its generic symplectic leaves. It is outside the size of this paper to fully discuss this issue here. However we think it is appropriate to display this transformation in the simple case of sl(2). This goes as follows.

2 This

observation is due to B. Dubrovin.

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We recall that the n matrices Ai can be explicitly parametrized by means of 3n coordinates hi , ei , fi , i = 1, . . . , n:  Ai =

ei

hi

fi −hi

 ,

with Lie-Poisson brackets given by {hi , e j} P = δije j,

{hi , f j} P = −δij f j,

{ei , f j} P = 2δijh j

Let us denote with λi , μi the set of separation variables associated with the Lax matrix L(λ) see (5.1) of the n-particles sl(2)-Gaudin model and with ζi , ρi those associated with the bihamiltonian picture discussed in this paper. We can assume that the coordinates associated with the global SL(2) invariance of the problem are the same; in particular, on the symplectic leaves of the Lie Poisson tensor P0 we have to consider the pair λ1 = ζ1 =

n 

n j=1

hj

j=1

fj

μ1 = ρ1 = n

f j,

j=1

According to the ‘Sklyanin’ recipe, the λa , a = 2, . . . , n are the zeroes of the (1, 2) entry of the rational Lax matrix (1.2), i.e. the roots of the polynomial: L (λ) ≡

n−1 

n−1 

1 λ Ck = n k

fi

i=1

k=0

n  λ (−1)n−k−1 sn−k−1 (a1 , . . . , a.j, . . . , an ) f j.

k=0

k

j=1

(5.3) Here the polynomials sn−k−1 (a1 , . . . , aˆi , . . . , an ) are the elementary symmetric polynomials in n − 1 letters b1 , . . . , bn−1 , defined by n−1 /

n−1  (λ − b j) = (−1)n− j−1 sn− j−1 (b1 , . . . , bn−1 )λ j,

j=1

where s0 ≡ 1,

(5.4)

j=0

evaluated for b1 = a1 , . . . , b j−1 = a j−1 , b j = a j+1 , . . . , bn−1 = an . We can express the ‘physical’ coordinates fi in terms of the λi as follows: n

j=1

 j 1 ai C j, k=i (ai − ak ) j=0 n−1

fi fj

=0

n 

f j = λ1 .

(5.5)

j=1

Since the coordinates ζi are rational functions of the fi alone, and namely, a−1 ζa = −

k=1

fa

fk

,

a = 2, . . . , n,

On separation of variables for homogeneous SL(r) Gaudin systems

253

we can explicitly find the transformation yielding the ζi in terms of the λi as:   n−1 j b −1 n− j−1  sn− j−1 (λ2 , . . . , λn ) l=b (ab − al ) j=0 ak (−1) ζb = − · n−1 j , (5.6) n− j−1 s l=k (ak − al ) n− j−1 (λ2 , . . . , λn ) j=0 a (−1) k=1

b

for b = 2, . . . , n, with, obviously ζ1 = λ 1 .

(5.7)

The variables μa , a = 2, . . . , n are the values of the (1, 1) entry of the rational Lax matrix (1.2) for λ = λa , while the ρa are be given by the values of the (1, 1) entry of the Lax matrix La in λ = ζa , divided by the normalizing factor ζa (ζa − 1). Explicitly: n n  hj j=1 h j μa = a = 2, . . . , n, μ1 = λ − aj λ1 j=1 a ρa =

ζa ha +

a−1 j=1

n

hj

a = 2, . . . , n,

ζa (ζa − 1)

ρ1 =

j=1

hj

ζ1

The transformation of coordinates connecting the ρi and the μi can be easily found noticing that they are connected to the coordinates hi by a linear transformation with coefficients depending on ζi and λi , respectively. Consequently, the transformation between the coordinates ρi and μi is a linear transformation with λi depending coefficients: ρi =

n 

Aij(λ1 , . . . , λn )μ j

(5.8)

j=1

and it follows that it must be the lifting of the transformation defined by (5.6– 5.7) among the ζi and the λi ; ρi =

n 

((J t )−1 )ijμ j

(5.9)

j=1

where J is the Jacobian of the transformation (5.6). Finally, we just mention some other problems which remain open. The first one is to compare our results with the picture of the generalized bending flows of [14]. The setting presented in that paper was aimed at providing a generalization of the previous paper [22], and, in our setting, should be obtained by reduction to the submanifold of matrices Ai having rank equal to one. The second one is the application of the scheme herewith presented to the quantum sl(r) case. Preliminary results for r = 3 indicate that this should be a viable procedure for giving explicit expressions to the quantum integrals whose existence has been proven in [8]. Work in both these directions is in progress.

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Acknowledgements We thank B. Dubrovin, J. Harnad, F. Magri, M. Pedroni, and O. Ragnisco for useful discussions. This work was partially supported by GNFM-INdAM, by the Italian MIUR under the project Geometry of Integrable Systems, and by the ESF project MISGAM and by the EC-FP6 Marie Curie RTN ENIGMA.

Appendix A: Proof of Proposition 3.4 The proof of Proposition 3.4 is divided into a couple of steps. We recall that we are considering a corank k affine bihamiltonian manifold (M, P1 − λP0 ), endowed with a transversal distribution Z , satisfying the requirements of Proposition 3.2. Z a , a = 1, . . . , k is a set of normalized flat generators for Z , and H (a) (λ) = λna H0a + · · · + Hnaa , are polynomial Casimirs of P1 − λP0 . Finally, the k2 polynomials Dab (λ) = Lie Z b H (a) (λ) = λna δba − Dba, 1 λna −1 − · · · − Dba, na .

(A.1)

are their deformations along the flat generators. Finally, we recall the definition !1 = P1 − P

k 

Xa ∧ Z a ,

Xa = P1 dH0a .

a=1

!1 and P0 on the deformation of the Casimirs of the Lemma 1 The actions of P pencil are related by the following formula: !1 dDa (λ) = λP0 dDa (λ) + P b b

k 

Dac (λ)P0 dDbc , 1 .

(A.2)

c=1

Proof We limit ourselves to sketch the proof of this Proposition, which is essentially contained in Section 7 of [13], although in a disguised form. We consider the characteristic property of a Casimir of the Poisson pencil, Pλ dH (a) (λ) = 0 and derive it w.r.t. Z b . We get: Lie Z b (Pλ ) dH (a) (λ) + Pλ dDab (λ) = 0. (A.3)  Since Lie Z b (Pλ ) = c [Z b , Xc1 ] ∧ Z c with Xc1 = P1 dH0c = P0 dH1c , we see that [Z b , Xc1 ] = Lie Z b (Xc1 ) = Lie Z b (P0 dH1c ) = −P0 dDbc , 1 . Thus (A.3) takes the form   Pλ dDab (λ) − Dac (λ)P0 dDbc , 1 − [Z b , Xc1 ], dH (a) (λ) · Z c = 0. c

c

(A.4)

On separation of variables for homogeneous SL(r) Gaudin systems

255

Let us consider the coefficient [Z b , Xc1 ], dH (a) (λ) in the last sum. This is, by definition, Lie[Z b ,Xc1 ] (H (a) (λ)) = Lie Z b Lie Xc1 (H (a) (λ)) − Lie Xc1 Lie Z b (H (a) (λ)). Since {H1c , H (a) (λ)}0 = 0 only the second term is non-identically vanishing, and equals −Lie Xc1 (Dab (λ)). Furthermore, thanks to the affinity of the GZ manifold, we see that all terms of the form Lie Z c (Dab (λ)) identically vanish. So, we see that (A.4) can be written as   Pλ dDab (λ) − Dac (λ)P0 dDcb , 1 − (Xc1 ∧ Z c ) · (dDab (λ)) = 0, (A.5) c

c

 

which, in view of (3.5), yields the statement. Proposition 5.1 Let D be a k × k polynomial matrix of the form Dab (λ) = λna δba − λna −1 Dba, 1 − · · · − Dba, na ,

a, b = 1, . . . , k,

(A.6)

where the Dab , p are smooth independent functions on a bihamiltonian manifold M, satisfying (A.2). Then: 1. Its determinant (λ) has the form (λ) = λν − 1 λν−1 + · · · + ν , with ν =

 a

(A.7)

na and satisfies !1 d (λ) = λP0 d (λ) + (λ)P0 d 1 . P

(A.8)

2. The roots λi of (λ) satisfy !1 dλi = λi dλi . P

(A.9)

! ! 3. Let D(λ) denote the classical adjoint matrix of D(λ), and let [ D(λ)] a,c be ! ! non-identically vanishing. Then any ratio ρ(λ) := [ D(λ)] /[ D(λ)] a,b a,c of ! elements belonging to the a-th row of D(λ), evaluated at the roots λi of (λ) satisfy the equation !1 dρ(λi ) = λi P0 dρ(λi ). P

(A.10)

Proof The power expansion (A.6) simply states that the (a, a) entry of Dab is a monic degree na polynomial, while all other entries in the ath row are of degree not exceeding na − 1. We preliminarily notice that (λ) =

k / a=1

Daa (λ) + O(λν−2 ),

whence 1 =

k  a=1

Daa, 1 .

(A.11)

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! to We multiply the matrix (A.2) say, on the left, by the classical adjoint D, get   !1 ( D !ac d(λ)Dc (λ)) − ! ac dDc (λ) = P λP0 D(λ) b b c

=



c

!ac (λ)Dc (λ)P0 dDd . D d b, 1

(A.12)

c,d

Recalling that

 c

!ac (λ)Dc (λ) = δad (λ) and D d  a !c (λ)dDac (λ) = Tr D(λ)dD ! D = d (λ), a,c

taking the trace of the matrix (A.12) and taking into account (A.11) we get the proof of the first item. To prove item # 2, we first notice that, for any function f on M, the evaluation of a polynomial (or rational) function F(λ) in the parameter λ, whose coefficients are themselves functions on M gives rise to a new function F( f ) on M. Its differential can be written as follows:  ∂ F(λ)   + d(F( f )) = dF(λ)  df λ= f ∂λ λ= f To clarify the notations, the first term in the RHS of the above equations means the differential of F(λ), with λ taken as a parameter, then evaluated for λ = f , and the second the partial derivative of F(λ) w.r.t. λ, subsequently evaluated for λ = f . Keeping this proviso in mind, we consider now F(λ) = (λ), and f = λi , i = 1, . . . , n. We have:  ∂ (λ)   + dλi , (A.13) d( (λi )) = d (λ)  λ=λi ∂λ λ=λi  where d (λ) = − j λ p− jd j. Taking into account the relation (A.13), we get !1 − λi P0 )d (λi ) = (λi )P0 d 1 + 0 = (P

∂ (λ)  !1 − λi P0 )dλi , (A.14) (P  ∂λ λ=λi

which implies the assertion, since (λi ) = 0,

while

∂ (λ)  = 0,  ∂λ λ=λi

thanks to the fact that being the coefficients a,i functionally independent, the roots are generically simple. The proof of the third assertion is basically contained in the proof of Proposition 8.4 of [13]. We limit ourselves to sketch it. By using the relations (A.2) and (A.8), together with the defining relation !ac (λ)Dc (λ) = δad (λ), one arrives at the matrix equation D d     !1 d D !ac − λP0 d D !ac Dc = (λ)P0 d 1 δ a − dDa . P (A.15) b b b ,1 c

On separation of variables for homogeneous SL(r) Gaudin systems

257

! If σ denotes one row of the adjoint matrix D(λ), we can rewrite the above equation as   !1 σ − λP0 σ D(λ) = (λ) · X P where X is the corresponding row of the RHS of eq. (A.15). If we consider the normalized row ρ = σ /σ j, we see that, since ρ D(λ) = (λ)σ /σ j, it holds 

 !1 ρ − λP0 ρ D = (λ) · Y, P

(A.16)

for some suitable Y whose form is irrelevant here. Evaluating this equation for λ = λi , we see that    !1 ρ − λP0 ρ D(λ) P =0 (A.17) λ=λi

Taking into account that has simple eigenvalues, we see that each row !1 dρ − λP0 dρ), evaluated at λ = λi must be proportional to the correspond(P ! ing row of D(λ), that is, there must exist vector fields X such that    !1 dρ − λP0 dρ |λ=λi = X · ρ λ=λi . P Since one element of ρ is normalized to 1, we thus see that X must vanish, whence the thesis.  

Appendix B: Proof of Proposition 3.5 The key point is the following observation on the (ordinary) Lie-Poisson brackets on a single copy of M = sl(r). The Poisson bracket of two functions ∂G ∂ F ∂G ∂F · [A, ]) = −Tr(A · [ , ]). Let F, G on M, is given by {F, G} = Tr( ∂A ∂A ∂A ∂A Aij denote the ijth entry of A, and consider the family of r − 1 vector fields ∂ on M defined by W α = , α = 1, . . . , r − 1, as well as the distribution ∂ Ar,α Z ⊂ T M defined by the W α . We notice that differentials of functions vanishing along Z admit a very simple matrix representation. Indeed W α is represented via its action on the matrix A as the elementary matrix eα,r , having 1 in the α-th place of the last  ∂F ∂F lies in column. So LieW α (F) = 0 iff = 0, a = 1, . . . , r − 1, i.e., iff ∂ A r,a ∂A the lower maximal parabolic subalgebra p− of sl(r). Let now W denote any element in Z , and let F, G be functions such that Lie Z F = Lie Z G = 0, and let us compute Lie Z ({F, G}). Thanks to the Leibniz

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property of the Lie derivative and the fact that Z is a constant vector field we have that 

 ∂ F ∂G LieW ({F, G}) = −Tr LieW (A) · , (B.1) ∂A ∂A which vanishes as well since p− is indeed a Lie subalgebra of sl(r). In the case of the n-particle sl(r) Gaudin model, whose phase space is parametrized by n matrices Ai , we consider the family of n · (r − 1) vector fields defined by LieWiα (A1 , . . . , An ) = (0, 0, · · · , eα,r , · · · , 0). "#

%$(B.2)

i-th place

The distribution Z generated by these vector fields is generically transversal to the symplectic leaves of the Lie-Poisson product structure on sl(r) N . We now prove that the space of functions vanishing along Z is a Poisson subalgebra for any affine Poisson tensor Q. The brackets {F, G} Q = dF, QdG are given by the multiple sum   N



 N   ∂F ∂G ∂G k k Tr · ci, j Ak , + di, j σk , {F, G} Q = ∂ Ai ∂ Aj ∂ Aj i, j,k=1

k=1

where σk denote constant matrices. Noticing that the differentials of functions F vanishing along Z are represented by n-tuples of matrices dF = ∂F ∂F ∂F ∂F , ,..., ) with ∈ p− , i = 1, . . . , N, we see that the Lie deriv( ∂ A1 ∂ A2 ∂ An ∂ Ai atives LieWiα {F, G} Q are given by multiple sums of terms like those of (B.1), and so vanish whenever LieWiα F = LieWiα G = 0.  

Appendix C: Proof of Proposition 3.6 The proof of Proposition 3.6 follows from a few elementary but important facts following from the definitions of the Casimirs of the Poisson pencil K(i) (λ) and of the normalized transversal vector fields Z iα , α = 1, . . . , r − 1, i = 1, . . . , n. Recall that we defined n2 matrices Dij by  α (C.1) Dij β = Lie Z β (Kα(i) (λ)), j

where the polynomial Casimirs Kα(i) are defined by Det(Mi (λ, ξ )) = ξ r +

r−1 

Kα(i) (λ)ξ r−α−1 .

(C.2)

α=1

One has:

  1. The matrix D11 is the identity. This trivially follows form our choice of L 1 = A1 .

On separation of variables for homogeneous SL(r) Gaudin systems

259

2. The n(r − 1) × n(r − 1) matrix of the deformations of the Casimirs w.r.t. the transversal vector fields has the following block form: ⎤ ⎡ ··· ··· 0 D11 0 ⎢ ⎥ ⎢ D D 0 ··· 0 ⎥ ⎥ ⎢ 21 22 ⎢ ⎥ (C.3) D=⎢ ⎥ . . .. ⎥ ⎢ .. . . ⎢. ⎥ 0 ⎣ ⎦ Dn1 Dk3 · · · Dnn This follows form the fact that, for j > i and every α, Lie Z αj L j(λ) = 0. Thanks to point 1 above, we now consider the non-trivial matrices Ma (λ, ξ ), a = 2, . . . , n, and the corresponding fields Z bα . Taking into account that: !a (λ, ξ )Lie Z α (La )); (a) Lie Z bα (Det(Ma (λ, ξ ))) = −Tr( M b (b) Lie Z aα (La ) = (λ − a + 2)Lie Z aα (Aa ); (c) The determinants of the diagonal blocks DetDaa = Da (λ) are monic polynomials of degree r(r + 1)/2 − 1 in λ; We can factorize Da (λ) as Da (λ) = (λ − a + 2)r−1 a (λ)

(C.4)

where a (λ) is a monic polynomial of degree r(r − 1)/2. 3. Thanks to the lower diagonal block form of the matrix D of (C.3), every diagonal block Daa satisfies Proposition (1). So its determinant satisfy, ! − λP)dDa (λ) = Da (λ)PdDa , and thanks according to Proposition 5.1 ( R 1 to the factorization property (C.4), we have   ! − λP d a (λ) = a (λ)Pd a1 . R (C.5) We now recall that the Casimir functions of P are given by the highest order (i) terms Kα,0 of the expansion of the Casimirs of the Poisson pencil Kα(i) (λ) in powers of λ. If we call α,β

Gi

(i) = LieWiα Kβ,0

(C.6)

the matrix of the deformations of the Casimirs of P with respect to the ( j) corresponding vector fields Wiα introduced in (3.10), noticing that LieWiα Kβ,0 α vanishes for j = i, we see that the normalized generators Z i and the ‘constant’ ones Wiα are related by  α,β β Wiα = Gi Z i i = 1, . . . , n. β

Thus, considering only the nontrivial indexes a = 2, . . . , n,  LieWaα Det(Ma (ξ, λ)) = Gaα,β Lie Z aα Det(Ma (ξ, λ)), β

(C.7)

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G. Falqui, F. Musso α,β

we can argue as follows. Since Ga are independent of λ, we see that the common solutions (ξa , λa ) of the two sets of r − 1-tuple of equations ⎧ ⎪ ⎨ LieWa2 Det(Ma (ξ, λ)) .. (C.8) . ⎪ ⎩ LieWar Det(Ma (ξ, λ)) and

⎧ ⎪ ⎨ Lie Z a2 Det(Ma (ξ, λ)) .. . ⎪ ⎩ Lie Z ar Det(Ma (ξ, λ))

(C.9)

coincide for every (fixed) a = 2, . . . , n. Expanding the RHS of the equations of the system (C.9) as Lie Z aα Det(Ma (ξ, λ)) =

r−1 

ξ r−1−β Lie Z aα Kβ(a) =

r−1 

β=1

ξ r−β (Da )αβ ,

(C.10)

β=1

we see that: (a) The roots of Da (λ), introduced in (C.4) are those values of λ for which the r − 1 equations, defined for a = 2, . . . , n, r−1 

ξ r−1−α (Da )αβ = 0

(C.11)

β=1

admit solutions. In particular, the roots of a (λ) define non-trivial elements λas , s = 1, . . . , r(r − 1)/2. (b) The values ξas corresponding to the roots λas of a (λ) are given by suitably !a , evaluated at λ = λas . normalized elements of the adjoint matrix D Thus, from the bihamiltonian theory, we can conclude that the only nonvanishing Poisson brackets between such functions admit the separate form: {λsb , ξat } P = δ st δab ϕas (ξas , λas ),

{λsb , ξat } R! = δ st δab λas ϕas (ξas , λas )

(C.12)

We now consider (C.8), taking into account the observation that the pairs (λsb , ξat ) are solutions of this system as well. We notice that from the definition of Waα , LieWaα Det(Ma (ξ, λ)) is nothing but the determinant of the minor of Ma (ξ, λ) relative to the α, r entry. Since the r(r − 1)/2 pairs (ξia , λia ), for every fixed a, annihilate the r − 1 minors of the matrix Ma (ξ, λ) relative to the entries (1, r), . . . (r − 1, r), they annihilate the minor relative to the (r, r) entry as well. Hence they annihilate the last row of the adjoint matrix of Ma (ξ, λ), and so satisfy the characteristic equation   Det Ma (ξas , λas ) = 0, a = 2, . . . , n, s = 1, . . . , r(r − 1)/2.

On separation of variables for homogeneous SL(r) Gaudin systems

261

References 1. Adams, M., Harnad, J., Hurtubise, J.: Darboux coordinates and Liouville-Arnold integration in loop algebras. Comm. Math. Phys. 155, 385–413 (1993) 2. Adams, M., Harnad, J., Hurtubise, J.: Darboux Coordinates on Coadjoint orbits of Lie Algebras. Lett. Math. Phys. 40, 41–57 (1996) 3. Ballesteros, A., Ragnisco, O.: A systematic construction of completely integrable Hamiltonians from coalgebras. J. Phys. A. 31, 3791–3813 (1998) 4. Błaszak, M.: Bi-Hamiltonian separable chains on Riemannian manifolds. Phys. Lett. A. 243, 25–32 (1998) 5. Degiovanni, L., Magnano, G.: Tri-hamiltonian vector fields, spectral curves, and separation coordinates. Rev. Math. Phys. 14, 1115–1163 (2002) 6. Diener, P., Dubrovin, B.: Algebraic geometrical Darboux coordinates in R-matrix formalism SISSA preprint 88/94/FM. (2004) 7. (See also the appendix of) Dubrovin, B., Mazzocco, M.: Canonical structure and symmetries of the Schlesinger equations. math.DG/0311261 8. Feigin, B., Frenkel, E., Reshetikhin, N.: Gaudin model, Bethe Ansatz and critical level. Comm. Math. Phys. 166, 27–62 (1994) 9. Falqui, G., Magri, F., Tondo, G.: Bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy. Theoret. Mat. Fiz. 122, 212–230 (2000) 10. Falqui, G., Magri, F., Tondo, G.: Bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy. Translation in Theoret. and Math. Phys. 122, 176– 192 (2000) 11. Falqui, G., Musso, F.: Gaudin models and bending flows: a geometrical point of view. J. Phys. A.: Math. Gen. 36, 11655–11676 (Also at nlin.SI/0306006) (2003) 12. Falqui, G., Pedroni, M.: On a poisson reduction for Gel’fand-Zakharevich manifolds. Rep. Math. Phys. 50, 395–407 (also at nlin.SI/0204050) (2002) 13. Falqui, G., Pedroni, M.: Separation of variables for bi-Hamiltonian systems. Math. Phys. Anal. Geom. 6, 139–179 (Also at nlin-SI/0204050) (2003) 14. Flaschka, H., Millson, J.: The moduli space of weighted configurations on projective space. math.SG/0108191 (2003) 15. Gaudin, M.: La Fonction d’ Onde de Bethe. Masson, Paris (1983) 16. Gekhtman, M.I.: Separation of variables in the classical SL(N) magnetic chain. Comm. Math. Phys. 167, 593–605 (1995) 17. Gel’fand, I.M., Zakharevich, I.: On the local geometry of a bi-Hamiltonian structure. In: Corwin, L., et al. (eds.) The Gel’fand Mathematical Seminars 1990–1992, pp. 51–112. Birkhäuser, Boston (1993) 18. Gel’fand, I.M., Zakharevich, I.: Webs, Lenard schemes, and the local geometry of bihamiltonian Toda and Lax structures. Selecta Math. (N.S.) 6, 131–183 (2000) 19. Harnad, J.: Loop groups, R-matrices and separation of variables. In: Harnad, J., et al. (eds.) Integrable Systems: From Classical to Quantum, CRM Proceedings & Lecture Notes, AMS, vol. 26, pp. 22–54. Providence, RI (2000) 20. Harnad, J., Hurtubise, J.: Multi-Hamiltonian strictures for r-matrix systems. math-ph/0211076 (2003) 21. Jurˇco, B.: Classical Yang-Baxter equations and quantum integrable systems. J. Math. Phys. 30, 1289–1293 (1989) 22. Kapovich, M., Millson, J.: The symplectic geometry of polygons in Euclidean space. J. Differential Geom. 44, 479–513 (1996) 23. Magri, F., Falqui, G., Pedroni, M.: The method of poisson pairs in the theory of nonlinear PDEs. In: Direct and Inverse Methods in Nonlinear Evolution Equations: Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5–12 1999, Lecture Notes in Physics, vol. 632, pp. 85–136. Springer, Berlin Heidelberg New York (2003) 24. Magri, F., Morosi, C.: Quaderno S 19/1984 del Dip. Di Matematica dell’Università di Milano. (1984) 25. Miller, Jr. W.: Multiseparability and superintegrability for classical and quantum systems. In: Harnad, J. et al. (eds.) Integrable Systems: from Classical to Quantum, CRM. Proceedings & Lecture Notes, vol. 26, pp. 129–156 AMS, Providence, RI (2000)

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26. Morosi, C., Tondo, G.: Quasi-Bi-Hamiltonian systems and separability. J. Phys. A.: Math. Gen. 30, 2799–2806 (1997) 27. Pedroni, M., Vanhaecke, P.: A Lie algebraic generalization of the Mumford system, its symmetries and its multi-Hamiltonian structure. Regul Chaotic Dyn 3, 132–160 (1998) 28. Reyman, A.G., Semenov-Tian-Shansky, M.A.: Group-theoretical methods in the theory of finite-dimensional integrable systems. In: Dynamical systems VII. Springer, Berlin Heidelberg New York (1994) 29. Scott, D.R.D.: Classical functional Bethe ansatz for SL(N): separation of variables for the magnetic chain. J. Math. Phys. 35, 5831–5843 (1994) 30. Sklyanin, E.K.: Separation of variables in the Gaudin model. J. Sov. Math. 47(2), 2473–2488 (1989) 31. Sklyanin E.K.: Separation of variables in the classical integrable SL(3) magnetic chain. Comm. Math. Phys. 150, 181–192 (1992) 32. Sklyanin, E.K.: Separation of variables: new trends. Progr. Theoret. Phys. Suppl. 118, 35–60 (1995)

Math Phys Anal Geom (2007) 9:263–290 DOI 10.1007/s11040-007-9014-7

On the Two Spectra Inverse Problem for Semi-infinite Jacobi Matrices Luis O. Silva · Ricardo Weder

Received: 4 January 2006 / Accepted: 29 December 2006 / Published online: 27 January 2007 © Springer Science + Business Media B.V. 2007

Abstract We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg–Marchenko theorem for Schrödinger operators on the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions. Key words semi-infinite Jacobi matrices · two-spectra inverse problem Mathematics Subject Classifications (2000) 47B36 · 49N45 · 81Q10 · 47A75 · 47B37 · 47B39 1 Introduction In the Hilbert space l2 (N) let us single out the dense subset l f in (N) of sequences which have a finite number of non-zero elements. Consider the

Research partially supported by Universidad Nacional Autónoma de México under Project PAPIIT-DGAPA IN 105799, and by CONACYT under Project P42553F. R. Weder is a fellow of Sistema Nacional de Investigadores. L. O. Silva · R. Weder Departamento de Métodos Matemáticos y Numéricos, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, México D.F. C.P. 04510, Mexico e-mail: [email protected] R. Weder (B) IIMAS Universidad Nacional Autónoma de México, Apartado Postal 20-726, México D.F. 01000, Mexico e-mail: [email protected]

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operator J defined for every f = { fk }∞ k=1 in l f in (N) by means of the recurrence relation (J f )k := bk−1 fk−1 + qk fk + bk fk+1 (J f )1 := q1 f1 + b1 f2 ,

k ∈ N \ {1}

(1.1) (1.2)

where, for every n ∈ N, bn is positive, while qn is real. J is symmetric, therefore closable, and in the sequel we shall consider the closure of J and denote it by the same letter. Notice that we have defined the Jacobi operator J in such a way that ⎛ ⎞ q1 b 1 0 0 · · · ⎜b q b 0 ···⎟ ⎜ 1 2 2 ⎟ ⎜ ⎟ ⎜ 0 b 2 q3 b 3 ⎟ (1.3) ⎜ ⎟ ⎜ ⎟ . ⎜ 0 0 b 3 q4 . . ⎟ ⎠ ⎝ .. .. .. .. . . . . is the matrix representation of J with respect to the canonical basis in l2 (N) (we refer the reader to [2] for a discussion on matrix representation of unbounded symmetric operators). It is known that the symmetric operator J has deficiency indices (1, 1) or (0, 0) [1, Chap. 4, Sec. 1.2] and [23, Corollary 2.9]. In the case (1, 1) we can always define a linear set D(g) ⊂ dom(J ∗ ) parametrized by g ∈ R ∪ {+∞} such that J ∗  D(g) =: J(g) is a self-adjoint extension of J. Moreover, for any self-adjoint extension (von Neumann extension)  J of J, there exists a  g ∈ R ∪ {+∞} such that J( g) =  J, [25, Lemma 2.20]. We shall show later (see the Appendix) that g defines a boundary condition at infinity. To simplify the notation, even in the case of deficiency indices (0, 0), we shall use J(g) to denote the operator J = J ∗ . Thus, throughout the paper J(g) stands either for a self-adjoint extension of the nonself-adjoint operator J, uniquely determined by g, or for the self-adjoint operator J. In what follows we shall consider the inverse spectral problem for the selfadjoint operator J(g). It turns out that if J = J ∗ (the case of indices (1, 1)), then for all g ∈ R ∪ {+∞} the Jacobi operator J(g) has discrete spectrum with eigenvalues of multiplicity one, i. e., the spectrum consists of eigenvalues of multiplicity one that can accumulate only at ±∞, [25, Lemma 2.19]. Throughout this work we shall always require that the spectrum of J(g), denoted σ (J(g)), be discrete, which is not an empty assumption only for the case J(g) = J. Notice that the discreteness of σ (J(g)) implies that J(g) has to be unbounded. For the Jacobi operators J(g) one can define boundary conditions at the origin in complete analogy to those of the half-line Sturm–Liouville

On the two spectra inverse problem for Jacobi matrices

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operator (see the Appendix). Different boundary conditions at the origin define different self-adjoint operators Jh (g), h ∈ R ∪ {+∞}. J0 (g) corresponds to the Dirichlet boundary condition, while the operator J∞ (g) has Neumann boundary condition. If J(g) has discrete spectrum, the same is true for Jh (g), ∀h ∈ R ∪ {+∞} (for the case of h finite see Section 2 and for h = ∞, Section 4). In this work we prove that a Jacobi operator J(g) with discrete spectrum is uniquely determined by σ (Jh1 (g)), σ (Jh2 (g)), with h1 , h2 ∈ R and h1 = h2 , and either h1 or h2 . If h1 , respectively, h2 is given, the reconstruction method also gives h2 , respectively, h1 . Saying that J(g) is determined means that we can recover the matrix (1.3) and the boundary condition g at infinity, in the case of deficiency indices (1, 1). We will also establish (the precise statement is in Theorem 3.2) that if two infinite real sequences {λk }k and {μk }k that can accumulate only at ±∞ satisfy (a) {λk }k and {μk }k interlace, i. e., between two elements of a sequence there is one and only one element of the other. Thus, we assume below that λk+1 . λk < μk < (b) The series k (μk − λk ) converges, so

(μk − λk ) =:  < ∞ . k

 μk − λn By (b) the product is convergent, so define λk − λn k=n

τn−1 :=

μn − λn  μk − λn .  λk − λn k=n

(c) The sequence {τn }n is such that, for m = 0, 1, 2, . . . ,

λ2m k

k

τk

converges.

(d) For a sequence of complex numbers {βk }k , such that the series

|βk |2 k

τk

converges

and

β k λm k

k

τk

= 0,

m = 0, 1, 2, . . .

it must hold true that βk = 0 for all k. Then, for any real number h1 , there exists a unique Jacobi operator J, a unique h2 > h1 , and if J = J ∗ , a unique g ∈ R ∪ {+∞}, such that σ (Jh2 (g)) = {λk }k and σ (Jh1 (g)) = {μk }k . Moreover, we show that if the sequences {λk }k and {μk }k are the spectra of a Jacobi operator J(g) with two different boundary conditions h1 < h2 (h2 ∈ R), then (a), (b), (c), (d) hold for  = h2 − h1 .

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Necessary and sufficient conditions for two sequences to be the spectra of a Jacobi operator J(g) with Dirichlet and Neumann boundary conditions are also given. Conditions (b) and (c) differ in this case (see Section 4). Our necessary and sufficient conditions give a characterization of the spectral data for our two spectra inverse problem. Our proofs are constructive and they give a method for the unique reconstruction of the operator J, the boundary condition at infinity, g, and either h1 or h2 . The two-spectra inverse problem for Jacobi matrices has also been studied in several papers [10, 14, 15, 24]. There are also results on this problem in [9]. We shall comment on these results in the following sections. The problem that we solve here is the discrete analogue of the two-spectra inverse problem for Sturm–Liouville operators on the half-line. The classical result is the celebrated Borg–Marchenko theorem [6, 20]. Let us briefly explain this result. Consider the self-adjoint Schrödinger operator, B f = − f (x) + Q(x) f (x),

x ∈ R+ ,

(1.4)

where Q(x) is real-valued and locally integrable on [0, ∞), and the following boundary condition at zero is satisfied, cos α f (0) + sin α f (0) = 0,

α ∈ [0, π ).

Moreover, the boundary condition at infinity, if any, is considered fixed. Suppose that the spectrum is discrete for one (and then for all) α, and denote by {λk (α)}k∈N the corresponding eigenvalues. The Borg–Marchenko theorem asserts that the sets {λk (α1 )}k∈N and {λk (α2 )}k∈N for some α1 = α2 uniquely determine α1 , α2 , and Q. Thus, the differential expression and the boundary conditions are determined by two spectra. Other results here are the necessary and sufficient conditions for a pair of sequences to be the eigenvalues of a Sturm–Liouville equation with different boundary conditions found by Levitan and Gasymov in [19]. Other settings for two-spectra inverse problems can be found in [3, 4, 11]. A resonance inverse problem for Jacobi matrices is considered in [7]. Recent local Borg–Marchenko results for Schrödinger operators and Jacobi matrices [13, 27] are also related to the problem we discuss here. Jacobi matrices appear in several fields of quantum mechanics and condensed matter physics (see for example [8]). The paper is organized as follows. In Section 2 we present some preliminary results that we need. In Section 3 we prove our results of uniqueness, reconstruction, and necessary and sufficient conditions (characterization) in the case where h1 and h2 are real numbers. In Section 4 we obtain similar results for the Dirichlet and Neumann boundary conditions. Finally, in the Appendix we briefly describe – for the reader’s convenience – how the boundary conditions are interpreted when J is considered as a difference operator.

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2 Preliminaries Let us denote by γ the second order symmetric difference expression (see (1.1), (1.2)) such that γ : f = { fk }k∈N → {(γ f )k }k∈N , by (γ f )k := bk−1 fk−1 + qk fk + bk fk+1 ,

k ∈ N \ {1},

(γ f )1 := q1 f1 + b1 f2 .

(2.1) (2.2)

Then, it is proven in Section 1.1, Chapter 4 of [1] and in Theorem 2.7 of [23] that dom(J ∗ ) = { f ∈ l2 (N) : γ f ∈ l2 (N)},

J ∗ f = γ f,

f ∈ dom(J ∗ ).

The solution of the difference equation, (γ f ) = ζ f,

ζ ∈ C,

(2.3)

is uniquely determined if one gives f1 = 1. For the elements of this solution the following notation is standard [1, Chap. 1, Sec. 2.1] Pn−1 (ζ ) := fn ,

n ∈ N,

where the polynomial Pk (ζ ) (of degree k) is referred to as the kth orthogonal polynomial of the first kind associated with the matrix (1.3). The sequence {Pk (ζ )}∞ k=0 is not in l f in (N) but it may happen that ∞

|Pk (ζ )|2 < ∞ ,

(2.4)

k=0

in which case ζ is an eigenvalue of J ∗ and f (ζ ) the corresponding eigenvector. Since the eigenspace is always one-dimensional, the eigenvalue of J ∗ is of multiplicity one . Moreover, since the (von Neumann) self-adjoint extensions of J, J(g), are restrictions of J ∗ , it follows that the point spectrum of J(g), g ∈ R ∪ {+∞}, has multiplicity one. The polynomials of the second kind {Qk (ζ )}∞ k=0 associated with the matrix (1.3) are defined as the solutions of bk−1 fk−1 + qk fk + bk fk+1 = ζ fk ,

k ∈ N \ {1},

under the assumption that f1 = 0 and f2 = b −1 1 . Then Qn−1 (ζ ) := fn ,

n ∈ N.

Qk (ζ ) is a polynomial of degree k − 1. By construction the Jacobi operator J is a closed symmetric operator. It is well known, [1, Chap. 4, Sec. 1.2] and [23, Corollary 2.9], that this operator has either deficiency indices (1, 1) or (0, 0). In terms of the polynomials of the first kind, J has deficiency indices (1, 1) when ∞

k=0

|Pk (ζ )|2 < ∞,

for ζ ∈ C \ R

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(this holds for all ζ ∈ C \ R if and only if it holds for one ζ ∈ C \ R), and deficiency indices (0, 0) otherwise. Since J is closed, deficiency indices (0, 0) mean that J = J ∗ . The symmetric operator J with deficiency indices (1, 1) has always self-adjoint extensions, which are restrictions of J ∗ . When studying the self-adjoint extensions of J in a more general context the self-adjoint restrictions of J ∗ are called von Neumann self-adjoint extensions of J [2, 23]. All self-adjoint extensions considered in this paper are von Neumann. Let us now introduce a convenient way of parametrizing the self-adjoint extensions of J in the nonself-adjoint case. We first define the Wronskian ∞ associated with J for any pair of sequences ϕ = {ϕk }∞ k=1 and ψ = {ψk }k=1 in l2 (N) as follows Wk (ϕ, ψ) := bk (ϕk ψk+1 − ψk ϕk+1 ),

k ∈ N.

Now, consider the sequences v(g) = {vk (g)}∞ k=1 such that ∀k ∈ N vk (g) := Pk−1 (0) + gQk−1 (0),

g∈R

(2.5)

and vk (+∞) := Qk−1 (0) .

(2.6)

All the self-adjoint extensions J(g) of the nonself-adjoint operator J are restrictions of J ∗ to the set [25, Lemma 2.20]

  ∗ D(g) := f = { fk }∞ lim Wn v(g), f = 0 k=1 ∈ dom(J ) : n→∞

  (2.7) = f ∈ l2 (N) : γ f ∈ l2 (N), lim Wn v(g), f = 0 . n→∞

Different values of g imply different self-adjoint extensions. If J is self-adjoint, we define J(g) := J, for all g ∈ R ∪ {+∞}; otherwise J(g) is a self-adjoint extension of J uniquely determined by g. We have defined the domains D(g) in such a way that g defines a boundary condition at infinity (see the Appendix). It is worth mentioning that if J = J ∗ then, for all g ∈ R ∪ {+∞}, J(g) has discrete spectrum. This follows from the fact that the resolvent of J(g) turns out to be a Hilbert–Schmidt operator [25, Lemma 2.19]. Let us now define the self-adjoint operator Jh (g) by Jh (g) := J(g) − h ·, e1 e1 ,

h ∈ R,

where {ek }∞ k=1 is the canonical basis in l2 (N) and ·, · denotes the inner product in this space. Clearly, J0 (g) = J(g).

 We define J∞ (g) as follows. First consider the space l2 (2, ∞) of square ∞ summable sequences { fn }∞ n=2 and the sequence v(g) = {vk (g)} k=1 given  by (2.5) and (2.6) with g fixed. Let us denote J∞ (g) the operator in l2 (2, ∞) such that J∞ (g) f = γ f , where (γ f )k is considered for any k  2 and f1 = 0 in the definition of (γ f )2 , with domain given by





  dom(J∞ (g)) := f ∈ l2 (2, ∞) : γ f ∈ l2 (2, ∞) , lim Wn v(g), f = 0 . n→∞

On the two spectra inverse problem for Jacobi matrices

269

Clearly, the matrix ⎛

q2 ⎜b ⎜ 2 ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎝ .. .

⎞ b2 0 0 · · · q3 b3 0 · · · ⎟ ⎟ ⎟ ⎟ b3 q4 b4 ⎟, .. ⎟ ⎟ . 0 b4 q5 ⎠ .. .. .. . . .

which is our original matrix (1.3) with the first column and row removed,

is the  matrix representation of J∞ (g) with respect to the canonical basis in l2 (2, ∞) . It follows easily from the definition of Jh (g) that if J(g) has discrete spectrum, the same is true for Jh (g) (h ∈ R ∪ {+∞}). Indeed, for h ∈ R this is a consequence of the invariance of the essential spectrum – that is empty in our case – under a compact perturbation [22]. We shall show in Section 4 that it is also true that J∞ (g) has discrete spectrum provided that σ (J(g)) is discrete. For the self-adjoint operator Jh (g), we can introduce the right-continuous resolution of the identity E Jh (g) (t), such that  tdE Jh (g) (t) . Jh (g) = R

Let us define the function ρ(t) as follows: ρ(t) := E Jh (g) (t)e1 , e1 ,

t ∈ R.

(2.8)

Consider the function (see [24] and [25, Chap. 2, Sec. 2.1]) mh (ζ, g) := (Jh (g) − ζ I)−1 e1 , e1 ,

ζ ∈ σ (Jh (g)) .

(2.9)

mh (ζ, g) is called the Weyl m-function of Jh (g). We shall use below the simplified notation m(ζ, g) := m0 (ζ, g). The functions ρ(t) and mh (ζ, g) are related by the Stieltjes transform (also called Borel transform):  dρ(t) . mh (ζ, g) = R t−ζ It follows from the definition that the Weyl m-function is a Herglotz function,i. e., Im mh (ζ, g) > 0, Im ζ

Im ζ > 0 .

Using the Neumann expansion for the resolvent (cf. [25, Chap. 6, Sec. 6.1]) (Jh (g) − ζI)−1 = −

N−1

k=0

(Jh (g))k (Jh (g)) N + (Jh (g) − ζ I)−1 , k+1 ζ ζN

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where ζ ∈ C \ σ (J(g)), one can easily obtain the following asymptotic formula mh (ζ, g) = −

1 q1 − h b 21 + (q1 − h)2 − − + O(ζ −4 ) , ζ ζ2 ζ3

(2.10)

as ζ → ∞ (Im ζ  ,  > 0). An important result in the theory of Jacobi operators is the fact that m(ζ, g) completely determines J(g) (the same is of course true for the pair mh (ζ, g) and Jh (g)). There are two ways for recovering the operator from the Weyl m-function. One way consists in obtaining first ρ(t) from m(ζ, g) by means of the inverse Stieltjes transform (cf. [25, Appendix B]), namely, 1 ρ(b ) − ρ(a) = lim lim δ↓0 ↓0 π



b +δ

(Im m(x + i, g)) dx .

a+δ

The function ρ is such that all the moments of the corresponding measure are finite [1, 23]. Hence, all the elements of the sequence {tk }∞ k=0 are in L2 (R, dρ) and one can apply, in this Hilbert space, the Gram–Schmidt procedure of orthonormalization to the sequence {tk }∞ k=0 . One, thus, obtains a sequence of polynomials {Pk (t)}∞ normalized and orthogonal in L2 (R, dρ). These k=0 polynomials satisfy a three term recurrence equation [23] t Pk−1 (t) = bk−1 Pk−2 (t) + qk Pk−1 (t) + bk Pk (t)

k ∈ N \{1}

t P0 (t) = q1 P0 (t) + b1 P1 (t) ,

(2.11) (2.12)

where all the coefficients bk (k ∈ N) turn out to be positive and qk (k ∈ N) are real numbers. The system (2.11) and (2.12) defines a matrix which is the matrix representation of J. We shall refer to this procedure for recovering J as the method of orthogonal polynomials. The other method for determining J from m(ζ, g) was developed in [12] (see also [24]). It is based on the asymptotic behavior of m(ζ, g) and the Ricatti equation [24], b2n m(n) (ζ, g) = qn − ζ −

1 m(n−1) (ζ, g)

,

n ∈ N,

(2.13)

where m(n) (ζ, g) is the Weyl m-function of the Jacobi operator associated with the matrix (1.3) with the first n columns and n rows removed. After obtaining the matrix representation of J, one can easily obtain the boundary condition at infinity which defines the domain of J(g) in the nonselfadjoint case. Indeed, take an eigenvalue, λ, of J(g), i. e., λ is a pole of m(ζ, g). Since the corresponding eigenvector f (λ) = { fk (λ)}∞ k=1 is in dom(J(g)), it must be that

 lim Wn v(g), f (λ) = 0 .

n→∞

On the two spectra inverse problem for Jacobi matrices

271



This implies that either limn→∞ Wn {Qk−1 (0)}∞ k=1 , f (λ) = 0, which means that g = +∞, or 

limn→∞ Wn {Pk−1 (0)}∞ k=1 , f (λ) .

g=− limn→∞ Wn {Qk−1 (0)}∞ k=1 , f (λ) If the spectrum of Jh (g) is discrete, say σ (Jh (g)) = {λk }k , the function ρ(t) defined by (2.8) can be written as follows

1 ρ(t) = , α λ t k k

where the coefficients {αk }k are called the normalizing constants and are given by ∞

|Pk (λn )|2 . (2.14) αn = k=0

√ Thus, αn equals the l2 norm of the eigenvector f (λn ) := {Pk (λn )}∞ k=0 corresponding to λn . The eigenvector f (λn ) is normalized in such a way that f1 (λn ) = 1. Clearly, 

1 dρ = . (2.15) 1 = e1 , e1 = αk R k

The Weyl m-function in this case is given by

1 . mh (ζ, g) = αk (λk − ζ )

(2.16)

k

From this we have that (λn − ζ )mh (ζ, g) = (λn − ζ )



1

k

αk (λk − ζ )

=

k=n

1 λn − ζ + . αk (λk − ζ ) αn

Therefore, αn−1 = lim (λn − ζ )m(ζ, g) = −Res m(ζ, g) . ζ →λn

ζ =λn

(2.17)

Let us now introduce an appropriate way for enumerating sequences that we shall use. Consider a pair of infinite real sequences {λk }k and {μk }k that have no finite accumulation points and that interlace, i. e., between two elements of one sequence there is one and only one element of the other. We use M, a subset of Z to be defined below, for enumerating the sequences as follows ∀k ∈ M

λk < μk < λk+1 ,

(2.18)

where (a) If infk {λk }k = −∞ and supk {λk }k = ∞, M := Z and we require μ−1 < 0 < λ1 .

(2.19)

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(b) If 0 < supk {λk }k < ∞, max M := {k}kk=−∞ , (kmax  1) and we require μ−1 < 0 < λ1 .

(2.20)

(c) If supk {λk }k  0, M := {k}0k=−∞ .

(2.21)

M := {k}∞ k=0 .

(2.22)

(d) If infk {μk }k  0, (e) If −∞ < infk {μk }k < 0, M := {k}∞ k=kmin ,

(kmin  −1)

and we require μ−1 < 0 < λ1 . (2.23)

Notice that, by this convention for enumeration, the only elements of {λk }k∈M and {μk }k∈M allowed to be zero are λ0 or μ0 .

3 Rank One Perturbations with Finite Coupling Constants In this section we consider a pair of operators Jh1 (g) and Jh2 (g), where h1 , h2 ∈ R, that is, rank one perturbations of the Jacobi operator J(g) with finite coupling constants. 3.1 Recovering the Matrix from Two Spectra Let g ∈ R ∪ {+∞} be fixed. Since Jh (g) is a rank one perturbation of J(g), the domain of J(g) coincides with the domain of Jh (g) for all h ∈ R. Moreover, since the perturbation is analytic in h, the multiplicity-one eigenvalues, λk (h), and the corresponding eigenvectors, are analytic functions of h [16]. Lemma 3.1 Let {λk (h)}k be the set of eigenvalues of Jh (g) (h ∈ R). For a fixed k the following holds d 1 λk (h) = − , dh αk (h)

(3.1)

where αk (h) is the normalizing constant corresponding to λk (h). Proof For the sake of simplifying the formulae, we write Jh and λ(h) instead of Jh (g) and λk (h), respectively (k is fixed). Let us denote by f (h) the eigenvector of Jh corresponding to λ(h). Take any δ > 0, taking into account that dom(Jh+δ ) = dom(Jh ) and that Jh is symmetric for any h ∈ R, we have that (λ(h + δ) − λ(h)) f (h + δ), f (h) = Jh+δ f (h + δ), f (h) − f (h + δ), Jh f (h) = (Jh+δ − Jh + Jh ) f (h + δ), f (h) − f (h + δ), Jh f (h) = (Jh+δ − Jh ) f (h + δ), f (h) = −δ .

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273

Therefore, lim

δ→0

λ(h + δ) − λ(h) 1 1 = − lim =− . δ→0 f (h + δ), f (h) δ αk (h)

 

The cornerstone of our analysis below is the Weyl m-function. Let us establish the relation between mh (ζ, g) and m(ζ, g). Consider the second resolvent identity [26]: (Jh (g) − ζ I)−1 − (J(g) − ζ I)−1 = (J(g) − ζ I)−1 (J(g) − Jh (g))(Jh (g) − ζ I)−1 ,

(3.2)

where ζ ∈ C \ {σ (J(g)) ∪ σ (Jh (g))}. Then, for h ∈ R,    mh (ζ, g) − m(ζ, g) = (Jh (g) − ζ I)−1 − (J(g) − ζ I)−1 e1 , e1   = (J(g) − ζ I)−1 (h ·, e1 e1 )(Jh (g) − ζ I)−1 e1 , e1   = h (Jh (g) − ζ I)−1 e1 , e1 (J(g) − ζ I)−1 e1 , e1 = hmh (ζ, g)m(ζ, g) . Hence, mh (ζ, g) =

m(ζ, g) . 1 − hm(ζ, g)

(3.3)

Remark 3.2 If J(g) has discrete spectrum, then m(ζ, g) is meromorphic and, by (3.3), so is mh (ζ, g). The poles of mh (ζ, g) are the eigenvalues of Jh (g). Since the poles of the denominator and numerator in (3.3) coincide, assuming that h = 0, the poles of mh (ζ, g) are given by the zeros of 1 − hm(ζ, g) and the zeros of mh (ζ, g) by the zeros of m(ζ, g). Thus, Jh1 (g) and Jh2 (g) have different eigenvalues, provided that h1 = h2 . Theorem 3.3 Consider the Jacobi operator J(g) with discrete spectrum. The sequences {μk }k = σ (Jh1 (g)) and {λk }k = σ (Jh2 (g)), h1 = h2 , together with h1 (respectively, h2 ) uniquely determine the operator J, h2 , (respectively, h1 ) and, if J = J ∗ , the boundary condition g at infinity. Proof Without loss of generality we can assume that h1 < h2 . Consider the Weyl m-function m(ζ, g) of the operator J(g). Let us define the function m(ζ, g) =

mh2 (ζ, g) , mh1 (ζ, g)

ζ ∈ C \ R.

(3.4)

Notice first that the zeros of m(ζ, g) are the eigenvalues of Jh1 (g) while the poles of m(ζ, g) are the eigenvalues of Jh2 (g). This follows from Remark 3.2 and (3.4). Let us now show that m(ζ, g) is a Herglotz or an anti-Herglotz function. Indeed, since m(ζ, g) is Herglotz, then m(ζ, g) =

1 − h1 m =1+ 1 − h2 m

−1 h2 h2 −h1

+

−1 (h2 −h1 )m(ζ,g)

.

(3.5)

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Therefore, m(ζ, g) is Herglotz or anti-Herglotz depending on the sign of h2 − h1 . Recall that if a function f is Herglotz, then, − 1f is also Herglotz. Since h2 − h1 > 0, m(ζ, g) is a Herglotz function. Thus, the zeros {μk }k of m(ζ, g) and its poles {λk }k interlace. Let us use the convention (2.18–2.23) for enumerating the zeros and poles of m(ζ, g). By this convention, if the sequence {λk }k (or {μk }k ) is bounded from below, the least of all zeros is greater than the least of all poles, while, if {λk }k is bounded from above, the greatest of all poles is less than the greatest of all zeros. It is easy to verify, using for instance (3.1), that this is what we have for the zeros and poles of m(ζ, g) when J(g) is semi-bounded. According to [18, Chap. 7, Sec.1, Theorem 1], the meromorphic Herglotz function m(ζ, g), with its zeros and poles enumerated as convened, can be written as follows    ζ − μ0  ζ ζ −1 m(ζ, g) = C 1− 1− , C > 0, (3.6) ζ − λ0 μk λk k∈M

where the prime in the infinite product means that it does not include the factor k = 0. From the asymptotic behavior of m(ζ, g), given by (2.10), one easily obtains that, as ζ → ∞ with Im ζ   ( > 0), m(ζ, g) = 1 + (h1 − h2 )ζ −1 + (h1 − h2 )(q1 − h2 )ζ −2 + O(ζ −3 ) .

(3.7)

Therefore, lim m(ζ, g) = 1 .

ζ →∞ Im ζ  

Then, using (3.6), we have C−1 = lim

 

ζ →∞ Im ζ   k∈M

1−

ζ μk

 1−

ζ λk

−1

,

 > 0.

(3.8)

Thus, m(ζ, g) is completely determined by the spectra σ (Jh1 (g)) and σ (Jh2 (g)). Having found m(ζ, g), we can determine h2 , respectively, h1 , by means of (3.7). Hence, from (3.5) one obtains m(ζ, g) and, using the methods introduced in the preliminaries, J is uniquely determined. In the case when J = J ∗ , we can also find the boundary condition g at infinity as indicated in Section 2.   In [24] (see also [9]) it is proven that the discrete spectra of Jh1 (g) and Jh2 (g), together with h1 and h2 uniquely determine J and the boundary condition g in the (1, 1) case. Our result shows that it is not necessary to know both h1 and h2 , one of them is enough. It turns out that if one knows the spectra σ (Jh1 (g)) and σ (Jh2 (g)) together with q1 , the first element of the matrix’s main diagonal, it is possible to recover uniquely the matrix, the boundary conditions h1 , h2 and the boundary condition at infinity, g, if any. Indeed, the term of order ζ −1 in the asymptotic

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275

expansion of m(ζ, g) (3.7) determines h1 − h2 . Since the coefficient of ζ −2 term is (h1 − h2 )(q1 − h2 ), if we know q1 one finds h2 , and then h1 . 3.2 Necessary and Sufficient Conditions Theorem 3.4 Given h1 ∈ R and two infinite sequences of real numbers {λk }k and {μk }k without finite points of accumulation, there is a unique real h2 > h1 , a unique operator J(g), and if J = J ∗ also a unique g ∈ R ∪ {+∞}, such that, {μk }k = σ (Jh1 (g)) and {λk }k = σ (Jh2 (g)) if and only if the following conditions are satisfied. (a) {λk }k and {μk }k interlace and, if {λk }k is bounded from below, mink {μk }k > mink {λk }k , while if {λk }k is bounded from above, maxk {λk }k < maxk {μk }k . So we use below the convention (2.18–2.23) for enumerating the sequences. (b) The following series converges

(μk − λk ) =  < ∞ . k∈M

 μk − λn is convergent, so define λk − λn

By condition (b) the product

k∈ M k = n

τn−1 :=

μn − λn  μk − λn ,  λk − λn

∀n ∈ M .

(3.9)

k∈ M k = n

(c) The sequence {τn }n∈M is such that, for m = 0, 1, 2, . . . , the series

λ2m k τk

converges.

k∈M

(d) If a sequence of complex numbers {βk }k∈M is such that the series

|βk |2 τk

converges

k∈M

and, for m = 0, 1, 2, . . . ,

β k λm k = 0, τk

k∈M

then βk = 0 for all k ∈ M. Proof We first prove that if {λk }k and {μk }k are the spectra of Jh2 (g) and Jh1 (g), with h2 > h1 , then (a), (b), (c), and (d) hold true. The condition (a) follows directly from the proof of the previous theorem. To prove that (b) holds, observe that (3.1) implies  h2 dh μk − λk = . h1 αk (h)

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Consider a sequence {Mn }∞ n=1 of subsets of M, such that Mn ⊂ Mn+1 and ∪n Mn = M, then, using (2.15), we have sn :=



(μk − λk ) =

k∈Mn

 k∈Mn

h2 h1

dh = αk (h)



h2 h1

k∈Mn

dh  h2 − h1 . αk (h)

The sequence {sn }∞ n=1 is then convergent and clearly

(μk − λk ) = lim sn = h2 − h1 . n→∞

k∈M

Thus,  = h2 − h1 . The convergence of the series in (b) allows us to write (3.8) as follows C−1 =

 λk  μk − ζ , lim μk ζ → ∞ λk − ζ

k∈M

 > 0.

Im ζ   k∈M

Now, using again (b), it easily follows that for any  > 0   μk − ζ   μk − λk lim = lim = 1. 1+ λk − ζ λk − ζ ζ →∞ ζ →∞ Im ζ   k∈M

Thus, C =

 k∈M

Im ζ   k∈M

μk /λk and by (3.6), m(ζ, g) =

 μk − ζ . λk − ζ

(3.10)

k∈M

Let us now find formulae for the normalizing constants in terms of the sets of eigenvalues for different boundary conditions. By (2.17), αn−1 (h2 , g) = lim (λn − ζ )mh2 (ζ, g) . ζ →λn

Using the second resolvent identity, as we did to obtain (3.3), we have that mh1 (ζ, g) =

mh2 (ζ, g) . 1 − (h1 − h2 )mh2 (ζ, g)

Therefore, m(ζ, g) =

mh2 (ζ, g) = 1 − (h1 − h2 )mh2 , mh1 (ζ, g)

ζ ∈ C \ R.

(3.11)

Then, the normalizing constants are given by αn−1 (h2 , g) = lim (λn − ζ ) ζ →λn

1 m(ζ, g) − 1 = lim (λn − ζ )m(ζ, g) . h2 − h1 h2 − h1 ζ →λn

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277

Now, lim (λn − ζ )m(ζ, g) = lim (λn − ζ )

ζ →λn

ζ →λn

 μk − ζ λk − ζ

k∈M

 μk − λn = (μn − λn ) . λk − λn

(3.12)

k∈ M k = n

Hence, αn−1 (h2 , g) =

μn − λn  μk − λn . h2 − h1 λk − λn

(3.13)

k∈ M k = n

Notice that, since  = h2 − h1 , it follows from (3.13) that τn = αn for all n ∈ M. Hence the spectral function ρ of the self-adjoint extension Jh2 (g) is given by the expression ρ(t) = λk t τk−1 . Thus (c) follows from the fact that all the moments of ρ are finite [1, 23]. Similarly, (d) stems from the density of polynomials in L2 (R, dρ), which takes place since ρ is N-extremal [1], [23, Proposition 4.15]. We now prove that conditions (a), (b), (c), and (d) are sufficient. Let {λk }k and {μk }k be sequences as in (a) and (b). Then, 0<

 μk − λn < ∞. λk − λn

(3.14)

k∈ M k = n

The convergence of this product allows us to define the sequence of numbers {τn }n∈M . Observe that for all n ∈ M, τn > 0. Indeed,  > 0 and (2.18–2.23) yield μn − λn > 0 for all n ∈ M. Thus, taking into account (3.14), we obtain τn > 0,

∀n ∈ M.

(3.15)

Let us now define the function ρ(t) :=

1 , τ λ t k

t ∈ R.

(3.16)

k

Since (3.15) holds, ρ is a monotone non-decreasing function and has an infinite number of points of growth. Notice also that ρ is right continuous. Now, we want to show that for the measure corresponding to ρ all the moments are finite and  dρ(t) = 1 . (3.17) R

The fact that the moments are finite follows directly from condition (c). Indeed, 

λm k tm dρ(t) = . τk R k∈M

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We show next that (3.17) holds true. Given the sequences {λk }k and {μk }k satisfying (a) and (b) we can define the function (ζ ) := m

 μk − ζ . λk − ζ

(3.18)

k∈M

Taking into account (3.9), one obtains that m(ζ ) − 1) = − Res( ζ =λn

 . τn

In view of (b), we easily find that lim ( m(ζ ) − 1) = lim

ζ →∞ Im ζ  

 μk − ζ −1 λk − ζ

ζ →∞ Im ζ   k∈M

= lim



ζ →∞ Im ζ   k∈M

1+

μk − λk λk − ζ

 − 1 = 0.

(3.19)

ˇ Thus, on the basis of Cebotarev’s theorem on the representation of meromorphic Herglotz functions [18, Chap. VII, Section 1 Theorem 2], one obtains (ζ ) − 1 = m

k∈M

(ζ ) := We now define the function m (ζ ) = m

 . (λk − ζ )τk

 (ζ )−1 m . 

Then, (3.20) yields



1

k∈M

τk (λk − ζ )

.

We next show that (ζ ) = −1 . lim ζ m

ζ →∞ Im ζ  

Indeed, (ζ ) m 1  μk − ζ =   λk − ζ k∈M  

 μk − ζ  1 = exp ln  λk − ζ k∈M  

 1 μk − λk exp = ln 1 +  λk − ζ k∈M ⎧ ⎫   p⎬ ∞ ⎨



1 μk − λk = exp (−1) p−1 . ⎩ ⎭  λk − ζ p=1 k∈M

(3.20)

(3.21)

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279

Thus, as ζ → ∞ with Im ζ   ( > 0), (ζ ) m 1 1 μk − λk = + + O(ζ −2 ) .    λk − ζ k∈M

Then, (ζ ) = lim ζ lim ζ m

ζ →∞ Im ζ  

ζ →∞ Im ζ  

=−

1 μk − λk  λk − ζ k∈M

1

(μk − λk ) = −1 .  k∈M

Also, from (3.21) one has (ζ ) = − lim ζ m

ζ →∞ Im ζ  

Therefore, 1=

1 . τk

k∈M



1 = dρ(t) . τk R

k∈M

Having found a function ρ with infinitely many growing points and such that (3.17) is satisfied and all the moments exist, one can obtain, applying the method of orthogonal polynomials (see Section 2), a tridiagonal semi-infinite matrix. Let us denote by  J the operator whose matrix representation is the obtained matrix. By what has been explained before, this operator is closed J + h2 ·, e1 e1 . and symmetric. Now, define h2 :=  + h1 and J :=  e , e

, where E If  J= J ∗ , we know that ρ(t) = E(t)  J 1 1 J (t) is the spectral decomposition of the self-adjoint Jacobi operator  J. Then, obviously,  J = Jh2 . If  J =  J ∗ , the Stieltjes transform of ρ is the Weyl m-function, we denote it by w(ζ ), of some self-adjoint extension of  J that we denote by  J. This is true because of the density of polynomials in L2 (R, dρ). Indeed, (d) means that the polynomials are dense in L2 (R, dρ). Thus, w(ζ ) lies on the Weyl circle, and then, it is the Weyl m-function of some self-adjoint extension of  J [1], [23, Proposition 4.15]. Therefore,  J + h2 ·, e1 e1 is a self-adjoint extension of J and hence,  J + h2 ·, e1 e1 = J(g) for some unique g ∈ R ∪ {∞}. Furthermore, we obviously have that,  J = Jh2 (g) and w(ζ ) = mh2 (ζ, g). We uniquely reconstruct m(ζ, g) from mh2 (ζ, g) using (3.3) and then, we uniquely reconstruct g as explained in Section 2. Notice that we have  dρ(t) (ζ ) . =m mh2 (ζ, g) = R t−ζ It remains to show that σ (Jh2 (g)) = {λk }k and σ (Jh1 (g)) = {μk }k . To this end consider the function m(ζ, g) for the pair Jh2 and Jh1 : m(ζ, g) =

mh2 (ζ, g) , mh1 (ζ, g)

ζ ∈ C \ R.

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Let the sequence {γk }k denote the spectrum of Jh1 . Then, arguing as in the proof of (3.10) we obtain that m(ζ, g) =

 γk − ζ . λk − ζ

k∈M

Since we have already proven that (a) and (b) are necessary conditions, we have that

(γk − λk ) =  < ∞ .

k∈M

Then, as in the proof of (3.19), it follows that lim (m(ζ ) − 1) = 0 .

ζ →∞ Im ζ  

ˇ Hence by Cebotarev’s theorem [18, Chap. VII, Section 1 Theorem 2], m(ζ, g) = 1 +

k∈M

h2 − h1 , (λk − ζ )αk (h2 , g)

where we compute the residues of m(ζ ) as in (3.12). Thus, since αk (h2 , g) = τk , ∀k ∈ M, m(ζ, g) = 1 +

k∈M

 (ζ, g) . =m (λk − ζ )τk

(ζ, g) and then, the eigenvalues But {λk }k and {μk }k are the poles and zeros of m   of Jh2 (g) and Jh1 (g), respectively. Remark 3.5 We draw the reader’s attention to the fact that the matrix associated with the function ρ, constructed in the proof of the previous theorem, may have deficiency indices (1, 1) [1, 23, 25]. If we drop the condition of the density of polynomials in L2 (R, dρ) and our reconstruction method yields a nonself-adjoint operator J, then the sequences {λk }k and {μk }k correspond to the spectra of some generalized self-adjoint extensions of Jh2 and Jh1 , respectively (see [23]). The generalized extensions of symmetric operators, which are not von Neumann extensions, were first introduced by Naimark (see Appendix I in [2] on Naimark’s theory). In [15] the case of Jacobi operators bounded from below is considered. A uniqueness result is proven, and some sufficient conditions for a pair of sequences to be the spectra of a Jacobi operator with different boundary conditions are given.

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281

4 Dirichlet–Neumann Conditions 4.1 Recovering the Matrix from Two Spectra In this section we shall consider the pair of Jacobi operators J0 (g) = J(g) and J∞ (g). Here, as before, we keep the convention of writing J(g) even if J = J ∗ . The matrix representation of J∞ (g) corresponds to the matrix representation of J(g) with the first column and row removed. From the Ricatti equation (2.13), taking into account that m(0) (ζ, g) = m(ζ, g) and m(1) (ζ, g) = m∞ (ζ, g), we have

m∞ (ζ, g) = −

1 b21

 (ζ − q1 ) +

 1 . m(ζ, g)

(4.1)

As before, we assume that the spectrum of J = J(g) is discrete. If m(ζ, g) is a meromorphic function, then, by (4.1), m∞ (ζ, g) is also meromorphic and the spectrum of J∞ (g) is discrete. The poles of m(ζ, g) are the eigenvalues of J(g), while the zeros of m(ζ, g) are the eigenvalues of J∞ (g). Since m(ζ, g) is always a Herglotz function, under our assumption on the discreteness of σ (J(g)), m(ζ, g) is a meromorphic Herglotz function. This implies that σ (J(g)) and σ (J∞ (g)) are interlaced, that is, between two successive eigenvalues of one operator there is exactly one eigenvalue of the other. Let the sequence {λk }k denote the eigenvalues of J(g) (the poles of m(ζ, g)). Furthermore, {μk }k will stand for the eigenvalues of J∞ (g) (the zeros of m(ζ, g)). It is worth remarking that, in contrast to the case of boundary conditions being rank one perturbations with finite coupling constant, here our convention for enumerating the sequences {λk }k and {μk }k does not work in the case when J(g) is semi-bounded from above. Indeed, it follows from the minimax principle [21] that if J(g) is bounded from below, the smallest of all poles is less than the smallest of all zeros of m(ζ, g), and if J(g) is bounded from above, the min–max principle applied to −J(g) implies that the greatest of all zeros is less than the greatest of all poles of m(ζ, g). So let us consider first the case when J(g) is not semi-bounded or semibounded from below and enumerate the sequences {λk }k and {μk }k by (2.18), (2.19), (2.22), and (2.23). Then, by the same theorem we used to obtain (3.6) [18], m(ζ, g) can be written as follows

m(ζ, g) = C

   ζ − μ0  ζ ζ −1 , 1− 1− ζ − λ0 μk λk

C > 0,

(4.2)

k∈M

where, as before, the prime in the infinite product means that it does not include the factor k = 0.

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If J(g) is bounded from above, then we are still able to use (2.18), (2.20) and (2.21) for enumerating the zeros and poles of the meromorphic Herglotz 1 . Thus, function − m(ζ,g)    1 ζ − λ0  ζ ζ −1  − , =C 1− 1− m(ζ, g) ζ − μ0 λk μk

> 0. C

(4.3)

k∈M

1 by our Notice that, since we have enumerated zeros and poles of − m(ζ,g) convention, we have now

∀k ∈ M,

μk < λk < μk+1 ,

(4.4)

and (a) If 0 < supk {μk }k < ∞, max , M := {k}kk=−∞

(kmax  1) requiring λ−1 < 0 < μ1 ,

(4.5)

(b) If supk {μk }k  0, M := {k}0k=−∞ .

(4.6)

Here again λ0 or μ0 are the only ones allowed to be zero. Equations (4.2) and (4.3) can be written in one formula    ζ ζ −1 ζ − μ0  1− 1− m(ζ, g) = K , ζ − λ0 μk λk

(4.7)

k∈M

where, if J(g) is not semi-bounded from above, K = C and {λk }k and {μk }k are −1 and {λk }k and enumerated by (2.18), (2.19), (2.22), and (2.23), while K = −C {μk }k are enumerated by (4.4–4.6) if J(g) is semi-bounded from above. We give now, for the reader’s convenience, a simple proof of a theorem that was proven by Fu and Hochstadt [10] for regular Jacobi operators (a regular Jacobi matrix is defined in [10]), and by Teschl [24] in the general case. Theorem 4.1 (Fu and Hochstadt, Teschl) Consider the Jacobi operator J(g) with discrete spectrum. The sequences {λk }k = σ (J(g)) and {μk }k = σ (J∞ (g)) uniquely determine the operator J and, if J = J ∗ , the boundary condition, g, at infinity. Proof From (2.10) we know that lim ζ m(ζ, g) = −1,

ζ →∞ Im ζ  

 > 0.

Then, if J(g) is not semi-bounded from above, (4.2) yields     ζ ζ −1 −1 C = − lim ζ 1− 1− , μk λk ζ →∞ Im ζ  

k∈M

 > 0,

(4.8)

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283

where {λk }k and {μk }k are enumerated by (2.18), (2.19), (2.22), and (2.23). On the other hand, in the semi-bounded from above case (4.3) implies     ζ ζ −1 −1 = lim 1 1− 1− C , λk μk ζ →∞ ζ Im ζ  

 > 0.

(4.9)

k∈M

where {λk }k and {μk }k are enumerated by (4.4–4.6). Thus, in any case, one can find K, the constant in (4.7), from the sequences {λk }k and {μk }k . Therefore, the spectra σ (J(g)) and σ (J∞ (g)) uniquely determine m(ζ, g). Having found m(ζ, g) we can, using the methods introduced in Section 2, determine J and, in the case when J = J ∗ , also find uniquely the boundary condition at infinity, g.   Remark 4.2 It turns out that, by (4.8) and (4.9), K can be written as K

−1

= − lim ζ ζ →∞ Im ζ  

  k∈M

ζ 1− μk



ζ 1− λk

−1

,

 > 0,

(4.10)

where the sequences {λk }k and {μk }k have been enumerated by (2.18), (2.19), (2.22), and (2.23), when J(g) is not semi-bounded from above and by (4.4–4.6), otherwise. In what follows the Weyl m-function will be written through (4.7) with K given by (4.10). From (4.7) one can obtain straightforward formulae for the normalizing constants (2.14) in terms of the sequences {λk }k and {μk }k . Indeed, when n = 0 lim (λn − ζ )m(ζ, g) = lim (λn − ζ )K

ζ →λn

ζ →λn

=K

ζ ζ − μ0  1 − μk ζ − λ0 1 − λζk k∈M

λn λn λn − μ0  1 − μk (μn − λn ) . μn λn − λ0 1 − λλnk k∈ M k = n

Formulae (2.17) and (4.10) then give , for n = 0, λn (μn μn

αn−1

=−

0 − λn ) λλnn−μ −λ0

 

1−

k∈ M k = n

lim ζ

ζ →∞ Im ζ  

  1− k∈M

ζ μk

λn μk





1−

1−

ζ λk

λn λk

−1

−1 .

(4.11)

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Analogously,  

(μ0 − λ0 ) α0−1 = −

λ0 μk

1−

k∈M

  1− lim ζ

ζ →∞ Im ζ  

ζ μk





λ0 λk

1−

1−

ζ λk

−1

−1 .

(4.12)

k∈M

4.2 Necessary and Sufficient Conditions The following result establishes necessary and sufficient conditions for two given sequences of real numbers to be the spectra of J(g) and J∞ (g). Theorem 4.3 Given two infinite sequences of real numbers {λk }k and {μk }k without finite points of accumulation, there is a unique operator J(g), and if J = J ∗ also a unique g ∈ R ∪ {+∞}, such that {λk }k = σ (J(g)) and {μk }k = σ (J∞ (g)) if and only if the following conditions are satisfied. (a) {λk }k and {μk }k interlace and, if {λk }k is bounded from below, mink {μk }k > mink {λk }k , if {λk }k is bounded from above, maxk {λk }k > maxk {μk }k . So we use below the convention (2.18), (2.19), (2.22), and (2.23) for enumerating the sequences when J(g) is not semi-bounded from above, and (4.4–4.6) otherwise. By condition (a) the product  

ζ μk

1−

k∈M

 1−

ζ λk

−1

converges uniformly on compact subsets of C (see the proof below and [18, Chap. 7, Section 1]. (b) The limit lim iξ

ξ →∞ ξ ∈R

  k∈M

iξ 1− μk



iξ 1− λk

−1 (4.13)

is finite and negative when the sequences {λk }k and {μk }k are not bounded from above, and it is finite and positive otherwise. (c) Let {τn }n∈M be defined by λn (μn μn

τn−1

0 − λn ) λλnn−μ −λ0

 

1−

k∈ M k = n

=− lim iξ

ξ →∞ ξ ∈R

  1− k∈M

iξ μk



λn μk

 1−

1−

iξ λk

λn λk

−1

−1

On the two spectra inverse problem for Jacobi matrices

285

for n ∈ M , n = 0, and (μ0 − λ0 ) τ0−1 = −

  1−

λ0 μk

k∈M

  1− lim iξ

ξ →∞ ξ ∈R

iξ μk



1−

 1−

λ0 λk

iξ λk

−1

−1 .

k∈M

The sequence {τn }n∈M is such that, for m = 0, 1, 2 . . . , the series

λ2m k τk

converges.

k∈M

(d) If a sequence of complex numbers {βk }k∈M , is such that the series

|βk |2 τk

converges

k∈M

and, for m = 0, 1, 2, . . . ,

β k λm k = 0, τk

k∈M

then βk = 0 for all k ∈ M. Proof We begin the proof by showing that the sequences σ (J(g)) = {λk }k and σ (J∞ (g)) = {μk }k satisfy (a), (b), (c), and (d). Since the Weyl m function is Herglotz, the eigenvalues of J(g) and J∞ (g) interlace as indicated in (a). To prove that (b) holds, consider first the case when J(g) is not semi-bounded or only bounded from below, then (4.8) yields (b). If J(g) is semi-bounded from above, (4.9) implies (b). On the basis of (4.11) and (4.12), τn coincides with the normalizing constant αn for all n ∈ M. Hence the spectral function ρ of the self-adjoint extension J(g) is given by the expression ρ(t) = λk t τk−1 . Thus (c) follows from the fact that all the moments of ρ are finite [1, 23]. Similarly, (d) stems from the density of polynomials in L2 (R, dρ), which takes place since ρ is N-extremal [1], [23, Proposition 4.15]. Let us now suppose that we are given two real sequences {λk }k and {μk }k that satisfy (a). It can be shown that 0<

  k∈ M k = n

λn 1− μk



λn 1− λk

−1

< ∞.

(4.14)

Indeed, the convergence of the infinite product follows from (a) and is part of the Theorem 1 in [18, Chap. 7, Sec.1] used to obtain (3.6). We give here, for

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the reader’s convenience, some details. The product in (4.14) converges if and only if      



 1 λn −1 λn −1 λn 1 1− 1− 1− − 1 = λn − <∞ , μk λk λk μk λk k∈ M k = n

k∈ M k = n

where prime means that the summand k = 0 is excluded. Thus, we have to prove that 

 1 1 − < ∞. λk μk k∈M

It will suffice to consider that in (a) the sequences are ordered by (2.18) with M given by (2.19). For any k ∈ N, (2.18) implies     1 1 1 1 0< − < − , ∀k ∈ N . λk μk λk λk+1   1 is convergent. Analogously, it can be proven that Clearly, k∈N λ1k − λk+1 

 1 1 − < ∞. λ−k μ−k k∈N

Having established the convergence of the the product in (4.14), its positivity follows easily. We have, therefore, a sequence of real numbers {τk }k∈M and let us now show that τn > 0, ∀n ∈ M. First notice that (2.18), (2.19), (2.22), and (2.23), yield λn − μ0 λn (μn − λn ) > 0 (n = 0) μn λn − λ0

and

μ0 − λ0 > 0 .

and

μ0 − λ0 < 0 .

On the other hand (4.4–4.6) imply λn − μ0 λn (μn − λn ) < 0 (n = 0) μn λn − λ0

From these last inequalities, taking into account (4.14) and condition (b) we obtain τn > 0, Let us now define the function ρ(t) :=

1 , τ λ t k

∀n ∈ M.

∀t ∈ R .

(4.15)

(4.16)

k

In view of (4.15), ρ is a monotone non-decreasing function and has an infinite number of points of growth. Now, we want to show that, for the measure corresponding to ρ, all the moments are finite and  dρ(t) = 1 . (4.17) R

On the two spectra inverse problem for Jacobi matrices

287

The fact that the moments are finite follows directly from condition (c). Indeed, 

λm k tm dρ(t) = . τk R k∈M

We show next that (4.17) is true. Given the sequences {λk }k and {μk }k satisfying (a) and (b), we can define the function   −1  ζ −μ0 1 − μζk 1 − λζk ζ −λ0 k∈M (ζ ) := − m (4.18)   −1 .  lim ξ → ∞ iξ 1 − μiξk 1 − λiξk k∈M

ξ ∈R

Now, arguing as in the proof of (4.11) and (4.12), we obtain (ζ ) = −τn−1 . Res m ζ =λn

On the other hand,

  −1 1 − μiξk 1 − λiξk k∈M (iξ ) = − lim lim m   −1 = 0 .  ξ →∞ ξ →∞ 1 − μiξk 1 − λiξk ξ ∈R ξ ∈ R iξ 

k∈M

ˇ Thus, using again Cebotarev’s theorem [18] we find that

1 (ζ ) = m . τk (λk − ζ )

(4.19)

k∈M

It follows from (4.18) that

 1− k∈M (iξ ) = − lim lim iξ m     ξ →∞ ξ →∞ 1 − μiξk 1 − ξ ∈R ξ ∈ R iξ iξ





1−

iξ μk

k∈M

iξ λk iξ λk

−1 −1 = −1 .

Also from (4.19) one has (iξ ) = − lim iξ m

ξ →∞ ξ ∈R

Therefore, 1=

1 . τk

k∈M



1 = dρ(t) . τk R

k∈M

We have found a function ρ(t) with infinitely many growing points, such that all the moments exist for the corresponding measure and (4.17) holds. Therefore one can obtain, applying the method of orthogonal polynomials (see Section 2), a tridiagonal semi-infinite matrix. Let us denote by J the operator whose matrix representation is the obtained matrix. As was mentioned before, J is symmetric and closed. Now, if J = J ∗ , we know that ρ(t) = E J (t)e1 , e1 , where E J (t) is the spectral decomposition of the self-adjoint Jacobi operator J. If J = J ∗ , then the Stieltjes transform of ρ(t) is the Weyl m-function m(ζ, g)

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of some self-adjoint extension of J with boundary conditions at infinity given by g, that is,  dρ(t) m(ζ, g) = . R t−ζ This last assertion is true because of the density of polynomials in L2 (R, dρ), which follows from (d). Hence ρ is N-extremal [1]. This implies that m(ζ, g) lies on the Weyl circle, and then it is the Weyl m-function of some self-adjoint extension J(g) [1], [23]. It remains to show that σ (J(g)) = {λk }k and σ (J∞ (g)) = {μk }k . So we start from (4.16) and find the Weyl m-function of J(g) using (4.19) 

dρ(t) 1 (ζ ) . = =m m(ζ, g) = t − ζ τ (λ k k − ζ) R k∈M

 and then the eigenvalues of But {λk }k and {μk }k are the poles and zeros of m   J(g) and J∞ (g), respectively. For Jacobi operators semi-bounded from below, necessary and sufficient conditions are given in [14]. Note that Remark 3.5 can also be made here. It is worth mentioning that, from (4.8) and (4.9), it follows that, when (b) is seen as a necessary condition, one could write     ζ ζ −1 ,  > 0, lim ζ 1− 1− μk λk ζ →∞ Im ζ  

k∈M

instead of (4.13).

Appendix Boundary Conditions for Jacobi Operators The difference expression γ defined by (2.1) and (2.2) can be written together in one equation with the help of some conditions. Indeed, consider the difference expression γ˜ given by (γ˜ f )k = bk−1 fk−1 + qk fk + bk fk+1 ,

k ∈ N (b0 = 1) .

(A.1)

Clearly, γ f is equal to γf ˜ provided that f0 = 0 .

(A.2)

This requirement can be considered as a boundary condition for the difference equation (A.1). Notice that, although f0 is not an element of the sequence { fk }∞ k=1 , it can be used to introduce boundary conditions for (A.1) which turn out to be completely analogous to the boundary conditions at the origin for the Sturm–Liouville operator on the semi-axis. We shall refer to (A.2) as the

On the two spectra inverse problem for Jacobi matrices

289

Dirichlet boundary condition. Thus, J is the closure of the operator which acts on sequences of l f in (N) by (A.1) with the Dirichlet boundary condition (A.2). Suppose that the deficiency indices of J are (1, 1) and consider now the following solution of (2.1) v˜k (β) := Qk−1 (0) cos β + Pk−1 (0) sin β,

β ∈ [0, π ) .

Let us define the set  f = { fk }∞ ˜ f) = 0 . k=1 ∈ l2 (N) : γ˜ f ∈ l2 (N), lim Wn (v(g), n→∞

(A.3)

Notice that D(g), defined by (2.7), coincides with (A.3) as long as g = cot β. As pointed out in Section 2, the domain of every self-adjoint extension of J is given by (A.3) for some β, and different β’s define different self-adjoint extensions [25]. Let us denote these self-adjoint extensions by J(g), as we did in Section 2, bearing in mind that g = cot β. The condition ˜ f ) = 0, lim Wn (v(g),

n→∞

f ∈ dom(J ∗ )

(A.4)

determining the restriction of J ∗ is considered to be a boundary condition at infinity. In analogy with the case of Sturm–Liouville operators one can define general boundary conditions at zero for the difference expression (A.1). To this end, consider the operator J(α, g) defined by the difference expression (A.1) with boundary condition at infinity (A.4) if necessary, and boundary condition ‘at the origin’ f1 cos α + f0 sin α = 0,

α ∈ [0, π ) .

(A.5)

Thus, if α ∈ (0, π ), J(α, g) = J(g) − cot α ·, e1 e1 . Therefore J(α, g) = Jh (g), provided that h = cot α. When α = 0, from (A.5), one has f1 = 0 and (A.1) is used to define the action of the operator for k  2. J(0, g) is said to be operator J(g) with Neumann boundary condition. For this case we have that J(0, g) is equal to J∞ (g). Acknowledgement

We thank Rafael del Rio for a hint on the literature.

References 1. Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Hafner, New York (1965) 2. Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover, New York (1993) 3. Aktosun, T., Weder, R.: Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrödinger equation. Inverse Problems 22, 89–114 (2006) 4. Aktosun, T., Weder, R.: The Borg–Marchenko theorem with a continuous spectrum. In: Recent Advances in Differential Equations and Mathematical Physics, Contemp. Math., 412, 15–30. AMS, Providence, RI (2006)

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5. Berezans’ki˘ı, J.M.: Expansions in Eigenfunctions of Selfadjoint Operators. Transl. Math. Monogr. Amer. Math. Soc. Providence, R.I. 17 (1968) 6. Borg, G.: Uniqueness theorems in the spectral theory of y + (λ − q(x))y = 0. In: Proceedings of the 11th Scandinavian Congress of Mathematicians. Johan Grundt Tanums Forlag, Oslo pp. 276–287 (1952) 7. Brown, B.M., Naboko, S., Weikard, R.: The inverse resonance problem for Jacobi operators. Bull. London Math. Soc. 37, 727–737 (2005) 8. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer, Berlin Heidelberg New York (1987) 9. Donoghue, Jr., W.F.: On the perturbation of spectra. Comm. Pure Appl. Math. 18, 559–579 (1965) 10. Fu, L., Hochstadt, H.: Inverse theorems for Jacobi matrices. J. Math. Anal. Appl. 47, 162–168 (1974) 11. Gesztesy, F., Simon, B.: Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators. Trans. Amer. Math. Soc. 348, 349–373 (1996) 12. Gesztesy, F., Simon, B.: m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices. J. Anal. Math. 73, 267–297 (1997) 13. Gesztesy, F., Simon, B.: On local Borg–Marchenko uniqueness results. Comm. Math. Phys. 211, 273–287 (2000) 14. Guse˘ınov, G.Š.: The determination of the infinite Jacobi matrix from two spectra. Mat. Zametki 23, 709–720 (1978) 15. Halilova, R.Z.: An inverse problem. Izv. Akad. Nauk Azerba˘ıdžan. SSR Ser. Fiz.-Tehn. Mat. Nauk 3–4, 169–175 (1967) 16. Kato, T.: Perturbation Theory of Linear Operators. Second Edition. Springer, Berlin Heidelberg New York (1976) 17. Koelink, E.: Spectral theory and special functions. In: Laredo Lectures on Orthogonal Polynomials and Special Functions. Adv. Theory Spec. Funct. Orthogonal Polynomials, pp. 45–84. Nova, Hauppauge, NY (2004) 18. Levin, B.Ja.: Distribution of zeros of entire functions. Transl. Math. Monogr. Am. Math. Soc. Providence 5 (1980) 19. Levitan, B.M., Gasymov, M.G.: Determination of a differential equation by two spectra. Uspekhi Mat. Nauk 19, 3–63 (1964) 20. Marchenko, V.A.: Some questions in the theory of one-dimensional linear differential operators of the second order. I, Tr. Mosk. Mat. Obš. 1, 327–420 (1952) [Amer. Math. Soc. Transl. (Ser. 2) 101, 1–104 (1973)] 21. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV Analysis of Operators. Academic, New York (1978) 22. Schechter, M.: Spectra of Partial Differential Operators. Second Edition. Applied Mathematics and Mechanics 14. North Holland, Amsterdam (1986) 23. Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203 (1998) 24. Teschl, G.: Trace formulas and inverse spectral theory for Jacobi operators. Comm. Math. Phys. 196, 175–202 (1998) 25. Teschl, G.: Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs. Amer. Math. Soc., Providence 72, 2000 26. Weidmann, J.: Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics 68. Springer, Berlin Heidelberg New York (1980) 27. Weikard, R.: A local Borg–Marchenko theorem for difference equations with complex coefficients. In: Partial differential equations and inverse problems. Contemp. Math., vol. 362, pp. 403–410. AMS, Providence, RI (2004)

Math Phys Anal Geom (2006) 9:291–333 DOI 10.1007/s11040-007-9018-3

The Canopy Graph and Level Statistics for Random Operators on Trees Michael Aizenman · Simone Warzel

Received: 11 July 2006 / Accepted: 12 March 2007 / Published online: 11 May 2007 © Springer Science + Business Media B.V. 2007

Abstract For operators with homogeneous disorder, it is generally expected that there is a relation between the spectral characteristics of a random operator in the infinite setup and the distribution of the energy gaps in its finite volume versions, in corresponding energy ranges. Whereas pure point spectrum of the infinite operator goes along with Poisson level statistics, it is expected that purely absolutely continuous spectrum would be associated with gap distributions resembling the corresponding random matrix ensemble. We prove that on regular rooted trees, which exhibit both spectral types, the eigenstate point process has always Poissonian limit. However, we also find that this does not contradict the picture described above if that is carefully interpreted, as the relevant limit of finite trees is not the infinite homogenous tree graph but rather a single-ended ‘canopy graph.’ For this tree graph, the random Schrödinger operator is proven here to have only pure-point spectrum at any strength of the disorder. For more general single-ended trees it is shown that the spectrum is always singular – pure point possibly with singular continuous component which is proven to occur in some cases. Keywords Random operators · Level statistics · Canopy graph · Anderson localization · Absolutely continuous spectrum · Singular continuous spectrum Mathematics Subject Classifications (2000) 47B80 · 60K40

M. Aizenman · S. Warzel (B) Departments of Mathematics and Physics, Princeton University, Princeton, NJ 08544, USA e-mail: [email protected]

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Contents 1 Introduction 1.1 1.2 1.3

An overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 The random operator on finite subgraphs of a regular tree . . . . . . . 294 The canopy operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

2 Conditions for Poisson statistics for tree operators 2.1 2.2

302

The decay of fractional moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Lower bound on the Lyapunov exponent . . . . . . . . . . . . . . . . . . . . . . 306 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

4 Proof of Poisson statistics for tree operators 4.1 4.2 4.3

298

The density bounds of Wegner and Minami . . . . . . . . . . . . . . . . . . . 298 Proof strategy and a sufficient condition . . . . . . . . . . . . . . . . . . . . . . . 299

3 Decay estimates of the Green function 3.1 3.2 3.3

293

309

Divergent fluctuations of the spectral measure in the bulk . . . . . . . 309 The intensity measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

5 Complete localization for random operators on the canopy graph

314

6 The spectra of random operators on single-ended trees

317

6.1 6.2

Absence of absolutely continuous spectrum . . . . . . . . . . . . . . . . . . . 317 Appearance of singular continuous spectrum . . . . . . . . . . . . . . . . . . 319

A Some basic properties of the canopy operator A.1 A.2

323

Existence of the canopy density of states measure . . . . . . . . . . . . . 323 Spectrum of the adjacency operator on the canopy graph . . . . . . . 324

B Negligibility in probability of the spectral measure within the singular spectrum 325 C A decorated Delyon-Kunz-Souillard bound C.1

327

Proof of Proposition 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

D Discussion

331

Canopy graph and level statistics for random operators on trees

293

1 Introduction 1.1 An Overview For random operators with extensive disorder it is generally expected that there is an interesting link between the nature of the spectra of the infinite operator and the statistics of energy gaps of the finite-volume restriction of the random operator. Extensively studied examples of operators with disorder include the Schrödinger operator with random potential [14, 36, 40] and the quantum graph operators, as in [6, 31]. The often heard conjecture (see e.g. [8, 19, 22, 39] and references therein) is that on the scale of typical energy spacing the energy levels will exhibit Poisson statistics throughout the pure point (pp) spectral regimes, and level repulsion through energy ranges for which the infinite systems has absolutely continuous (ac) spectrum. The presence of pp spectra for random Schrödinger operators on 2 (Zd ) or 2 L (Rd ) is now thoroughly investigated. In this context, the conjectured Poisson statistics has been established throughout the localization regime for the lattice cases [33], and also for the d = 1 continuum operators [35], which exhibit only pure point spectra. For dimensions d > 2 it is expected that random operators will exhibit also ac spectra. However, so far the only cases of operators with extensive disorder for which the existence of an ac spectral component was proven are operators on tree graphs [5, 23, 28, 29]. Attempting to analyze the conjecture in that context we encountered two surprises, on which we would like to report in this note: 1. For random operators on trees, under an auxiliary technical assumption which is spelled below, the level distribution is given by Poisson statistics through the entire spectral regime. In particular, the statistics of the neighboring levels is typically free of level repulsion even throughout the spectral regimes where the infinite tree operator has ac spectrum. 2. For the purpose of the level statistics of finite tree graph operators, as observed within energy windows scaled by a volume factor, the relevant infinite graph is not the regular tree graph, but another one, which is called here the canopy graph. This graph is isomorphic to the horoball subgraphs of the regular tree, in the terminology explained in [44]. The second point is related to the known result, proven in [41, 42, Thm.1.1], concerning the density of states of Schrödinger operators on hyperbolic spaces (see Section 1.3). The main surprise (1.) is then somewhat reconciled with the above general expectation by the next result: 3. The corresponding random operator on the (infinite) canopy graph, has only pp spectrum at any level of extensive disorder. In the above statement, the absence of an absolutely continuous component is readily explained by the fact that the canopy graph has exactly one end in the sense (see, e.g. [44]) that from each point on it emanates exactly one

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infinite path. However, some more detailed analysis is required to prove that the spectrum is pure point. To make that clear we also prove the following, which may be of some independent interest: 4. There are tree graphs with exactly one end on which the spectrum of the corresponding Schrödinger operator almost surely has a singular continuous spectral component. We shall now make those statements more explicit. 1.2 The Random Operator on Finite Subgraphs of a Regular Tree Let T denote the vertex set of a rooted tree graph for which all vertices have K neighbors in directions away from the root 0, for some fixed K  2. Out of the infinite tree T we carve an increasing sequence of finite trees of depth L, denoting: T L := {x ∈ T : dist(0, x)  L} ,

(1.1)

Here dist(·, ·) refers to the natural distance between two vertices in T . The adjacency operator on the Hilbert space of square-summable functions ψ ∈ 2 (T L ) is given by (Aψ) (x) :=



ψ(y) .

(1.2)

y∈T L : dist(x,y)=1

In the notation for A we omit the index (T L ) indicating on what 2 -space the operator acts. We will be concerned with random perturbations of the adjacency operator, namely self-adjoint operators of the form HTL := A + V + B,

(1.3)

acting in 2 (T L ), with V a random potential and B a boundary term, both given by multiplication operators: (Vψ) (x) := ωx ψ(x) ,  (Bψ) (x) :=

b ψ(x) if dist(0, x) = L 0 otherwise .

(1.4) (1.5)

Here {ωx }x∈T stands for a collection of independent identically distributed (iid) random variables, and b ∈ R is a fixed number. The latter serves as a control parameter, in effect allowing to vary the boundary conditions at the outer boundary, a term by which we refer to the set ∂ T L := {x ∈ T : dist(0, x) = L}.

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295

Throughout this discussion we restrict ourselves to random potentials whose probability distribution meets the following condition: Assumption A1: The distribution of the potential variables ωx is of bounded  density,  ∈ L∞ (R), and satisfies R |ω0 |τ (ω0 )dω0 < ∞ for some τ > 0. The main object of interest will be the random point process of eigenvalues of HTL , seen on the scale of the mean level spacing. For a finite operator, the expected number of eigenvalues in an interval is proportional to the number of sites of the finite graph, |T L | (see the Wegner estimate (2.1) below). It is therefore natural to consider the point process of the eigenvalues as seen under the magnification by the volume. Thus, for a given energy E ∈ R we consider the random point measure  μ LE := δ |TL |(En (TL )−E) , (1.6) n

where {En (T L )} denotes the sequence of random eigenvalues of HTL , counting multiplicity. Our main results are derived under the additional assumption:    −1   Assumption A2: The expectation values E ln  δ0 , HTL − E δ0  are equicontinuous functions of E ∈ I over some Borel set I ⊂ R. An explicit example, which satisfies both Assumptions A1 and A2 for I = R, is the Cauchy distribution. In that case, Cauchy integration allows to equate the above expectation value with the resolvent at an energy off the real axis, and then A2 is easily seen to be valid. For the general case, through a Thouless-type formula one gets    −1  ν L (E)  dE, (1.7) E ln  δ0 , HTL − z δ0  = Re R E−z

  where ν L (E) := E Tr P(−∞,E) (HTL ) − K E Tr P(−∞,E) (HTL−1 ) defines the spectral shift function related to the removal of the root in T L . The symbol Tr refers to the trace and P I denotes the spectral projection onto the Borel set I ⊂ R. Assumption A2 is therefore connected to the regularity of this spectral shift function. Such regularity may be deduced from some of the results in [1] which address distributions ‘near’ the Cauchy case. The main result of the present paper is Theorem 1.1 (Poisson statistics) Let H be a random Schrödinger operator, as in (1.3), for which the conditions A1 and A2 hold for some interval I ⊂ R. Then for Lebesgue almost every E ∈ I the random point measures μ LE converges to a Poisson point measure μ E as L → ∞. The intensity of the limiting Poisson point process μ E is given by the Lebesgue measure times the canopy density of states dC (E), which is the topic of Subsection 4.2. In particular, it is shown there that this intensity is non-zero

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in some energy regimes. The proof of Theorem 1.1 is provided in Section 4 below. The convergence refers to the usual notion of weak convergence of random point measures [27]. As explained above, at first glance Theorem 1.1 may appear to be surprising, since it is known that random Schrödinger operators on regular infinite trees exhibit also spectral regimes where the spectrum is ac [5, 23, 28, 29]. Furthermore, the cases for which this result was established include some for which both assumptions are satisfied, and the ac spectrum was even shown to be pure in the present setting [29]. Thus, the result may appear to fly in the face of the oft repeated expectation that ac spectra of the infinite volume limit should be linked with level repulsion of the finite subsystems. However, that discrepancy is resolved by the observations presented next. 1.3 The Canopy Operator It may seem natural to take the line that the infinite-volume limit of the sequence of finite trees T L is the infinite tree T . That is indeed what the graph converges to when viewed from the perspective of the root, or from any site at fixed distance from the root. However, if one fixes the perspective to be that of a site at the outer boundary of T L , the limit which emerges is different. We use the term canopy tree to describe that limiting graph. More explicitly, the rooted canopy tree C is recursively defined in terms of a hierarchy of infinite layers of vertices: starting from an infinite outermost boundary layer, ∂ C , each layer is partitioned into sets of K elements, and the elements of each component are joined to a common site in the next layer; see Fig. 1. Two remarks apply: 1. The canopy graph can be imbedded in the regular tree. It is isomorphic to a horoball, the canopy’s outermost boundary layer corresponding to a horosphere and its different layers to horocycles – in the terminology explained, e.g., in [44]. 2. The observation that a given nested sequence of graphs may have different limits applies also to other graphs. In particular, for the sequence [−L, L]d ∩ Zd analogs of the canopy construction yield the graphs N × Z(d−1) , and also Nk × Z(d−k) for any 0  k  d. In view of the multiplicity of the limiting graphs, one may ponder which of them is of relevance for a given question. If the question concerns an Fig. 1 The canopy graph C for K = 2. The dots indicate that the boundary layer ∂C as well as any layer below is infinite. The vertices x0 , x1 , x2 , . . . mark the points on the unique path P(x0 ) of x0 to ‘infinity’

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 extensive quantity, e.g. Tr F(H L ) = x δx , F(H L ) δx , where H L is a finiterange operator and F some smooth function, then the answer depends on how the environment appears from the perspective of a point which is chosen at random uniformly within the finite graph. In this respect there is a fundamental difference between the finite cubic subgraphs of Zd , [−L, ..., L]d , and the finite subgraphs of a regular tree. In the former case, for L → ∞, under the uniform sampling the distance from the boundary regresses to infinity, and Zd is the natural limit. However, for the tree graphs T L the distribution of the distance to the boundary converges to the exponential distribution: the fraction of points whose distance to the outer boundary exceeds n, decays as K−n . In this case it is the canopy graph which captures the limit. For an explicit formulation of the statement we introduce the canopy operator acting on 2 (C ), HC := A + V + B ,

(1.8)

Here, A is the adjacency operator on 2 (C ) which is defined similarly to (1.2), and B is a boundary term acting as in (1.5), with the same b ∈ R. Moreover, the iid random variables {ωx }x∈C underlying the random multiplication operator V are supposed to satisfy A1. Associated to HC is the following density of states (dos) measure given by ∞

nC (I) :=

 K − 1  −n K E δxn , P I (HC ) δxn  , K n=0

(1.9)

where the sum ranges over all vertices x0 , x1 , . . . on the unique path P (x0 ) of a given vertex x0 ∈ ∂ C to infinity, see Fig. 1. Note that nC does depend on the choice of the boundary conditions parameter b ∈ R, however it is independent of the choice of x0 ∈ ∂ C on the boundary. Theorem 1.2 The density of states of the finite tree T L is asymptotically given by nC in the sense that, with probability one: for any bounded continuous F ∈ Cb (R) lim |T L |−1 Tr F(HTL ) = nC (dE) F(E) . (1.10) L→∞

R

The statement reflects the fact that on trees, asymptotically, almost all points are located not far from the surface. The proof is given in Appendix A.1. The fact that for non-amenable graphs like trees bulk averages as in (1.10) do not converge to corresponding infinite-volume quantities is well known. In particular, a continuum analogue of Theorem 1.2 was presented in [41, 42, Thm.1.1] where it is shown that the finite-volume density of states (dos) of a Laplacian plus Poissonian random potential on hyperbolic space converges to the dos on a horoball. Analogous statements apply to the dos of periodic Schrödinger operators on hyperbolic spaces [2]. Part of the surprise of Theorem 1.1 is now removed by the following result, which is proven in Section 5.

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Theorem 1.3 (Localization of canopy states) If the conditions A1 and A2 hold for I ⊂ R, then the random canopy operator HC has almost surely only pure point spectrum in I. It may also be of interest to note the following curious property of C . Theorem 1.4 (Spectrum of the adjacency operator) The spectrum of the adjacency operator with (constant) boundary conditions, A + B on 2 (C ), is only pure point with compactly supported eigenfunctions. A more detailed description of the spectrum of the adjacency operator can be found in Appendix A.2. The considerations yielding Theorem 1.4 are essentially as in the analysis of the regular tree in [7]. We postpone further comments on possible directions for studies of the relation which was explored in this work to the concluding section, Appendix D.

2 Conditions for Poisson Statistics for Tree Operators 2.1 The Density Bounds of Wegner and Minami Key information on the point process which describes the eigenvalues of the random operator as seen under the magnification by the volume factor |T L | is provided in the following two essential estimates. The Wegner estimate implies that the mean density of states is bounded relative to the Lebesgue measure. The Minami bound guarantees that the energy levels are non-degenerate on the scale of the mean level spacing. Proposition 2.1 Under the assumption A1, for every bounded Borel set I ⊂ R and every L ∈ N 

 P Tr P I (HTL )  1  E Tr P I (HTL )  |I| |T L |  ∞ , (2.1) (the Wegner estimate), and ∞  

  P Tr P I (HTL )  m  E Tr P I (HTL ) Tr P I (HTL ) − 1 m=2

 π 2 |I|2 |T L |2  2∞ ,

(2.2)

(the Minami estimate). Here Tr P I (HTL ) stands for the trace of the spectral projection of HTL onto I. A proof of the Wegner bound (2.1) can be found in [36, 43]. Minami’s estimate (2.2) is presented in [33, Lemma 2, Eq. (2.48)], see also [24]. Although it is stated there for Zd only, its derivation clearly applies to all graphs.

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2.2 Proof Strategy and a Sufficient Condition The main effort in the proof of the convergence of the energy level process to a Poisson limit is to establish infinite divisibility. In the background are the following observations. 1. The Wegner estimate (2.1) implies tightness of the collection of random variables {μ LE (I)} for all intervals I ⊂ R and every E ∈ R. This in turn guarantees that for every E ∈ R the sequence of measures {μ LE }, is tight with respect to the vague topology on the space of Borel measures on the real line. Since the subspace of point measures is closed with respect to this topology, all accumulation points of the above sequence are point measures [27]. 2. In order to show that any accumulation point is a Poisson measure, it is sufficient to prove that each such point is infinitely divisible and almost surely has no double points. The latter is guarantied already by the Minami estimate. 3. Once the divisibility property is established, for convergence of the point process it suffices to show that the intensity measure of any accumulation point is given by some common measure [27]. In our case, that measure is the canopy mean density of states nC . Turning to the divisibility, one may note that for random operators on 2 (Zd ) the divisibility and convergence of the energy level process to a Poisson process were proven by Minami under a natural localization condition (the fractional moment characterization of the pp spectral regime [33]). However, Minami’s proof does not extend to tree graphs, since it makes use of the fact that cubic regions in Zd have the van Hove property, which is that most of the volume is, asymptotically, far from the surface. While this approach does not apply to trees, or hyperbolic spaces, with positive Cheeger isoperimetric constant, for trees there is another pathway towards infinite divisibility of any accumulation point of {μ LE }. In order to show K N -divisibility at some arbitrary N ∈ N, we cut the finite tree T L below the Nth generation. This leaves us with a ‘tree stump’ and the subtrees T L (x) which are forward to vertices x in the Nth generation. Associated with the above collection of forward subtrees is the collection of iid point measures E := μx,L



δ |TL |(En (TL (x))−E) .

(2.3)

n

 E For the sum dist(0,x)=N μx,L to be asymptotically equal to μ LE as L → ∞, so that any of its accumulation points is K N -divisible, it suffices that the spectral measures associated with the roots of the subtrees satisfy the following fluctuation condition. For any site x ∈ T L the spectral measure is defined for Borel sets I ⊂ R by

 σx,L (I) := δx , P I (HTL ) δx .

(2.4)

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 By a Wegner-type estimate the averaged spectral measure, E σx,L , is seen to be ac with a density bounded uniformly in L ∈ N. The condition we require for the proof of divisibility is that the typical value of σx,L on the scale of its mean, |T L |−1 , is much smaller than the average value. More explicitly: Definition 2.1 For a fixed site x ∈ T and energy E ∈ R, the sequence of spectral measures {σx,L } is said to be negligible in probability iff for all w > 0  P −lim |T L | σx,L E + |T L |−1 (−w, w) = 0 ,

(2.5)

L→∞

where the limit refers to distributional convergence. Several remarks apply: 1. The prelimit quantity in (2.5) compares the spectral measure σx,L to the blown-up Lebesgue measure of the corresponding interval. In terms of the normalized eigenfunctions ψn (T L ) ∈ 2 (T L ) of HTL , with the corresponding eigenvalues En (T L ), one has: σx,L (I) =





  δx , ψn (T L ) 2 .

(2.6)

En (T L )∈I

By Wegner’s estimate (2.1) the mean number of levels within any given Borel set with Lebesgue measure proportional to |T L |−1 is bounded uniformly in L. If the corresponding eigenfunctions are spread uniformly over the volume, and the relevant spectral density is non-zero, then the above condition is not satisfied, since whenever an eigenvalue falls within the interval the rescaled spectral measure is not smaller than order 1. Thus, condition (2.5) is equivalent to the statement that either i. the probability of finding an eigenvalue in the energy window vanishes, or ii. the corresponding eigenfunctions are spread very unevenly over the 

2 volume, so that typically  δx , ψn (T L )  × |T L | << 1 even though the average of the quantity over the volume (or over n) is 1. In other words, on the scale of their mean the eigenfunctions exhibit divergent fluctuations. 2. We shall show below, in Appendix B, that the above scenario i. occurs 

if for some energy range the (weak) limiting measure, δx , P· (HT ) δx = lim L→∞ σx,L for x ∈ T , is purely singular. In that case, due to the mutual singularity, the spectral measures underperforms at Lebesgue - chosen E ∈ R, in small intervals of arbitrary scale. It is more of an issue to verify that (2.5) holds throughout the regime of ac spectrum of HT . This will be proven in Subsection 4.1 below, where we show that for canopy graph scenario ii. is in effect. The criterion for Poisson statistics may now be formulated as follows. Theorem 2.2 (Condition for Poisson statistics) Suppose that the sequence of spectral measures at the root, {σ0,L }, is negligible in probability at E ∈ R. Then

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 E for any N ∈ N the sum dist(0,x)=N μx,L converges weakly to the same limit as E 1 μ L , i.e., for all ψ ∈ L+ (R)       E E   lim E e− dist(0,x)=N μx,L (ψ) − E e−μL (ψ)  = 0 . (2.7) L→∞

As a consequence, all accumulation points of μ LE are random Poisson measures. Proof Since the set of functions ϕz := Im( · − z)−1 with z ∈ C+ are dense in L1+ (R), it suffices to verify (2.7) for such functions. It is easy to see that the latter follows from the distributional convergence        E E  P −lim  μx,L (ϕz ) − μ L (ϕz ) = 0 . (2.8) L→∞   dist(0,x)=N Abbreviating ξ L := E + z |T L |−1 the prelimit in (2.8) can be written as       −1 −1    1  Im Tr HTL (x) − ξ L − Im Tr HTL − ξ L  |T L |   dist(0,x)=N



  −1

1  Im δ y , HTL − ξ L δ y + |T L | |y|
+

1 |T L |

    −1   −1   δ y , HTL (x) − ξ L δ y − δ y , HTL − ξ L δ y  . 

 dist(0,x)=N

y∈T L (x)

(2.9) The first term on the right side converges to zero in distribution as L → ∞. Using the resolvent identity twice the modulus in the second term is seen to be equal to     −1   −1   −1   δ y , HTL (x) − ξ L δx δx− HTL − ξ L δx− δx , HTL (x) − ξ L δ y   y∈T L (x)

−2     −1   δx ,  HTL (x) − ξ L  δx  δx− HTL − ξ L δx−  ,

(2.10)

where x− is the backward neighbor of x in T L . The second term in (2.10) is bounded in probability as L → ∞ as is seen from the fractional moment bound (3.1) below. Thanks the fluctuation condition and Lemma 2.3 below the first term, when dividing by |T L |, converges to zero in this limit. 

The previous proof was based on the following Lemma 2.3 Either of the following is equivalent to the statement that the sequence of spectral measures {σx,L } is negligible in probability, at E ∈ R:   −2

1. For all α ∈ R: P −lim |T L |−1 δx , HTL − E − α |T L |−1 δx = 0. L→∞

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2. For all z ∈ C+ :

  −1

P −lim Im δx , HTL − E − z |T L |−1 δx = 0. L→∞

Proof 1. ⇒ 2. ⇒ ‘negligibility in probability’: These two implications are a consequence of the following chain of elementary inequalities   −1

 

2 (Im z)−1 δx , P I(Re z,Im z) HTL , δx  Im δx , HTL − z δx   −2

 Im z δx , HTL − Re z δx , (2.11) valid for all z ∈ C+ , where I(E, w) := E + (−w, w) denotes the open interval centred at E of width 2w > 0. ‘Negligibility in probability’ ⇒ 1.: We split the prelimit in 1. into two terms by inserting a spectral projection onto the interval I L := I(E, w|T L |−1 ) and its complement. Abbreviating ξ L := E + α |T L |−1 the first term is then estimated as follows 

  −2

−1 −1 δx , P I L (HT L )δx (2.12) |T L | δx , HTL − ξ L P IL (HTL ) δx  |T L |  2 . dist σ (HTL ), ξ L Using Wegner’s estimate (2.1) and (2.5), this term is seen to converge in distribution to zero as L → ∞ for any w > 0. The remaining second term is   −2

|T L |−1 δx , HTL − ξ L P ILc (HTL ) δx   −1

 2w −1 Im δx , HTL − ξ L + iw |T L |−1 δx . (2.13) The imaginary part of the resolvent is bounded in probability. Therefore the probability that the right side in (2.13) is greater than any arbitrarily small constant is arbitrarily small for w large enough. 

3 Decay Estimates of the Green Function As was shown in [3, Thm. II.1], fractional moments of the Green function of rather general random operators are uniformly bounded. Proposition 3.1 (Fractional moment bounds) Under the assumption A1, for any s ∈ (0, 1),    −1 s   Cs := sup sup sup E  δx , HTL − z δ y  {x, y}c < ∞ , (3.1) z∈C L∈N x,y∈T L

  where the average E · {x, y}c is the conditional expectation with respect to the sigma-algebra the generated by {ωv }v∈T \{x,y} . The main aim of this section is to prove that fractional moments of the Green function of HTL are not only bounded but decay exponentially along any ray in the tree.

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Theorem 3.2 (Exponential decay I) Assume A1 and A2 holds for a bounded Borel set I ⊂ R. Then there exists s ∈ (0, 1), δ, C ∈ (0, ∞) such that for all E ∈ I, L ∈ N and all x ∈ T L which are in the future of y ∈ T L      √ −1 s   E  δ y , HTL − E δx   C exp −s δ + ln K dist(x, y) .

(3.2)

Several remarks apply: 1. Unlike on Zd , for trees the exponential decay (3.2) does not imply complete localization, i.e. dense pure point spectrum at all energies. In fact, the infinite-volume operator HT has a regime with delocalized eigenstates [5, 23, 28, 29]. 2. The rate of decay in (3.2) is related to a Lyapunov exponent of the infinite-volume operator HT , cf. Subsection 3.2 below. Note that in the unperturbed case where HT = A, the decay rate in (3.2) would be given √ by ln K. It is important for us that the decay rate in (3.2) is strictly larger. 3.1 The Decay of Fractional Moments Our proof of Theorem 3.2 is based on reasonings similar to that which applies in the one-dimensional setup [13], the reason being that any pair of sites on the tree is connected through a single path. As in one dimension, the Green function decays exponentially at a rate characterized by a Lyapunov exponent. In order to relate the decay rate of the fractional-moment of the Green function to that exponent, the following simple lemma will be of help. Lemma 3.3 Let (ξ j) Nj=1 be a collection of independent, positive random variables   2   with c := max j E ln ξ j ξ j + 1 /2 < ∞. Then X := Nj=1 ξ j satisfies E [X]  exp (E [ln X] + Nc ) .

(3.3)

Proof This is a straightforward consequence of the assumed independence and the elementary inequalities eα  1 + α + α 2 (eα + 1) /2 and 1 + β  eβ valid 

for all α, β ∈ R. Lemma 3.4 Let I ⊂ R be a bounded Borel set and assume A1. Then for every ε > 0 there exists sε ∈ (0, 1/2) and Lε ∈ N such that for all s ∈ (0, sε ), E ∈ I, L  Lε and x ∈ T L with dist(0, x)  Lε   −1 s   ln E  δ0 , HTL − E δx     −1    ε dist(0, x) + s E ln  δ0 , HTL − E δx  .

(3.4)

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The proof is based on the factorization of the Green function on a tree, which we recall from [29, Eq. (2.8)],   −1

δ0 , HTL − E δx =

dist(0,x) 

 j,L

j=0

  −1

with  j,L := δx j , HTL (x j ) − E δx j .

(3.5)

Here 0 =: x0 , x1 , . . . , xdist(0,x) := x are the vertices on the unique path connecting the root 0 with x. Moreover, T L (x j) is that subtree of T L which is rooted at and forward to x j. Proof of Lemma 3.4 The idea is to group together subproducts of (3.5) and use certain independence properties in order to apply Lemma 3.3. To do so we pick L0 ∈ N \ {1} and express the distance of x to the root modulo L0 , dist(0, x) = Nx L0 + Lx

(3.6)

with suitable Nx ∈ N0 and Lx ∈ {0, . . . , L0 − 1}. Thanks to the factorization (3.5) we may thus write  N −1  x    −1 s  Xk Yk R (3.7)  δ0 , HTL − E δx  = k=0

with

Xk Yk :=

(k+1)L 0 −1

   j,L s ,

R :=

j=kL0

dist(0,|x|) 

   j,L s .

(3.8)

j=Nx L0

Each product Xk Yk may now be split into two terms by setting Yk equal to the modulus of a diagonal element of the operator corresponding to the forward subtree T L (xkL0 ),   −1

s  (3.9) Yk :=  δx(k+1)L0 −1 , HTL (xkL0 ) − E δx(k+1)L0 −1  . Nx −1 The point is that in this way we obtain a collection (Xk )k=0 of independent, positive random variables. Moreover,

1. Each random variable Xk is independent of the value of the potential at vertex x j with j = (k + 1)L0 − 1 for some k ∈ {0, . . . , Nx − 1} or Nx L0  j  dist(0, |x|). 2. The random variable Yk is independent of the value of the potential at vertex x j with 0  j < kL0 . One may therefore successively integrate the product in (3.7) by first conditioning on the potential at x L0 −1 thereby integrating Y0 , then conditioning on x2L0 −1 thereby integrating Y1 and so forth until we reach x Nx L0 −1 and integrate Y Nx −1 . Thanks to (3.1) these integrals are all uniformly bounded, Ex(k+1)L0 −1 [Yk ]  Cs .

(3.10)

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Moreover, conditioning on the values of the potential at x Nx L0 and x, the fractional-moment bound (3.1) also yields Ex Nx L0 ,x [R]  Cs .

(3.11)

 Nx −1 One is then left with the integral of the product k=0 Xk , which can be bounded with the help of Lemma 3.3. Its assumption is satisfied since

 2c := max E (ln Xk )2 (Xk + 1) k



     4 1/4   1/4 2 + E (ln Yk )4 × sL0 max E ln  j,L 



j

  1/2 × E Xk2 + 2Xk + 1  s2 L20 C .

(3.12)

The above result is based on the Cauchy–Schwarz inequality and norm subadditivity. Moreover, the last inequality uses (3.1) which also proves that the expectations of powers of logarithms of diagonal Green functions are uniformly bounded by Lemma 3.6. In applying Lemma 3.3 it is also useful to note that  N −1  x     −1   E ln Xk − s E ln  δ0 , HTL − E δx  k=0

=−

N x −1 

E [ln Yk ] − E [ln R]

k=0

    Nx max |E [ln Yk ]| + sL0 max E ln  j,L    s C(Nx + L0 ) , k

j

(3.13)

where we have again used the fact that expectations of logarithms of diagonal Green functions are uniformly bounded. Summarizing the above estimates we obtain the bound      −1 s  −1    ln E  δ0 , HTL − E δx  − s E ln  δ0 , HTL − E δx   (Nx + 1) ln Cs + Nx s2 L20 C + s C(Nx + L0 )  2  dist(0, x) 2L−1 0 ln Cs + s L0 C + 2s C ,

(3.14)

where the last inequality holds provided dist(0, x)  L0 . Consequently, for a given ε > 0 we may then pick L0 = Lε large enough and sε small enough such that the right side in (3.14) is smaller that ε dist(0, x) for every s ∈ (0, sε ). We close this subsection by compiling two elementary estimates on expectations of functions of the diagonal of the Green function. The first bounds concern fractional moments of the Green function going back to [3].

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Lemma 3.5 Assume A1 and let s ∈ (0, 1), z ∈ C and L ∈ N. Then   −1 s   E  δ0 , HTL − z δ0   Cs and

  −1 −s   E  δ0 , HTL − z δ0   |ξ |s (ξ )dξ + |z|s + K Cs , R

(3.15)

(3.16)

where Cs is the constant appearing in (3.1). Proof The first inequality is an immediate consequence of the fractional moment bound (3.1). The second one is a consequence of the first and the recursion relation which the diagonal of the resolvent is well-known satisfy, cf. [29],      −1  −1 −1 δx , HTL (x) − z δx δ0 , HTL − z δ0 = ω0 − z − (3.17) dist(x,0)=1

where we recall that T L (x) is that subtree of T L which is forward to x.



Lemma 3.5 in particular implies that any moment of the logarithm of the Green function is uniformly bounded. Lemma 3.6 Assume A1 and let I ⊂ R be a bounded Borel set and n ∈ Z. Then     −1 n    sup sup E  ln  δ0 , HTL − E δ0  < ∞ . (3.18) E∈I L∈N

Proof This estimate immediately follows from Lemma 3.5 and the fact that |ln ξ |  ξ τ + ξ −τ for any ξ > 0 and τ = 0. 

3.2 Lower Bound on the Lyapunov Exponent In [5] we considered a Lyapunov exponent for the operator HT on the infinite regular rooted tree with branching number K  2, defining it as  √   −1   K  δ0 , HT − z δ0  . (3.19) γ (z) := −E ln It was shown there [5, Thm. 3.1 and Thm. 4.1] that 1. γ (z) is a positive harmonic function of z ∈ C+ and hence its boundary values γ (E + i0) with E ∈ R define a locally integrable function. 2. For all z ∈ C+ and all α ∈ (0, 1/2)   α2 δ |0 (z)|−2 , α 2 , 2 32(K + 1)   −1

where 0 (z) := δ0 , HT − z δ0 and δ(·, ·) is defined as follows. γ (z) 

(3.20)

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Definition 3.1 (Relative width) For α ∈ (0, 1/2] the relative α-width of a positive random variable X is given by δ(X, α) := 1 −

ξ− (X, α) . ξ+ (X, α)

(3.21)

where ξ− (X, α) := sup{ ξ , P (X < ξ )  α} and ξ+ (X, α) := inf{ ξ , P (X > ξ )  α}. Our next task is to further estimate the right side of (3.20) from below. This will be done with the help of the following lemma. Lemma 3.7 Let X be a positive random variable with probability measure P. Suppose 1. There exists σ ∈ (0, 1] and Cσ < ∞ such that P (X ∈ I)  Cσ |I|σ for all Borel sets I ⊂ [0, ∞) with |I|  1. 2. There exists τ > 0 such that E [X τ ] < ∞. Then for all α ∈ (0, 1/2)

  1/τ   1 − 2α 1/σ α δ(X, α)  min 1, . Cσ E [X τ ]

(3.22)

 Proof The first assumption implies that 1 − 2α = P{X ∈ ξ− (X, α), ξ+ (X,  σ α) }  Cσ ξ+ (X, α) − ξ− (X, α) provided ξ+ (X, α)− ξ− (X,   α)  1. From the second assumption we conclude that α  P X ∈ ξ+ (X, α), ∞  E [X τ ] /ξ+ (X, α)τ by a Chebychev inequality. Inserting these two estimates into (3.21) completes the proof. 

Lemma 3.8 Let I ⊂ R be a bounded Borel set and  2/τ  1 − 2α α α2

 min 1, l(I) := sup 2 2  ∞ E sup E∈I |0 (E + i0)|−τ α∈(0,1/2) 32(K + 1) (3.23) where τ is the constant appearing in Assumption A1. Then γ (E + i0)  l(I) > 0 for any E ∈ I. Proof In order to apply Lemma 3.7 to the right side in (3.20) we need to check its assumptions. We first note that by the Krein formula |0 (z)|−2 = (ω0 − a)2 + b 2 with suitable a, b ∈ R. An elementary computation shows that for every Borel set I ⊂ [0, ∞) with |I|  1 !  ∞ ! (ξ ) 1{(ξ −a)2 +b 2 ∈I} dξ  1{ξ b 2 } dξ  2  ∞ |I| . (3.24) ξ − b2 R I

 Moreover, Lemma 3.5 guarantees that sup E∈I E |0 (E + i0)|−τ < ∞. 

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Associated with γ (z) is the following finite-volume approximation  √   −1   γ L (z) := −E ln K  δ0 , HTL − z δ0  .

(3.25)

It is easy to see that γ L (z) also defines a harmonic function of z ∈ C+ . Moreover, its boundary values γ L (E) are defined everywhere by setting z = E ∈ R in (3.25). Strong resolvent convergence implies that lim L→∞ γ L (z) = γ (z) for every z ∈ C+ . Assumption A2 guarantees that this convergence holds and is locally uniform also for real arguments. Lemma 3.9 Suppose A2 holds for a bounded Borel set I ⊂ R. Then lim sup |γ L (E) − γ (E + i0)| = 0 .

L→∞ E∈I

(3.26)

Proof Since γ L (E) are uniformly bounded for E ∈ I, cf. Lemma 3.6, the Arzela–Ascoli theorem Assumption A2 thus implies that every subsequence of γ L has a uniformly convergent subsequence. The claim (3.26) then follows by showing that any pointwise limit of γ L (E) coincides with γ (E + i0). This is derived from the above mentioned strong resolvent convergence and the dominated convergence theorem, which imply that for any bounded and compactly supported function φ ∈ L∞ c (R) γ (E + i0)φ(E)dE = lim γ L (E)φ(E)dE = lim γ L (E)φ(E)dE . L→∞ R

R

R L→∞

Provided lim L→∞ γ L (E) exists for Lebesgue-almost all E ∈ R.

(3.27) 

3.3 Proof of Theorem 3.2 Proof of Theorem 3.2 If x = y is in the future of y, the Green function factorizes according to −1   −1   −1

  δ y , HTL − E δx = δ y , HTL − E δ y δv , HTL (v) − E δx (3.28) where v is that forward neighbor of y which lies on the unique path connecting x and y. We may therefore suppose without loss of generality that y coincides with the root in T L . In this case, Lemma 3.4 bounds the fractional moment of the Green function by an exponential involving dist(y,x)      √  −1  γ L− j(E) + ln K , E ln δ y , HTL − E δx = −

(3.29)

j=0

where we the last equality results from (3.5), stationarity and the definition of the finite-volume Lyapunov exponent in (3.25). According to Lemma 3.9, for a given ε > 0 there exists Lε ∈ N such that γ L (E)  γ (E + i0) − ε  l(I) − ε for all E ∈ I and L  Lε , where l(I) > 0 was defined in Lemma 3.8. The proof is completed by choosing ε small enough in the last estimate and in Lemma 3.4.

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4 Proof of Poisson Statistics for Tree Operators We will follow the general strategy outlined in Subsection 2.2: Poisson statistics are established through the proof of the negligibility in probability condition (2.5), which holds due to the divergence of fluctuations of the value at a fixed site of the normalized eigenfunctions. 4.1 Divergent Fluctuations of the Spectral Measure in the Bulk The following theorem allows to conclude that for any x ∈ T the sequence of spectral measures {σx,L } has divergent fluctuations in the sense of Definition 2.1, at any E ∈ R for which the average over the disorder of theses measures is non zero. Theorem 4.1 (Negligibility in probability) Assume A1 and A2 holds for a bounded Borel set I ⊂ R. If I L ⊂ I are bounded Borel sets such that lim sup L→∞ |I L ||T L | < ∞, then for any x ∈ T P −lim |T L | σx,L (I L ) = 0 .

(4.1)

L→∞

For the proof we now fix x ∈ T , and for L ∈ N large enough so that x ∈ T L and every y ∈ T L we define the ratio   −1 2   δx , HTL − E δ y  g y,L (E) :=   (4.2) −2 . δ y , HTL − E δ y It is well-defined for Lebesgue-almost all E ∈ R. Moreover, by the rank-one perturbation formula and the spectral theorem it is seen to enjoys the following properties: 1. g y,L (E) is independent of the value of the potential at y ∈ T L . 2. The function E → g y,L (E) has a continuous extension on R. Moreover, if the eigenvalue En (T L ) of HTL is non-degenerate, then the corresponding eigenfunction satisfies 

  δx , ψn (T L ) 2 = lim g y,L (E) (4.3) E→En (T L )

Theorem 4.1 will now be a consequence of the following result. Lemma 4.2 Under the assumptions of Theorem 4.1 for any ε > 0    lim E σ y,L E ∈ I L : g y,L (E)  ε |T L |−1 = 0 . L→∞

(4.4)

y∈T L

Proof The proof is based on the spectral averaging principle (cf. [14, 38]), (ω y ) σ y,L (I)dω y   ∞ σ y,L (I) dω y   ∞ |I| (4.5) R

R

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for all bounded Borel sets I ⊂ R. Using this inequality and the fact that g y,L (E) does not depend on ω y , the prelimit in (4.4) can be bounded from above by   (4.6)  ∞ P g y,L (E)  ε |T L |−1 dE . y∈T L

IL

We now pick N ∈ N and split the summation in (4.6) into two terms. The first term collects all contributions corresponding to T L−N ⊂ T L ,      P g y,L (E)  ε |T L |−1 dE  T L−N   I L  . (4.7) y∈T L−N

IL

In the limit L → ∞ this term is arbitrarily small for N large  enough.  To estimate the remaining second term, we abbreviate α y,L (E) := δ y , HTL − −2

E δ y and write      −1 2  P g y,L (E)  ε |T L |−1 = P |T L |  δx , HTL − E δ y   ε α y,L (E) 

    −1 2   P |T L |  δx , HTL − E δ y   εα + P α y,L (E) < α (4.8) where the last inequality holds for any α ∈ (0, ∞). The first term on the right side of (4.8) gives rise to the following contribution to the sum in (4.6),      −1 2 |T L |s  P |T L |  δx , HTL − E δ y   εα dE  s s s(L−N) × εα K IL y∈T \T L

L−N

× sup |I L | E∈I



K

s dist(x,y)

  −1 2s   E  δx , HTL − E δ y  ,

(4.9)

y∈T L \T L−N

where I ⊂ R is some bounded Borel set which contains eventually all I L . While the prefactors on the right side of (4.9) remains finite in the limit L → ∞, the supremum converges to zero in this limit, since it is bounded by |I L ||T L | C exp (−2sδ (L − N)) for sufficiently small s by Theorem 3.2. To complete we note that the second term in (4.8), converges to zero as α ↓ 0, uniformly in E ∈ I, L ∈ N and y ∈ T L . This follows from the bound  " "−2  P α y,L (E) < α  P " HTL − E δ y " < α " "2s

  α s " (A + B − E) δ y " + E |ω y |2s , where the last step requires 2s < min(1, τ ).

(4.10) 

Proof of Theorem 4.1 Wegner’s bound (2.1) implies that I L carries only a finite number of eigenvalues  lim sup P Tr P IL (HTL )  N = 0 . (4.11) N→∞ L∈N

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It therefore remains to prove that for any ε > 0 ⎡ lim E ⎣

L→0

⎤ & %

2 1  δx , ψn (T L )   ε |T L |−1 ⎦ = 0 ,



(4.12)

En (T L )∈I L

where 1{· · · } stands for the indicator function. Using the fact that HTL has almost surely no degenerate eigenfunctions (cf. Proposition 2.1) and (4.3), the left side in (4.12) is seen to be equal to the left side in (4.4). 

  Theorem 2.2 now guarantees that any accumulation point of μ LE is a random Poisson measure. The uniqueness of the accumulation point will be proven by establishing uniqueness of the intensity measure, which is defined next. 4.2 The Intensity Measure For a random point measure μ the intensity measure is defined as the average μ := E [μ] .

(4.13)

Thus μ LE are the intensity measures of the random point measures μ LE . We shall E also use the symbol  Eμ for the intensity measure of a given accumulation point of the sequence μ L . Let us proceed with a more explicit representation for μ LE . For any Borel set I ⊂ R we have

 μ LE (I) = E Tr P E+I/|TL | (HTL ) =





 E δx , P E+I/|TL | (HTL ) δx

x∈T L

=

L 



 K L−n E δxn , P E+I |TL |−1 (HTL ) δxn ,

(4.14)

n=0

where we used the fact that the expectation in the second line does not depend on x as long as dist(0, x) is constant. Moreover, xn denotes any vertex with dist(xn , ∂ T L ) = n. In view of Lemma A.1 in the Appendix, the above calculation (4.14) suggests that the intensity measure μ LE converges for Lebesgue almost all E ∈ R to Lebesgue measure times the canopy density of states given by dC (E) :=

∞

 K − 1  −n −1  K π E Im δxn , (HC − E − i0)−1 δxn . K n=0

(4.15)

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Theorem 4.3 (Limiting intensity measure) Under assumption A1 and A2 (or alternatively A1’ below) for Lebesgue-almost all E ∈ R the intensity measure μ E of any weak accumulation point μ E of the sequence μ LE is given by μ E (I) = lim μ LE (I) = dC (E) |I| L→∞

(4.16)

for all bounded Borel sets I ⊂ R. Proof As an immediate consequence of Wegner’s estimate (2.1) and the first line in (4.14) we have that for Lebesgue-almost all E ∈ R and all L ∈ N the measures μ LE are absolutely continuous with bounded density, μ LE (dξ )   ∞ . dξ

(4.17)

The same applies to any accumulation point μ E . As a consequence, the linear  E functional given by μ L (ψ) := R ψ(ξ ) μ LE (dξ ) is uniformly equicontinuous on the space of non-negative integrable functions on the real line, ψ ∈ L1+ (R). More precisely, (4.17) yields  E  μ (φ) − μ E (ψ)   ∞ φ − ψ 1 (4.18) L L for all φ, ψ ∈ L1+ (R). Using this and the fact that the functions ϕz := π −1 Im(· − z)−1 with z ∈ C+ are dense in L1+ (R) implies that it suffices to check (4.16) if the indicator function of I is replaced by ϕz . Moreover, elementary considerations show that it suffices to verify  E  μ ϕz − d(E) ϕz 1  dE = 0 lim (4.19) L L→∞ R

with z ∈ C+ fixed but arbitrary. A computation similar to (4.14) and the fact that ϕz 1 = 1 then proves that this derives from    −1

 lim E Im δx , HTL − E − z |T L |−1 δx − L→∞ R



 − Im δx , (HC − E − i0)−1 δx  dE = 0

(4.20)

for x ∈ C with dist(x, ∂ C ) ∈ N0 fixed but arbitrary. For a proof of (4.20), we appeal to Riesz’s theorem which guarantees that the claimed L1 -convergence follows from the almost sure convergence of the integrand in (4.20) with respect to the product of the probability measure and Lebesgue measure, and the equality of the integrals    −1  1 lim E Im δx , HTL − E − z |T L |−1 δx dE L→∞ π R



 1 E Im δxn , (HC − E − i0)−1 δxn dE = 1 . (4.21) = π R

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In fact, we only need to show that the integrand in (4.20) converges in distribution with respect to the product measure. To prove the latter we first note that one has the non-tangential limit 



lim δx , (HC − E − z |T L |−1 )−1 δx = δx , (HC − E − i0)−1 δx

L→∞

(4.22)

for Lebesgue-almost all E ∈ R. Moreover, using the resolvent identity twice, we obtain the inequality  

 

  δx , HT − E − z |T L |−1 −1 δx − δx , HC − E − z |T L |−1 −1 δx  L   −1  −1      δx , HTL − E − z|T L |−1 δ0L δ0L , HTL − E − z |T L |−1 δx  ×   −1   ×  δ0−L , HC − E − z |T L |−1 δ0−L  , (4.23) where 0 L is the root in T L and 0− L is its backward neighbor. The right side converges to zero in distribution with respect to the product of the probability measure and Lebesgue measure on any bounded interval. This follows from Theorem 3.2 (or alternatively Proposition 6.2 below) and the fact that the factional-moment bound (3.1) implies that the probability that the last term in (4.23) is large is bounded. 

4.3 Proof of Theorem 1.1 Theorem 1.1 may be stated using the characterisation of the Poisson process in terms of its characteristic functional. Namely, the random measure μ E is Poisson if for any bounded Borel set I ⊂ R and t  0   

 E E e−tμ (I) = exp −E μ E (I) 1 − e−t .

(4.24)

Given Theorems 2.2 and 4.1, the proof of (4.24) is basically a repetition of wellknown arguments how to conclude the Poisson nature of accumulation points from infinite divisibility and the exclusion of double points [27]. Proof of Theorem 1.1 Let μ E be an accumulation point of {μ LE }. Theorem 2.2 implies that for any N ∈ N and any bounded Borel set I ⊂ R ⎡

⎛

lim E ⎣ exp ⎝−t

L→∞



⎞⎤ E μx,L (I)⎠⎦

dist(0,x)=N

    E E = lim E e−tμL (I) = E e−tμ (I) . L→∞

(4.25)

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Since the measures in the left side of (4.25) are iid, the expectation factorizes into a K N -fold product of ∞     E N E e−tμx,L (I) = e−tm P μx,L (I) = m m=0

E  (I) 1 − e−t + Rx,L (I) , = 1 − E μx,L

(4.26)

where 0  Rx,L (I) :=

∞     E P μx,L (I) = m m 1 − e−t + e−tm − 1 m=2



∞ 

∞  E   E (m − 1) P μx,L (I) = m = P μx,L (I)  m

m=2

(4.27)

m=2

By (2.2) this term is arbitrarily small in the limit L → ∞ provided N is large enough. The second term in (4.26) converges,  

E

(I) = E μ E (I) . (4.28) lim K N E μx,L L→∞

The claim now follows  by taking the subsequent limit N → ∞ in (4.25) from the fact that limn→∞ 1 + xn /n)n = ex for any complex-valued sequence with 

limn→∞ xn = x.

5 Complete Localization for Random Operators on the Canopy Graph We shall now prove Theorem 1.3, which asserts that, under assumptions A1 and A2, on the canopy graph the random Schrödinger operator has only pure point spectrum, i.e., a complete set of square integrable eigenfunctions. The argument is based on the Simon–Wolff criterion [38], for which a sufficient condition is that for every energy the Green function be almost surely square summable, when summed over one of its arguments. (Through spectral averaging a.s. properties of the Green function are shared by the eigenfunctions.) Applying the above criterion, an intuitive reason for localization on the canopy graph is that the number of points at distance n from x0 grows there as Kn/2 , which is square root of the corresponding number for the regular tree. By Theorem 3.2, the Green function decays at a rate which – if one could ignore large deviations, would yield square summability. That in itself is not enough since typically the sum of the Green function is much larger that the sum of the typical values – otherwise, the result would be valid also for the full homogeneous tree. While the corresponding statement is not valid in that case, square summability is missed there rather marginally. Thus it may be not that surprising that the significant reduction in the number of sites suffices to yield the required summability. We establish that with the help of a fractional moment estimate, and making use of the two lemmas which follow.

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We now regard T L as being embedded into C in such a way that the outer boundary ∂ T L is embedded into ∂ C for every L ∈ N. Lemma 5.1 Assume A1 and A2 holds for a bounded Borel set I ⊂ R. Then there exists s ∈ (0, 1) such that for all x ∈ C and Lebesgue-almost all E ∈ I   −2 s  sup sup E δx ,  HTL − E − iη δx < ∞ , (5.1) η=0 L Lx

where Lx := min{L ∈ N : x ∈ T L }. Proof We first note that the inequality −2   −2

  δx ,  HTL − E − iη δx  δx ,  HTL − E δx ,

(5.2)

implies that we only need to bound the 2 (T L )-norm in (5.1) for η = 0. The expectation of the fractional-moment of this 2 (T L )-norm is split into two contributions. One involves all terms corresponding to the finite subtree   C (x) := y ∈ C : y is forward (in the direction of ∂ C ) or equal to x , (5.3) which has x as its root, andthe other collects all remaining terms. Employing the elementary inequality ( j α j)s  j α sj , which is valid for any s ∈ (0, 1) and any collection of non-negative numbers α j, we thus obtain ⎡⎛ ⎞s ⎤    −1 2 E ⎣⎝ (5.4)  δx , HTL − E δ y  ⎠ ⎦  S1 + S2 , y∈T L

where

S1 :=



  −1 2s   E  δx , HTL − E δ y 

y∈C (x)

⎡⎛

S2 := E ⎣⎝



⎞s ⎤ 2  

−1    δx , HTL − E δ y  ⎠ ⎦ .

y∈T L \C (x)

By the fractional-moment bound (3.1) the first terms, S1 , is bounded for any s ∈ (0, 1/2) by a constant, |C (x)|C, which is independent of L  Lx and z ∈ C+ . To bound the second term, S2 , we use the fact that the Green’s function factorizes,   −1   −1   −1

(5.5) δx , HTL − E δ y = δx , HTL − E δv δw , HC (w) − E δ y , where v is the first joint ancestor of x and y, and w is that neighbor of v which has the least distance from y. We may therefore organize the summation in S2 as follows. We sum over the vertices on the unique path in P (x) ⊂ C which connects x and ‘infinity,’ cf. Fig. 1. For each vertex along this path we then collect terms of the form    −1 2 S(w) := (5.6)  δw , HC (w) − E δ y  , y∈C (w)

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which stem from the K − 1 neighbors w of v, which are not in P (x). Consequently, the second term in (5.4) is bounded according to S2 

   −1 2s   E  δx , HTL − E δv  1 +

 v∈P (x)∩T L







S(w)s



dist(w,v)=1 w∈P (x)



   −1 4s 1/2  1/2    E  δx , HTL − E δv  1 + E S(w)2s

v∈P (x)∩T L dist(w,v)=1 w∈P (x)

      −1 4s 1/2 2s dist(x,v)   C(s, K) E K ×  δx , HTL − E δv  

v∈P (x)∩T L

  × 1+ E

1 S(w)2s |C(w)|2s

1/2  ,

(5.7)

where C(s, K) < ∞ is independent of w and v, and w is any of the (K − 1) neighbors of v with w ∈ / P (x). According to Lemma 5.2 below, the last term in the right side of (5.7) is bounded from above by a constant which is independent of w. Lemma 3.2 then proves that the remaining sum over v ∈ P (x) ∩ T L in (5.7) is bounded from above by a constant which is independent of L ∈ N. 

Lemma 5.2 Under assumption A1 for any s ∈ (0, 1/4)  sup sup E

z∈C+ L∈N

1 |T L |s

   −2 s   δ0 ,  HTL − z δ0  < ∞ .

(5.8)

Proof A combination of (2.12), (2.13) (with w = 1) and (2.11) below yields for all z ∈ C+ and L ∈ N −2

  −1

  |T L |−1 δ0 ,  HTL − z δ0  Im δ0 , HTL − Re z − i |T L |−1 δ0 ×   −2  × 2 + |T L |−2 dist σ (HTL ), z .

(5.9)

We now take the fractional-moment and apply the Cauchy–Schwarz inequality. The claim then follows from the fractional-moment bound (3.1) and Wegner’s estimate (2.1). 

Proof of Theorem 1.3 We pick an arbitrary bounded Borel set I ⊂ R. By the strong resolvent convergence, " −1  −1 " lim " HTL − z δx − HC − z δx " = 0,

L→∞

(5.10)

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for all x ∈ C and all z ∈ C+ , and monotone convergence, it follows from (5.1) that for Lebesgue-almost all E ∈ I  s  " −1 " " "2 E lim " HC − E − iη δx " < ∞, (5.11) η↓0

with the same s as in (5.1). Since the conditional distribution of ωx – conditioned on the sigma-algebra generated by {ω y } y=x – has a bounded density, , the Simon–Wolff localization criterion [38, Thm. 8] is thus satisfied and yields the assertion. 

6 The Spectra of Random Operators on Single-ended Trees On the canopy tree C from any site there is a unique path to infinity, i.e., in the terminology of [44] it is a single-ended tree. The purpose of this section is to clarify that this point in itself implies part of above localization statement, but not all of it. To place Theorem 1.3 in a more general context we prove here that on single-ended trees random operators of the kind considered here have no absolutely continuous spectrum, but singular continuous spectrum can occur (though not in the specific case of the canopy graph). 6.1 Absence of Absolutely Continuous Spectrum The main result of the present subsection is proven for the class of graphs defined next. Definition 6.1 For a graph G a backbone B is a connected path, indexed by either Z, N, or {1, ..., L}, whose deletion transforms G into a collection of finite disconnected sets. Graphs with a backbone are referred to as backbone graphs. Not every graph has a backbone, and in case it does, the backbone is not unique (except in the double-ended case) as can be seen by considering the canopy graph. We shall consider below self-adjoint random operators HG ,B = A + W + V,

(6.1)

acting on the Hilbert space 2 (G ) over a backbone graph, where A denotes the adjacency operator and W stands for an arbitrary multiplication operator. Moreover, V denotes the random multiplication operator which acts only along the backbone with values at sites x ∈ B given by random variables {ωx }x∈B which we assume to be independent and identically distributed, with a distribution satisfying the following assumption. Assumpton A1’: The distribution of the variables ωx is of bounded density,   ∈ L∞ (R), and satisfies R (1 + |ω0 |)2 (ω0 )2 dω0 < ∞. Note that Assumption A1 with τ = 2 implies Assumption A1’.

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Theorem 6.1 (Absence of ac spectrum) Assuming A1’, for any operator HG ,B = A + W + V on the Hilbert-space 2 (G ) associated with a backbone graph the absolutely continuous component of the spectrum is empty. In the proof we shall make use of the following extension of a bound which was proven for one dimensional random operators in the work of Delyon, Kunz and Souillard [17]. For this result we assume the structure described above, except that it is not necessary for the backbone to extend to infinity. Proposition 6.2 (Exponential decay II) On a finite graph G with a backbone B , let HG ,B = A + W + V be a random operator of the form described above. Assuming A1’ holds, for any bounded interval I ⊂ R and s ∈ (0, 1/2) there exists C(s, I) < ∞ such that for any pair of sites along the backbone x, y ∈ B :   −1 s   E δx , HG ,B − E δ y  dE  C(s, I) exp (−s λ dist(x, y)) , (6.2) I

where (as in [17, Eq. (1.8)]) λ :=

inf

η>0 40η| log η|<1

-



α(η) −2 log 1 − (1 − 40η| log η|)2 25

 ,

(6.3)

ˆ )|, and ˆ the Fourier transform of the single-site with α(η) := 1 − sup|ξ |>η |(ξ density. Although it is not stated there in the above form, this exponential bound readily follows from the one-dimensional analysis of [17]. We present this reduction in Appendix C. Assuming Proposition 6.2 we turn now to the derivation of the main result of this section. Proof of Theorem 6.1 We shall give the proof for B single-ended; the case of a double-ended backbone follows similarly and, in case B is finite there is nothing to show. Let x0 ∈ B be arbitrary and pick x L ∈ B in the direction towards infinity with dist(x0 , x L ) = L. By removing the bond between x L and x L+1 , we cut G and its backbone into finite parts G L , B L and the infinite remainders. Let HGL ,BL denote the restriction of HG ,B to 2 (G L ). The resolvent identity and the fact that Im δx0 , (HGL ,BL − E)−1 δx0  = 0 for almost all E ∈ R implies  2 Im δx0 , (HG ,B − E − i0)−1 δx0   δx0 , (HGL ,BL − E)−1 δxL  ×   × δxL+1 , (HG ,B − E − i0)−1 δxL+1  .

(6.4)

Since the ac component of the spectral measure of HG ,B associated with δx0 is supported on those E ∈ R for which the left-hand side is finite and strictly positive, it remains to show that the right-hand side converges as L → ∞ to zero in distribution with respect to the product of the probability measure associated with {ωx } and Lebesgue measure for E ∈ I, where I ⊂ R is an

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319

arbitrary bounded interval. For a proof of the latter we note that the second term on the right-hand side of (6.4) is seen to be bounded in probability using, for example, a fractional moment estimate, cf. (3.1). Moreover, Proposition 6.2 implies that the first term converges to zero in distribution with respect to the above product measure. 

6.2 Appearance of Singular Continuous Spectrum We shall now focus on backbone graphs which are obtained by decorating an infinite path B with trees, as in Fig. 2. The canopy graph is within this class but our main result applies to graphs where the trees attached to the line grow at a much faster rate. The goal is to prove that there are single-ended trees G for which random operators of the form HG ,B = A + V

(6.5)

have a singular continuous component in their spectrum. Here and in the following, we will assume that V is the multiplication operator corresponding to independent and identically distributed random variables {ωx }x∈G on the whole tree. Theorem 6.3 (Singular continuous spectrum) Fix a bounded, open interval I ⊂ cont (HT ) where HT = A + V with random variables satisfying A1. Then there exist a single-ended tree G for which HG ,B = A + V has singular continuous spectrum on I. Two remarks apply: 1. The existence of continuous spectrum, cont (HT ) = ∅ for the random operator HT = A + V on the infinite rooted regular tree T for small disorder is ensured by [5, 23, 28, 29]. 2. It has been noted before that random operators with decaying potential [16] or the Laplacian on certain (tree) graphs [12, 37] may exhibit singular continuous spectrum. The construction of trees G in Theorem 6.3 is based on the following observation.

Fig. 2 Example of the construction of a single-ended tree G obtained by gluing finite trees to the sites of a backbone

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Lemma 6.4 Assume A1 and let I ⊂ cont (HT ) be a bounded Borel set and s ∈ (0, τ/2]. Then

 lim E δ0 , (HTL − E)−2 δ0 −s dE = 0 . (6.6) L→∞

I

Proof We pick η > 0. The spectral theorem implies that for every L ∈ N

−1 −1 δ0 , (HTL − E)−2 δ0 −1  δ0 , (HTL − E)2 + η2 δ0   δ0 (HTL − E)2 δ0  + η2  4K + ω02 + E2 + η2 .

(6.7)

Using the dominated convergence theorem and strong resolvent convergence we thus conclude

 lim E δ0 , (HTL − E)−2 δ0 −s dE L→∞

I





−1 −s  E δ0 , (HT − E)2 + η2 δ0  dE .



(6.8)

I

The claim follows with the help of the dominated convergence theorem from the fact that for Lebesgue almost all E ∈ cont (HT )

−1 −1 δ0  = 0 . limδ0 , (HT − E)2 + η2 η↓0

(6.9) 

The proof of Theorem 6.3 is also based on the following Simon-Wolff type criterion, which may be deduced from [15]. Proposition 6.5 (cf. [15]) Let H be a random operator on the Hilbert space over a graph, with iid random potential whose single site distribution is ac and bounded. Assume there is a Borel set I such that 1. H has no absolutely continuous spectrum in I. 2. For Lebesgue almost every E ∈ I almost surely the Green function is not square summable: lim δx , [(H − E)2 + η2 ]−1 δx −1 = 0 . η↓0

(6.10)

Then, in the space for which δx is a cyclic vector, almost surely H has only singular continuous spectrum in I. Finally, our construction is also based on the following lower bound on the decay of the Green function of single-ended graphs G along the backbone B . It is important for us that the decay rate is controlled independently of the depth of the trees glued to B .

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Lemma 6.6 Assume A1 and let E0 > 0 and s ∈ (0, min{τ, 1/2}]. There exist some constant C(s, E0 ) < ∞ such that for all x0 , x ∈ B    −s  sup sup E δx0 , HG ,B − E − iη δx   {ωx }x∈{y:dist(y,B)1} |E| E0 η∈(0,1)

 C(s, E0 ) eλ(s,E0 ) dist(x0 ,x) .

Here



(6.11)

λ(s, E0 ) := 1 + log 1 + E0 +





|ω0 | (ω0 )dω0 + K Cs s

,

(6.12)

where K is the maximal number of vertices neighboring B , and Cs is the constant appearing in (3.1). Proof Similarly as in (3.5) (cf. [29]) we factorize the Green function into a product, 0 ,x)   −1 dist(x j δx0 , HG ,B − E − iη δx =

(6.13)

j=0

  −1

with  j := δx j , HG (x j ) − E δx j . Here x0 , x1 , . . . , xdist(0,x) := x are the vertices on B connecting x0 with x. Moreover, G (x j) is that infinite subtree of G which is forward to x j in the direction away from x0 . The factors in (6.13) satisfy the following relation (cf. [29])  −1 j = Vx j − E − iη −  j+1 − G j where

G j :=



(6.14)

  −1

δw , HG (w) − E δw ,

dist(w,x j )=1 w∈B

and the sum is over all neighboring vertices of x j which are not on B and each term involves the finite subtree tree G (w) which is rooted at w and extends away from the backbone. We now integrate the product in (6.13) step by step starting with estimating the conditional expectation, conditioning on all random variables aside from x0 and its neighbors which do not belong to P (x0 ), 

 E |0 |−s {ωx0 , ωw }c{w∈B:dist(w,x0 )=1} 

|ω0 |s (ω0 )dω0 + |E| + η + |1 |−s + K Cs

 exp(λ(s, E0 ) − 1) + |1 |−s .

(6.15)

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Here the last factor in the second line stems from estimating the expectation of the terms contributing to G0 using (3.1). Iterating this bound yields the result. 

Proof of Theorem 6.3 Let E I = max{|E| : E ∈ I} and τ  := min{τ, 1/2}/2. We define a sequence of finite regular trees T Ln through the requirement     E δ0 , (HTLn − E)−2 δ0 −τ dE  exp −2λ(2τ  , E I ) n , (6.16) I

for every n ∈ N0 , where λ(τ  /4, E I ) is the constant appearing in (6.12). Note that such a sequence exists thanks to Lemma 6.4. Given a half-infinite line B we glue to every vertex xn ∈ B with dist(x0 , xn ) = n another edge which connects xn with the root of the tree T Ln ; cf. Fig. 2. To conclude that the spectral measure of HG ,B associated with δx0 is purely singular continuous in I, we use Proposition 6.5 and first note that for any L ∈ N and Lebesgue – almost every E ∈ R lim inf δx0 , [(HG ,B − E)2 + η2 ]−1 δx0  η↓0

 lim inf η↓0

   −1 2 δ x  δx0 , HG ,B − E − iη x∈B L

L  2   −1   δxn  (1 + S(n)) δx0 , HG ,B − E − i0

=

(6.17)

n=0

where

S(n) := δ0 , (HTLn − E)−2 δ0  .

We now average an inverse power of the above quantity and integrate over I. Jensen’s inequality thus yields   2 2 −1 −τ  /2 E lim sup δx0 , [(HG ,B − E) + η ] δx0  dE η↓0

I



1 Lτ  +1

L  n=0

I

  −τ   −1   −τ  /2 E δx0 , HG ,B − E − i0 δxn  S(n) dE . (6.18)

The Cauchy-Schwarz inequality together with (6.16) and Lemma 6.6 imply that the right-hand side in (6.18) is bounded from above by 1 Lτ  +1

C(2τ  , E0 )1/2

 n exp −λ(2τ  , E0 ) , 2 n=0

L 

which converges to zero as L → ∞. This yields the claimed result.

(6.19) 

The lengths of the regular trees T Ln are defined via (6.16) in a rather indirect way. In particular, no estimates are given. To answer the question about the

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minimal growth of Ln as n → ∞, which is sufficient for the production of singular continuous spectrum, one needs to estimate the growth of the quantity δ0 , (HTL − E)−2 δ0  as L → ∞ for E ∈ cont (HT ).

Appendix For completeness, in the following appendix sections we shall briefly sketch proofs of some spectral properties of the canopy graph which are of relevance to our discussion and which are derived by arguments which are already in the literature. We also add some observations and discussion. A Some Basic Properties of the Canopy Operator A.1 Existence of the Canopy Density of States Measure Following is a brief sketch of the proof of Theorem 1.2. Proof of Theorem 1.2 We embed T L into C so that ∂ T L ⊂ ∂ C . The trace in (1.10) can be decomposed into contributions from layers with a fixed distance to the outer boundary, K − 1  −n K Tn,L (F) K n=0 L

|T L |−1 Tr F(HTL ) = where

Tn,L (F) := Kn+1−L



(A.1)

δx , F(HTL ) δx  .

x : dist(x,∂ T L )=n

Each contribution Tn,L (F) is normalized to one for F = 1 and, more generally, Tn,L (F)  F ∞ . Thanks to dominated convergence, it is therefore enough to prove the following almost-sure convergence for each n ∈ N0 

lim Tn,L (F) = E δxn , F(HC ) δxn  , (A.2) L→∞

where xn ∈ C is an arbitrary vertex with dist(xn , ∂ C ) = n, cf. Fig. 1. The proof of (A.2) boils down to the Birkhoff–Khintchin ergodic theorem [27] and an approximation argument. Since the functions ϕz = (· − z)−1 with z ∈ C+ are dense in Cb (R) and the linear functionals in both sides of (A.2) are (uniformly) continuous on Cb (R), it is sufficient to prove (A.2) for F = ϕz . By truncating T L at a layer n + L0 below the outer boundary, we may approximate the sum Tn,L (ϕz ) by K L−n−L0 stochastically independent terms of the form  1 δ y , ϕz (HTn+L0 ) δ y  . (A.3) K L0 dist(0,y)=L0

The approximation error can be kept arbitrarily small by taking L0 ∈ N large. The approximating average of K L−n−L0 stochastically independent terms

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satisfies the assumptions of the Birkhoff-Khintchin ergodic theorem for iid random variables. As L → ∞, it therefore converges almost surely to ⎡ ⎤  

1 E⎣ L δ y , ϕz (HTn+L0 ) δ y ⎦ = E δxn , ϕz (HTn+L0 ) δxn  , (A.4) 0 K dist(0,y)=L0

where xn is an arbitrary vertex in the nth layer below the surface ∂ Tn+L0 . Taking L0 → ∞, the last term converges to the right side in (A.2) by the dominated convergence theorem. 

The standard Wegner estimate allows to conclude some regularity of nC . Lemma A.1 Under the assumption A1 the canopy density of states measure nC is absolutely continuous, with bounded density satisfying dC (E) =

nC (dE)   ∞ . dE

(A.5)

Proof This readily follows from the definition of the density of states measure (1.10) and the Wegner estimate (2.1). 

A.2 Spectrum of the Adjacency Operator on the Canopy Graph We will now give a brief sketch of the proof of the following assertion: Proposition A.2 (cf. [7]) The spectrum of the adjacency operator with boundary condition A + B on 2 (C ) consists of infinitely degenerate eigenvalues coinciding with the union of all eigenvalues of the adjacency operator (with constant boundary condition b ∈ R) on 2 ({1, 2, . . . , n}) with n ∈ N arbitrary. The corresponding eigenfunctions are compactly supported. Proof The basic construction is simplest to describe for K = 2. In that case, for each eigenfunction ψ of the adjacency operator on 2 ({1, 2, . . . , n}) eigenfunctions can be constructed on the canopy graph which are supported on the forward trees corresponding to any site x which is at distance n + 1 from the outer boundary ∂ C . The functions are defined so they are antisymmetric with respect to the exchange of the two forward trees of length n which lie between x and the boundary ∂ C , and on each of the two subtrees are given by a radially ’fanned out’ versions of ψ (cf. [5, Proof of Prop. A.1]). It is easy to verify that the construction yields a complete orthonormal collection of eigenfunctions. To determine the spectrum of A + B on 2 (C ) with a general K one may use – analogously to [7] – a decomposition of the Hilbert space into invariant subspaces:  .  . Qx , with Qx := S y  Sx (A.6) 2 (C ) = x∈C

y∈C (x) dist(x,y)=1

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325

where Sx denotes the subspace of symmetric functions on the forward subtree C (x), cf. (5.3): 

 y → δ y , ψ is supported on C (x) Sx := ψ ∈  C : and constant on each generation of C (x) 2

 .

The orthogonal decomposition (A.6) reduces the operator A + B on 2 (C ) to an orthogonal sum of operators on Qx , each of which is unitarily equivalent to the orthogonal sum of K − 1 operators on S y where y is one of the forward neighbors of x. In turn, each operator on S y is unitary equivalent to the adjacency operator (with constant boundary condition b ∈ R) on the Hilbert space 2 ({1, 2, . . . , dist(y, ∂ C )}). 

As an aside, we note that other examples of discrete operators with finitely supported eigenfunctions can be found in [18, 25].

B Negligibility in Probability of the Spectral Measure within the Singular Spectrum It may be of some interest to observe that, as is proven below, the condition which by Theorem 2.2 implies Poisson statistics holds throughout the singular spectrum of the infinite-volume operator. To avoid confusion, let us note that although the reference here is to the spectral measure at the root, the infinity divisibility which this implies is of the spectral measure whose density is given by the canopy dos, weighted as in (1.9). This observation is not used for our main result since we establish the negligibility in probability thorough another mechanism, which is valid throughout the entire spectrum. The relevant statement is valid not only in the tree setup and is based on the following measure theoretic statement. Theorem B.1 Suppose the operator HT has almost surely only singular spectrum in a given Borel set I. Then for Lebesgue – almost all E ∈ I the condition in Definition 2.1 is satisfied. Proof As L → ∞ the spectral measure σx,L converges vaguely to δx , P· (HT )δx , which is finite and purely singular on I. The subsequent lemma thus implies that P – almost surely for every ε > 0 and w > 0       lim sup E ∈ I : |T L | σx,L E + |T L |−1 (−w, w) > ε  = 0 .   L→∞

(B.1)

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By Fubini–Tonelli’s theorem this implies that for every ε > 0 and every w > 0 there exists a subset J(ε, w) ⊂ sing (HT ) of full Lebesgue measure such that for all E ∈ J(ε, w)    lim P |T L | σx,L E + |T L |−1 (−w, w) > ε L→∞

     −1  P lim sup |T L | σx,L E + |T L | (−w, w) > ε = 0 .

(B.2)

L→∞

Since the event in the right-hand side is monotone in both ε and w, we may / pick any two monotone sequences n → 0 and wm → ∞ and define J := n,m J(εn , wm ), a set of full Lebesgue measure, on which the claimed convergence (2.5) holds for all ε > and w > 0. 

Following is a rather general observation for singular measures. Lemma B.2 Let σ be a purely singular measure on I ⊂ R, suppose that limn→∞ σn = σ vaguely, and let {ξn }∞ n=0 be a null sequence. Then for every ε > 0 and w > 0 the sequence of sets An (ε, w) := {E ∈ I : σn (E − w ξn , E + w ξn ) > ε ξn } satisfies    ∞ ∞    0 1   lim sup An (ε, w) =  A (ε, w)  = 0. n    n→∞  m=n

(B.3)

n=0

Proof We prove the assertion by contradiction. Suppose there exists ε > 0, w > 0, M ∈ N such that  ∞  1    An (ε, w) > 0 . (B.4)    m=M

/ This implies that there exists an open ball B ⊂ ∞ m=M An (ε, w). By assumption σ is purely singular on this ball, such that for every δ > 0 there exists a finite Nδ collection of disjoint closed intervals {Ikδ }k=1 , each of which is contained in B, such that [27]   N  Nδ   0 0δ  δ δ Ik  < δ and σ Ik < δ . (B.5) B \   k=1

k=1

Since the above intervals are closed, vague convergence implies σ

N 0δ k=1

 Ikδ

 lim sup σn n→∞

N 0δ k=1

 Ikδ

=

Nδ  k=1

 lim sup σn Ikδ . n→∞

(B.6)

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/ Since the intervals are contained in ∞ m=M An (ε, w), it follows by a covering argument that for every δ > 0 and k ∈ {1, . . . , Nδ }  ε  lim sup σn Ikδ   Ikδ  . 2 n→∞ Inserting this inequality in (B.6), we thus obtain δ > contradiction for δ small enough.

(B.7)

ε 2

(|B| − δ), which yields a 

C A Decorated Delyon–Kunz–Souillard Bound It is rather generally true that eigenfunctions of one-dimensional random operators decay exponentially. In particular, Delyon, Kunz and Souillard [17] (see also [30]) presented a proof of localization which does not require translational covariance. Their result can be used in a fairly straightforward way to imply Proposition 6.2, which asserts a generalization of their result to exponential decay of the Green function on graphs which arise from an arbitrary decorations of a line with finite graphs. The statement which is more directly related to the result of [30] is the exponential decay of the eigenfunction correlation. We shall first prove that and then use it to derive Proposition 6.2. Proposition C.1 In the setting of Proposition 6.2 there exists C < ∞ such that for any bounded interval I ⊂ R and any x, y ∈ B : ⎡ E⎣



⎤ |ψ E (x)| |ψ E (y)|⎦ dE  C |I| exp (−λ dist(x, y))

(C.1)

E∈I∩specHG ,B

with λ > 0 as in (6.3), which is independent of I. Proof We proceed by relating Proposition C.1 with a result in [17]. In essence, the point is that for a graph G with random potential along the backbone B , the restriction of an eigenfunction to B coincides with an eigenfunction of a one dimensional operator for which the rest of G provides an energy dependent potential. The analysis of [17] is carried pointwise in energy, and hence it is applicable also to the backbone graphs which are considered here. Let us start by recalling a change of variable formula, for which B can be any finite subgraph of a graph G . Let T be an arbitrary self-adjoint operator in 2 (G ) and suppose that V a random multiplication operator whose potential variables {ωx }x∈B are independently distributed with densities x ∈ L1 (R). Moreover, let ψ E be the 2 (G )-normalized eigenfunctions of HG ,B = T + V at

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eigenvalues E ∈ specHG ,B . The the following relation holds between probability averages:  x∈B

=



x (ωx )dωx

|ψ E (x0 )|2 (· · · )

E∈spec(HG ,B )



x (ωx )dωx

 dE x0 Vx0 (E, {ωx }x=x0 ) (· · · )

(C.2)

x=x0

where in the last integral Vx0 is regarded as a function of E and {ωx }x=x0 , defined so that E ∈ specHG ,B , i.e., Vx0 (E, {ωx }x=x0 ) := −δx0 , (HG(0),B − E)−1 δx0 −1

(C.3)

where HG(0),B is a precursor of HG ,B with ωx0 = 0. The relation between HG(0),B and HG ,B is such that the Green function of the former appears as the eigenfunction of the latter when Vx0 is chosen as above, and in particular: δx0 , (HG ,B − E)−1 δx  ψ E (x) = . ψ E (x0 ) δx0 , (HG ,B − E)−1 δx0 

(C.4)

Furthermore, the ratio on the right does not depend on the value of Vx0 . For the correlation of eigenfunctions at energies in a Borel set I ⊂ R, which is the quantity of interest for us, the above considerations yield:  E

 |ψ E (x0 )| |ψ E (x)| = S(x0 , x; E) dE .



(C.5)

I

E∈spec(HG ,B )∩I

with    δx0 , (HG − E)−1 δx       x Vx (E, {ω y } y=x ) S(x0 , x; E) :=   y (ω y )dω y . 0 0 0  −1 δx0 , (HG − E) δx0  y=x 0

(C.6)     This representation is obtained by averaging  ψψEE(x(x)0 )  1 I (E) with respect to the probability measure in (C.2). In case of a backbone graph (in the sense of Definition 6.1) one may now further change variables in the integral in (C.6) and introduce the Riccatti variables associated with x ∈ B \ {x0 }  ψ E (x−1) r x :=

ψ E (x) ψ E (x+1) ψ E (x)

,

x > x0 ,

,

x < x0 .

(C.7)

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For HG ,B = A + W + V these variables are related to the random variables {ωx }x=x0 through the Schrödinger equation ωx = E − Wx − ψψE (x+1) − ψψE (x−1) . E (x) E (x) This implies that  −1 −1 S(x0 , x; E) = |r1 | |r2 |−1 · · · |r x |−1 x0 (E − Wx0 − r−1 − r1−1 ) × ×



 y (E − W y − r y − r−1 y+1 ) dω y ×

y>x0

×



 y (E − W y − r y − r−1 y−1 ) dω y .

(C.8)

y<x0

The above integral is identical to that in [17, Eq. (2.11)]. It is proven there for the case of interest to us, with the random identically distributed, that there exists C < ∞ such that for any E ∈ I S(x0 , x; E)  C e−λ dist(x0 ,x) .

(C.9) 

This completes the proof. C.1 Proof of Proposition 6.2

Proposition 6.2 which concerns the decay of the factional moments of the Green function will be deduced from Proposition C.1 through rather general considerations relating that quantity to the eigenfunction correlators. Lemma B.2 Let H be a self-adjoint operator with purely discrete spectrum with eigenvalues E1  E2  . . . (counting multiplicity) and corresponding orthonormal set {ψ En } of eigenfunctions. Then: 1. For any pair of bounded intervals I, J ⊂ R, and s ∈ (0, 1), ⎛ ⎞s    δx , P J (H) (H − E)−1 δ y s dE  Cs (I) ⎝ |ψ En (x)| |ψ En (y)|⎠ , I

En ∈J

   where Cs (I) := 4s R ts−1 min |I|, t−1 dt < ∞. 2. For any Borel J ⊂ R and E ∈ J with dist(E, ∂ J) > 0 one has

(C.10)

  δx , P J c (H) (H − E)−1 δ y 2 

 m∈Z: Im ∩J=∅

with Im := (m, m + 1].

 1 |ψ En (x)| |ψ En (y)| dist(E, Im )2 E ∈I n

m

(C.11)

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Proof Abbreviating g(E) := δx , P J (H) (H − E)−1 δ y , we start from the representation ∞ s |g(E)| dE = s (C.12) ts−1 |{E ∈ I : |g(E)| > t}| dt . I

0

 In terms of the spectral representation: g(E) = En ∈J (En − E)−1 ψ En (x) ψ En (y). After a decomposition of ψ En (x) ψ En (y) into four terms corresponding to positive, respectively non-positive, real and imaginary parts, one may apply Boole’s lemma [11] to obtain: ⎧ ⎫ ⎨ ⎬  |{E ∈ I : |g(E)| > t}|  4 min |I|, t−1 |ψ En (x)| |ψ En (y)| . (C.13) ⎩ ⎭ En ∈J

Substitution in (C.12) yields the first assertion. For a proof of the second bound we use the spectral decomposition and estimate  |ψ E (x)| |ψ E (y)   n n δx , P J c (H) (H − E)−1 δ y   |En − E| c En ∈J

⎛

⎞1/2  |ψ E (x)| |ψ E (y) n n ⎠ × ⎝ 2 |E − E| n c E ∈J n

⎛ ×⎝



⎞1/2 |ψ En (x)| |ψ En (y)|⎠

(C.14)

En

 Using the Cauchy–Schwarz inequality and the fact that En |ψ En (x)|2 = 1, the last term is seen to be bounded by one. The claim then follows by decomposing the set J c into a the union of sets Im ∩ J c and estimating the denominator in the first factor on each of those sets. 

Proof of Proposition 6. 2 We pick J ⊃ I such that dist(∂ J, ∂ I) > 0 and decompose the Green function into the two parts according to the previous lemma. The estimates presented there yield the bound   s  E δx , (HG ,B − E)−1 δ y  dE I

⎛ ⎡  C sup ⎝E ⎣ m∈Z



⎤⎞s/2 |ψ En (x)| |ψ En (y)|⎦⎠

(C.15)

En ∈(m,m+1]

with the constant C depending on I and s. Thus Proposition 6.2 can be deduced from Proposition C.1.

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It may be noted that a converse relation also holds – the expectation in the right-hand side of (C.15) can be bounded in terms of a suitably averaged fractional moment of the Green function [4]. D Discussion Our main result, Theorem 1.1, shows that regular tree graphs do not provide an example of the expected relation between the presence of ac spectrum for the infinite graph and random-matrix like statistics in the spectra of the corresponding finite volume restrictions. Let us therefore comment on a number of other directions in which it is natural to look for examples of such a relation. As we saw, the negative result concerning the above relation reflects the fact that a finite tree is mostly surface. By implication, bulk averages of local quantities yield results representing the local mean not at sites deep within a tree but at sites near the canopy. In physicists discussions, the term ‘Bethe lattice average’ is intended to reflect the average at sites deep within the tree, and a standard device is used for extracting it from the bulk sum. For an extensive quantity such as F L = Tr F(HHL ) where H denotes the homogeneous tree in which also the root has K + 1 neighbors, the ‘Bethe lattice average’ F BL is obtained not by taking lim L→∞ F L /|H L |, which gives the weighted canopy average (1.10), but rather (as in [32]) through the limit: F BL = lim (F L − K F L−1 ) /2 . L→∞

(D.1)

It would be of interest to see an adaptation of this approach for some separation of the statistics of eigenvalues corresponding to regions deep within the tree from the canopy average. However, even for the mean density of states, averaged over the disorder, it remains to be shown that the corresponding limit exists, and is given by a positive spectral measure. Furthermore, it is not clear how to use an analog of (D.1) for specific realizations of an operator with disorder, as the latter ruins the homogeneity. Alternatively, one may look for graphs which have local tree structure without an obvious surface. Let us briefly comment on results which relate to ˝ two such cases: the random regular and the random Erdos–Rényi graph also known as the sparse random matrix ensemble. The ensemble of random c-regular graphs [10] consists of the uniform probability measure on graphs on N ∈ N vertices where each vertex has c neighbors. It is known that as N → ∞ the girth of the graph (the minimal loop length) diverges in probability and numerical simulations suggest [26] that for large c the eigenvalue spacing distribution of the adjacency operator approaches that of the Gaussian Orthogonal Ensemble (GOE). ˝ The ensemble of Erdos–Rényi graphs results from the complete graph on N ∈ N vertices by removing bonds with probability 1 − p. This ensemble is known to have a percolation transition with an infinite tree-like connected component appearing as N → ∞ if the average connectivity c := pN is bigger than one. The adjacency operator on these graphs is believed to exhibit a

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quantum percolation transition, i.e., the existence of extended eigenstates, at some value c > 1. Numerical [9, 20, 21] and theoretical-physics calculations [34] suggest that the eigenvalue spacing distribution of the adjacency operator approaches GOE at least for large values of c (possibly depending on N). Since the graphs in both ensembles do not show a an obvious surface for finite N, they may offer a natural setting for the study of the relation between the extendedness of eigenstates of a finite volume random Schrödinger operator and its level statistics (a point which was also made in private discussions by Spencer). Acknowledgements It is a pleasure to thank R. Sims and D. Jacobson for stimulating discussions of related topics. We also thank the referee for useful references concerning the appearance of the canopy graph in other studies. Some of the work was done at the Weizmann Institute (MA), at the Department of Physics of Complex Systems, and at University of Erlangen-Nürnberg (SW), Department of Physics. We are grateful for the hospitality enjoyed there. This work was supported in part by the NSF Grant DMS-0602360 and the Deutsche Forschungsgemeinschaft (Wa 1699/1).

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19. Disertori, M., Rivasseau, V.: Random matrices and the Anderson model. Avalable at mathph/0310021 (2003) 20. Evangelou, S.N., Economou, E.N.: Spectral density singularities, level statistics, and localization in a sparse random matrix ensemble. Phys. Rev. Lett. 68, 361–364 (1992) 21. Evangelou, S.N.: A numerical study of sparse random matrices. J. Statist. Phys. 69, 361–364 (1992) 22. Efetov, K.B.: Supersymmetry in Disorder and Chaos. Cambridge University Press, Cambridge (1997) 23. Froese, R., Hasler, D., Spitzer, W.: Absolutely continuous spectrum for the Anderson model on a tree: a geometric proof of Klein’s theorem. Comm. Math. Phys. 269, 239–257 (2007) 24. Graf, G.M., Vaghi, A.: A remark on an estimate by Minami. Available at math-ph/0604033 (2006). 25. Grigorchuk, R.I., Zuk, A.: The lamplighter group as a group generated by a 2-state automaton and its spectrum. Geom. Dedicata 87, 209–244 (2001) 26. Jacobson, D., Miller, S.D., Rivin, I., Rudnick, Z.: Eigenvalue spacing for regular graphs. In: Hejhal D.A. et al. (eds.), Emerging Applications in Number Theory. Spinger, Berlin (1999) 27. Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002) 28. Klein, A.: The Anderson metal-insulator transition on the Bethe lattice. In: Iagolnitzer, D. (ed.), Proceedings of the XIth International Congress on Mathematical Physics, Paris, France, July 18-23, 1994. International Press, Cambridge, MA (1995) 29. Klein, A.: Extended states in the Anderson model on the Bethe lattice. Adv. Math. 133, 163– 184 (1998) 30. Kunz, H., Souillard, B.: Sur le spectre des operateurs aux difference finies aleatoire. Comm. Math. Phys. 78, 201–246 (1980) 31. Kottos, T., Smilansky, U.: Periodic orbit theory and spectral statistics for quantum graphs. Ann. Physics 274, 76–124 (1999) 32. Miller, J.D., Derrida, B.: Weak disorder expansion for the Anderson model on a tree. J. Statist. Phys. 75, 357–388 (1993) 33. Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Comm. Math. Phys. 177, 709–725 (1996) 34. Mirlin, A.D., Fyodorov, Y.V.: Universality of the level correlation function of sparse random matrices. J. Phys. A, Math. Gen. 24, 2273–2286 (1991) 35. Molchanov, S.A.: The local structure of the spectrum of the one-dimensional Schrödinger operator. Comm. Math. Phys. 78, 429–446 (1981) 36. Pastur, L., Figotin, A.: Spectra of Random and Almost-periodic Operators. Springer, Berlin (1992) 37. Simon, B.: Operators with singular continuous spectrum, IV: Graph Laplacians and LaplaceBeltrami operators. Proc. Amer. Math. Soc. 124, 1177–1182 (1996) 38. Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Comm. Pure Appl. Math. 39, 75–90 (1986) 39. Shklovskii, B.I., Shapiro, B., Sears, B.R., Lambrianides, P., Shore, H.B.: Statistics of spectra of disordered systems near the metal-insulator transition. Phys. Rev. B 47, 11487–11490 (1993) 40. Stollmann, P.: Caught by Disorder: Bound States in Random Media. Birkhäuser, Boston, MA (2001) 41. Sznitman, A.-S.: Lifshitz tail and Wiener sausage on hyperbolic space. Comm. Pure Appl. Math. 17, 1033–1065 (1989) 42. Sznitman, A.-S.: Lifshitz tail on hyperbolic space: Neumann conditions. Comm. Pure Appl. Math. 18, 1–30 (1990) 43. Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B 44, 9–15 (1981) 44. Woess, W.: Random walks on infinite graphs and groups. In: Cambridge Tracts in Mathematics, vol. 138. Cambridge University Press, Cambridge (2000)

Math Phys Anal Geom (2006) 9:335–352 DOI 10.1007/s11040-007-9015-6

Waveguides with Combined Dirichlet and Robin Boundary Conditions P. Freitas · D. Krejˇciˇrík

Received: 27 February 2006 / Accepted: 29 December 2006 / Published online: 16 February 2007 © Springer Science + Business Media B.V. 2007

Abstract We consider the Laplacian in a curved two-dimensional strip of constant width squeezed between two curves, subject to Dirichlet boundary conditions on one of the curves and variable Robin boundary conditions on the other. We prove that, for certain types of Robin boundary conditions, the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Laplacian in a Dirichlet-Robin annulus determined by the geometry of the strip. Moreover, we show that an appropriate combination of the geometric setting and boundary conditions leads to a Hardy-type inequality in infinite strips. As an application, we derive certain stability of the spectrum for the Laplacian in Dirichlet–Neumann strips along a class of curves of sign-changing curvature, improving in this way an initial result of Dittrich and Kˇríž (J. Phys. A, 35:L269–275, 2002). Key words Dirichlet and Robin boundary conditions · eigenvalues in strips and annuli · Hardy inequality · Laplacian · waveguides Mathematics Subject Classifications (2000) 35P15 · 58J50 · 81Q10

P. Freitas (B) Department of Mathematics, Faculdade de Motricidade Humana (TU Lisbon) and Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal e-mail: [email protected] D. Krejˇciˇrík Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, ˇ near Prague, Czech Republic 250 68 Rež e-mail: [email protected]

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1 Introduction The Laplacian in an unbounded tubular region  has been extensively studied as a reasonable model for the Hamiltonian of electronic transport in long and thin semiconductor structures called quantum waveguides. We refer to [10, 29] for the physical background and references. In this model, it is more natural to consider Dirichlet boundary conditions on ∂ corresponding to a large chemical potential barrier (cf [10, 17, 20]). However, Neumann boundary conditions or a combination of Dirichlet and Neumann boundary conditions have been also investigated. We refer to [27, 28] for the former and to [9, 25, 30] for the latter. Moreover, these types of boundary conditions are relevant to other physical systems (cf [8, 13, 21]). Although we are not aware of any work in the literature where more general boundary conditions have been considered in the case of quantum waveguides, it is possible to think also of Robin boundary conditions as modelling impenetrable walls of  in the sense that there is no probability current through the boundary. Furthermore, Robin boundary conditions may in principle be relevant for different types of interphase in a solid. Moreover, the interplay between boundary conditions, geometry and spectral properties is an interesting mathematical problem in itself. To illustrate this, let us recall that it has been known for more than a decade that the curved geometry of an unbounded planar strip of uniform width may produce eigenvalues below the essential spectrum. We refer to the pioneering work [17] of Exner and Šeba and the sequence of papers [4, 10, 20, 25, 32] for the existence results under rather simple and general geometric conditions. However, it has not been noticed until the recent letter [9] of Dittrich and Kˇríž that the existence of eigenvalues in fact depends heavily on the geometrical setting provided the uniform Dirichlet boundary conditions are replaced by a combination of Dirichlet and Neumann ones. In particular, the discrete spectrum may be eliminated provided the Dirichlet–Neumann strip is ‘curved appropriately,’ i.e., the Neumann boundary condition is imposed on the ‘locally shorter’ boundary curve. Recently, it has also been shown that the discrete spectrum may be eliminated by adding a local magnetic field perpendicular to a planar Dirichlet strip [3, 11], by embedding the strip into a curved surface [24] or by twisting a three-dimensional Dirichlet tube of non-circular cross-section [12]. The aim of the present paper is to examine further the interplay between boundary conditions, geometry and spectral properties in the case of  being a planar strip with a combination of Dirichlet and (variable) Robin boundary conditions on ∂. Our main result is a lower bound to the spectral threshold of the Laplacian in a (bounded or unbounded) Dirichlet–Robin strip. This enables us to prove quite easily nonexistence results about the discrete spectrum for certain waveguides, and generalize in this way the results of Dittrich and Kˇríž [9]. Moreover, we show that certain combinations of boundary conditions and geometry lead to Hardy-type inequalities for the Laplacian in unbounded strips. These inequalities are new in the theory of quantum waveguides with combined boundary conditions. As an application, we further extend the class of Dirichlet–Neumann strips with empty discrete spectrum.

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2 Scope of the Paper In this section we precise the problem we deal with in the present paper and state our main results. 2.1 The Model Given an open interval I ⊆ R (bounded or unbounded), let  ≡ ( 1 ,  2 ) : I → R2 be a unit-speed C2 -smooth plane curve. We assume that  is an embedding. The function N := (−˙ 2 , ˙ 1 ) defines a unit normal vector field along  and the couple ˙ N) gives a distinguished Frenet frame (cf [23, Chap. 1]). The curvature of  (, ˙ ); ¨ it is a continuous is defined through the Serret–Frenet formulae by κ := det(, function of the arc-length parameter. We assume that κ is bounded. It is worth noticing that the curve  is fully determined (except for its position and orientation in the plane) by the curvature function κ alone (cf [26, Sec. II.20]). Let a be a given positive number. We define the mapping   L : I × [−a, a] → R2 : (s, t) → (s) + N(s) t (1) and make the hypotheses that κ∞ a < 1

and

L is injective.

(2)

Then the image    := L I × (−a, a)

(3)

has a geometrical meaning of an open non-self-intersecting strip, contained between the parallel curves ± := L(I × {±a})  at the distance a from , and, if ∂ I is not   empty, the straight lines L− := L {inf I} × (−a, a) and L+ := L {sup I} × (−a, a) . The geometry is set in such a way that κ > 0 implies that the parallel curve + is locally shorter than − , and vice versa. We refer to [12, App. A] for a sufficient condition ensuring the validity of the second hypothesis in (2). Given a bounded continuous function α˜ : + → R, let −κ,α denote the (nonnegative) Laplacian on L2 (), subject to uniform Dirichlet boundary conditions on the parallel curve − , uniform Neumann boundary conditions on L− ∪ L+ (i.e. none if ∂ I is empty) and the Robin boundary conditions of the form ∂u + α˜ u = 0 ∂N

on

+ ,

(4)

  where u ∈ D(−κ,α ). Hereafter we shall rather use α := α˜ L(·, a) , a function on I. Notice that the choice α = 0 corresponds to uniform Neumann boundary conditions

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on + and α → +∞ approaches uniform Dirichlet boundary conditions on + ; for this reason, we shall sometimes use ‘α = +∞’ to refer to the latter. The Laplacian −κ,α is properly defined in Section 3 below by means of a quadratic-form approach. 2.2 A Lower Bound to the Spectral Threshold If the curvature κ is a constant function, then the image  can be identified with a segment of an annulus or a straight strip. We prove that, in certain situations, this constant geometry minimizes the spectrum of −κ,α , within all admissible functions κ and α considered as parameters. More precisely, let us denote by D(r) the open disc of radius r > 0 and let A(r1 , r2 ) := D(r2 ) \ D(r1 ) be an annulus of radii r2 > r1 > 0. Abusing the notation for κ and α slightly, we introduce a function λ : (−a, a) × R → R by means of the following definition: Definition 1 Given two real numbers α and κ, with κ in (−1/a, 1/a), we denote by λ(κ, α) the spectral threshold of the Laplacian on    A |κ|−1 − a, |κ|−1 + a if κ = 0 , Aκ := R × (−a, a) if κ = 0 , subject to uniform Dirichlet boundary condition on ⎧ −1 ⎪ if κ > 0 , ⎨∂ D(κ + a) R × {−a} if κ = 0 , ⎪ ⎩ ∂ D(|κ|−1 − a) if κ < 0 , and uniform Robin boundary conditions of the type (4) (with α constant and N being the outward unit normal on ∂ Aκ ) on the other connected part of the boundary. The most general result of the present paper reads as follows: Theorem 1 Given a positive number a and a bounded continuous function κ, let  be the strip defined by (3) with (1) and satisfying (2). Let α be a bounded continuous function. Then inf σ (−κ,α )  λ(inf κ, inf α) provided κ  0 or α  0.

(5)

The lower bound λ(κ, α) as a function of curvature κ for certain values of α is depicted in Fig. 1. We prove the following properties which are important for (5): Theorem 2 λ satisfies the following properties: (1) (2) (3) (4) (5)

∀κ ∈ (−1/a, 1/a), α  → λ(κ, α) : R → R is continuous and increasing, ∀α ∈ R, κ  → λ(κ, α) : (−1/a, 1/a) → R is continuous, ∀α ∈ R, κ  → λ(κ, α) : (−1/a, 0] → R is increasing, ∀α ∈ (−∞, 0], κ  → λ(κ, α) : (−1/a, 1/a) → R is increasing, ∀α ∈ R, lim λ(κ, α) = ν(α), lim λ(κ, α) = ν(+∞), κ→−1/a

κ→1/a

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λ (κ,α) 8

α=+

2

α = 0.78

1

α=3

α=0

κ -1

α = −0.5

-0.5

0.5

1

-1 α = −0.9

-2 α = −1.3 Fig. 1 Dependence of the lower bound λ(κ, α) on the curvature κ for a = 1 and different values of α. (All curves meet at κ = 1/a, the small gap for the curve with α = +∞ is due to a numerical inaccuracy)

where ν(α), with α ∈ R ∪ {+∞}, denotes the first eigenvalue of the Laplacian in the disc D(2a), subject to uniform Robin boundary conditions of the type (4) if α ∈ R (with α constant and N being the outward unit normal on ∂ D(2a)) or uniform Dirichlet boundary conditions if α = +∞. Of course, ν(+∞) = j 20,1 /(2a)2 , where j0,1 denotes the first zero of the Bessel function J0 , and ν(0) = 0. Theorem 1 is a natural continuation of efforts to estimate the spectral threshold in curved Dirichlet tubes [2, 14]. More specifically, in the recent article [14], Exner and the present authors established a lower bound of the type (5) for the case α = +∞, i.e. for pure Dirichlet strips (the results in that paper are more general in the sense that the tubes considered there were multi-dimensional and of arbitrary crosssection). Namely,   inf σ (−κ,+∞ )  λ κ∞ , +∞ , where λ(κ, +∞) is the spectral threshold of the Dirichlet Laplacian in Aκ . It is also established in [14] that κ  → λ(κ, +∞) is an even function, decreasing on [0, 1/a), reaching its maximum π 2 /(2a)2 for κ = 0 (a straight strip) and approaching its infimum ν(+∞) as κ → 1/a (a disc). The style and the main idea (i.e. the intermediate lower bound (14) below) of the present paper are similar to that of [14]. However, we have to use different techniques to establish the properties of λ (Theorem 2), and consequently (5).

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2.3 A Hardy Inequality in Infinite Strips Theorem 1 is optimal in the sense that the lower bound (5) is achieved by a strip (along a curve of constant curvature). On the other hand, since the minimizer is bounded if the curvature is non-trivial, a better lower bound is expected to hold for unbounded strips. Indeed, in certain unbounded situations, we prove that the lower bound of Theorem 1 can be improved by a Hardy-type inequality. Let us therefore consider the infinite case I = R in this subsection. Let α0 be a given real number. If κ is equal to zero identically (i.e.  is a straight strip) and α is equal to α0 identically, it is easy to see that   (6) σ (−0,α0 ) = σess (−0,α0 ) = λ(0, α0 ), ∞ . Although the results below hold under more general conditions about vanishing of κ and the difference α − α0 at infinity (cf Section 7 below), for simplicity, we restrict ourselves to strips which are deformed only locally in the sense that κ and α − α0 have compact support. Under these hypotheses, it is easy to verify that the essential spectrum is preserved:   σess (−κ,α ) = λ(0, α0 ), ∞ . (7) A harder problem is to decide whether this interval exhausts the spectrum of −κ,α , or whether there exists discrete eigenvalues below λ(0, α0 ). On the one hand, Dittrich and Kˇríž [9] showed that the curvature which is negative in a suitable sense creates eigenvalues below the threshold λ(0, 0) in the uniform Dirichlet–Neumann case (i.e. in the case α = 0 identically). For instance, using, in analogy to [9], a modification of the ‘generalized eigenfunction’ of −0,α0 corresponding to λ(0, α0 ) as a test function, it is straightforward to extend a result of [9] to the case of uniform Robin boundary conditions: Proposition 1 Let I = R. If α(s) = α0 for all s ∈ R and



R κ(s) ds

< 0, then

inf σ (−κ,α0 ) < λ(0, α0 ) . In particular, Proposition 1 together with (7) implies that the discrete spectrum of −κ,α0 exists if the strip is appropriately curved and asymptotically straight. Notice also that the discrete spectrum may be created by variable α even if  is straight (cf [15, 16] for this type of results in a similar model). On the other hand, Dittrich and Kˇríž [9] showed that the spectrum of −κ,0 coincides with the interval (7) with α0 = 0 provided the curvature κ is non-negative and of compact support. More precisely, they proved that inf σ (−κ,0 )  λ(0, 0)

provided

κ  0,

(8)

which implies the result in view of (7). Of course, not only the lower bound (8) is contained in our Theorem 1, but the latter also generalizes the former to variable Robin boundary conditions: Corollary 1 Let I = R and assume that κ and α − α0 have compact support. Under the hypotheses of Theorem 1,   σ (−κ,α ) = σess (−κ,α ) = λ(0, α0 ), ∞ if κ  0, α0  α  0 .

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Apart from this generalization, Theorem 1 provides an alternative and, we believe, more elegant, proof of (8). Indeed, the proof of Dittrich and Kˇríž in [9] is very technical, based on a decomposition of −κ,0 into an orthonormal basis and an analysis of solutions of Bessel type to an associated ordinary differential operator, while the proof of Theorem 1 does not require any explicit solutions whatsoever. Furthermore, we obtain a stronger result, namely, that a Hardy-type inequality actually holds true in positively curved Dirichlet-Robin strips: Theorem 3 Let I = R. Given a positive number a and a bounded continuous function κ, let  be the strip defined by (3) with (1) and satisfying (2). Let α be a bounded continuous function such that α0  α  0. Assume that κ is non-negative and that either one of κ or α − α0 is not identically equal to zero. Then, for any s0 such that κ(s0 ) > 0 or α(s0 ) > α0 , we have −κ,α  λ(0, α0 ) +

c (ρ ◦ L−1 )2

(9)

in the sense of quadratic forms (cf (30) below). Here c is a positive constant which depends on s0 , a, κ and α, ρ(s, t) := 1 + (s − s0 )2 and L is given by (1). It is possible to find an explicit lower bound for the constant c; we give an estimate in (29) below. Theorem 3 implies that the presence of a positive curvature or of suitable Robin boundary conditions represents a repulsive interaction in the sense that there is no spectrum below λ(0, α0 ) for all small potential-type perturbations having a sufficiently fast decay at infinity. This provides certain stability of the spectrum of the type established in Corollary 1. Moreover, in the uniform Dirichlet-Neumann case, we use Theorem 3 to show that the spectrum is stable even if κ is allowed to be negative: Corollary 2 Given a positive number a and a bounded continuous function κ of compact support, let  be the strip defined by (3) with (1) and satisfying (2). Assume that |κ− |  ε

with

ε  0,

while κ+ is independent of ε and not identically equal to zero. Then there exists a positive number ε0 such that for all ε  ε0 we have   σ (−κ,0 ) = σess (−κ,0 ) = λ(0, 0), ∞ . Here ε0 depends on a, κ+ and I. Corollary 2 follows as a consequence of (7) and the Hardy inequality (32) below. This generalizes a result of [9] to strips with sign-changing curvature. 2.4 Contents The present paper is organized as follows. In the following section we introduce the Laplacian −κ,α in the curved strip  by means of its associated quadratic form and express it in curvilinear coordinates

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defined by (1). We obtain in this way an operator of the Laplace–Beltrami form in the straight strip I × (−a, a). In Section 4 we show that the structure of the Laplace-Beltrami operator leads quite easily to a ‘variable’ lower bound (14), expressed in terms of the function λ of Definition 1. We call this lower bound ‘intermediate’ since this and Theorem 2 imply Theorem 1 at once. In Section 5 we prove Theorem 2 using a combination of a number of techniques, such as the minimax principle, the maximum principle, perturbation theory, etc. Section 6 is devoted to infinite strips, namely, to the proofs of Theorem 3 and Corollary 2. The former is based on an improved intermediate lower bound, Theorem 2 and the classical one-dimensional Hardy inequality. In the closing section we discuss possible extensions and refer to some open problems. 3 The Laplacian The Laplacian −κ,α is properly defined as follows. We introduce on the Hilbert space L2 () the quadratic form Qκ,α defined by   Qκ,α [u] := |∇u(x)|2 dx + α(σ ˜ ) |u(σ )|2 dσ , 

+

   u ∈ D(Qκ,α ) := u ∈ W 1,2 ()  u(σ ) = 0 for a.e. σ ∈ − ,

(10)

where u(σ ) with σ ∈ + ∪ − is understood as the trace of the function u on that part of the boundary ∂ (cf Remark 1 below). The associated sesquilinear form is symmetric, densely defined, closed and bounded from below (the latter is not obvious unless α˜  0, but it follows from the results (14) and (17) below). Consequently, Qκ,α gives rise (cf [22, Sec. VI.2]) to a unique self-adjoint bounded-from-below operator which we denote by −κ,α . It can be verified that −κ,α acts as the classical Laplacian with the boundary conditions described in Section 1 provided  is sufficiently regular. It follows from assumptions (2) that L : I × (−a, a) →  : {(s, t)  → L(s, t)} is a C1 diffeomorphism. Consequently,  can be identified with the Riemannian manifold I × (−a, a) equipped with the metric Gij := (∂i L) · (∂ jL), where i, j ∈ {1, 2} and the ˙ one dot denotes the scalar product in R2 . Employing the Frenet formula N˙ = −κ , easily finds that (Gij) = diag(gκ2 , 1), where gκ (s, t) := 1 − κ(s) t

(11)

is the Jacobian of L. It follows that gκ (s, t) ds dt is the area element of the strip, L2 () can be identified with the Hilbert space   (12) L2 I × (−a, a), gκ (s, t) ds dt and −κ,α is unitarily equivalent to the operator Hκ,α on (12) associated with the quadratic form   2  2 hκ,α [ψ] := gκ−1 ∂1 ψ κ + ∂2 ψ κ + α(s) |ψ(s, a)|2 gκ (s, a) ds , R

    ψ ∈ D(hκ,α ) := ψ ∈ W 1,2 R × (−a, a)  ψ(s, −a) = 0 for a.e. s ∈ R .

(13)

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Here  · κ stands for the norm in (12) and ψ(s, ±a) means the trace of the function ψ on the part of the boundary I × {±a} (cf Remark 1 below). In fact, if the curve  is sufficiently smooth, then Hκ,α acts as the Laplace-Beltrami operator −G−1/2 ∂i G1/2 Gij∂ j, where (Gij) = (Gij)−1 and G := det(Gij), but we will not use this fact. Finally, let us notice that the first assumption of (2) yields 0 < 1 − κ∞ a  gκ (s, t)  1 + κ∞ a < 2 uniformly in (s,t) ∈ I × (−a, a), and that is actually why  we can indeed write  W 1,2 I × (−a, a) instead of W 1,2 I × (−a, a), gκ (s, t) ds dt in (13).   Remark 1 The traces of ψ ∈ W 1,2 I × (−a, a) on the boundary of the strip I × (−a, a) are well defined and square integrable (cf [1]). In particular, the boundary integral appearing in (13) is finite (recall that α is assumed to be bounded). To ensure that the traces and the boundary integral appearing in (10) are well defined too, it is sufficient to notice that one can construct traces of u ∈ W 1,2 () to + ∪ − by means of the diffeomorphism L, the trace operator for the straight strip I × (−a, a) and inverses of the boundary mappings L(·, {±a}). The latter exists due to the second hypothesis in (2), which is in fact a bit stronger than an analogous assumption in the uniform Dirichlet case [10, 14] (there it is enough to assume that L  I × (−a, a) is injective). In this context, one should point out that the approach used by Daners in [6] makes it possible to deal with Robin boundary conditions with positive α on arbitrary bounded domains, without using traces.

4 An Intermediate Lower Bound In this section, we derive the central lower bound of the present paper, i.e. inequality (14) below, and explain its connection with Definition 1. Neglecting in (13) the ‘longitudinal kinetic energy’, i.e. the term gκ−1 ∂1 ψκ in the expression for hκ,α [ψ], and using Fubini’s theorem, one immediately gets   (14) inf σ (Hκ,α )  inf λ κ(s), α(s) , s∈I

where λ(κ, α) denotes the first eigenvalue of the self-adjoint one-dimensional operator Bκ,α on   Hκ := L2 (−a, a), (1 − κ t)dt associated with the quadratic form  a |ψ  (t)|2 (1 − κ t)dt + α |ψ(a)|2 (1 − κ a) , bκ,α [ψ] := −a

   ψ ∈ D(bκ,α ) := ψ ∈ W 1,2 ((−a, a))  ψ(−a) = 0 .

(15)

With a slight abuse of notation, we denote by κ ∈ (−1/a, 1/a) and α ∈ R given constants now. One easily verifies that κ ψ  (t) , 1−κt    ψ ∈ D(Bκ,α ) = ψ ∈ W 2,2 ((−a, a))  ψ(−a) = 0 & ψ  (a) + α ψ(a) = 0 . (16) (Bκ,α ψ)(t) = −ψ  (t) +

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Note that the values of ψ and ψ  at the boundary points of (−a, a) are well defined due to the Sobolev embedding theorem. Bκ,α is clearly a positive 0. Furthermore, using the elementary a operator for α  a inequality |ψ(a)|2  ε −a |ψ  (t)|2 dt + ε −1 −a |ψ(t)|2 dt with ε > 0, it can be easily shown that λ(κ, α)  −α 2

(1 + |κ| a)2 , (1 − |κ| a)2

(17)

i.e., Bκ,α is bounded from below in any case. This and (14) prove that Hκ,α (and therefore −κ,α ) is bounded from below a fortiori. Using coordinates analogous to (1) and the circular (respectively straight) symmetry, it is easy to see that Bκ,α is nothing else than the ‘radial’ (respectively ‘transversal’) part of the Laplacian on L2 (Aκ ) if κ = 0 (respectively κ = 0) in Definition 1. (We refer to [14, Lemma 4.1] for more details on the partial wave decomposition in the case α = +∞.) This shows that the geometric Definition 1 of λ and the definition via (15) are in fact equivalent. In view of (14), it remains to establish the monotonicity properties of λ stated in Theorem 2 in order to prove Theorem 1. This will be done in the next section.

5 Dirichlet–Robin Annuli Using standard arguments (cf [19, Sec. 8.12]), one easily shows that λ(κ, α), as the lowest eigenvalue of Bκ,α , is simple and has a positive eigenfunction. We denote the latter by ψκ,α and normalize it to have unit norm in the Hilbert space Hκ . 5.1 Dependence on α The first property of Theorem 2 follows directly from the variational definition of λ(κ, α). In detail, using ψκ,α+δ with any δ > 0 as a test function for λ(κ, α), we get λ(κ, α)  λ(κ, α + δ) − δ ψκ,α+δ (a)2 (1 − κ a) < λ(κ, α + δ) ,

(18)

i.e. α  → λ(κ, α) is increasing. Note that the strict monotonicity is a consequence of  the fact that ψκ,α+δ ∈ D(Bκ,α+δ ); indeed, ψκ,α+δ (a) = 0 would imply that ψκ,α+δ (a) = 0 also, giving a contradiction. Using now ψκ,α as a test function for λ(κ, α + δ), we get λ(κ, α + δ) ≤ λ(κ, α) + δ ψκ,α (a)2 (1 − κ a) −−→ λ(κ, α) , δ→0

(19)

which, together with (18), gives the continuity of λ in the second variable. 5.2 Dependence on κ Not all of the other properties of Theorem 2 are so obvious from the variational definition of λ(κ, α) via Bκ,α because the Hilbert space Hκ depends on κ. To overcome this, we introduce the unitary transformation U κ : Hκ → H0 :

  1 ψ  → (1 − κ t) 2 ψ

(20)

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and the unitarily equivalent operator Bˆ κ,α := U κ Bκ,α U κ−1 associated with the transformed form bˆκ,α [·] := bκ,α [U κ−1 ·]. Given any φ ∈ D(bˆκ,α ), we insert ψ = U κ−1 φ into (15), integrate by parts and finds    a  a κ κ2 2 bˆκ,α [φ] = |φ  (t)|2 dt − |φ(t)| dt + α + |φ(a)|2 . (21) 2 2(1 − κ a) −a −a 4(1 − κ t) We also verify that κ2 φ(t) , 4(1 − κ t)2   φ ∈ D( Bˆ κ,α ) = φ ∈ W 2,2 ((−a, a))  φ(−a) = 0    κ & φ  (a) + α + 2(1−κ φ(a) = 0 . a) ( Bˆ κ,α φ)(t) = −φ  (t) −

(22)

It is important to notice that while D(Bκ,α ) is not invariant under U κ , one still has D(bˆκ,α ) = D(bκ,α ). 5.2.1 Continuity Following [22, Sec. VII. 4], κ  → bˆ κ,α forms a holomorphic family of forms of type (a) and κ  → Bˆ κ,α forms a self-adjoint holomorphic family of operators of type (B). In particular, κ  → λ(κ, α) is continous, which proves (2) of Theorem 2. Moreover, denoting by φκ,α := U κ ψκ,α the eigenfunction of Bˆ κ,α corresponding to λ(κ, α), we get that κ  → φκ,α is continous in the norm of H0 . In view of (20), it then follows that also κ  → ψκ,α is continuous in the norm of H0 . 5.2.2 Monotonicity κ Since the function f : κ  → 1−κ is increasing on (−1/a, 1/a) for any t ∈ [−a, a], one t easily verifies Theorem 2.(3) by means of the variational definition of λ(κ, α) via Bˆ κ,α and an argument similar to that used in Section 5.1. However, the above argument fails to prove (4) of Theorem 2 because − f 2 is decreasing on [0, 1/a), so that one gets an interplay between the increasing boundary term and decreasing potential in (21) for positive curvatures. Therefore we come back to the initial operator (16) and calculate the derivative of κ  → λ(κ, α):

Lemma 1 ∀κ ∈ (−1/a, 1/a), ∀α ∈ R, ∂λ (κ, α) = ∂κ



a

−a

 ψκ,α (t) ψκ,α (t) dt. 1−κt

(23)

Proof Throughout this proof, we omit the dependence of λ and the corresponding eigenfunction on α. We write the eigenvalue equation for Bκ,α with ψκ and λ(κ) as   − ψκ (t) (1 − κ t) = λ(κ) ψκ (t) (1 − κ t) (24) and consider the analogous equation at κ + δ, with δ ∈ R \ {0} so small that |κ +δ| a is less than 1. Multiplying (24) by ψκ+δ , integrating by parts, combining the result

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with the result coming from analogous manipulations applied to the problem at κ +δ, dividing by δ, integrating by parts once more and using the eigenvalue equation for Bκ,α , we arrive at  λ(κ + δ) − λ(κ) a ψκ (t) ψκ+δ (t) (1 − κ t) dt δ −a  a ψκ (t) ψκ+δ (t) t dt − = λ(κ + δ)  −

−a

a

−a

 ψκ (t) ψκ+δ (t) t dt − α a ψκ (a) ψκ+δ (a)

  = λ(κ + δ) − λ(κ)



a

−a

 ψκ (t) ψκ+δ (t) t dt +

a

−a

ψκ (t) ψκ+δ (t) dt . 1−κt

Letting δ go to zero yields the desired result by means of the continuity of κ  → λ(κ) and κ  → ψκ established in Section 5.2.1.   Lemma 1 yields (4) of Theorem 2 whenever the integral on the right hand side  of (23) is positive. In particular, this is the case when ψκ,α is non-negative: Lemma 2 ∀κ ∈ (−1/a, 1/a), ∀α ∈ (−∞, 0], t  → ψκ,α (t) : (−a, a) → R

is increasing.

Proof Throughout this proof, we omit the dependence of λ and the corresponding eigenfunction on κ and α. Since ψ is a positive eigenfunction and ψ(−a) = 0, respectively ψ  (a) = −αψ(a), we know that ψ  (−a) > 0, respectively ψ  (a)  0. Recall also that ψ(a) > 0. We claim that ψ  > 0 on (−a, a). Case λ < 0. The eigenvalue problem for (16) implies that if ψ  (t) = 0 for some t ∈ (−a, a), then ψ  (t) > 0, i.e. ψ has a local minimum at t. Consequently, if there exists a t1 ∈ (−a, a) such that ψ  (t1 ) = 0, then, since ψ  (−a) > 0, there must also be a t2 ∈ (−a, t1 ) such that ψ has a local maximum at t2 , a contradiction. Case λ > 0. The eigenvalue problem for (16) implies that if ψ  (t) = 0 for some t ∈ (−a, a], then ψ  (t) < 0, i.e. ψ has a local maximum at t. Consequently, if there exists a t1 ∈ (−a, a) such that ψ  (t1 ) = 0, then, since ψ  (a)  0, there must also be a t2 ∈ (t1 , a] such that ψ  (t2 ) = 0 and ψ  < 0 on (t1 , t2 ), i.e. ψ does not have a local maximum at t2 , a contradiction. a Case λ = 0. Integrating (24), we get ψ  (t) = −α 1−κ ψ(a) > 0 for all t ∈ [−a, a] 1−κ t (the equality would imply a trivial eigenfunction).   5.2.3 Boundary Values Using the geometrical meaning of λ(κ, α) (cf Definition 1) and since Aκ converges (e.g., in the sense of metrical convergence [31]) to the disc D(2a) with the central point removed as |κ| → 1/a, the limits in Theorem 2.(5) are natural to expect. We prove each of them separately.

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The negative limit The limit value for λ(κ, α) as ε := −(κ −1 + a) → 0 follows from Flucher’s paper [18], where an approximation formula for eigenvalues in domains with spherical holes is found. The only difference is the fact that in our case the boundary of the domain also changes as ε goes to zero. We overcome this complication by transforming the eigenvalue problem for the Laplacian on Aκ into ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

−u = λε (αε )u

u=0 ⎪ ⎪ ⎪ ⎪ ⎩ ∂u + αε u = 0 ∂N

on

  A ε(2a + ε)−1 , 1 ,   ∂ D ε(2a + ε)−1 ,

on

∂ D(1) ,

in

(25)

where λε (αε ) := (2a + ε)2 λ(−(a + ε)−1 , α), αε := (2a + ε)α and N is the outward unit normal on ∂ D(1). By the minimax principle,     λε α−(sgn α)ε0  λε (αε )  λε α(sgn α)ε0 for any fixed ε0 ∈ (ε, 2a), where λε (α±ε0 ) denotes the eigenvalue of the problem (25) with αε being replaced by α±ε0 . Then it is clear that λε (αε ) → (2a)2 ν(α) as ε → 0 because it is true for λε (α±ε0 ) by [18] and ε0 can be chosen arbitrarily small. The positive limit If α > 0, the limit value for λ(κ, α) as κ −1 → a could be derived by means of a paper by Dancer and Daners, [5], where they study domain perturbations for elliptic equations subject to Robin boundary conditions. However, since they restrict to positive α and we do not know about a similar perturbation result for α < 0, we establish the limit value by rather elementary considerations. Assuming κ = 0, the eigenvalue problem for Bκ,α is explicitly solvable in terms of the Bessel functions J0 and Y0 (cf [34, Chap. 7]) and the eigenvalue λ(κ, α) is then determined as the smallest (in absolute value) zero λ of the implicit equation J0

 √   √ √ √ λ(1 + κa)/κ λY1 λ(1 − κa)/κ + αY0 λ(1 − κa)/κ  √ √   √ √ = Y0 λ(1 + κa)/κ λJ1 λ(1 − κa)/κ + α J0 λ(1 − κa)/κ . (26)

Although the case λ(κ, √ α) = 0 should be treated separately, a formal asymptotic expansion of (26) around λ = 0 also gives the correct condition for a zero eigenvalue: λ(κ, α) = 0

⇐⇒

κ = α (1 − κ a) log

1−κa . 1+κa

(27)

In particular, the condition yields that for any α < −1/(2a) there always exists κ0 ∈ (0, 1/a) such that λ(κ0 , α) = 0. This and the properties (1), (2) and (4) of Theorem 1 imply that limκ→1/a λ(κ, α) > 0 for any α ∈ R. We also know that the limit is bounded because λ(κ, α) < λ(κ, +∞) by the minimax principle and λ(κ, +∞) tends to the first eigenvalue of the Dirichlet Laplacian in the disc D(2a), i.e. ν(+∞) ≡ j 20,1 /(2a)2 , as κ → 1/a by known convergence theorems (cf one of [7, 31, 33]). Applying the limit to (26), we get a bounded value on the right hand side, while the left hand side admits

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√ √ √ −1 the asymptotic expansion − πλ J0 ( λ 2a) λ (1 − κ a)/(2κ) + O(κ −1 − a). That √ is, λ 2a necessarily converges to the first zero of the Bessel function J0 as κ → 1/a.

6 Infinite Strips Let I = R throughout this section. The proof of Theorem 3 is based on the following two lemmata. Firstly, Theorem 2 implies: Lemma 3 Assume the hypotheses of Theorem 3. Then the function μ : R → R defined by   s  → μ(s) := λ κ(s), α(s) − λ(0, α0 ) is continuous, non-zero and non-negative. Hereafter we shall use the same notation μ for the function μ ⊗ 1 on R × (−a, a). Secondly, we shall need the following Hardy-type inequality for a Schrödinger operator in a strip with the potential being a characteristic function:   Lemma 4 For any ψ ∈ W 1,2 R × (−a, a) ,      ρ −2 |ψ|2  16 |∂1 ψ|2 + 2 + 64/|J|2 R×(−a,a)

where ρ(s, t) := point of J.

R×(−a,a)

|ψ|2 , J×(−a,a)

1 + (s − s0 )2 , J is any bounded subinterval of R and s0 is the mid-

This lemma can be established quite easily by means of the classical onedimensional Hardy inequality R x−2 |v(x)|2 dx  4 R |v  (x)|2 dx valid for any v ∈ W 1,2 (R) with v(0) = 0 and Fubini’s theorem; we refer the reader to [12, Sec. 3.3] or [24, proof of Lem. 2] for more details. 6.1 Proof of Theorem 3 Let ψ belong to the dense subspace of D(hκ,α ) given by C∞ -smooth functions on R × (−a, a) which vanish in a neighbourhood of R × {−a} and which are restrictions of functions from C0∞ (R2 ). Assume the hypotheses of Theorem 3 so that the conclusions of Lemma 3 hold. Let J be any closed subinterval of R on which μ defined in Lemma 3 is positive. The first step is to come back to the intermediate lower bound (14); we also use the definition of λ via (15), but we do not neglect the ‘longitudinal kinetic energy’: 2  2  hκ,α [ψ] − λ(0, α0 ) ψ2κ  gκ−1 ∂1 ψ κ + μ1/2 ψ κ

 2   gκ−1 ∂1 ψ κ +  (1 − κ∞ a) min μ J

|ψ|2 . J×(−a,a)

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349

Here  ∈ (0, 1] is arbitrary for the time being. Applying Lemma 4 to the last integral, we arrive at hκ,α [ψ] − λ(0, α0 ) ψ2κ   16  (1 − κ∞ a) min J μ 1 −  |∂1 ψ|2 + 1 + κ∞ a 2 + 64/|J|2 R×(−a,a)   (1 − κ∞ a) min J μ ρ −2 |ψ|2 . + 2 + 64/|J|2 R×(−a,a) Choosing now  as the minimum between 1 and the value such that the first term on the right hand side of the last estimate vanishes, we finally get  2 hκ,α [ψ] − λ(0, α0 ) ψ2κ  c ρ −1 ψ κ

(28)

with 

1 (1 − κ∞ a) min J μ  , c := min  2 16 (1 + κ∞ a)2 2 + 64/|J| (1 + κ∞ a)

 .

(29)

In view of Section 3, we conclude that (28) is equivalent to  2 Qκ,α [u] − λ(0, α0 ) u2L2 ()  c (ρ ◦ L)−1 u L2 ()

(30)

for all u ∈ D(Qκ,α ), which is the exact meaning of (9). 6.2 Proof of Corollary 2 Let ψ be as in the previous section. The present proof is based on an algebraic comparison of hκ,0 [ψ] − λ(0, 0) ψ2κ with hκ+ ,0 [ψ] − λ(0, 0) ψ2κ+ and a usage of (28). For every (s, t) ∈ R × (−a, a), we have 1 − fε (s) 

gκ (s, t)  1 + fε (s) gκ+ (s, t)

with

fε (s) :=

ε a χ I (s) , 1 − κ+ ∞ a

where χ I denotes the characteristic function of the set I × (−a, a). Hereafter we assume ε  (1 − κ+ ∞ a)/(2a) so that the lower bound is greater or equal to 1/2. Using the same notation fε for the functions fε ⊗ 1 on R × (−a, a), we have hκ,0 [ψ] − λ(0, 0) ψ2κ   (1 + fε )−1 gκ−1 |∂1 ψ|2 + + R×(−a,a)



+

R

  ds 1 − fε (s)

− λ(0, 0)

 R×(−a,a)



a −a

  dt gκ+ (s, t) |∂2 ψ(s, t)|2 − λ(0, 0) |ψ(s, t)|2 −

2 fε gκ+ |ψ|2 .

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Recalling the definition of λ via (15) and Lemma 3, it is clear that the term in the second line after the inequality sign is non-negative. Consequently, hκ,0 [ψ] − λ(0, 0) ψ2κ 

 1 hκ+ ,0 [ψ] − λ(0, 0) ψ2κ+ − 2  2 fε gκ+ |ψ|2 . − λ(0, 0) R×(−a,a)

Using (28) with α being equal to 0, with κ being replaced by κ+ and with s0 being from the support of κ+ , we finally obtain 2  (31) hκ,0 [ψ] − λ(0, 0) ψ2κ  w 1/2 ψ κ , where w(s, t) :=

ε a χ I (s) c/4 − λ(0, 0) 2 1 + (s − s0 ) 1 − κ+ ∞ a

is positive for all sufficiently small ε. Equivalently, −κ,0  λ(0, 0) + w ◦ L−1

(32)

in the sense of quadratic forms on L2 (). This concludes the proof of Corollary 2.

7 Remarks and Open Questions It follows immediately from the minimax principle that the lower bound of Theorem 1 also applies to other boundary conditions imposed on L± , e.g., Dirichlet, periodic, certain Robin, etc. Of course, it is also possible to impose Robin boundary conditions on − instead of Dirichlet. Then the lower bound of the type (14) still holds and the problem is translated to the study of properties of the first eigenvalue in a Robin-Robin annulus. The techniques of the present paper will also apply to certain values of the parameters in such a case. However, we refrained from doing so to keep the statement of results as simple as possible. It follows from Theorem 2 that ν(α) gives a uniform lower bound to the spectral threshold of −κ,α provided α  0 or κ  0. We conjecture this to be always the case, but were not able to prove it in general. In this context, it would be desirable to prove that κ  → λ(κ, α) does not possess local minima for any α ∈ R. We proved the fact that κ  → λ(κ, α) is increasing on (0, 1/a) only for nonpositive α. It is clear from the limiting Dirichlet problem (cf [14]) that this property will not hold for large positive α. However, formula (23) suggests that this is still true for small values of α. Numerical results show (cf Fig. 1) that the critical value is approximately 0.78 for a = 1. To transfer the numerical results of Fig. 1 for different values of a, it is sufficient to notice that λ scales as: λ(κ, α; a) = a−2 λ(κa, αa; 1). Proposition 1 contains just one example of sufficient condition which guarantees the existence of discrete eigenvalues in infinite curved strips. Further results can be obtained in the spirit of [9, 25]. An open question is, e.g., whether the discrete spectrum exists for certain strips with κ > 0 and α > 0. Let us recall that this is always the case for α = +∞.

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For simplicity, we assumed that κ and α − α0 had compact support when we considered infinite strips. However, the claim of Corollary 1 holds whenever the essential spectrum (7) is preserved, and this might be checked under much less restrictive conditions about the decay of κ and α − α0 at infinity. For instance, modifying the approach of [25], it should be enough just to require that the limits at infinity are equal to zero. In fact, Theorem 3 holds without any condition about the decay of κ and α − α0 at infinity, but it is of interest only in the case the essential spectrum does not start above λ(0, α0 ). In any case, a fast decay of curvature at infinity is needed to prove Corollary 2; namely, κ(s) = O(s−2 ) as |s| → ∞. This quadratic decay is related to the decay of the Hardy weight in Theorem 3, which is typical for Hardy inequalities involving the Laplacian, and cannot be therefore improved by the present method. Under suitable global geometric conditions about the reference curve , the intrinsic distance |s − s0 | which appears in the function ρ of Theorem 3 can be estimated by an exterior one. For instance, if  is an embedded unit-speed curve with compactly supported curvature, then it is easy to see that there exists a positive number δ such that ∀s, s ∈ R,

δ |s − s |  |(s) − (s )|  |s − s | .

Corollary 2 extends the class of strips from [9] with empty discrete spectrum. An open question is to decide whether an analogous result holds for other α satisfying α0  α  0. Acknowledgements This work was partially supported by FCT (Portugal) through projects POCI/MAT/60863/2004 (POCI2010) and SFRH/BPD/11457/2002. The second author (D.K.) was also supported by the Czech Academy of Sciences and its Grant Agency within the projects IRP AV0Z10480505 and A100480501, and by the project LC06002 of the Ministry of Education, Youth and Sports of the Czech Republic.

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12. Ekholm, T., Kovaˇrík, H., Krejˇciˇrík, D.: A Hardy inequality in twisted waveguides. Arch. Rational Mech. Anal. (preprint on math-ph/0512050 (2005)) (in press) 13. Evans, D.V., Levitin, M., Vassiliev, D.: Existence theorems for trapped modes. J. Fluid Mech. 261, 21–31 (1994) 14. Exner, P., Freitas, P., Krejˇciˇrík, D.: A lower bound to the spectral threshold in curved tubes. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 460(2052), 3457–3467 (2004) 15. Exner, P., Krejˇciˇrík, D.: Quantum waveguides with a lateral semitransparent barrier: spectral and scattering properties. J. Phys. A 32, 4475–4494 (1999) 16. Exner, P., Krejˇciˇrík, D.: Waveguides coupled through a semitransparent barrier: a Birman– Schwinger analysis. Rev. Math. Phys. 13(3), 307–334 (2001) 17. Exner, P., Šeba, P.: Bound states in curved quantum waveguides. J. Math. Phys. 30, 2574–2580 (1989) 18. Flucher, M.: Approximation of Dirichlet eigenvalues on domains with small holes. J. Math. Anal. Appl. 193(1), 169–199 (1995) 19. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin Heidelberg New York (1983) 20. Goldstone, J., Jaffe, R.L.: Bound states in twisting tubes. Phys. Rev. B 45, 14100–14107 (1992) 21. Johnson, E.R., Levitin, M., Parnovski, L.: Existence of eigenvalues of a linear operator pencil in a curved waveguide – localized shelf waves on a curved coast. SIAM J. Math. Anal. 37(5), 1465–1481 (2006) 22. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin Heidelberg New York (1966) 23. Klingenberg, W.: A Course in Differential Geometry. Springer, Berlin Heidelberg New York (1978) 24. Krejˇciˇrík, D.: Hardy inequalities for strips on ruled surfaces. J. Inequal. Appl. 2006, Article ID 46409, 10 (2006) 25. Krejˇciˇrík, D., Kˇríž, J.: On the spectrum of curved quantum waveguides. Publ. RIMS, Kyoto University 41(3), 757–791 (2005) 26. Kreyszig, E.: Differential Geometry. University of Toronto Press, Toronto (1959) 27. Kuchment, P., Zeng, H.: Convergence of spectra of mesoscopic systems collapsing onto a graph. J. Math. Anal. Appl. 258, 671–700 (2001) 28. Kuchment, P., Zeng, H.: Asymptotics of spectra of Neumann Laplacians in thin domains. In: Advances in Differential Equations and Mathematical Physics, Birmingham, AL, 2002. Contemporary Mathematics, vol. 327, pp. 199–213. American Mathematical Society, Providence, RI (2003) 29. Londergan, J.T., Carini, J.P., Murdock, D.P.: Binding and Scattering in Two-dimensional Systems, LNP, vol. m60. Springer, Berlin Heidelberg New York (1999) 30. Olendski, O., Mikhailovska, L.: Localized-mode evolution in a curved planar waveguide with combined Dirichlet and Neumann boundary conditions. Phys. Rev. E 67, art. 056625 (2003) 31. Rauch, J., Taylor, M.: Potential and scattering theory on wildly perturbed domains. J. Funct. Anal. 18, 27–59 (1975) 32. Renger, W., Bulla, W.: Existence of bound states in quantum waveguides under weak conditions. Lett. Math. Phys. 35, 1–12 (1995) 33. Stollmann, P.: A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains. Math. Z. 219, 275–287 (1995) 34. Wang, Z.X., Guo, D.R.: Special Functions. World Scientific, Teaneck, NJ (1989)

Math Phys Anal Geom (2006) 9:353–369 DOI 10.1007/s11040-007-9016-5

Generalized Weierstrass Relations and Frobenius Reciprocity Shigeki Matsutani

Received: 9 November 2006 / Accepted: 13 February 2007 / Published online: 25 April 2007 © Springer Science + Business Media B.V. 2007

Abstract This article investigates local properties of the further generalized Weierstrass relations for a spin manifold S immersed in a higher dimensional spin manifold M from the viewpoint of the study of submanifold quantum mechanics. We show that the kernel of a certain Dirac operator defined over S, which we call a submanifold Dirac operator, gives the data of the immersion. In the derivation, the simple Frobenius reciprocity of Clifford algebras S and M plays an important role. Keywords Generalized Weierstrass relation · Frobenius reciprocity Mathematics Subject Classifications (2000) Primary 53C42 · 53A10; Secondary 53C27 · 15A66

1 Introduction This article is a sequel to our previous paper [22]. We study a connection between the generalized Weierstrass relation and Frobenius reciprocity, which is partially described in [22], and obtain a further generalized Weierstrass relation over a spin manifold S immersed in a higher dimensional spin manifold M. The generalized Weierstrass relation is a generalization of the Weierstrass relation appearing in minimal surface theory [7], which gives data of immersion of a conformal surface in higher dimensional flat spaces, e.g., Euclidean space. Although similar relations appeared in [7] and were obtained by Kenmotsu [16], the generalized Weierstrass relation was mainly studied in the 1990s by

S. Matsutani (B) 8-21-1 Higashi-Linkan, 228-0811, Sagamihara, Japan e-mail: [email protected]

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Konopelchenko [17] and Landolfi [18], Pedit and Pinkall [28], Taimanov [30], and so on. Their studies are, basically, in the framework of geometrical interpretations of an integrable system. In the studies a certain Dirac operator appears and global solutions of its Dirac equation provide information on the immersion of surfaces; in this article we shall call the Dirac operator (equation) the submanifold Dirac operator (equation). T. Friedrich investigated the relations for a surface immersed in the Euclidean three-space R3 from the viewpoint of studying the Dirac operator [9]. V. V. Varlamov also studied the relations from the point of view of Clifford algebra [33]. In [18], surfaces in flat n-space are treated and those in Riemann spaces were mentioned. Further, L. V. Bogdanov and E. V. Ferapontov generalized the relation to a surface in projective space [3]. Recently, I. A. Taimanov presented an extensive survey of related topics and open problems in [32]. The author has studied the submanifold Dirac operator since 1990 in the framework of quantum mechanics over submanifolds, which we call submanifold quantum mechanics; in the framework, we deal with a restriction of a differential operator, a hamiltonian, defined over a manifold to one over its submanifold and then we find a non-trivial structure in the operator due to the half-density [13, 23]. In [19, 26] I and my coauthor investigated the Dirac operator over curves in flat space and showed that the Dirac operator is identified with the operator of the Frenet–Serret relation and a natural linear operator in soliton theory. The latter gives a geometrical interpretation of an integrable system. When we apply the scheme developed in [19, 26] to the immersed surface case ([20] and reference therein), we encounter the same situation; the Dirac operator coincides with the Dirac operator appearing in the generalized Weierstrass relations and with a natural linear operator of a two-dimensional soliton equation. Further, the analytic torsion of the submanifold Dirac operator is also connected with globally geometrical properties [20, 21], as the Dirac operator with gauge fields exhibits geometrical properties of its related principal bundle via the analytic torsion in the framework of the Atiyah–Singer index theorem and so on [4]; the submanifold Dirac operator is also directly associated with the global geometry. In a series of works, the author has considered why the Dirac operator given in the framework of submanifold quantum mechanics appears in the generalized Weierstrass relation and expresses geometrical properties of the submanifold. In other words, our motivation is to clarify what is the submanifold quantum mechanics and what is the generalized Weierstrass relation from the viewpoint of a study of submanifold quantum mechanics. In fact, shape effect in quantum mechanics has recently come to play a more important role in physics due to the development of nanotechnology. The submanifold Schrödinger operator in submanifold quantum mechanics is applied to more realistic geometrical objects [8, 12, 24, 25]. There is thus a need to reveal the mathematical (analytic, geometrical and algebraic) structure of the submanifold quantum mechanics. On the submanifold Schrödinger operator, its essentially algebraic nature was clarified in [23].

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This article is the final one in our studies of the construction and the local properties of the submanifold Dirac operators. We find the answers to the problem why the submanifold Dirac operator constructed in the framework of submanifold quantum mechanics represents an immersed geometry; this means a local aspect of the generalized Weierstrass relation. Although, of course, the global feature of the Dirac equation might be more interesting than local ones, the essence of the answer is based on local properties, which are connected with the simple Frobenius relations in a local chart. Thus it is not difficult to generalize the submanifold Dirac operator defined over a surface immersed in Rn to one over more general geometrical situations, at least locally. If there is no obstruction, it might determine a submanifold globally. Here we note that the definition and construction of our submanifold Dirac operator differs from that of Bär [2], though both forms coincide. On similar lines, Ginoux and Morel [10], and d’Oussama and Zhang [27] also investigated eigenvalues of the submanifold Dirac operator. However, their construction is not directly associated with our requirement. Thus we concentrate on using the scheme of submanifold quantum mechanics. Section 2 of the paper shows our conventions of the Clifford algebra, while Section 3 provides our geometrical assumptions and conventions. After we consider the Dirac operator in a manifold in Section 4, we construct a Dirac operator in its submanifold and investigate it in Section 5. Our main theorem is Theorem 5.1.

2 Local Expression of Clifford Algebra This section briefly reviews the Clifford algebra [1, 5, 11] in order to show the conventions used. The Clifford Algebra CLIFF(Rm ) is introduced as a quotient ring of a tensor algebra, T(Rm )/((v, u)Rm − 1), where u, v are elements of m-dimensional vector space Rm and (v, u)Rm is the natural inner product. With respect to the degree of a tensor product, we have a natural filtration F  CLIFF(Rm ) ⊃ F −1 CLIFF(Rm ), where F 0 CLIFF(Rm ) = R and F p CLIFF (Rm ) = 0 for p< 0, with a graded algebra CLIFF p (Rm ) := F p CLIFF1 (Rm )/F p−1 CLIFF1 (Rm ). Let its subalgebra with even degrees be denoted by CLIFFeven p m (Rm ) = ∪m p=even CLIFF (R ). The exterior algebra ∧Rm = ⊕mj=1 ∧ j Rm , is isomorphic to CLIFF(Rm ) as Rm vector space, ∧ p Rm → CLIFF p (Rm ) and thus let the isomorphism, γ (m) : Rm → CLIFF1 (Rm ).

(2.1)

For the basis of Rm denoted by (e(m),i )i=1,··· ,m , let operator ∗ be the involution in CLIFF(Rm ) such that (γ (m) (e(m),i1 ) · · · γ (m) (e(m),i j ))∗ := (γ (m) (e(m),i j ) · · · γ (m) (e(m),i1 )).

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Let Cliff(Rm ) be a left CLIFF(Rm )-module whose endomorphism END(Cliff(Rm )) is isomorphic to CLIFFC (Rm ) (≡ CLIFF(Rm ) ⊗ C) as 2[n/2] dimensional C-vector space representation; m : CLIFFC (Rm ) → END(Cliff(Rm )). Let Cliff ∗ (Rm ) be a right CLIFF(Rm )-module which is isomorphic to Cliff(Rm ); ϕ : Cliff(Rm ) → Cliff(Rm )∗ ; for C ∈ CLIFF(Rm ) and c ∈ Cliff(Rm ), ϕ(Cc) = ϕ(c)C∗ and let c := ϕ(c). We may find bases (c(m),a )a=1,··· ,2[m/2] ∈ Cliff(Rm ) such that for c(m),a = ϕ(c(m),a ), c(m),a c(m),b = δa,b . Every ψ (m) ∈ Cliff(Rm ) is expressed as ψ (m) = 2[m/2] (m) (m),a 2[m/2] (m) (m),a . For φ (m) = a=1 φa c and ψ (m) , we will introduce a a=1 ψa c natural pairing:  Cliff(Rm ) : Cliff(Rm )∗ × Cliff(Rm ) → C,

(2.2)

by φ (m) , ψ

(m)

Cliff(Rm ) =

[m/2] 2

φa(m) ψa(m) .

a=1

For a multiplicative group of CLIFFeven (Rm ), CLIFFeven,× (Rm ), the Clifford group CG(Rm ) is defined by {τ ∈ CLIFFeven× (Rm ) | for ∀v ∈ CLIFF1 (Rm ), τ vτ ∗ ∈ CLIFF1 (Rm )}.  For representations m and m , there exist τ ∈ CG(Rm ) and an action Aτ on  ’s such that Aτ m (C) = m (τ Cτ −1 ) for C ∈ CLIFF(Rm ). Due to γ (m) : Rm → CLIFF1 (Rm ) and (2.2), we have

 , Cliff(Rm ) : Cliff(Rm )∗ × γ (m) (Rm ) × Cliff(Rm ) → C.

(2.3)

This is a linear map from Rm to C. Let the coproduct be m : Cliff(Rm ) → Cliff(Rm ) × Cliff(Rm ), (ψ → (ψ, ψ)). A domain restricted to its inverse image of R ⊂ C, (2.3) with the operator ‘, γ (m) (·) ◦ ϕ ⊗ 1 ◦ m’ can be regarded as HomR (Rm , R) ≈ Rm . Hence we have the following lemma. Lemma 2.1 There exists a subset Cliff pr (Rm ) of Cliff(Rm ) which is isomorphic to Rm as R-vector space such that for i : Rm → Cliff pr (Rm ) ⊂ Cliff(Rm ),

(v → ψvpr ),

j :=  , γ (m) (·) ◦ ϕ ⊗ 1 ◦ m ⊗ 1 ◦ i ⊗ 1 : Rm × Rm → R

is identified with the inner product (, )Rm : Rm × Rm → R, (u, v)Rm ≡ j(u, v), i.e., ψv , γ (m) (w)ψvpr Cliff(Rm ) = (v, w)Rm . pr

pr This lemma shows that there exist elements (φe(m) (Rm ); for (m), j ) j=1,··· ,m of Cliff m i (m), j b = j=1 je , (m),i

(m)

i φ e(m), , γ (m) (b (m),i )φe(m) (m), Cliff(Rm ) =  .

(2.4)

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357

This correspondence is well-known to physicists; it is, of course, independent of the coordinate system and gives the data of SO(Rm ) × R. Here let us consider an embedding Rk into Rn (k < n): ιn,k : Rk → Rn and πk,n : Rn −→ Rk such that for u(n) ∈ Rn and v (k) ∈ Rk , (ιn,k u(n) , v (k) )Rn ≡ (u(n) , πn,,k v (k) )Rk . In this article we are concerned with the moduli of the embedding or Grassmann manifold Grn,k := SO(n)/SO(k)SO(n − k). The embedding ιn,k corresponds to a point q of Grn,k = SO(Rn )/SO(Rn )SO(Rn−k ). Using the Clifford module we will deal with them as in [1]. The following proposition is obvious due to [1]. Proposition 2.1 (1) For k < n, CLIFF(Rk ) is a subalgebra CLIFF(Rn ) by the natural inclusion  of generators. ιn,k : CLIFF(Rk ) → CLIFF(Rn ). (2) For k < n, CG(Rk ) is a natural subgroup of CG(Rn ). The ιn,k and πk,n give an induced representation and a restriction representation: There exists an element τq in CG(Rn ) such that ⎛ ⎡ [n/2] ⎤⎞ [k/2] 2 2 ψa(k) ⎣ τq ab c(n),b ⎦⎠ , Indτq nk : Cliff(Rk ) → Cliff(Rn ), ⎝Indτq nk ψ (k) := b =1

a=1

⎛ Resτq nk : Cliff(Rn ) → Cliff(Rk ), ⎝Resτq nk ψ (n)

⎡ [n/2] ⎤ ⎞ [k/2] 2 2 b ⎣ := ψ (n) τq−1 ⎦ c(k),a ⎠ . b

a=1

b =1

a

The Frobenius reciprocity gives for ψ (k) ∈ Cliff(Rk ) and φ (n) ∈ Cliff(Rn ), n

Resτq k ψ (n) , φ (k) Cliff(Rk ) = ψ (n) , Indτq nk φ (k) Cliff(Rn ) .

(2.5)

Using the relation (2.5), we will consider in (2.4) and its relation to the point q of the Grassmannian Grn,k . For u(n) ∈ Rn and v (k) ∈ Rk , let ψu(n) (n) = τq n (n) n i(u(n) ) and ψu(k) := Res ψ using τ ∈ CG( R ), and then we have the relaq (n) k u(n) τq n (k) (k) tion, γ (n) (ιn,k (v (k) ))ψu(n) (v )ψu(k) (n) = Ind k γ (n) . The Frobenius reciprocity (2.5) gives (n) τq n (k) (k) (k) (k) ψu(k) (v )ψu(k) (v )ψu(k) (n) , γ (n) Cliff(Rk ) = ψu(n) , Ind k γ (n) Cliff(Rn ) (n) = ψu(n) (ιn,k (v (k) ))ψu(n) (n) , γ (n) Cliff(Rn )

= (ιn,k (v (k) ), u(n) )Rn .

(2.6)

Every pair (u(n) , v (k) ) recovers the point q in Grn,k . This relation (2.6) has an alternative expression using another reference embedding ιon,k : Rk → Rn associated to a base point of o ∈ Grn,k and τo ∈ CG(Rn ). For given τq and τo of CG(Rn ), we find an element τ ∈ CG(Rn ) such

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that τq = τ −1 τo . When one wishes to consider τq as a representation of Grn,k , the element τ can be dealt with by fixing τo ; Indτq nk = τ −1 Indτo nk ,

Resτq nk = Resτo nk τ.

(n) (k) For the situations of (2.6), let φu(n) (n) := τ ψu(n) and then we have ψu(n) = Resτo nk φu(n) (n) . (2.6) becomes (n) (k) (k) (n) o ψu(k) (v )ψu(k) (ιn,k (v (k) ))φu(n) (n) , γ (n) Cliff(Rk ) = φu(n) , γ (n) Cliff(Rn )

= (ιn,k (v (k) ), u(n) )Rn .

(2.7)

This also provides the data of Grn,k and the immersion, which essentially comes from (2.5) and Lemma 2.1. We will use latter relation (2.7) for the generalized Weierstrass relations. 3 Geometrical Preliminary This section provides a geometrical preliminary. As we use primitive facts in sheaf theory [15], we first show our conventions as follows. For a fiber bundle A over a paracompact differential manifold X and an open set U ⊂ X, let A X denote a sheaf given by a set of smooth local sections of the fiber bundle A, e.g., CrX is a sheaf given by smooth local sections of complex vector bundle over X of rank r, and A X (U) ≡ (U, A X ) sections of A X over U. Further, for open sets U ⊂ V ⊂ X, the restriction of a sheaf A X is denoted by rU V . Using the direct limit for {U | pt ∈ U ⊂ X}, we have a stalk A pt of A X by setting A pt ≡ ( pt, A X ) := limU→ pt A X (U). Similarly for a compact subset K in X, i K : K → X and for {U | K ⊂ U ⊂ X}, we have (K, A X ) := limU→K A X (U) and r K,U A X . On the other hand, for a topological subset Y of X, iY : Y → X, there is an −1 inverse sheaf, i−1 Y A X given by the sections iY A(U) = (iY (U), A X ) for U ⊂ Y. When Y is a compact set we have an equality (iY Y, A X ) = (Y, i−1 Y AX ) (Theorem 2.2 in [15]) and we identify them in this article. Further, c (U, A X ) denotes the set of smooth sections of A X whose support is compact in U. For a compact subset K of X,  K (X, A X ) is a set of global sections of A X whose support is in K. Let (M, g M ) be an n spin manifold, which is acted on by a Lie transformation group G as its isometory. The metric g M of M is a global section of sheaf HomR ( M ,  M ), where  M and  M are sheaves of tangent and cotangent spaces as C ∞ -modules: g M ( , ) :  M ×  M → R M . Let us consider a locally closed k spin manifold S embedded in M [14, 34], ι M,S : S → M, so that for every point p in S, there is a subgroup H of G satisfying T p M = T p (H ◦ p) ⊕ T p S.

(3.1)

We identify ι M,S (S) with S. H may depend on the position p in general. Since ι M,S −1  M can be regarded as a subsheaf of the (n,k) Grassmannian sheaf Gr(n,k) over S, fixing a section Gr(n,k) corresponds to determining the S S

Generalized Weierstrass relations and Frobenius reciprocity

359

immersion ι M,S up to global symmetry like Euclidean moves. We consider (n,k) −1 can be realized as the ι M,S −1  M and ι M,S ∗  S . Let ⊥ S := ι M,S  M / S ; Gr S (n,k) −1 quotient of orthogonal group sheaves Gr S = SO(ι M,S  M )/SO( S )SO(⊥ S ). For example, as r S,M is defined by a direct limit of open sets of M to ι M,S (S). we should consider its vicinity in M. We prepare a tubular neighborhood T S of S in M; π S,T S : T S → S and ι M,T S : T S → M. As our theory is local and we use only germs at a point in the vicinity of ι M,S (S), we consider a sufficiently small open set U in M such that U ∩ S = ∅ instead of M and S; Without loss of generality we assume that M and S are diffeomorphic to Rn and Rk , respectively; there exists a compact subset K of M such that S ⊂ K, and later we may sometimes identify M with T S . Further, due to the group action H, we assume that T S and S satisfy the following conditions. (1) T S behaves as a normal bundle π S,T S : T S → S, (2) There exist the base b (n),α˙ (α˙ = k + 1, · · · , n) of T S and ⊥ S , its dual base (n) α˙ b α˙ , and q := (q )α=k+1,··· ˙ ,n the normal coordinate of T S such that 1) for X ∈  S (S) and the Riemannian connection ∇ X in M, ∇ X b (n) α˙ belongs to  S (S) (See proof of Lemma 3.1) and 2) every point pt ∈ T S is expressed by pt = π S,T S pt + qα˙ b (n) α˙ . (3) T S and S have local parameterization. u : T S → Rk × Rn−k such that u = (s, q) and s : S → Rk ; u = (uμ )μ=1,··· ,n = (sα , qα˙ )α=1,··· ,k,α=k+1,··· ˙ ,n . (We use the Einstein convention.) Let Sq be u−1 (Rk × {q}) for fixing q. {Sq }q has a foliation structure. As a result of (3.1), S could be interpreted as an analytic manifold; S ≡ {(s, q) ∈ T S | q = 0}. For every sheaf AT S of T S , we have a sheaf A S of S and a restriction map r S,M : AnT S → AnS by substituting q = 0 into f (s, q). Hereafter we use the symbol r S,M with this meaning. Due to the above assumption, the metric gT S of T S at (s, q) induced from M is given as

g Sq 0 , (3.2) gT S = 0 1 where g Sq is a metric Sq given by proof of Lemma 3.1. We also introduce objects and maps for Sq as for S, e.g., ι M,Sq . Lemma 3.1 Let gT S and g S be induced metrics of g M and α˙ /k be the mean curvature vector field along b (n)α˙ ([34] p.119), det gT S = ρ det g S ,

˙

ρ = (1 + α˙ qα˙ + O(qα˙ qβ ))2 .

(3.3)

⊥ Proof In general, we consider more general normal unit vectors b˜ α(n) ˙ ∈ T pt S at β pt ∈ S. At a point in S, we find the Christoffel symbol  βα ˙ over S as a relation, the equation of Weingarten ([34] p.119), β (n) ˜ α˙ ˜ (n) =  βα ∇α b˜ (n) ˙ b β +  α β˙ b α˙ . β˙

(3.4)

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S. Matsutani

Here ∇α is the Riemannian connection of M for the direction ∂/∂sα of T S, β˙ ⊥ β and b (n) β := ∂/∂s of T S. Let α˙ be a section of SO( S ) such that its Lie ˜ α˙ ˜ α˙ ˜ β˙ algebraic parameter θα, ˙ β˙ satisfies ∂α θα, ˙ β˙ =  ˙ noting  ˙ = − α α˙ . Since S αβ

αβ

is diffeomorphic to Rk , we can find such a parameter θα, ˙ β˙ by solving the differential equation. β˙ ˜ (n) Let b (n) α˙ = α˙ b β˙ . Then (3.4) is reduced to β

(n) ∇α b (n) =  βα ˙ bβ . β˙

For a point pt in T S , the moving frame e(n),i = dxi ∈ ( pt, T S ) is expressed (n),i α i α˙ β α by e(n),i α ds = (π S,T S ∗ (eα ) + q  αα ˙ b β )ds . The metric in T S and its determinant are given by γ

γ

˙

α˙ δ α˙ β g Sq αβ = g Sαβ + [ αα ˙ g Sγβ + g Sαγ  αβ ˙ g Sδγ  ββ ˙ ]q + [ αα ˙ ]q q ,

(3.5)

where g Sαβ := g M i, jeiα eβ ; Let  β˙ :=  αβα ˙ over S; ( β˙ )/k is the mean curvature j

˙

vector of b (n)β ([34] p.119).

 

⊥ There is an action of SO(⊥ S ) on  S . Obviously (3.3) is invariant for the action SO(⊥ ). S

4 Dirac System in M For the above geometrical situation, we will consider a Dirac equation over M ([4] 3.3) here. We first introduce a paring given by the pointwise product , Cliff M for the germs of the Clifford module Cliff M over M and its natural hermite conjugate Cliff ∗M ; ϕ pt is the hermite conjugate operator which gives the isomorphism from Cliff M to Cliff ∗M and ψ M,1 ψ M,2 Cliff M ∈ ( pt, C M ). We deal with a Dirac equation over M as an equation over another preHilbert space H = (c (M, Cliff ∗M ) × c (M, Cliff M ), , , ϕ). Here , M is the L2 -type pairing, for (ψ M,1 , ψ M,2 ) ∈ c (M, Cliff ∗M ) × c (M, Cliff M ),  ψ M,1 , ψ M,2 M = dvol M ψ M,1 , ψ M,2 Cliff M (4.1) M

Here in T S , the measure of M is decomposed to dvol M = ρ(det g S )1/2 dk sdn−k q, ∧sα=1 dsα ,

n−k

∧nα=k+1 dqα˙ . ˙

(4.2)

and d q = Further in this article, we express the d s= preHilbert space using the triplet with the inner product (◦, ·) M := ϕ◦, · M . For an operator P over Cliff M , let Ad(P) be defined by the relation, ψ 1 , Pψ2 M = ψ 1 Ad(P), ψ2 M if exists. Further for ψ ∈ c (M, Cliff M ), P∗ is defined by P∗ ψ = ϕ −1 (ϕ(ψ)Ad(P)). Let the sheaf of the Clifford ring over M be denoted by CLIFF M . As a model of (2.1) let γ M be a morphism from  M to CLIFF1M . k

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361

The Dirac operator is a morphism between the Clifford module  D M : Cliff M → Cliff M but as a differential operator, we could extend its domain and region to, [n/2]

 D M : C2M

[n/2]

→ C2M .

Since Cliff M (U) contains zero section, we may consider that Ker( D M ) as a [n/2] subset of germs of C2M means a subset of germs of Cliff M . Then there are a set of germs {caM }a=1,··· ,2[n/2] of Cliff M (M) and caM := ϕ(caM ) which hold relations at each point, caM cbM Cliff M = δ a,b ,

for a, b = 1, · · · , 2[n/2] .

(4.3)

A of solutions of Dirac equation  D M ψ = 0 is expressed by ψ =  germ a a a ψ c a M M for ψ M ∈ ( pt, C M ) at a point pt ∈ M. Lemma 2.1 gives pr

pr

Proposition 4.1 There is a subsheaf Cliff M ⊂ Cliff M satisfying the following: pr

(1) Cliff M is isomorphic to  M as vector sheaves via the following j M , i.e., there pr is a morphism i :  M → Cliff M (i(u(n) ) = ψu(n) ). (2) j M whose model is j in Lemma 2.1 gives an equivalence j M = g M (, ), i.e., pr for v (n) , u(n) ∈ ( pt,  M ), every ψu(n) ∈ ( pt, Cliff M ) satisfies ψ u(n) γ M (g M (v (n) ))ψu(n) Cliff M = g M (u(n) , v (n) ). We call this relation R × SO(n)-representation in this article. Due to the Proposition, for ij ∈ ( pt, SO(n) × R)), and v (n),i := ije(n), j ∈ pr ( pt,  M )), there is a pair of germs (ψe(n),i )i=1,··· ,n in ( pt, Cliff M ) of the Clifford module and its dual pair ψ e(n),i := ϕ pt (ψe(n),i ) which hold a relation, ψ e(n), γ M (g M (v (n),i ))ψe(n), Cliff M = i

(not summed over ).

Every sheaf AT S over T S is determined by AT S = r T S ,M A M for every sheaf A M over M and in our conditions these properties preserves over T S . Remark 4.1 Using a C-valued smooth compact function b ∈ c (M, C M ) over a M such that b ≡ 1 at U ⊂ M and its support is in M, b ψ M , b ψe(n),k and their ∗ partners belong to c (M, Cliff M ) and c (M, Cliff M ). Hereafter we assume that a ψM , ψe(n),k and their partners are sections of c (M, Cliff M ) and c (M, Cliff ∗M ) in the sense. The Dirac operator restricted over T S is explicitly given by  DT S = γT S (duμ )(∂μ + ωT S ,μ ). where ∂μ := ∂/∂uμ and ωT S ,μ is a spin connection.

(4.4)

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5 Submanifold Dirac Operator over S in M In this section we define the submanifold Dirac operator over S in M and investigate its properties. Since T S is homeomorphic to Rn , CT S is soft (Theorem 3.1 in [15]). Hence we have the following proposition. [n/2]

Proposition 5.1 Cliff T S and C2T S

are soft.

Proof Cliff T S is considered as a sheaf of C-vector bundle with 2[n/2] rank. From the proof of Theorem 3.2 in [15], it is justified.   Due to Proposition 5.1, at each point pt in S and for a germ ψ pt ∈ ( pt, Cliff T S ), there exists ψc ∈ c (T S , Cliff T S ) and ψo ∈ (T S , Cliff M ) such that ψ pt = ψc and ψ pt = ψo around pt. Thus an element of ( pt, Cliff T S ) need not be distinguished whether it comes from c (T S , Cliff T S ) or (T S , Cliff M ). From here, every AT S is again identified with A M . We continue to consider the action of H along the fiber direction in the framework of the unitary representation of the Clifford module and we wish to consider kernel of ∂α˙ √ := ∂/∂qα˙ (α˙ = k + 1, · · · , n) ([23] and references therein). However, pα˙ := −1∂α˙ is not self-adjoint, p∗α˙ = pα˙ in general due to the existences ρ in (4.1) and (4.2). Let us follow the techniques in the pseudo-regular representation. We intro˜ ∗ ) × c (T S , Cliff ˜ T ), , sa , ϕ) duce another preHilbert space H ≡ (c (T S , Cliff ˜ S TS so that pα˙ ’s become self-adjoint operators there. Using the half-density (Theorem 18.1.34 in [13]), we construct self-adjointization: ηsa : H → H by, ηsa (ψ) := ρ 1/4 ψ,

ηsa (ψ) := ρ 1/4 ψ,

ηsa (P) := ρ 1/4 Pρ −1/4 .

Here since ρ does not vanish in T S , ηsa gives an isomorphism ηsa : Cliff ∗T S × ˜ ∗ × Cliff ˜ T . Here this transformation is also essentially the Cliff T → Cliff S

TS

S

same as that in the radical Laplace operator, e.g., in Theorem 3.7 of [14].1 For ˜ ∗ ) × c (T S , Cliff ˜ T ), by letting ϕ˜ := ηsa ϕη−1 , the pairing (ψ 1 , ψ2 ) ∈ c (T S , Cliff S TS sa is defined by  ψ 1 , ψ2 sa := (det g S )1/2 dk sdn−k q ψ 1 , ψ2 Cliff M . (5.1) TS −1 −1 ◦, ηsa · M , 2) for an Here we have the properties of ηsa that 1) ◦, · sa = ηsa −1 operator P of Cliff T S , ηsa (P) = ηsa Pηsa , and 3) pα˙ ’s themselves become selfadjoint in H , i.e., pα˙ = p∗α˙ . Noting ρ = 1 at a point in S, for (ψ, ψ) ∈ (S, Cliff ∗T S ) × (S, Cliff T S ), we have

r S,M ηsa (ψ) = r S,M ψ,

1 Our

ρ 1/2 corresponds to δ in p.261 in [14].

and r S,M ηsa (ψ) = r S,M ψ.

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363

Further we have the following proposition. Proposition 5.2 By letting pq := aα˙ pα˙ for real generic numbers aα˙ , the projection, ˜ ∗ × Cliff ˜ T → Ker(Ad( pq )) × Ker( pq ), π pq : Cliff S TS induces the projection in the preHilbert space, i.e., (1) For an open set U ⊂ T S , ϕ| ˜ Ker( pq ) : (U, Ker( pq )) → (U, Ker(Ad( pq ))) is isomorphic as vector space. We simply express ϕ| ˜ Ker( pq ) by ϕ˜ hereafter. (2) H pq := (c (T S , Ker(Ad( pq ))) × c (T S , Ker( pq )), , sa , ϕ) ˜ is a preHilbert space. (3)  pq := π pq |Cliff ˜ T induces a natural restriction of pointwise multiplication S

pt

for a point in Ts , H pq := (( pt, Ker(Ad( pq ))) × ( pt, Ker( pq )), ·, ϕ˜ pt ) becomes a preHilbert space. The hermite conjugate map ϕ˜ pt is still an isomorphism. 2 ∗ Proof By letting  pq := π pq |Cliff ˜ T , we have  pq =  pq =  pq in H pq . In S ∗ fact since pα˙ is self-adjoint, Ker( pq ) = Ker( pq ) and Ker( pq ) is isomorphic to Ker(Ad( pq )), i.e., ϕ( pq ψ) = ϕ(ψ)Ad( pq ).  p∗q ψ = ϕ −1 (ϕ(ψ)Ad( pq )) gives  pq =  p∗q .  

Remark 5.1 We remark that deformation of preHilbert space by the action of ηsa makes  pq a projection operator in the sense of ∗-algebra. This is the essence of the scheme of the submanifold quantum mechanics [23], which provides non-trivial quantum mechanics [12, 24, 25]. It is absolutely a non-trivial fact, but the same idea appeared in the computation of Hydrogen atom in [6]. [n/2]

[n/2]

Further, we consider pq as a morphism between C2T S → C2T S and its [n/2] [n/2] kernel KerC pq ⊂ C2T S . We are concerned with r S,M KerC pq ⊂ r S,M C2T S , but [n/2] it is obvious that r S,M KerC pq can be identified with C2S , because its element is a function only of S. Then we have a similar relation of KerC pq in Proposition 5.2. After we suppress a normal translation freedom in H pq , we might choose a position q and make q vanish. Thus we will give our definition of the submanifold Dirac operator. Definition 5.1 We define the submanifold Dirac operator over S in M by,  DS →M := r S,M (ηsa ( D M )|Ker( pq ) ), ˜ M , i.e., as an endomorphism of Clifford submodule r S,M Ker( pq ) ⊂ r S,M Cliff  DS →M : r S,M Ker( pq ) → r S,M Ker( pq ). Further we extend its domain and region to r S,M KerC pq or C2S

[n/2]

[n/2]

 DS →M : C2S

[n/2]

→ C2S

.

;

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Here we note that the first restriction |Ker( pq ) is as an operator but the second one r S,M is associated with a sheaf theory [15]. [n/2] In order to find the extension for  DS →M over C2S we need an explicit representation of the Dirac operator. For the case that M is the Euclidean space, we find a natural frame to represent the Clifford objects explicitly. However, a local parameter of M is not privileged in general. Thus we introduce another Clifford ring sheaf isomorphic to r S,M CLIFF M and find its explicit isomorphism using an element of the Clifford group. Let us introduce a vector sheaf RnS related to G-action and a sheaf morphism ιRn ,S :  S → RnS and an isomorphism μRn ,M : r S,M  M → RnS . Using this, we will investigate the Clifford objects over S and ones over M with r S,M before we deal with the Dirac operator. Using the vector sheaf RnS , we construct a Clifford ring sheaf CLIFF(RnS ) over S generated by a linear sheaf morphism γRnS : RnS → CLIFF1 (RnS ). We could define its representation module Cliff(RnS ) and its Clifford groups CG(RnS ) similarly. We have an isomorphism μRn ,M  : r S,M CLIFF M → CLIFF(RnS ) and one between the Clifford groups CG(RnS ) and r S,M CG M . By identifying CG(RnS ) with r S,M CG M , μRn ,M  is realized as μRn ,M  (c) = τ −1 cτ for c ∈ CLIFF M and τ ∈ r S,M CG M . Then we also have its representation Cliff(RnS ), and an isomorphism μRn ,M  : r S,M Cliff M → Cliff(RnS ). The ιRn ,S induces a ring homomorphism ιRn ,S  : CLIFF S → CLIFF(RnS ) by its generator corresponding to u(k) ∈  S by γ S (u(k) ) → γ (ιRn ,S (u(k) )). Similarly we have ι M,S  : CLIFF S → r S,M CLIFF M . The ιRn ,S  = μRn ,M  ι M,S  and ι M,S  induce the induced and restricted representations, modeling ones in Section 2 such that IndιRn ,S RS : Cliff S → Cliff(RnS ), n

Indι M,S SM : Cliff S → Cliff M ,

ResιRn ,S RS : Cliff(RnS ) → Cliff S , n

Resι M,S SM : Cliff M → Cliff S ,

are connected by natural relations, Indι M,S SM = τ −1 IndιRn ,S RS , n

Resι M,S SM = ResιRn ,S RS τ. n

For every u ∈ ( pt, r S,M  M ), v ∈ ( pt,  S ), ψu := i M (u), and ψ S,u := Resι M,S SM ψu , as we showed in (2.6), the Frobenius reciprocity shows g M (ι M,S (v), u) = ψ u , γ M (g M (ι M,S ∗ (v)))ψu Cliff M = ψ S,u , γ S (g S (v))ψ S,u Cliff S .

(5.2)

As in (2.7), by letting ψτ u := τ ψu ∈ ( pt, r S,M Cliff RnS ), we have ψ S,u = n ResιRn ,S RS ψτ u and (5.2) becomes ψ τ u , γRn (gRn (ιRn S∗ (v)))ψτ u Cliff(RnS ) = ψ S,u , γ S (g S (v))ψ S,u Cliff S = g S (v, π SM (u)),

(5.3)

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where π S,M : r S,M  M →  S is given by g M (ι M,S (v), u) = g S (v, π SM (u)). This is the simplest Frobenius reciprocity; we use its lift to the Clifford modules. These give the data of Gr(n,k) and immersion ι M,S , which are our purpose. S As we find relations among the Clifford objects over S and ones over M with r S,M , we pass to the consideration of the Dirac operator. In order to obtain the relation (5.3), we will use the Dirac operator  DS →M . However, we have not yet given its explicit representation. In order to determine an explicit representation of the Dirac operator, using τ ∈ CG(RnS ) which connects CLIFF(RnS ) and r S,M CLIFF M as mentioned above, we will define the Dirac operator defined over Cliff(RnS ) ι

R ,S  DS →M := τ  DS →M τ −1 .

n

Proposition 5.3 The submanifold Dirac operator of S in M can be expressed by 1 ιRn ,S  DS →M = ιRn ,S  ( DS ) + γ α˙ μRn ,M ∗ α˙ . 2

(5.4)

where  DS is the proper Dirac over S, α˙ /k is the mean curvature vector of b (n)α˙ ([34] p.119) of S and γ α˙ := γRnS (μRn ,M ∗ (dqα˙ )). Proof First we note that ηsa ( D M ) has a decomposition, 

ηsa ( D M ) =D M + D  ⊥ M, 

α˙ α˙ where D  ⊥  M does not include the normal derivative M := γ M (dq )∂/∂q and D ⊥ pα˙ .  D M vanishes at Ker( pq ) and at KerC ( pq ). Due to the constructions, ι M,S (γ S (e(k),α )) and γ M (dqα˙ ) become generator of the CLIFFT S at a sufficiently close vicinity of S. A direct computation shows that the following relation holds

  1  r S,M D  M − τ −1 ιRn ,S  ( DS )τ = r S,M (γ M (dqα )α ) . 2 The geometrical independence due to (3.2) and direct computations give the   above result. Using ιRn ,S and μRn ,M , we have the result. Remark 5.2 √ (1) −1ιRn ,S  ( DS ) is a formal self-adjoint for a L2 -type √ integral of the Clifford module over S because from the definition, −1 DS is selfadjoint√for the integral over S and ιRn ,S  is ∗-morphism. On the other ιRn ,S is not self-adjoint because of the extra term and the hand, −1  DS →M √ self-adjointness of −1ιRn ,S  ( DS ). (2) Here we comment on the submanifold Dirac operators defined by C. ˜ in [2] corresponds to our Bär ([2], Lemma 2.1). The Dirac operator D √  ˆ in [2] corresponds to our ( DS ) whereas the Dirac operator D √−1ιRn ,S ιRn ,S −1  DS →M . In [9] the generalized Weierstrass relation is studied using the Dirac operator which is the same as Bär’s. Further we note that in [2],

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√ ˜ is mainly investigated, whereas we consider −1  DιRn ,S , which is not D S →M self-adjoint. (3) Ginoux and Morel and Oussama and Zhang [27] dealt with the √ [10], ιRn ,S same operator −1  DS →M but their studies started from the definition √ ιRn ,S . They did not explain why they employ the definition in of −1  DS →M detail, at least, from of the submanifold quantum mechanics. √ viewpoint ιRn ,S (4) It is clear why −1  DS →M has an extra non-trivial term. It appears due to the requirement that the projection  pq should be the selfadjointness, which is the same as the requirement that the isomorphism ϕ should preserve for the action of  pq . These are essential to submanifold quantum mechanics [23]. Now we give our main theorem: Theorem 5.1 Fix the data of Cliff M i.e., its base caM a=1,··· ,2[n/2] , and a morphism [n/2] pr i :  M → Cliff M . Let a point pt be in S immersed in M. Let C2S be a sheaf [n/2] of complex vector bundle over S with rank 2[n/2] . A set of germs of ( pt, C2S ) satisfying the submanifold Dirac equation, √ ιRn ,S −1  DS →M ψ = 0 at pt, is given by {b a ψ a | a = 1, · · · , 2[n/2] , b a ∈ C} such that elements satisfy the orthonormal relation as C-vector space; ϕ pt (ψ a )ψ b = δa,b

at

pt.

Then the following hold: (1) ψ a a=1,··· ,2[n/2] is a base of ( pt, Cliff(RnS )). There exists an isomorphism μRn ,M  : r S,M Cliff M → Cliff(RnS ) related to τ ∈ ( pt, r S,M CG M ) satisfying ψ a = τ caM (a = 1, · · · , 2[n/2] ) by identifying Cliff(RnS ) with r S,M Cliff M . τ corresponds to an element of SO(r S,M  M ) as a representative element of . Gr(n,k) S pr (2) For every u ∈ ( pt, r S,M  M ), let ψu := i M (u) ∈ ( pt, r S,M Cliff M ), ψu,S := pr τ ψu ∈ ( pt, r S,M Cliff M ) using τ of (1), and ψ u,S := ϕ(ψu )τ −1 ∈ pr

( pt, r S,M Cliff M ). Then for every v ∈ ( pt, r S,M  S ), the following relation holds: ψ u,S [ιRn ,S  (γ S (g S (v)))]ψu,S Cliff(RnS ) = g M (ι M,S (v), u).

(5.5)

This value brings us the local data of immersion ι M,S . ι

R ,S Proof Since  DS →M is the 2[n/2] rank first order differential operator and has no [n/2] singularity over S due to the construction, a germ of its kernel in ( pt, C2S ) ι Rn ,S is given by 2[n/2] dimensional vector space at each point of S. Since  DS →M [n/2] is defined as an endomorphism of KerC ( pq ) ≈ C2S . The kernel of the Dirac [n/2] n ι R ,S operator, KerC ( DS →M ) of C2S has an injection into Cliff(RnS ). There exist ιRn ,S −1 n  τ ∈ CG(R S ) such that μRn ,M : r S,M KerC ( DS →M ) → Cliff M .

n

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367

Let D  S ⊥ := τ −1 γ α˙ ∂α˙ τ at S. From the construction, we have ι M,S ∗  DS →M + D  S ⊥ = r S,M (ηsa ( D D )). ι

R ,S ) is a subset of a kernel of τ (r S,M (ηsa ( D M )))τ −1 ⊂ r S,M C2M . Hence KerC ( DS →M pt Noting Proposition 5.2, ϕ˜ pt is an isomorphism and H pq gives (5.2) and (5.3). The proof is thus done.  

[n/2]

n

Remark 5.3 (1) The final result does not depend upon a choice of ιRn ,S . (2) This theorem is based upon the Frobenius reciprocity of Clifford ring sheaves on category of differential geometry as shown in (5.2) and (5.3). n We have compared IndιRn ,S RS Cliff S , which is obtained by using the Dirac operator, with Cliff M as each germ in Theorem 5.1. (3) We have assumed that M and S are homeomorphic to Rn and Rk , respectively. However, as our arguments are local, the theorem could be extended to spin manifolds S and M under assumptions on the group action if there is no geometrical obstruction. (4) With Remark 5.1 and 5.2 (4), it is obvious that the submanifold Dirac operator given in submanifold quantum mechanics represents local immersed geometry. Its essence is that the restriction of the Dirac operator preserving ϕ in Definition 5.1 consists of the Frobenius reciprocity. This is the answer to the question mentioned in Introduction. (5) If M and S have natural parameterization (xi )i=1,··· ,n and (sα )α=1,··· ,k and S is an analytic submanifold such that  s dxi (s) xi (s) = S

represents an immersion S in M, it can be expressed as  s i x (s) = g S,α,β ψ ∂xi ,S [ι M,S  (γ S (dsα ))]ψ∂xi ,S Cliff(RnS ) dsβ , S

(6)

(7) (8) (9)

where ∂xi := ∂/∂ xi using above ψ. This is the generalized Weierstrass relation. When M ≡ Rn and k = 2, the theorem is reduced to the generalized Weierstrass relation [9, 17, 18, 28]. In this case, ιRn ,S is properly determined and identify RnS with Rn . These are closely related to the twoιRn ,S dimensional integrable system. Especially, when  DS →M is identified ¯ which correspond to minimal surface cases, it becomes with  DS and ∂, the original Weierstrass relation ([7], p.260–7). As mentioned in [22], we can put the Frenet–Serret torsion field into the Dirac operator. For case k = 1, the Theorem is merely the Frenet–Serret relation [19, 20]. As we showed in [19, 20], the Dirac operator might also give the global properties of the immersion of S, i.e., its topological properties, though we mentioned only local properties in this article. Thus we should

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investigate the global properties using the submanifold Dirac operator as a generalization of [19, 20] in future. (10) When S is a conformal surface, we may consider the relations along the line of arguments presented in [3, 17, 28, 30–32]. For example, we could classify the immersions using the Dirac operator. Furthermore, when S has holomorphic properties, we may also give similar arguments. Acknowledgements The author thanks Professor K. Tamano, Professor N. Konno and Dr. H. Mitsuhashi for encouragements on this work, especially Dr. H. Mitsuhashi for his lecture on Frobenius reciprocity.

References 1. Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modues. Topology 3, 3–38 (1964) 2. Bär, C.: Extrinsic bounds for eigenvalues of the Dirac operator. Ann. Global Anal. Geom. 16, 573–596 (1998) 3. Bogdanov, L.V., Ferapontov, E.V.: Projective differential geometry of higher reductions of the two-dimensional Dirac equation. J. Geom. Phys. 52, 328–352 (2004) 4. Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Springer, Berlin (1996) 5. Chevalley, C.: The Algebraic Theory of Spinors and Clifford Algebras. Springer, Berlin (1997) 6. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, Oxford, UK (1958) 7. Eisenhart, K.P.: A Treatise on the Differential Geometry of Curves and Surfaces. Ginn and Company, Boston (1909) 8. Encinosa, M.: Electron wave functions on T 2 in a static magnetic field of arbitrary direction. Physica, E, Low-dimens. Syst. Nanostruct. 28, 209–218 (2005) 9. Friedrich, T.: On the spinor representation of surfaces in Euclidean 3-space. J. Geom. Phys. 28, 143–157 (1998) 10. Ginoux, N., Morel, B.: On the eigenvalue estimates for the submanifold Dirac operator. Int. J. Math. 13, 533–548 (2002) 11. Goodman, R., Wallach, N.R.: Representations and Invariants of the Classical Groups. Cambridge University Press, Cambridge (2003) 12. Gravesen, J., Willatzen, M., Lew Yan Voon, L.C.: Schrödinger problems for surfaces of revolution – the finite cylinder as a test example. J. Math. Phys. 46, 012107 (2005) 13. Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlin (1985) 14. Helgason, S.: Groups and Geometric Analysis. Academic, New York (1984) 15. Iversen, B.: Cohomology of Sheaves. Springer, Berlin (1986) 16. Kenmotsu, K.: Weierstrass formula for surfaces of prescribed mean curvature. Math. Ann. 245, 89–99 (1979) 17. Konopelchenko, B.G.: Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy. Ann. Global Anal. Geom. 16, 61–74 (2000) 18. Konopelchenko, B.G., Landolfi, G.: Generalized Weierstrass representation for surfaces in multi-dimensional Riemann spaces. J. Geom. Phys. 29, 319–333 (1999) 19. Matsutani, S.: Anomaly on a submanifold system – new index theorem related to a submanifold system. J. Phys. A 28, 1399–1412 (1995) 20. Matsutani, S.: Immersion anomaly of Dirac operator of of surface in R3 . Rev. Math. Phys. 11, 171–186 (1999) 21. Matsutani, S.: Generalized Weierstrass relation for a submanifold S in En arising from the submanifold Dirac operator. Adv. Stud. Pure Math. (to appear) 22. Matsutani, S.: Submanifold Dirac operators with torsion. Balk. J. Geom. Appl. 9, 1–5 (2004) 23. Matsutani, S.: On the essential algebraic aspect of submanifold quantum mechanics. J. Geom. Symm. Phys. 2, 18–26 (2004)

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24. Meyer, G.J., Blick, R.H., Knezevic, I.: Curvature-Dependent Conductance Resonances in Quantum Cavities. (2005) 25. Mott, L., Encinosa, M., Etemadi, B.: A numerical study of the spectrum and eigenfunctions on a tubular arc. Physica, E, Low-Dimens. Syst. Nanostruct. 25, 532–529 (2005) 26. Matsutani, S., Tsuru, H.: Physical relation between quantum mechanics and solitons on a thin elastic rod. Phys. Rev. A 46, 1144–1447 (1992) 27. d’Oussama, H., Zhang, X.: Lower bounds for the eigenvalues of the Dirac operator, part II. The submanifold Dirac operator. Ann. Global Anal. Geom. 19, 163–181 (2001) 28. Pedit, F., Pinkall, U.: Quaternionic analysis on Riemann surfaces and differential geometry. Doc. Math. J. DMV., Extra Vol. ICM II, 389–400 (1999) 29. Serre, J.-P.: Linear Representations of Finite Group. Springer, New York (1977) 30. Taimanov, I.A.: The Weierstrass representation of closed surfaces in R3 . Funct. Anal. Appl. 32, 258–267 (1998) 31. Taimanov, I.A.: Surfaces in the four-space and the Davey–Stewartson equations. J. Geom. Phys. 56, 1235–1256 (2006) 32. Taimanov, I.A.: Two-dimensional Dirac operator and the theory of surfaces. Russian Math. Surveys 61, 79–159 (2006) 33. Varlamov, V.V.: Generalized Weierstrass representation for surfaces in terms of DiracHestenes spinor field. J. Geom. Phys. 32, 241–251 (2000) 34. Willmore, T.J.: Riemannian Geometry. Clarendon Press, Oxford (1993)

Math Phys Anal Geom (2006) 9:371–388 DOI 10.1007/s11040-007-9017-4

Form-preserving Transformations for the Time-dependent Schrödinger Equation in (n + 1) Dimensions Axel Schulze-Halberg

Received: 16 July 2006 / Accepted: 13 February 2007 / Published online: 16 March 2007 © Springer Science + Business Media B.V. 2007

Abstract We define a form-preserving transformation (also called point canonical transformation) for the time-dependent Schrödinger equation (TDSE) in (n + 1) dimensions. The form-preserving transformation is shown to be invertible and to preserve L2 -normalizability. We give a class of timedependent TDSEs that can be mapped onto stationary Schrödinger equations by our form-preserving transformation. As an example, we generate a solvable, time-dependent potential of Coulombic ring-shaped type together with the corresponding exact solution of the TDSE in (3+1) dimensions. We further consider TDSEs with position-dependent (effective) masses and show that there is no form-preserving transformation between them and the conventional TDSEs, if the spatial dimension of the system is higher than one. Keywords (n + 1)-dimensional time-dependent Schrödinger equation · Exact solutions · Form-preserving transformation PACS 03.65.Ge · 03.65.Ca

1 Introduction Form-preserving transformations (FPT), sometimes called point canonical transformations, are major tools for finding exact solutions and solvable potentials of the time-dependent Schrödinger equation (TDSE). An FPT changes the dependent and independent variables of a TDSE in such a way that the

A. Schulze-Halberg (B) Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, C.P. 28045, Colima, Colima, México e-mail: [email protected]

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transformed equation maintains the form of a TDSE, but for a potential different from the original one. Therefore, an FPT establishes a mapping between two TDSEs for different potentials. If one of the TDSEs is solvable, then the other TDSE is solvable too, and its solution can be computed by means of the FPT. Besides factorization methods like the Darboux transformation (also called supersymmetrical factorization), the FPT is the method most used for generating solvable cases of the TDSE, and there is a large number of works in which it was employed. In particular, FPTs have been used to derive exact solutions for quadratic potentials [8], including harmonic oscillators [1], linear potentials [2] and harmonic oscillators with damping [10], just to name a few. FPTs were employed in systematic searches for non-central solvable potentials [4] and for PT-symmetric potentials [7]. Furthermore, FPTs are fundamental for the generation and classification of quasi-exactly solvable potentials, see the monograph [11] and references therein. Recently, FPTs were also defined for TDSEs with position dependent (effective) masses [9]. Properties of FPTs like reality conditions for the transformed potential [8], interplay with Darboux transformations and special cases of FPTs between TDSEs and stationary Schrödinger equations [3] are well known. However, there is a restriction to the FPTs used in the previously mentioned articles: they are only defined in (1 + 1) dimensions. Results in higher dimensions were only obtained by separating the TDSE (e.g. in spherical coordinates), and then applying the FPT to the separated equations (e.g. the radial equation in spherical coordinates). In general, if the TDSE cannot be reduced to scalar equations, the FPT cannot be applied. Therefore, in order to fill this gap, in the present note we shall define the FPT for TDSEs in (n + 1) dimensions. We will see that the most important properties of the (1 + 1)-dimensional case are maintained in arbitrary spatial dimension, e.g. invertibility of the FPT and the fact that it preserves the L2 -normalizability. In the next Section 2, we give a summary of our results on the FPT in (n + 1) dimensions. These results will be proved in Section 3. In order to illustrate our results, we give three applications. In the first application (Section 4) the FPT is applied to a stationary Coulombic ring-shaped potential in three dimensions, generating a solvable modified Coulombic ring-shaped potential and the corresponding exact solutions of the TDSE. The second application in Section 5 states a class of time-dependent potentials, that can be mapped onto stationary potentials by means of the FPT. This application is a generalization of a result in [3]. The last application in Section 6 proves that there are no FPTs between conventional TDSEs and effective mass TDSEs, if the spatial dimension of the system is two or higher. This stands in contrast to the (1 + 1)-dimensional case [9], where an FPT can be constructed explicitly. 2 The FPT in (n + 1) Dimensions In this section we summarize our results, giving the explicit form of the FPT, the transformed potential and the transformed solution. All results stated in this section will be proven in the subsequent section.

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The original TDSE Consider the TDSE in (n + 1) dimensions: i t +

1 x  − V  = 0, 2m

(1)

where  = (x1 , ..., xn , t) is the wavefunction, V = V(x1 , ..., xn , t) is the potential, and m represents the constant mass. As usual, the Laplacian has the form n  ∂2 x = . ∂ x2k k=1 The index x of the Laplacian denotes that the derivatives are to be taken with respect to the spatial coordinates xk , k = 1, ..., n. Explicit form of the FPT Let arbitrary functions λ = λ(t), λk0 = λk0 (t) and f0 = f0 (t) be given. We substitute the solution  in the TDSE (1) as follows:   n    λ λk0 2 (x1 , ..., xn , t) = exp −im x + xk + i f0 × 2λ k λ k=1

× (u1 (x1 , t), ..., un (xn , t), s(t)),

(2)

where uk = λ xk + λk0 ,

k = 1, ..., n

(3)

s = s(t),

(4)

where s is an invertible, but otherwise arbitrary function. The transformed TDSE: potential and solution The FPT (2–4) transforms the TDSE (1) into another TDSE in the coordinates (u1 , ..., un , s): i s +

1 u  − Uˆ  = 0. 2m

(5)

ˆ 1 , ..., un , s) reads The transformed potential Uˆ = U(u   n  1 2 ˆ ˆ U=  A uk + B k uk + C k V+ λ(n+1)0

,

(6)

|t=t(s)

k=1

where the functions A = A(t), Bk = Bk (t) and Ck = Ck (t) are given in (33–35), respectively, and λ(n+1)0 = λ(n+1)0 (t) is arbitrary. The solution  of the transformed TDSE (5) reads   n  2 (u1 , ..., un , s) = exp −i m D u k + E k u k + Fk × k=1

× (x1 (u1 , s), ..., xn (un , s), t(s)),

|t=t(s)

(7)

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where the functions D = D(t), Ek = Ek (t) and Fk = Fk (t) are given in (37–39), respectively. Furthermore, xk is the inverse of uk in (3), and s → t(s) is the inverse of s in (4). Reality condition The transformed potential (6) is a real-valued function, if Im(V) +

λ − f0 = 0, 2λ

provided the functions λ, λk0 and the mass m are real-valued. Remark We will now give some comments and list basic properties of the FPT. (a) The explicit expressions for the FPT (2–4) and the transformed potential (6) reduce for n = 1 to the well known expressions in (1 + 1) dimensions [8]. (b) The change of coordinates (3), (4) is invertible, as each xk → uk (xk ) is invertible, and t → s(t) is invertible. It then follows from (2) that the FPT is invertible. (c) The FPT preserves L2 -normalizability, if the mass m and the functions D and Ek in (7) are real-valued. This is especially true, if the functions λ and λk0 are real-valued. (d) Besides mapping TDSEs onto TDSEs, the FPT can also map stationary Schrödinger equations onto TDSEs. This is important, as in general much more is known about solvability of stationary potentials than about solvability of time-dependent potentials. In Section 3 we will perform an example of the latter type. (e) As for the inverse problem of (d), there is a class of time-dependent potentials, which the FPT transforms into stationary potentials. This class will be given in Section 5.

3 Derivation of the FPT We now prove the results given in the previous section. To this end, we first perform a general multiplicative transformation and a change of coordinates. Afterwards, we impose constraints on the transformed equation, such that it takes the form of a TDSE. The multiplicative transformation Consider the TDSE in (n + 1) dimensions: i t +

1 x  − V  = 0. 2m

(8)

We now apply the following multiplicative transformation to the wavefunction  in (8):  = exp( f ) χ,

(9)

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375

where f = f (x1 , ..., xn , t) is a transformation function, the explicit form of which will be determined later. The function χ = χ(x1 , ..., xn , t) is the solution of the equation that we will obtain from the TSDE (8) by substitution of (9). We obtain for the first and second derivative of : xk = (exp( f ) χ)xk  xk xk

= exp( f ) ( fx χ + χx )   = exp( f ) fx2 χ + fxx χ + 2 fx χx + χxx .

Now, on using these, we substitute (9) into (8) and arrive after collecting terms at the following equation for χ:   n   n  1 1  1 i χt + fxk χxk + i ft + fx2k + fxk xk −V χ = 0. (10) x χ + 2m m 2m k=1

k=1

The change of coordinates Next, we perform a general change of coordinates, (x1 , ..., xn , t) → (u1 , ..., un , s),

(11)

taking the original coordinates x1 , ..., xn , t into the new set of coordinates u1 , ..., un , s. In other words, the uk , k = 1, ..., n, will be the new spatial variables, and s will be the new time variable. Thus, the new coordinates are functions depending on the old coordinates, that is, uk = uk (x1 , ..., xn , t) for all k = 1, ..., n and s = s(x1 , ..., xn , t). Furthermore, we require the mapping (x1 , ..., xn , t) → (u1 , ..., un , s) to be invertible. We now want to rewrite (10) in the new coordinates. To this end, let us define (u1 , ..., un , t) = χ(x1 , ..., xn , t) and, for the sake of convenient notation, un+1 = s. The partial derivatives of χ then change as follows: χ xk =

n+1 

u j (u j)xk

j=1

χ xk xk =

n+1 

ul u j (u j)xk (ul )xk + ul (ul )xk xk .

j,l=1

On using the latter forms of the partial derivatives, we perform the change of coordinates on (10) and obtain i

n+1 

uk (uk )t +

k=1

n n+1 1  ul u j (u j)xk (ul )xk + 2m k=1 j,l=1

+ ul (ul )xk xk + 

n n+1 1  fxk u j (u j)xk + m j=1 k=1

   n  1 2 + i ft + fxk + fxk xk −V  = 0. 2m k=1

(12)

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Note that the coefficients of  and its derivatives are still expressed in the old coordinates, which will make it easier to calculate with them later. Setting up the constraints Now we collect terms in (12) with respect to  and its derivatives in order to extract their coefficients. We can then impose constraints on these coefficients, such that (12) takes the form of a TDSE. A careful inspection of (12) yields these coefficients that are displayed in Table 1. Now, if we want (12) to take the form of a TDSE, the coefficients of  and its derivatives must fulfill the following necessary (not sufficient) constraints, the origin of which we will discuss now. ⎛

⎞ n  1 ⎝ (uk )2x j ⎠ = 0 for all k, l = 1, ..., n 2 m j=1

(13)

xl

1 2m

n 

(un+1 )2x j = 0

(14)

j=1

n 1  (ul )x j (uk )x j = 0 for all k, l = 1, ..., n + 1 with k = l. m j=1

(15)

Let us explain where the constraints come from. The coefficient of uk uk (part of the Laplacian of  in the transformed equation), as can be seen from Table 1, must be independent of the spatial variables x1 , ..., xn , because in a TDSE this coefficient always has the constant value 1/2m. Therefore, only if all partial

Table 1 Coefficients from the inspection of (12)

Derivative of  (l, k = 1, ..., n + 1)

u k u k

Coefficient

n 1  (uk )2x j 2m j=1

ul uk (k  = l) u k

n 1  (ul )x j (uk )x j m j=1

i (uk )t +

n n 1  1  (uk )x j x j + fx j (uk )x j 2m m j=1



i ft +

1 2m



n  k=1

j=1

 fx2k + fxk xk

− V.

Form-preserving transformations for the time-dependent Schrödinger equation in (n + 1)

377

derivatives with respect to the spatial variables vanish, as required in (13), the coefficient of uk uk can be a constant. The second constraint (14) says that the coefficient of un+1 un+1 = ss must vanish, since a TDSE does not have second order derivatives with respect to the time variable. Finally, the third constraint (15) guarantees that all terms with mixed derivatives of  vanish, because a TDSE does not have such terms. Solution of the constraints Let us now evaluate the constraints (13–15), beginning with the first constraint. This constraint can be resolved, if the coordinates uk are affine linear functions in the xk , that is, if each uk for k = 1, ..., n + 1 is given by uk =

n 

λkj x j + λk0 ,

(16)

j=1

where λkj = λkj(t) for all j = 0, ..., n. Clearly, if uk is of the form (16), then all its second partial derivatives in the spatial variables vanish, and therefore the constraint (13) is fulfilled. Let us now solve the second constraint (14). If the first constraint holds, then we know already that un+1 = s is of the form (16), that is, un+1 =

n 

λ(n+1) j x j + λ(n+1)0 .

j=1

Inserting this into the second constraint (14) gives  n 2 n 1   λ(n+1)l xl + λ(n+1)0 0= 2 m j=1 l=1

xj

 n 2 n 1   = λ(n+1)l (xl )x j 2 m j=1 l=1

 n 2 n 1   = λ(n+1)l δ jl 2 m j=1 l=1

n 1  2 = λ . 2 m j=1 (n+1) j

The symbol δ jl denotes the kronecker delta. This constraint can be fulfilled, if λ(n+1) j = 0 for all j = 1, ..., n. Consequently, un+1 simplifies to un+1 = λ(n+1)0 .

(17)

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A. Schulze-Halberg

Hence, the new time coordinate s = un+1 depends only on the original time coordinate t. Next, we solve the last constraint (15) by inserting the explicit form (16) of the functions uk : ⎛ ⎞ ⎛ ⎞ n n n  1  ⎝ 0= λlp x p + λl0 ⎠ ⎝ λkp x p + λk0 ⎠ m j=1 p=1 p=1 xj

xj

⎛ ⎞⎛ ⎞ n n n  1  ⎝ = λlp (x p )x j ⎠ ⎝ λkp (x p )x j ⎠ m j=1 p=1 p=1 ⎛ ⎞⎛ ⎞ n n n  1  ⎝ = λlp δ jp ⎠ ⎝ λkp δ jp ⎠ m j=1 p=1 p=1 =

n 1  λlj λ jk m j=1

(18)

In order to solve this constraint, we further simplify the form (16) of the functions uk , k = 1, ..., n: uk = λkk xk + λk0 .

(19)

This is equivalent to saying λkj = λkjδjk for all j, k = 1, ..., n. Now, on taking into account (19), we can evaluate the right hand side of (18): k=l

λlj λkj = λlj λkj δjl δ jk = 0. Thus, if (19) holds, then the third constraint (15) is satisfied. Now suppose the constraints (13–15) are fulfilled, that is, the settings (17) and (19) hold. The transformed TDSE (12) then takes the following form: ⎛ ⎞ n n n   1 1  2 ⎝i (uk )t + λkk uk uk + fx (uk )x j ⎠ uk + i (un+1 )t un+1 + 0= 2m m j=1 j k=1

k=1



 n    1 2 + i ft + f xk + f xk xk − V  2m k=1

 n n  1  2 λkk   fx uk + i λ(n+1)0 un+1 + = λkk uk uk + i λkk xk + i λk0 + 2m m k k=1

k=1

 n    1 2 + i ft + f xk + f xk xk − V  2m 

k=1

(20)

Form-preserving transformations for the time-dependent Schrödinger equation in (n + 1)

379

The last equation does not yet have the form of a TDSE due to the remaining terms ∼ uk . In order to remove them, we have to fulfill n equations, namely, the coefficient of each uk we must equate to zero. For a k with 1  k  n we have 0 = i λkk xk + i λk0 + ⇒ fxk = −i ⇒ f = −i

λkk f xk m

(21)

 m   λkk xk + λk0 λkk m λkk

 λkk 2 xk + λk0 xk + c, 2

where c = c(x1 , ..., xk−1 , xk+1 , ..., xn ), that is, c does not depend on xk , but can depend on all the other original coordinates. Thus, we can incorporate the solutions of (21) for all indices j = k into c. Hence, on setting

 n  m λkk 2  x + λk0 xk + i f0 , f = −i (22) λkk 2 k k=1

with f0 = f0 (t) a constant of integration. Note that the imaginary unit in front of f0 is not necessary, but will simplify calculations later. Now, (20) takes after division by λ(n+1)0 and using un+1 = s the following form: n  1 1 i s + λ2kk uk uk +  × 2 m λ(n+1)0 λ(n+1)0 k=1

   n  1 fx2k + fxk xk − V  = 0. × i ft + 2m 

(23)

k=1

Finally, we convert the latter equation into a TDSE by adjusting the coefficient of uk uk : λ2kk 1 =  2 m λ(n+1)0 2m ⇒ λkk =

 λ(n+1)0 .

(24)

This means that all λkk are the same function for k = 1, ..., n. The transformed potential We simplify the notation by abbreviating λkk = λ, implying λ(n+1)0 = λ2 for k = 1, ..., n. After insertion of (24), our (23) takes its final form: i s +

1 u  − U  = 0, 2m

(25)

380

A. Schulze-Halberg

where the transformed potential U reads ft 1 U = −i 2 − λ 2 m λ2

 n 

 fx2k

+ f xk xk

+

k=1

1 V. λ2

(26)

This can be expressed more explicitly, since we know f from (22). Let us first compute the derivatives of f :  ft = −i

n  m λ k=1

λ

2

x2k

+

λk0





xk + i f0 t



 2  2   n  xk λk0 λ λ λ λ = −i m − + − k02 xk + i f0 λ λ 2 λ λ

(27)

k=1

fxk = −i m fxk xk = −i m

λ λ xk + k0 λ λ

 (28)

λ . λ

(29)

On substituting (27–29) into the potential (26), we obtain the following expression:  

 2 

 n  2 m λ λk0 m λk0 λ 1 m λ U = 2 V+ +m + − x2k + − xk + λ 2λ λ λ λ2 k=1

m + 2

λk0 λ

2

 i λ  + − f0 . 2 λ

(30)

Now we express this potential through the new coordinates uk , given by the relation (19), and un+1 = s. To this end, we first solve the relation (19) for the original coordinates xk : xk =

1 λk0 uk − . λ λ

(31)

After inserting these xk into the potential (30) and simplification, we obtain ˆ 1 , ..., un , t): with U(x1 , ..., xn , t) = U(u 1 Uˆ = 2 λ

 V+

n  k=1

 A

u2k

+ B k uk + C k

, |t=t(s)

(32)

Form-preserving transformations for the time-dependent Schrödinger equation in (n + 1)

381

where the functions A = A(t), Bk = Bk (t) and Ck = Ck (t) are given by A=− Bk =

m λ m (λ )2 + 3 2λ λ4

(33)

m λk0 2 m λ λk0 2 m λ λk0 2 m (λ )2 λk0 − − + λ3 λ4 λ2 λ3

(34)

m λ λ2k0 m (λ )2 λ2k0 m λk0 λk0 2 m λ λk0 λk0 + + − + 2 λ3 λ4 λ2 λ3

2 m λk0 i λ + + − f0 . 2 λ 2 λ

Ck = −

(35)

The transformed solution Next, we determine explicitly the relation between the solution  of the original TDSE (8) and the solution  of the transformed TDSE (25). From (9) and the change of coordinates (11) we know that (u1 , ..., un , s) = exp (− f (x1 (u1 , s), ..., xn (un , s), t(s))) (x1 (u1 , s), ..., xn (un , s), t(s)),

(36)

such that we just need the explicit form of f in the new coordinates. Set fˆ(u1 , ..., s) = f (x1 , ..., xn , t), then we find by inserting (31) into (22):  n   D u 2 + E k u k + Fk , fˆ = −i m k

k=1

where the functions D = D(t), Ek = Ek (t) and Fk = Fk (t) are given by λ 2 λ3 λ λ λk0 Ek = − 3 + k0 λ λ2   2 λ λk0 λ λk0 Fk = − k0 2 + i f0 . 3 2λ λ Therefore, together with (36) we have     n  2 (u1 , ..., un , s) = exp i m D uk + Ek uk + Fk − i f0 D=

k=1

(37) (38) (39)

× |t=t(s)

× (x1 (u1 , s), ..., xn (un , s), t(s)).

4 Application 1: Coulombic Ring-shaped Potential We will now illustrate our results proven in the last section by applying the FPT to the stationary Schrödinger equation with Coulombic ring-shaped potential in three dimensions. The latter Schrödinger equation admits an infinite set of

382

A. Schulze-Halberg

bound state solutions [12], out of which we will generate solutions for a TDSE with a modified, time-dependent Coulombic ring-shaped potential. The stationary Schrödinger equation and its solution The three-dimensional, stationary Coulombic ring-shaped (or Hartmann) potential has the form VH = − 

V1 x21 + x22 + x23

+

x21

V2 , + x22

(40)

where V1 and V2 are positive constants. The corresponding stationary Schrödinger equation 1 x ψ + (Elmn − V H ) ψ = 0, 2m

(41)

has an infinite set of bound state eigenvalues Elmn , parametrized by the usual quantum numbers l, m, n. The explicit form of the Elmn is not of interest here, they can be looked up in [12]. The eigenfunctions for these eigenvalues in spherical coordinates r, ϕ and θ are of the form  

V1 V1 r+i m ϕ sinm (θ)L2l+1 r Pnl(l+1) (cos(θ )), ψ = Clmn rl+1 exp − n−l−1 2 V2 n V2 n (42) where Clmn is a constant depending on the quantum numbers, L denotes a generalized Laguerre polynomial, and P stands for a Gegenbauer (or ultraspherical) polynomial [12] [5]. The TDSE i t +

1 x  − V H  = 0, 2m

(43)

with potential VH as given in (40), then admits the following solution:  = exp (−i Elmn t) ψ,

(44)

where ψ is the solution (42) of the stationary Schrödinger equation in (41). The transformed TDSE We now apply the FPT (2– 4) to the TDSE (43), which transforms (43) into the following TDSE: i s +

1 u  − Uˆ H  = 0, 2m

(45)

where the transformed potential Uˆ H = Uˆ H (u1 , ..., un , s) and the transformed solution  = (u1 , ..., un , s) will be computed now. We start with the

Form-preserving transformations for the time-dependent Schrödinger equation in (n + 1)

383

transformed potential, the general form of which is given in (6). For the sake of simplicity, let us make the following setting in (3) and (35): λk0 = 0, i f0 = 2

k = 1, ..., n 

λ dx λ

i log(λ). 2 This gives simply Bk = Ck = 0 for all k = 1, ..., n in the transformed potential (6). Now, on employing the latter setting, the transformed potential (6) takes the following form: ⎡ ⎤ V V 1 2 Uˆ = ⎣−  + 2 + A (u21 + u22 + u23 )⎦ , (46) 2 u + u 2 2 2 1 2 λ u +u +u =

1

2

3

|t=t(s)

where A = A(t) is expressed through λ as m (λ )2 m λ + . 2 λ5 λ6 The transformed potential reads in spherical coordinates

 V1 V2 2 Uˆ = − + . + A r λ r r2 sin2 (θ) |t=t(s) A=−

Now we construct the explicit solution of the TDSE with potential (46). We need the results (7) and (44), which give the following relation between the solution  in (44) of the original TDSE and the solution  of the transformed TDSE (45):    m (u1 , u2 , u3 , s) = λ 2 exp −i m D (u21 + u22 + u23 ) − iElmn t ×  × ψ(x1 (u1 , s), x2 (u2 , s), x3 (u3 , s))

|t=t(s)

.

(47)

Note that in (7) Ek = Fk = 0 due to λk0 = 0. In order to get an explicit form of the solution (47), we have to express the solution ψ in (42) through the new coordinates uk . To this end, let us first recall the relation between spherical coordinates and cartesian coordinates in three dimensions: r= ϕ=



x21 + x22 + x23

(48)

⎧ ⎨ arccos √ x21

⎫ if x2  0, ⎬

x1 +x22

⎩ 2π − arccos √ x21

x1 +x22

if x2 < 0 ⎭

(49)

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A. Schulze-Halberg

θ=

π x3 . − arctan  2 2 x + x2 1

(50)

2

We now express the right hand sides of the latter equations through the coordinates uk by substituting (31), recall that λk0 = 0. It is immediate to see that (49) and (50) are invariant under this change of coordinates, whereas in (48) we get a factor 1/λ in front of the square root. Thus, the solution (47) has the following form in spherical coordinates: 

  1 m  = Clmn λ 2 exp −i m D r2 − i Elmn t l+1 rl+1 × λ  V1 r+imϕ × × exp − 2 V2 n λ

× sinm (θ) L2l+1 n−l−1

 V1 r Pl(l+1) (cos(θ )) n V2 n λ

. t=t(s)

The latter solution can be written in cartesian coordinates by inserting (48–50).

5 Application 2: Mapping to a Stationary Schrödinger Equation In the last section we have seen how solvable TDSEs can be generated out of solvable stationary Schrödinger equations. Now we consider the opposite question, that is, we want to find out which TDSEs allow a reduction to stationary Schrödinger equations. More precisely, we are interested in a condition on the potential of the TDSE, such that the FPT transforms the TDSE into a TDSE with a time-independent potential, thus permitting reduction to a stationary Schrödinger equation. In the (1 + 1)-dimensional case, a class of potentials is known, for which the TDSE can be mapped onto a stationary Schrödinger equation [3]. In the following, we give a straightforward generalization of the result in [3] to the (n + 1)-dimensional case. In particular, we prove the following criterion. The criterion Consider the TDSE in (n + 1) dimensions, i t +

1 x  − V  = 0, 2m

(51)

where the potential V is given by V=

n 

α

k=1

x2k

4 + βk x + γk + exp m



 a dt

 Gk .

(52)

Form-preserving transformations for the time-dependent Schrödinger equation in (n + 1)

385

Here the following notation has been used: 2 a2 m 2 a bk βk = b k − m α = a −

γk = −

b 2k 2m ⎡

2 Gk = G ⎣exp m



 a dt

⎛



1 xk + m

2 exp ⎝ m

t

⎞

⎤

a dt ⎠ b k dt⎦ .

(53)

The functions a = a(t), b k = b k (t), and G are arbitrary, note that a is not allowed to depend on k. Then the FPT (2–4) with the following parameters

2 λ = exp m

λk0

1 = m



a dt ⎛

2 exp ⎝ m

 s=





exp

4 m

t

(54) ⎞ a dt ⎠ b k dt

(55)



 a dt

dt,

(56)

transforms the TDSE (51) with potential (52) into the TDSE with stationary potential  n   1 u  − i s + Gk  = 0. 2m k=1

Thus, on setting (u1 , ..., un , t) = exp(−i E t) ψ(u1 , ..., un ), the function ψ solves the stationary Schrödinger equation   n  1 u ψ + E − Gk ψ = 0. 2m k=1

Thus, if the potential of the TDSE can be written in the form (52), then it can be mapped onto a stationary potential. Proof of the criterion We prove this criterion by simply computing the transformed potential generated by the FPT. To this end, we shall use the form (30) of the transformed potential, because this will facilitate calculations. We employ the settings (54 – 56) and insert them into the transformed potential

386

A. Schulze-Halberg

(30). Due to the length of the expressions involved, let us perform the calculation in several parts. We find

 2 λ m λ 2 a2 +m − a − = 2λ λ m 2 m λ λk0 m λk0 2 a bk + − b k = 2 λ λ m

2 b2 m λk0 i λ a + − f0 = k + i − f0 . 2 λ 2 λ 2m m −

We remove the imaginary term in the last row by setting  i f0 = a dt. m On comparing the last results with the functions α, βk and γk in (52), we find

 2 λ m λ +m − = −α (57) 2λ λ

−

m 2

2 m λ λk0 m λk0 + = −βk λ λ2

λk0 λ

2 +

i λ − f0 = −γk . 2 λ

(58)

(59)

Now we insert the original potential (52) into the transformed potential (30). On taking into account (57–59) and the definition of λ in (54), we get  n 

  4 1  U= 2 exp a dt Gk λ m k=1

=

n 

Gk .

(60)

k=1

If we now write the latter potential in the new coordinates (u1 , ..., un , s), we infer from the definition of uk in (3) and from (53) that ⎡ ⎛ ⎤ ⎞

   t 2 1 2 Gk = G ⎣exp a dt xk + exp ⎝ a dt ⎠ b k dt⎦ = G(uk ). m m m Therefore, Gk depends only on the spatial variables uk , but not on s. Hence, the potential (60), written in the coordinates (u1 , ..., un , s), becomes a stationary potential and the proof of our criterion is complete.

Form-preserving transformations for the time-dependent Schrödinger equation in (n + 1)

387

6 Application 3: Effective Mass TDSE An effective mass TDSE is a generalization of the TDSE for a positiondependent (effective) mass. Effective masses are used to model transport phenomena in semiconductors, for details the reader may consult [6] or [9] and references therein. The effective mass TDSE has the following form: 1 1 x  − (61) (∇x M) (∇x ) − V  = 0, 2M 2 M2 where the mass is allowed to depend on all variables, that is, M = M (x1 , ..., xn , t). Furthermore, the symbol ∇x denotes the gradient operator with respect to the variables xk , k = 1, ..., n. Obviously, if the mass is constant or depends only on t, (61) reduces to the conventional TDSE. It is known [9] that in (1 + 1) dimensions there exists an invertible FPT between the TDSE and the effective mass TDSE (61). The purpose of this section is to show that in spatial dimensions higher than one such an FPT cannot exist. To this end, consider the conventional TDSE (8) and apply the FPT as done in Section 3. We then arrive at the transformed (12). The difference between an FPT to a conventional TDSE and to an effective mass TDSE lies in the constraints on the coefficients (see Table 1) of the transformed equation. For an effective mass TDSE, the coefficient of the second spatial derivatives is not the constant 1/(2m), but the function 1/2M – see (61) – where M can depend on all the coordinates. Thus, the first constraint (13) must read i t +

n 1  1 , for all k = 1, ..., n (uk )2x j = 2 m j=1 2M

(62)

where M = M(x1 , ..., xn , t) is the effective mass in the transformed equation. Note that we cannot take the same solution (16) for the uk that we used for transforming into a conventional TDSE; if the uk were affine linear functions, then the left hand side of (62) would be independent of the spatial variables, and hence, M would not be an effective mass. Thus, the uk must have a more complicated dependency on the spatial variables x j. Keeping this in mind, we continue with the third constraint (15) that maintains its form, because in effective mass TDSEs there are no mixed spatial derivatives: n 1  (ul )x j (uk )x j = 0 for all k, l = 1, ..., n + 1 with k = l. m j=1

(63)

This time the uk are not affine linear, and therefore the constraint cannot be fulfilled as before in (19). Geometrically, the constraint (63) can be seen as a scalar product between gradients: 1 (64) (∇x ul ) (∇x uk ) = 0 for all k, l = 1, ..., n + 1 with k = l, m and this constraint says that the gradient vectors are pairwise orthogonal. Next, after a short reflection it becomes clear that in n spatial dimensions the constraint (64) consists of n(n+1) equations, counting all possible combinations of 2

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k and l. This means that n(n+1) vectors are supposed to be pairwise orthogonal 2 in n spatial dimensions. Since only n vectors can be pairwise orthogonal, (64) contains n(n + 1) n(n − 1) −n= (65) 2 2 constraints that cannot be fulfilled. We see that only for n = 1, that is, in one spatial dimension, the constraint (64) can be satisfied. Consequently, there is no FPT between a conventional TDSE and an effective mass TDSE in (n + 1) dimensions, if n  2.

7 Concluding Remarks In this note we have generalized the FPT for the TDSE to the case of (n + 1) dimensions, which turned out to be equivalent to n one-dimensional FPTs with respect to each coordinate. This is due to the particular choice of the new variables (16). In fact, it is possible that there are more complicated solutions of (13), leading to more general FPTs. However, the goal of this paper was not to track down all possible solutions of the constraints (13–15), but to present an extension of the one-dimensional FPTs and to show which properties of the one-dimensional FPT persist in higher dimensions.

References 1. Ciftja, O.: A simple derivation of the exact wavefuction of a harmonic oscillator with timedependent mass and frequency. J. Phys. A 32, 6385–6389 (1999) 2. Feng, M.: Complete solution of the Schrödinger equation for the time-dependent linear potential. Phys. Rev. A 64 034101, p. 3 (2001) 3. Finkel, F., Gonzalez-Lopez, A., Kamran, N., Rodriguez, M.A.: On form-preserving transformations for the time-dependent Schrodinger equation. J. Math. Phys. 40, 3268–3274 (1999) 4. Gonul, B., Kocak, M.: Systematic search of exactly solvable non-central potentials. Modern Phys. Lett. A 20(5), 355–361 (2005) 5. Gradstein, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products. Academic, San Diego (1980) 6. Landsberg, G.T.: Solid State Theory: Methods and Applications. Wiley-Interscience, London (1969) 7. Levai, G., Znojil, M.: Systematic search for PT-symmetric potentials with real energy. J. Phys. A 33(40), 7165–7180 (2000) 8. Ray, J.R.: Exact solutions to the time-dependent Schrödinger equation. Phys. Rev. A (3) 26(2), 729–733 (1982) 9. Schulze-Halberg, A.: Quantum systems with effective and time-dependent masses: formpreserving transformations and reality conditions. Cent. Eur. J. Phys. 3(4), 591–609 (2005) 10. Um, C.I., Yeon, K.H., George, T.F.: The quantum damped harmonic oscillator. Phys. Rep. 362, 63–192 (2002) 11. Ushveridze, A.G.: Quasi-Exactly Solvable Models in Quantum Mechanics. Institute of Physics, Bristol (1994) 12. Yasuk, F., Berkdemir, C., Berkdemir, A.: Exact solutions of the Schrödinger equation with non-central potential by the Nikiforov–Uvarov method. J. Phys. A, Math. Gen. 38(29), 6579–6586 (2005)