Mathematical Physics, Analysis and Geometry - Volume 3

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Mathematical Physics, Analysis and Geometry 3: 1–31, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

1

Asymptotic Distribution of Eigenvalues of Weakly Dilute Wishart Matrices A. KHORUNZHY1 and G. J. RODGERS2

1 Institute for Low Temperature Physics, Kharkov 310164, Ukraine.

e-mail: [email protected] 2 Department of Mathematics and Statistics, Brunel University, Uxbridge, Middlesex UB8 3PH, U.K. e-mail: [email protected] (Received: 30 September 1998; in final form: 18 October 1999) Abstract. We study the eigenvalue distribution of large random matrices that are randomly diluted. We consider two random matrix ensembles that in the pure (nondilute) case have a limiting eigenvalue distribution with a singular component at the origin. These include the Wishart random matrix ensemble and Gaussian random matrices with correlated entries. Our results show that the singularity in the eigenvalue distribution is rather unstable under dilution and that even weak dilution destroys it. Mathematics Subject Classifications (2000): 15A52, 82B44. Key words: matrices, random, dilute, Wishart.

1. Introduction Random matrices of large dimensions play a central role in a number of theoretical physics applications, such as statistical nuclear physics, solid state physics, statistical mechanics, including neural network theory and quantum field theory (see, e.g., the monographs and reviews [8 – 10, 13, 24, 27]). In this work, most interest is attached to the various ensembles of random matrices whose entries are all of the same order of magnitude. This corresponds to the situation when the elements of a system all strongly interact with one another. In the last decade, however, systems in which some of the links between the different elements are broken have been studied in a variety of applications. This effect is particularly important in neural network theory where the total number of neurons is several orders of magnitude greater than the average number of connections per neuron [2, 5, 13]. Such matrices known as dilute (or sparse) ones are also important in other applications, such as the theory of random graphs and linear programming [21].

2 1.1.

A. KHORUNZHY AND G. J. RODGERS

STRONG DILUTION AND SEMICIRCLE LAW

In papers [17, 19] we have studied the limiting eigenvalue distribution of large random matrices that are strongly diluted. These are determined as the N-dimensional matrices that have, on average, pN nonzero elements per row and 1 pN N as N → ∞. We proved that under natural conditions the limiting eigenvalue distribution of strongly dilute random matrices exists and coincides with the Wigner’s famous semicircle law [32]. The semicircle law is also valid when the entries of the dilute random matrix are statistically dependent random variables [16, 20]. This case is of special interest in applications (see, for instance, [1, 7, 10, 13]). In the pure (nondilute) regime these matrices have singularities in the eigenvalue distribution. The strong dilution removes this singularity because the density of the semicircle distribution is bounded. It should be noted that the Wigner’s semicircle distribution is typical for large random matrices with jointly independent entries. Therefore we have conjectured that the semicircle law arises in the ensembles of [16, 20] because the strong dilution eliminates the statistical dependence between random matrix entries. The same reasoning can explain the disappearance of the singularity in the eigenvalue distribution. However, the last conjecture is not true. 1.2.

WEAKLY DILUTE WISHART MATRICES

To study the transition to the semicircle law under dilution, we pass to the case of weakly dilute random matrices. This means that we are now interested in the asymptotic regime when pN = qN, q > 0 as N → ∞. In this case the statistical dependence between random matrix entries persists, provided it exists in the pure (nondilute) ensemble. We consider two random matrix ensembles with different types of statistical dependence between the entries. These are the Wishart random matrices HN and Gaussian random matrices AN with correlated entries. The first ensemble is widely known in multivariate statistical analysis (see, e.q., [1]). Recent applications of these matrices are related with the the theory of disordered spin systems of statistical mechanics [14, 26, 24] and learning algorithms of memory models of neural network theory [2, 13, 14]. The entries {HN (x, y)} are statistically dependent (but uncorrelated) random variables. The degree of the dependence between HN (x, y) and HN (s, t) does not relate to the ‘distance’ |x − s| + |y − t|. In contrast, correlations between matrix elements AN (x, y) and AN (s, t) in the second ensemble we consider decay when the distance between them increases. Due to this property, {AN } can be regarded as the ensemble intermediate betweem random matrix models with strongly correlated entries (see, e.g., [7, 10]) and random matrices with independent entries. 1.3.

MAIN RESULTS AND STRUCTURE OF ARTICLE

We study the limit of the normalized eigenvalue counting function of weakly dilute real symmetric matrices HN and AN as N → ∞. We derive explicit equations for

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

3

the Stieltjes transform of the limiting distribution functions and determine recurrent equalities for their moments. Basing on these relations, we study the properties of the limiting eigenvalue distributions. We show that both these distributions are different from the semicircle law. We prove that, nevertheless, the singularity disappears from the limiting eigenvalue density and that this happens for arbitrary values of q < 1. Thus, our principal conclusion is that the singularity of the eigenvalue distribution is rather unstable under dilution and is destroyed even when this dilution is weak. To complete this introductory section, let us note that our results can be regarded as generalizations of the statements proved for strongly dilute random matrices in [19] and [20]. In this paper we use the technique developed in [20]. However, the case of weak dilution studied here is more complicated and requires more accurate analysis. The paper is organised as follows. In Section 2 we present our main results for the Gaussian random matrices AN with correlated entries. In Section 3 we consider the weak dilution of the Wishart matrices HN . We prove the existence of the limiting eigenvalue distributions σ (i), i = 1, 2 in the limit N → ∞ (Theorems 2.1 and 3.1). At the second part of each of these sections we formulate Theorems 2.2 and 3.2 concerning the properties of the respective σ (i) . In Sections 4 and 5 we derive the main equations that determine the eigenvalue distributions of AN and HN , respectively, as N → ∞. Section 6 contains the proofs of Theorems 2.2 and 3.2. We also present there the recurrent relations for the moments of σ (i) and make conclusions about the support of the measure dσ (i) . In Section 7 we give a summary of our results.

2. Gaussian Random Matrices with Correlated Entries Let us consider N × N symmetric random matrices 1 AN (x, y) = √ a(x, y), N

x, y = 1, . . . , N,

(2.1)

where random variables {a(x, y), x 6 y, x, y ∈ N} have a joint Gaussian distribution. We assume that {a(x, y)} satisfy the following conditions: Ea(x, y) = 0, Ea(x, y)a(s, t) = V (x − s)V (y − t) + V (x − t)V (y − s),

(2.2a) (2.2b)

where the sign E represents the mathematical expectation with respect to the measure generated by the a(x, y)’s and V (x) is a nonrandom function such that V (−x) = V (x) and V is nonnegatively defined. Then the right-hand side satisfies conditions for the covariance of random variables (see Lemma 4.5 at the end of Sec-

4

A. KHORUNZHY AND G. J. RODGERS

tion 4). The eigenvalue distribution of the random matrix ensemble (2.1), (2.2), where V (x) satisfies the condition X |V (x)| ≡ Vm < ∞, (2.3) x

was studied in [18]. This case is known as weakly correlated random variables. Indeed, condition (2.3) implies decay of the correlations between random matrix entries AN (x, y) and AN (s, t) that are situated far enough from each other in the matrix. 2.1.

ENSEMBLE AND MAIN EQUATIONS

In this section we consider the ensemble of real symmetric random matrices 1 (q) AN = √ a(x, y)πxy , N

x, y = 1, . . . , N,

(2.4)

where a(x, y) are the same as in (2.1) and the random variables {πxy , x 6 y} are both independent between themselves and independent from {a(x, y)}. We assume that πyx = πxy and the random variables have the common probability distribution 1 1, with probability q, (2.5) πxy = √ q 0, with probability 1 − q. Our main result concerns the normalized eigenvalue counting function (NCF) of (q) AN given by the formula (q) 6 λ N −1 , (2.6) σ λ; AN = # λ(N) j (q)

6 · · · 6 λ(N) where λ(N) 1 N are eigenvalues of AN . THEOREM 2.1. Assume that (2.3) holds. Then (q)

(i) given q ∈ (0, 1) the NCF σ (λ; AN ) weakly converges in probability as N → ∞ to a nonrandom function σq (λ); R (ii) the Stieltjes transform fq (z) = (λ − z)−1 dσq (λ), Imz 6= 0 can be found from the the system of equations Z 1 g˜q (p; z) dp, (2.7a) fq (z) = 0

g˜q (p; z) =

− z − (1 − q)v fq (z) − q V˜ (p)

Z

2

0

where V˜ (p) =

X x∈Z

V (x)E{2π ipx},

1

V˜ (r)g˜q (r; z) dr

−1 , (2.7b)

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

and

Z v ≡ V (0) =

1

5

V˜ (p) dp;

0

(iii) system (2.7) is uniquely solvable in the class 0 of functions g(p; z), p ∈ (0, 1), z ∈ C± analytical in this region and such that Im g(p; z)Im z > 0, z ∈ C± .

(2.8)

Remarks. (1) Here and below we mean by the weak convergence of nonnegative nondecreasing functions σ (λ; AN ) the weak convergence of the corresponding measures Z ∞ Z ∞ ϕ(λ) dσ (λ; AN ) = ϕ(λ) dσ (λ), ϕ ∈ C0∞ (R). lim N→∞

−∞

−∞

Generally the convergence of integrals can be regarded as convergence in average, in probability or with probability 1. (2) Each function gˆR∈ 0 determines a nonnegative nondecreasing function σˆ (λ) such that [11] g(z) ˆ = (λ − z)−1 dσˆ (λ) and Z 1 b Img(µ ˆ + iη) dµ. (2.9) σˆ (a) − σˆ (b) = lim η↓0 π a Relation (2.9), known as the inversion formula for the Stieltjes transform, is valid for all a, b such that σˆ is continuous at these points. Theorem 2.1 is proved in Section 4. Basing on (2.7), one can study the properties of σq (λ). 2.2.

LIMITING EIGENVALUE DISTRIBUTION

To discuss the consequences of Theorem 2.1, let us note first that relations (2.7) considered with q = 0 can be reduced to the equation f0 (z) =

1 . −z − v 2 f0 (z)

(2.10)

This equation is uniquely solvable and determines the Wigner semicircle distribution σ0 (λ) [32] with the density √ 1 4v 2 − λ2 , if |λ| 6 2v, 0 %0 (λ) ≡ σ0 (λ) = (2.11) 2π v 0, otherwise. This observation shows that Theorem 2.1 generalizes the results of paper [20], where the eigenvalue distribution of the ensemble (2.4) has been studied under con(N) that take values N 1/2 p −1/2 dition that random variables πxy are replaced by πˆ xy with probability p/N and 0 with probability 1 − p/N. Then the limit p, N → ∞

6

A. KHORUNZHY AND G. J. RODGERS

considered in [20] corresponds to subsequent limiting transitions N → ∞ and q → 0. Another limiting transition q → 1 in (2.7) leads to equations Z 1 g1 (p; z) dp, f1 (z) = 0

g1 (p; z) =

− z − V˜ (p)

Z

V˜ (r)g1 (r; z) dr

−1 .

This system has been derived in [18] the the limit f1 (z) of the Stieltjes transforms of the NCF σ (λ; AN ) of the ensemble determined by (2.1), (2.2), and (2.3) (see also [3, 4] for others and more general ensembles). It should be noted that if Z 1 dp = ∞, (2.12) 0 V˜ (p) then corresponding to f1 (z), measure σ1 (dλ) has an atom at the origin. Indeed, one can easily derive from (2.7) and (2.9) that if (2.12) holds, then lim Imf1 (iε) = ∞ ε↓0

(see also [4]). Relation (2.12) is known in stochastic analysis as the interpolation condition for the infinite sequence of Gaussian random variables {γ (x), x ∈ Z} with zero average Eγ (x) = 0 and covariance Eγ (x)γ (y) = V (x − y) [22]. If (2.12) holds, then the sequence {γ (x)} can be regarded as insufficiently random. Apparently, random dilution makes the sequence of such random variables ‘more disordered’. This observation can be regarded as a heuristic explanation of the following proposition. THEOREM 2.2. If q < 1, then the density %q (λ) = σq0 (λ) exists and is bounded √ everywhere by 1/(π v 1 − q). Theorem 2.2 is proved in Section R 6. There we also derive and analyse recurrent relations for the moments Lk = λk dσq (λ). Basing on these relations, we study the support of the measure dσq (λ). 3. Weak Dilution of Wishart Matrices In this section we study the eigenvalue distribution of symmetric matrices (q) HN (x, y)

N 1 X = ξµ (x)ξµ (y)πxy , N µ=1

x, y = 1, . . . , N,

(3.1)

where {ξµ (x), x, µ ∈ N} are independent random variables having joint Gaussian distribution. We assume that these random variables satisfy conditions Eξµ (x) = 0,

Eξµ (x)ξν (y) = δxy δµν ,

(3.2)

7

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

where δxy is the Kronecker δ-symbol; 1, if x = y, δxy = 0, if x 6= y.

We assume also that πxy = πyx and {πxy , x 6 y are i.i.d. random variables (independent of {Hµ}) that have probability distribution (2.5). Thus, (3.1) represents the weak dilution of random matrices m 1 X ξµ (x)ξµ (y), HN,m (x, y) = N µ=1

x, y = 1, . . . , N

(3.3)

known since 30s in the multivariate statistical analysis as the Wishart matrices [1]. Being at present of considerable importance in this field, the ensemble (3.3) is extensively studied in the statistical mechanics of the disordered spin systems (see, e.g., [6, 26, 31] for rigorous results). Another important application of (3.3) is related with the neural network theory, where HN are used as the interation matrix of learning algorithms modelling auto-associative memory. In this approach, N-dimensional vectors ξµ are regarded as the patterns to be memorised by the system [13]. Dilution versions of (3.3) are important in this field of applications as the models that can be tuned to give more precise correspondence with real systems (see, e.g., [2]). These models are mostly studied in the regime of strong dilution [5, 30]. The following statement concerns the normalized eigenvalue counting function (2.6) of of weakly dilute random matrices (3.1), (3.2). (q)

THEOREM 3.1. For each fixed q ∈ (0, 1) the NCF σ (λ; HN ) converges in the limit N, m → ∞, m/N → c > 0 to a nonrandom function σq,c . The Stieltjes transform fq,c (z) of σq,c (λ) satisfies equation −1 √ cu2 q 4 fq,c (z) = − z − cu (1 − q)fq,c + . (3.4) √ 1 + u2 qfq,c (z) This equation is uniquely solvable in the class of functions 0 determined in Theorem 2.1 and satisfying (2.8). We prove Theorem 3.1 in Section 5. Regarding (3.4), one can easily observe that, in complete analogy with (2.7), this equation determines a ‘mixture’ of two equations: the one for the semicircle distribution with q = 0 (cf. with (2.10)) f0,c (z) =

1 −z − cu4 f0,c (z)

and the equation (q = 1) −1 cu2 f1,c (z) = − z + 1 + u2 f1,c (z)

(3.5)

8

A. KHORUNZHY AND G. J. RODGERS

derived in [23] for the Stieltjes transform of σ (λ; HN,m ) in the limit N, m → ∞, m/N → c > 0. Corresponding to (3.5) eigenvalue distribution has the density given by the formula [23] q 2 1 dσ1,c (λ) = [1 − c]+ δ(λ) + 4cu2 − λ − (c + 1)u2 , (3.6) 2 dλ 2π u λ where [x]+ = max(0, x) and δ(x) is the Dirac delta function. Let us stress that if c < 1, then the density of σ1,c (λ) has the singular component at the origin. The following statement shows that this singularity disappears in the weak dilution regime. THEOREM 3.2. If q < 1 then√the density of σq,c (λ) determined by fq,c (z) (3.4) is bounded from above by 1/u2 c(1 − q) for all c > 0. We prove Theorem 3.2 in Section 6. In this section we also derive recurrent R relations for the moments Lk = λk dσq,c (λ). 4. Proof of Theorem 2.1 In this section we use the resolvent method developed in a series of papers (see, e.g., [18]) and improved in [15]. This method is based on the derivation of relations for the moments of the normalized trace of the resolvent of random matrix AN 1 1 X fN (z) ≡ Tr GN (z) = GN (x, x; z), GN (z) = (AN − z)−1 . (4.1) N N x=1 The important and in certain sense characteristic property of random matrices is that EfN (z) converges as N → ∞ to the variable f (z) and one can derive closed equations for it. The variance of fN (z) vanishes as N → ∞ that means that the normalized trace (4.1) is the self-averaging random variable. The case of weakly diluted random matrices AN,q (2.7) is more complicated. The average EGN (x, y) is expressed in terms of the generalized ‘trace’ N 1 X TN (x, y) = πxr G(r, s)Vrs πsy , N r,s=1

(4.2)

where we denoted G(x, y) ≡ GN (z) = (AN,q − z)−1 (q)

and Vxy ≡ V (x − y). Relations for the limit of EGN (x, y) involve the limit of the average ETN (x, y). Our main observation is that this pseudo-trace TN (x, y) is also the selfaveraging variable as N → ∞. We obtain expression for t (x, y) = limN→∞ ETN (x, y).

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

9

In contrast with the strong dilution, matrix t (x, y) is nonzero and its entries take two different values depending on whether x = y or not. These two limits t1 and t2 involve explicitly parameter q and t1 − t2 vanishes for q → 1. This determines the difference between equations that we obtain for weakly dilute random matrices and those derived for the corresponding pure (nondilute) ensemble. To make the derivation self-consistent, we recall the basic elements of the method developed in [15, 18]. Given two symmetric (or Hermitian) operators H and Hˆ acting in the same ˆ = (Hˆ − z)−1 space, the resolvent identity holds for G = (H − z)−1 and G ˆ = −G(H − Hˆ )G. ˆ G−G

(4.3)

This identity leads to two important observations; that the dependence of the resolvent on the random matrix can found explicitly and that this dependence is expressed in terms of the resolvent (see formulas (4.7) and (4.9)). These two properties make the resolvent approach a fairly powerful tool in the spectral theory of random matrices. We prove Theorem 2.1 by showing the convergence of the trace fN,q (z) = (q) −1 N Tr GN (z); lim EhfN,q (z)i = fq,V˜ (z),

(4.4)

lim E |fN,q (z) − EhfN,q (z)i|2

(4.5)

N→∞

and N→∞

for all z ∈ 3q = z ∈ C, |Imz| > 2Vm q −1 + 1 ,

(4.6)

where we denoted by h·i the mathematical expectation with respect to the measure generated by random variables {πxy }. On this way, we derive equation for fq,V˜ (z) that determines the limiting eigenvalue distribution. Taking into account that the normalized trace of the resolvent (4.1) is the Stieltjes transform of the NCF, we conclude that relations (4.4) and (4.5) imply the weak convergence in probability of σ (λ; AN,q ), in the limit N → ∞, to a nonrandom function σq (λ; V˜ ). This can be proved by the usual arguments of the theory of Herglotz functions. The reasonong is based on weak compactness of the family σ (λ; AN,q ) and the Helly theorem [11] (see, e.g., [18] for more details). Thus, relations (4.4) and (4.5) prove items (i) and (ii) of Theorem 2.1. Item (iii) is proved in Lemma 4.4 (see the end of this section). We split the remaining part of this section into three subsections. In the first one we derive main relations that lead to the equation for fq,V˜ (z). In the second subsection we derive relations leading to the proof of (4.5). The third subsection contains the proofs of the auxiliary facts and estimates.

10 4.1.

A. KHORUNZHY AND G. J. RODGERS

DERIVATION OF MAIN RELATIONS

Let us consider identity (4.3) with H = AN,q and Hˆ = 0; X −1/2 EG(x, s)a(s, y)π(s, y) , EhG(x, y)i = ζ δxy − ζ N

(4.7)

s

where ζ = −z−1 . In (4.7) and below, we omit the varable z and subscripts q, N and do not indicate limits of summations, if no confusion can arise. To compute the average EG(x, s)a(s, y), we use the following elementary facts (see also [18]). It is related to the Gaussian random vector γ = (γ1 , . . . , γk ) with zero average: Eγj F (γ1 , . . . , γk ) =

k X l=1

∂F , Eγj γl E ∂γl

(4.8)

where F is a nonrandom function such that all integrals in (4.8) exist. This formula can be proved by using the integration by parts technique. We will also use the formula that is a direct consequence of identity (4.3); ∂G(x, s) 1 = − √ G(x, p)G(r, s)πpr . ∂a(p, r) N

(4.9)

Now we can write that 1 X EG(x, s)a(s, y) = − √ Vps Vry + Vpy Vrs EG(x, p)G(r, s)πpr . N x,p Substituting this relation into (4.7), we obtain equality X

EhG(x, y)i = ζ δxy + ζ EhG(x, p)T (p, y)iVpy + ζ E ψN(1) (x, y) , (4.10) p

where T is determined in (4.2) and we denoted ψN(1) (x, y) =

1 X G(x, p)πpr G(r, s)Vps πsy Vry . N p,r,s

(4.11)

In the last part of this section we prove that ψN(1) vanishes in the limit N → ∞ (see Lemma 4.1). Turning back to (4.7), we can write for the average EhG(x, y)i ≡ g(x, y) the following relation X

g(x, p)t (x, y)Vpy + ζ E ψN(1) (x, y) + g(x, y) = ζ δxy + ζ p

+

ζ ψN(2)(x, y),

(4.12)

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

11

where t (x, y) ≡ EhT (x, y)i and X {EhG(x, y)T (x, y)i − g(x, y)t (x, y)}Vpy . ψN(2) (x, y) = p

Given a random variable γ with finite mathematical expectation, let us introduce the centered random variable γ 0 ≡ γ − Eγ . We will also use denotation [γ ]0 for more complex expressions. In what follows (see Subsection 4.2), we prove that

(4.13) lim sup E |G0 (x, y)|2 = 0 for all z ∈ 3q . N→∞ x,y

Let us note that the last inequality of (4.11) implies that supx,y |T (x, y)| 6 Vm q −1 η−1 . This estimate together with (4.13) leads to relation lim sup ψN(2)(x, y) = 0. (4.14) N→∞ x,y

It should be noted also that (4.13) implies (4.5). To derive the final equation (2.7b) from (4.12), it remains to compute the limit of the average t (x, y) = EhT (x, y)i. Basing on (4.2), one can write that EhT (x, y)i =

1 X hπxr πsy ig(r, s)Vrs + ψN(3) (x, y), N r,s

(4.15)

where the term ψN(3) (x, y) =

1 X

E πxr G0 (r, s)πsy Vrs N r,s

vanishes due to the self-averaging property (4.13). Examining the first term on the right-hand side of (4.15), we observe that it depends only on whether x = y: 1 X t , if x = y, hπxr πsy ig(r, s)Vrs = 1 t N r,s 2 , if x 6= y, where t1 = (1 − q)gV ¯ 0 + q g¯V ,

t2 = q g¯V ,

and N 1 X g¯ = g(x, x), N x=1

N 1 X g¯V = g(x, y)Vxy . N x,y=1

12

A. KHORUNZHY AND G. J. RODGERS

Let us consider the equation g(x, ˆ y) = ζ δxy

N X ˆ ˆ ˆ + ζ t1 − t2 g(x, ˆ y)V (x, y) + ζ t2 g(x, ˆ p)Vpy ,

(4.16)

p=1

where x, y = 1, . . . , N and tˆi are determined by the same formulas as ti with g(x, y) replaced by g(x, ˆ y). It is not hard to prove that (4.16) has a unique solution for z ∈ 3q (4.6) (see Lemma 4.3 at the end of this section). Also it is easy to see that N 1 X g(x, x) − g(x, ˆ x) = 0. N→∞ N x=1

lim

(4.17)

Some elementary calculations based on the finite-difference form of Vxy = V (x − y) show that Equation (4.16) leads in the limit N → ∞ to Equation (2.7b) and therefore N 1 X g(x, ˆ x) = fq,V˜ (z), lim N→∞ N x=1

where fq,V˜ (z) is given by (2.7a). Relation (4.4) is proved. 4.2.

SELFAVERAGING PROPERTY

Let us prove relation (4.13) that obviously implies (4.5). We consider, at the same time with G(x, y) ≡ G(x, y; z) the resolvent G0 (x, y) ≡ G(x, y; z0 ), and study the average

(4.18) SN (x, y) = E G0 (x, y)G0 (x, y) . Loosely speaking, the main idea is to derive relation of the form SN (x, y) = BS(N) (x, y) + 8N (x, y) where B is certain expression involving SN and term 8N vanishes as N → ∞. The crucial observation used in this approach is that BS(N) can be estimated by SN itself multiplied by coefficients depending on η = |Imz|−1 (see, e.g., [15]). This lead to relations (4.13) provided η is large enough. 4.2.1. Selfaveraging of S(x, y) Applying (4.3) to the last factor of (4.18), we obtain equality X

S(x, y) = −ζ 0 N −1/2 E G0 (x, y)G0 (x, t)a(t, y)πty , t 0

0

where ζ = −1/z . Using (4.8) and (4.9), we derive relation X

E G0 (x, y)G0 (x, p)T 0 (p, y) Vpy + 0N (x, y). S(x, y) = ζ 0 p

(4.19)

13

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

Here we denoted

0N (x, y) = ζ 0 E G0 (x, y)φN0 (x, y) + 1 X G(x, p)πpr G(r, y)G0 (x, t)πt,y (Vpt Vry + Vpy Vrt ), + N p,r,t

where φN0 (x, y) is given by (4.11) with G replaced by G0 . Using identity

E ξ10 ξ2 ξ3 = E ξ10 ξ2 Ehξ3 i + E ξ10 ξ2 ξ30 , we can rewrite (4.19) in the form X

E [G(x, y)]0 T 0 (p, y) E G0 (x, p) Vpy + S(x, y) = ζ 0 p

+ ζ0

X

E [G(x, y)]0 [G0 (x, y)]0 T 0 (p, y) Vpy + ζ 0 0N (x, y). (4.20) p

Repeating the reasoning used to estimate (4.11) (see Lemma 4.1), one can easily show that for z, z0 ∈ 3q (4.6) |0N | ≡ sup |0N (x, y)| = O N −1/2 + N −1/2 |SN | , x,y

where |SN | ≡ sup |SN (x, y)|. x,y

Substituting this inequality into (4.20), we obtain that for z0 = z¯ |SN |2 6 |SN ||UN |η−2 Vm + q −1 η−2 Vm2 |SN |2 + η−1 |0N |,

(4.21)

where

1/2 |UN | ≡ sup E |T 0 (x, y)|2 . x,y

Now it is clear that (4.21) implies (4.13) provided the estimate |UN | = o(1) as N → ∞

(4.22)

is true. 4.2.2. Selfaveraging of T (x, y) To prove (4.23), we treat the average

0 1 X 0 0 0 UN (x, y) ≡ E T (x, y)T (x, y) = E T (x, y) πxr G (r, s)Vrs πsy N r,s

14

A. KHORUNZHY AND G. J. RODGERS

by the same procedure as is used to study SN (x, y). We apply to G0 (r, s) the resolvent identity (4.3) and use (4.8) and (4.9) to compute the mathematical expectations. As a result, we obtain relations

UN (x, y) = ζ 0 E T 0 (x, y)ρ(x, y) V0 + X 0 −1 0 0 0 + ζ N E T (x, y) πxr Vrs G (r, p)T (p, s)Vps πs,y + p,r,s

+

ζ 0 θN(1) (x, y),

(4.23)

where we have denoted 1 X ρ(x, y) = πxr πry N r and θN(1) (x, y) =

1 X πxα G(α, l)πlj G(j, β)πβy × N 2 α,β,p,r,s × Vpl Vsj + Vpj Vsl πxr G0 (r, p)πps Vrs πsy .

Standard computations (see, e.g., Lemma 4.1) show that 4 (1) θ ≡ sup θ (1) (x, y) 6 2Vm . N N q 3 η3 x,y

Let us rewrite (4.23) in the following form

UN (x, y) = ζ 0 E T 0 (x, y)ρ(x, y) V0 + X + ζ 0 N −1 E T 0 (x, y) πxr Vrs G0 (r, p)[T 0 (p, s)]0 Vps πs,y + p,r,s

X 0 −1 0 0 0 0 + ζ N E T (x, y) πxr Vrs [G ] (r, p)Vps πs,y iEhT (p, s) + p,r,s

X 0 −1 0 πxr Vrs Vps πs,y EhG0 (r, p)iEhT 0 (p, s)i + + ζ N E T (x, y) p,r,s

+

ζ 0 θN(1) (x, y).

Then using a-priori estimate |T (x, y)| 6 Vm /(qη) and taking z0 = z¯ , we derive inequality |UN |2 6

0 2 1/2 vVm Vm2 sup E ρ (x, y) + |UN |2 + qη2 x,y qη2 1/2 X Vm3 −1 0 [πxr πsy ] Vrs + + 2 2 |UN | |SN | + |UN | N q η r,s + η−1 θN(1) .

(4.24)

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

15

Now we use elementary estimates 2 1/2

= O N −1/2 sup E ρ 0 (x, y) x,y

and

X 0 2 1 πxr Vrs πsy = O N −1 E N r,s

and derive from (4.24) that |UN |2 6

Vm2 Vm3 2 |U | + |UN ||SN | + O N −1/2 . N 2 2 2 qη q η

This inequality regarded jointly with (4.21) for z ∈ 3q (4.6) implies (4.22). Relation (4.5) is derived. We complete this section with the outline of the proofs of the following auxiliary statements. 4.3.

PROOF OF THE AUXILIARY FACTS

LEMMA 4.1. Relation

sup |φN (x, y)| = O Vm2 η−2 q −1 N −1/2 .

(4.25)

x,y

holds with probability 1. Proof. Taking into account that |πxy | < q −1/2 with probability 1, one can write the inequality X (1) ψ (x, y) 6 1 |G(x, p)|Vps G(r, s)Vry |. N qN p,r,s Let us look at |G(x, p)| ≡ Gx (p) and |G(s, r)| ≡ Gr (s) for given fixed x and r as vectors in l 2 (N) and observe that the norm of the linear operator Vˆ in l 2 (N) with the kernel Vˆ (p − s) = |Vps | is bounded by Vm (2.3). Then using the estimate X |G(x, p)|2 = |Gex |2 6 |G|2 6 η−2 , (4.26) p

where we introduced the unit vectors ex with components ex (j ) = δxj , we obtain that 2 1/2 Vm X X |G(s, r)| · |Vry | |φN (x, y)| 6 qηN s r 1/2 Vm2 X X 2 6 |G(s, r)| . qηN s r

16

A. KHORUNZHY AND G. J. RODGERS

This estimate, together with inequalities (4.26) leads us to (4.25).

2

LEMMA 4.2. Equation (4.16) has a unique solution in the class G of matrices {g(x, y; z), x, y = 1, . . . , N} such that (4.27) sup g(x, y; z) 6 2η−1 , η = |Imz| > 2Vm . x,y

Proof. The proof is based on the use of the method of subsequent approximations. We introduce the sequence of matrices {g (k) (x, y)}, k = 1, 2, . . . by the relations ζ = −z−1 , g (k+1) (x, y) = ζ δxy + ζ t2(k) g (k) V (x, y) + + ζ t1(k) − t2(k) g (k) (x, y)V (x, y),

g (1) (x, y) = ζ δxy ,

where t1(k) = V0 N −1 Tr g (k) + q N −1 Tr g (k)V − V0 N −1 Tr g (k) and t2(k) = qN −1 Tr g (k) V . It is easy to see that if g (k) satisfies (4.27), then g (k+1) also satisfies (4.27). Then simple computation shows that there exists such α(z) that 1k+1 ≡ sup g (k+1) (x, y) − g (k) (x, y) < α(z)1k x,y

and |α(z)| < 1 for z ∈ 1q . This completes the proof of the Lemma 4.2.

2

LEMMA 4.3. Relation (4.17) holds for z ∈ 3q (4.6). Proof. Subtracting (4.17) from (4.15), we obtain for the difference δg (x, y) = g(x, y) − b g (x, y) relation g V (x, y) + ζ t2 δg V (x, y) + δg (x, y) = ζ δt(2) b + ζ δt(1) − δt(2) b g (x, y)V (x, y) + ζ t1 − t2 δg (x, y)V (x, y) + + ζ ψN0 (x, y), (4.28) where δg (x, y) = g(x, y) − b g (x, y), δt(1) = N −1 Trδg + q N −1 Tr δg V − V0 N −1 Trδg , δt(2) = qN −1 Trδg V

17

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

and

sup x,y ψN0 (x, y) = O(1) as N → ∞.

(4.29)

Now we regard (4.28) as the matrix relation δg = Lζ,V δg + ζ δt(2)b g V + ζ δt(1) − δt(2) g(x, y)V (x, y) + ζ ψN0 , where L is a linear operator X (Lζ,V δ)(x, y) = ζ t2 δ(x, p)V (p, y) + ζ(t1 − t2 )δ(x, y)V (x, y). p

It is easy to show that for large enough values of |Imz| there exists β(z) < 1 such that kLζ,V k1 < β(z),

where the norm k · 1 is determined as

(4.30)

kδk1 = sup |δ(x, y)|. x,y

A priori estimates for g(x, y) and ti |g(x, y)| 6 η−1 ,

|ti | 6 Vm η−1

and estimates (4.27) and (4.29) allow one to deduce from (4.30) that kδg k1 = o(1)

as N → ∞. 2

This proves Lemma 4.3.

LEMMA 4.4. Item (iii) of Theorem 2.1 is true. Proof. We introduce a sequence {g (k), k = 0, 1, 2, . . .} of functions g (k) (p; z), p ∈ (0, 1), z ∈ C by the formulas g (0) (p; z) = −1/z and Z g (k+1) (p; z) = −z − (1 − q)v 2 g (k) (r; z) dr − q V˜ (p) × Z ×

V˜ (r)g (k) (r; z) dr

−1 .

It is easy to verify that if g (k) (p; z) satisfies conditions (2.8), then g (k+1) (p; z) also does. Next, it is not hard to deduce that inequality sup g (k) (p; z) 6 2η−1 , z ∈ 3q (4.31) p

implies the same for g (k+1) (p; z).

18

A. KHORUNZHY AND G. J. RODGERS

The next step is to show that the sequence {g (k) } is the Cauchy one that determines, in the limit k → ∞, the function γ (p; z) ∈ 0 that satisfies (2.7b). This function satisfies (4.31) that proves uniqueness of the solution of (2.7b). 2 LEMMA 4.5. Due to positivity of Vxy = V (x − y) the matrix M(x, y; s, t) = Vxs Vyt + Vxt Vys is also positively determined on the vectors ψ with complex components ψ(x, y), x, y = 1, 2, . . . , n; n X

M(x, y; s, t)ψ(s, t)ψ(x, y) > 0.

(4.32)

x,y,s,t =1

Proof. Let us consider the Gaussian random variables {γ (x), x ∈ N} such that Eγ (x) = 0,

Eγ (x)γ (y) = V (x − y).

Then one can derive with the help of (4.8) that Eγ (x)γ (y)γ (s)γ (t) = M(x, y; s, t). Therefore the right-hand side of (4.32) can be rewritten in the follpwing form X 2 n X n Eγ (x)γ (y)γ (s)γ (t)ψ(s, t)ψ(x, y) = E γ (x)γ (y)ψ(x, y) . x,y,s,t =1

This proves the lemma.

x,y=1

2

5. Proof of Theorem 3.1 To prove Theorem 3.1, we again use the resolvent approach of Section 3. However (q) the present case of HN is more complicated because of the bilinear structure of H (x, y) (3.3) with respect to the random variables {ξ µ (x)}. We modify the general (q) (q) scheme and regard the resolvent GN (x, y; z) of matrices HN (3.1) first as a function of the random variables {ξ µ (x) and derive equations for the mathematical (q) expectation EGN . Then we prove the selfaveragenessin the limit N → ∞ of the (q) random variable GN (x, y; z) as a function of random variables {πxy }. As before, our tools are the resolvent identity (4.3) and the version of (4.8) applied for jointly independent Gaussian random variables {γj } ∂F . E γj F (γ1 , . . . , γk ) = Eγj2 E ∂γj

(5.1)

19

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES (q)

We will also use the consequence of (4.3) applied for GN (we omit the super- and (q) subscripts in GN ): 1 X ∂G(x, s) = − G(x, r)G(t, s) + G(x, t)G(r, s) ξ ρ (r)πrt . ρ ∂ξ (t) N r

5.1.

(5.2)

DERIVATION OF MAIN RELATIONS

We start with the resolvent identity (4.3) written in the form EhG(x, y; z)i = ζ δxy − ζ EhM(x, y; z)i,

ζ = −z−1 ,

where we denoted M(x, y) =

1 X G(x, s)ξ µ (s)ξ µ (y)πsy . N s,µ

Here and below we omit the variable z when it is not important. To compute the mathematical expectation of M(x, y), we can use (5.1) Taking into account (5.2), we obtain the following relation EhM(x, y)i =

mu2

E G(x, y)πyy − N u2 X

E G(x, y)πyt H (t, s)G(t, s)πsy + ϑN(1), − N s,t

(5.3)

where H is the same as in (3.3) and ϑN(1) =

u2 X

E G(x, t)πty H (t, s)G(y, s)πsy . N s,t

(5.4)

In Lemma 5.1 (see the second part of this section) we prove that the variable ϑN(1) vanishes in the limit N → ∞. The √proof is based on the observation that all the moments of the random variables NH (t, s) and H (t, t) are bounded for all N. We also use elementary inequality (4.26). Let us introduce variable 1 X πxs H (s, t)G(s, t)πty (5.5) L(x, y) = N s,t that resembles the generalized trace T (x, y) (4.2). Taking into account that hπyy = q and denoting cN = mN −1 , we rewrite (5.3) in the form EhM(x, y)i = cN qu2 EhG(x, y)i − u2 EhG(x, y)iEhL(y, y)i − − ϑN(1)(x, y) + ϑN(2) (x, y),

(5.6)

20

A. KHORUNZHY AND G. J. RODGERS

where

ϑN(2) (x, y) = cN u2 E G0 (x, y)πyy + u2 E G0 (x, y)L0 (y, y)

(5.7)

and the subscripts 0 denote the centered random variables. In Lemma 5.2 we prove that ϑN(2) vanishes in the limit N → ∞. Let us note that Equation (5.6) is similar to Equation (4.12). Following the same ideas as of Section 4,we derive relations for the variable EhL(x, y) . To do this, we employ once more (5.1) and obtain relation (cf. (4.15)) X

E πxs G(s, s)πsy − EhL(x, y)i = cN u2 s

u X

− E L(x, t)G(t, t)πty + ϑN(2) , N t 2

(5.8)

where ϑN(3) = −

u2 X

E πxs H (s, t)G(s, t)πsr G(r, s)πty . 2 N s,t,r

(5.9)

Denoting lN (x, y) = EhL(x, y)i and gN (x, y) = EhG(x, y)i and taking into account definition (2.8), we can rewrite (5.6) and (5.7) as the system of relations √ gN (x, y) = ζ δxy − ζ cu2 qgN (x, y) + ζ u2 gN (x, y)lN (y, y) + + ϑN(1)(x, y) + ϑN(2)(x, y), (5.10a) √ X 2 u q lN (x, t)gN (t, t) + lN (x, y) = cu2 gˆN δxy + (1 − δxy )q − N t + ϑN(3) + ϑN(4), (5.10b) P −1 where gˆN = N x gN (x, x) and u2 X 0 u2 X

E πxs G0 (s, s)πsy − 2 E L (x, t)G(t, t)πty − ϑN(4) (x, y) = cN N s N t −

0 u2 X

E L(x, t) E G (t, t)π . ty N2 t

In Lemma 5.2 we prove that sup ϑ (i) (x, y) = o(1) x,y

N

as N → ∞, i = 3, 4.

Let us consider the system of matrix equations √ RN (x, y) = ζ δxy − ζ cu2 qRN (x, y) + ζ u2 RN (x, y)SN (y, y), SN (x, y) = cu2 RN (x, y) δxy + (1 − δxy )q − √ N u2 q X SN (x, t)RN (t, t). − N t =1

(5.11)

(5.12a)

(5.12b)

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

21

The subscript N indicates that we consider N-dimensional matrices. It is not hard to √ show that this system is uniquely solvable provided z ∈ 3q (4.6) with Vm = 2u c. See the proof in Lemma 5.3 at the end of this section. Under this condition it is not hard to show that vanishing of terms ϑN(i) , i = 1, 2, 3, 4 leads to relation N 1 X EhgN (x, x)i − RN (x, x) = 0. N→∞ N x=1

lim

(5.13)

This can be proved by the same reasoning as used Lemma 4.3 of the previous section. The next observation is that the system (5.12) admits the solution such that the diagonal elements RN = RN (x, x) do not depend on x and SN (x, y) = δxy SN(1) + (1 − δxy SN(2) Therefore we conclude that (5.10) is equivalent to the scalar system √ RN = ζ − ζ cu2 qRN + ζ u2 RN SN(1) , (5.14a) 1 (1) N − 1 (2) √ (5.14b) S + SN , SN(1) = cu2 RN − u2 qRN N N N 1 (1) N − 1 (2) (2) 2 2√ SN = cqu RN − u qRN (5.14c) S + SN N N N that naturally has the unique solution (Lemma 5.3). Now elementary computations show that RN satisfies the same Equation (3.4) that determines fq,c (z). Then we come to the conclusion that lim gˆN − fq (z) = 0. (5.15) N→∞

P Since the normalized trace gˆN = N −1 x gN (x, x) represents the Stieltjes transform of certain measure and lim N→∞ gˆN is unique, then the Helly theorem and (5.15) imply that fq,c (z) also represents the Stieltjes transform of certain measure σq,c (λ). Thus, to complete the proof of Theorem 5.1, it remains to show that 2 X 1 N 0 (5.16) GN (x, x) = 0. lim N→∞ E N x=1 According to the general scheme, relations (5.15) and (5.16) imply convergence in probability of the measure σ (λ; HN(π)) to the limit σq,c (λ). Relation (5.16) is proved in Lemma 5.2. 5.2.

PROOF OF THE AUXILIARY FACTS

LEMMA 5.1. Variables (5.7) and (5.9) admit the following estimate for all z ∈ 3q (4.6) sup ϑ (i) (x, y) = O N −1/2 as N → ∞ x,y

N

for i = 1, 3.

22

A. KHORUNZHY AND G. J. RODGERS

Proof. We can estimate |ϑN(1) (x, y)| by the sum u2 X u2 X Eh|G(x, t)H (t, s)G(y, s)|i + Eh|G(x, s)H (s, s)G(y, s)|i.(5.17) qN s6=t qN s For the first term of this sum we have inequality u2 X Eh|G(x, t)H (t, s)G(y, s)|i qN s6=t 6

2 1/2 √ u2 X

2 2 1/2 E |G(x, t)| |G(y, s)| NH (t, s) . qN 3/2 s6=t

√ Let us note that all the moments of the random variables N H (t, s) and H (t, t) are bounded for all N and do not depend on x and y. Taking into account that 1/2 X 1/2 1 X

E |G(x, t)|2 |G(y, s)|2 6E |G(x, t)|2 |G(y, s)|2 N s6=t s6=t and a priori estimate (4.26), we deduce that the first term of (5.17) is of the order O(N −1/2 ) as N → ∞. The second one is also of the same order of magnitude. Let us consider the case of i = 3. As in the previous case, we can write inequality (3) ϑ (x, y) 6 N

XX

√ u2 NH (r, t) + E |G(t, s)G(s, r)| · 3/2 5/2 q N r6=t s +

u2 η X Eh|G(t, s)| · |H (t, t)|i. q 3/2 N 2 s,t

Now it is easy to see that estimates (5.4) and (5.5) complete the proof of the lemma. 2 LEMMA 5.2. Regarding the centered random variables G0 (x, y) = G(x, y) − EG(x, y)

and

L0 (x, y) = L(x, y) − EL(x, y),

the following relations hold in the limit N → ∞ 2

sup E G0 (x, x) = o(1),

(5.18)

x

and 2

sup E L0 (x, y) = o(1). x,y

(5.19)

23

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

Remark. Relations (5.18) reflect the selfaveraging property of the random variables G(x, x) and L(x, y). It is easy to see that (5.18) and (5.19) imply estimates (5.11). Let us also note that (5.18) implies relation (5.16). Proof. We prove (5.18) using again the resolvent identity (4.3) and relations (5.1) and (5.2). According to the definition of the centered random variable, we can write relation

0 1 X 0 (5.20) πxr H (r, t)G(r, t)πty . E L (x, y)L(x, y) = E L (x, y) N r,t Let us apply relation (4.4) to the last average in (5.20). With the help of (5.2) we obtain equality

mu2 0 E L0 (x, y)L(x, y) = E L (x, y)πxt G(t, t)πty − N 1 X 2 0 0 − u E L (x, y) L(x, t)G(t, t)L (t, y) + N t + φN(1) (x, y) + φN(2) (x, y), where φN(1) (x, y)

(5.21)

1 X 0 = −E L (x, y) 2 πxt G(t, s)πsr H (t, r)G(r, s)πsy N r,s,t

and φN(2)(x, y) can be estimated by inequality (2) φ (x, y) 6 N

1 X Eh|G(l, p)[H (x, t) + H (l, t)]G(t, p)|i + q 2 N 2 l,p,t + +

X 1 Eh|G(k, l)H (r, t)H (k, p)G(r, p)G(t, l)|i + q 3 N 3 k,l,p,r,t 1 q3N 3

X

Eh|G(k, r)H (r, t)H (k, p)G(l, p)G(t, l)|i.

k,l,p,r,t

It is not hard to find, with the help of the estimates (5.4) and (5.5), that 2 1/2 u4 η2 sup φN(1) (x, y) 6 L0 √ . x,y q N A little more cumbersome computations lead to the estimate (2) 1 sup φN (x, y) = O 3 3/2 . q N x,y

(5.22)

(5.23)

24

A. KHORUNZHY AND G. J. RODGERS

Taking into account the identity 1 X 0 L (x, y) L(x, t)G(t, t)πty N t 1 X 0 0 = L (x, y) L(x, t) G(t, t)πty + N t

1 X 0 G(t, t)πty L(x, t) , + L (x, y) N t we derive from (5.21) inequality

0 L (x, y)L(x, y) 6

2 1/2

η mu2 0 2 1/2 sup G(x, y) + |PN (x, y)| + √ L0 (x, y) N q t 2 1/2 u4 η2 √ + η q sup |QN (x, t)| + √ L0 (x, y) + t q N + φ (3)(x, y) |ψN (x, y)|, N

(5.24)

where

X 0 1 πxt G(t, t)πty , PN (x, y) = L N t

and 2 1/2

G(t, t)0 2 1/2. QN (x, y, t) = L0 (x, y)G(t, t) 6 L0 (x, y) ˆ that are The term ψN contains the terms arising when one passes from L0 G to Lˆ 0 G 0 independent from random variables {πxy } and back to L G, after averaging over πty . In what follows, we derive inequalitites

PN (x, y) 6 √η L0 (x, y) 2 1/2 + √η L0 (x, y) 2 1/2 sup L0 (x, t) 2 1/2 + q t N (4) (5.25) + η sup QN (x, t) + φ (x, y), t

and

N

2 √

0

G (t, t) 2 6 mηu q G0 (t, t) 2 + η G0 (t, t) 2 h|L(x, x)|i + N

+ η2 G0 (t, t)L0 (t, t) + 8ηqu2 N −1/2 ,

where the variable |φN(4)(x, y)| is of the order O(N −1/2 ) as N → ∞.

(5.26)

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

25

Then elementary calculations show that (5.24), (5.25) and (5.26) imply (5.18) and therefore (5.19). Let us consider PN (x, y). Using the resolvent identity (2.10), we can write that 1 X PN (x, y) = ζ¯ E L0 πxt πty − N t X 0 1 ¯ − ζE L πxt G(t, k)H (k, t)πkt πty , (5.27) N t,k where we denoted ζ¯ = −1/z and L0 ≡ L0 (x, y). It is easy to derive that 0 2 1/2 X

0 2 1/2 1 X E L0 1 πxt πty 6 L πxt πty N t N t 1 2 1/2 6 √ L0 . N Turning to the second term in the right-hand side of (5.27), we use (5.2) to compute the average. Then we obtain X 0 1 πxt G(t, k)H (k, t)πkt πty E L N t,k X 0 m =E L 2 πxt G(t, t)πkt πty − N t,k X 0 1 −E L 2 πxt G(t, t)H (t, t)πkt πty + φN(5) (x, y), N t,k where φ (5) N (x, y), in common with (5.24) and (5.25), includes the terms of the order N −1/2 h|L0 |2 i1/2 and those that are of the order O(N −3/2 ). Now, repeating arguments used to derive expressions (5.22) and (5.23) we arrive at (5.19). Let us derive (5.18). Using the resolvent identity (4.3) and formula (5.2), we obtain the equality

m

E G0 (x, x)G(x, x) = −ζ¯ u2 E G0 (x, x)G(x, x)πxx + N

+ ζ¯ E G0 (x, x)G(x, x)L(x, x) + ζ¯ X 0 E G (x, x)G(x, t)H (s, t)πt x G(s, x)πsx + + N s,t +

2ζ¯ X

E G(x, t)H (s, t)πt x G(x, x)G(x, s)πsx . N s,t

Then (5.18) easily follows from (5.19). Lemma is proved.

2

26

A. KHORUNZHY AND G. J. RODGERS

LEMMA 5.3. The system of equations (5.12) is uniquely solvable and (5.15) holds. Proof. Let us introduce recurrence relations √ RN(k+1)(x, y) = ζ δxy − ζ cu2 qRN(k)(x, y) + ζ u2 RN(k) (x, y)SN(k) (y, y), u2 √ X (k) SN(k+1) (x, y) = cu2 RN(k) (x, y) δxy + q 1 − δxy − q SN (x, t)RN(k) (t, t) N t for k > 0 and RN(0) (x, y) = SN(0) (x, y) = 0. Then one can easily show that for all z ∈ 31 (2.9b)

(k)

R ≡ sup R (k)(x, y) 6 2η−1 (5.28) N N x,y

and the same relation holds for RN(k). Then elementary computations show that the differences 1k+1 (x, y) ≡ RN(k+1)(x, y) − RN(k) (x, y) and δk+1 (x, y) ≡ SN(k+1) (x, y) − SN(k) (x, y) are such that

k k1k+1 k 6 4(c + 1)u2 η−k−1 k10 k

and

k kδk+1 k 6 4(c + 1)u2 η−k−1 kδ0 k.

This proves the first statement of the lemma. Using the estimates (5.28), it is easy to prove (5.15). Lemma is proved. 2

6. Properties of the Eigenvalue Distribution In this section we prove Theorems 2.2 and 3.2. The reasonings based on the inversion formula (2.9) are similar in both cases and use Equations (2.7) and (3.4), respectively. Proof of Theorem 3.2. Let us denote I (z) = Im fq,c (z) and R(z) = Re fq,c (z). It is easy to derive from (3.6) that √ u2 qI I 4 2√ = Im z + cu (1 − q)I + cu q . √ I 2 + R2 (1 + u2 qR)2 + u4 qI 2 Taking into account inequality I (λ + iε) > 0, we derive the following estimate 1 > ε + cu4 (1 − q)I (λ + iε) I (λ + iε)

(6.1)

27

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

for all values of ε > 0 and λ. Using (2.9), one can easily deduce from (6.1) the estimate mentioned in Theorem 3.2. 2 Proof of Theorem 2.2. Taking into account that g˜q (p; z) posesses property (2.8), one can easily derive from (2.7b) inequality Im g˜q (p; λ + iε) 6

1 ε + (1 −

q)v 2 Im f

q (λ

+ iε)

.

Integrating both parts of this relation over p, one obtains inequality [Im fq (λ + iε)]2 6

1 . (1 − q)v 2 2

This estimate added by formula (2.9) completes the proof.

We continue the study of the distributions σq (λ) and σq,c (λ) and derive from Equations (2.7) and (3.4), respectively, relations for the moments Z Z Mk = λk dσq (λ) and Lk = λk dσq,c (λ). Let us note first that item (iii) of Theorem 2.1 implies that g˜q (p; z) can be represented in the form g˜q (p; z) = −

∞ X mk (p) k=0

zk+1

.

(6.2)

R Then (2.7a) implies that Mk = mk (p) dp. R Writing down similar to (6.2) expressions for fq (z), g˜q (r; z)V˜ (r) dr and their products, one can easily derive from (2.7b) the system m0 (p) = 1,

m1 (p) = 0, X mk+1 (p) = (1 − q)v 2 m(0) i mj (p) + q i+j =k−1

X

˜ m(1) i mj (p)V (p),

i+j =k−1

R where m(l) mi (r)V˜ l (r) dr, l = 0, 1 and the sum is taken over i, j > 0. This i = system can be reduced to the system Z (l) (6.3a) m0 = Vl ≡ V˜ l (r) dr, m(l) 1 = 0, X X (l) (l+1) 2 m(0) m(1) . (6.3b) m(l) i mj + q i mj k+1 = (1 − q)v i+j =k−1

i+j =k−1

THEOREM 6.1. (i) Distribution σq (λ) is even in the sence that M2k+1 = 0, ∀k ∈ N;

28

A. KHORUNZHY AND G. J. RODGERS

(ii) the support of the measure dσq (λ) is bounded; p supp dσq (λ) ⊂ (−2l, 2l), l = v[(1 − q)v + qVm ].

(6.4)

Proof. Item (i) obviously follows from (6.3). We start the proof of item (ii) with the following observation. Since matrix V with enries V (x − y) is positively determined, then obviously m(l) 0 > 0. Now it is (l) (l) easy to show that if ms > 0, then ms+2 > 0. The second step is the following statement. PROPOSITION 6.1. For all s > 0 and l > 0 6 Vm m(l) m(l+1) s s .

(6.5)

Proof. Relation (6.5) obviously holds for s = 0. It is easy to deduce from the form of (6.3b) that if (6.5) is valid for m(l) s with s 6 t then (6.5) is true also for . 2 m(l) t +1 Using Proposition 6.1, we can derive from (6.3b) inequality X (1) (l) m(l) (1 − q)v 2 m(0) mj . k+1 6 i + qVm mi

(6.6)

i+j =k−1

Now let us introduce the numbers m ˆ (l) k by the following recurrent relations Z V˜ l (r) dr, m ˆ (l) (6.7a) m ˆ (l) 0 = Vl ≡ 1 = 0, X (l) (1 − q)v 2 m ˆ (0) ˆ (1) m ˆj . (6.7b) m ˆ (l) i + qVm m i k+1 = i+j =k−1

It follows from Proposition 6.1 and (6.6) that ˆ (l) m(l) k 6m k .

(6.8)

Let us determine the support of the measure dσˆ q (λ) given by its moments m(0) k . To do this, we introduce the function ∞ X m(l) . gˆ (z) = − zk+1 k=0 (l)

It is not hard to derive from (6.7) that it satisfies the system of equations m(l) 0 gˆ (z) = , −z − (1 − q)v 2 gˆ (0) (z) − qVm gˆ (1) (z) (l)

l = 0, 1, 2, . . . .

(6.9)

ˆ (0)(z), m(1) It follows from (6.9) that gˆ (1)(z) = m(1) 0 g 0 ≡ v and then (cf. (2.10)) gˆ (0) (z) =

1 −z − [(1 −

q)v 2

+ qvVm ]gˆ (0) (z)

.

(6.10)

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

29

It is clear that dσˆ q (λ) determined by its Stieltjes transform gˆ (0) (z) (6.10) is the semicircle distribution (2.11) with v 2 replaced by (1 − q)v 2 + qvVm . Thus, its support is given by the interval from the right-hand side of (6.4). Relation (6.8) implies that supp dσq (λ) ⊂ supp dσˆ q (λ). 2

This observation completes the proof of Theorem 6.1. We finish this section with relation for the moments Lk . Using representation fq,c (z) = −

∞ X Lk , k+1 z k=0

one can easily derive from (3.6) the system L0 = 1, Lk+1

L1 = bc,

L2 = ac + (1 + c)bL1 , X X = (c − 1)bLk + b Li Lj + ac Li Lj − − abc

X

i+j =k

Li Lj Ll ,

i+j =k−1

for k > 2,

(6.6)

i+j +l=k−2

√ where a = u4 (1 − q) and b = u2 q. Analysis of this system is more complicated than that of (6.3). 7. Summary We have studied the eigenvalue distribution of weakly diluted N × N random matrices in the limit N → ∞. The basic ensemble represents dilution of the Wishart random matrices HN which are widely known in multivariate statistical analysis. The second ensemble is the dilute version of the Gaussian random matrices AN with weakly correlated entries. In the pure (nondilute) case both of these ensembles, {HN } and {AN }, have limiting eigenvalue distributions that can have a singular component at the origin. We derived explicit equations determining the limiting eigenvalue distributions of the dilute versions of {HN } and {AN }. We showed that in the case of weak dilution, when each matrix row contains, on average, qN, 0 < q < 1 nonzero entries, the density of the eigenvalue distribution is bounded by const.(1 − q)−1 . Thus, we can conject that in general the singularities (if any) in the spectra of random matrices are rather unstable and disappear when the dependence between matrix entries is waived. Taking into account our results, one can say that this perturbation can be fairly weak, as it is represented by the weak random dilution.

30

A. KHORUNZHY AND G. J. RODGERS

Acknowledgements A.K. would like to thank the Department of Mathematical Sciences at Brunel University for hospitality while this work was performed and both authors are grateful to the Royal Society, London for the financial support.

References 1. 2. 3. 4.

5. 6.

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

14. 15. 16. 17.

18.

19.

Anderson, T. W.: An Introduciton to Multivariate Statistical Analysis, John Wiley, New York, 1984. Amit, D. J.: Modeling Brain Function, Cambridge University Press, Cambridge, 1989. Boutet de Monvel, A. and Khorunzhy, A.: On the norm and eigenvalue distribution of large random matrices, Ann. Probab. 27(2) (1999), 913–944. Boutet de Monvel, A., Khorunzhy, A. and Vasilchuk, V.: Limiting eigenvalue distribution of eigenvalues of random matrices with correlated entries, Markov Proc. Rel. Fields 2 (1996), 607–636. Bovier, A. and Gayrard, V.: Rigorous results on the thermodynamics of the dilute Hopfield model, J. Stat. Phys. 69(1–2) (1993), 597–627. Bovier, A. and Gayrard, V.: The Hopfield model as a generalized random mean field model, In: A. Bovier and P. Picco (eds), Mathematical Aspects of Spin Glasses and Neural Networks, Progr. Probab. 41, Birkhäuser, Boston, 1997, pp. 3–89. Brouwer, P. W. and Beenakker, C. W. J.: Diagrammatic method of integration over the unitary group, J. Math. Phys. 37(10) (1996), 4904–4934. Brody, T. A., Flores, J., French, J. B., Mello, P. A., Pandey, A. and Wong, S. S. M.: Randommatrix physics: Spectrum and strength fluctuations, Rev. Modern Phys. 53(3) (1981), 385–479. Crisanti, A., Paladin, G. and Vulpiani, A.: Products of Random Matrices in Statistical Physics, Springer Ser. Solid-State 104, Springer-Verlag, Berlin, 1993. Di Francesco, P., Ginsparg, P. and Zinn-Justin, J.: 2D gravity and random matrices, Phys. Rep. 254 (1995), 1–133. Donoghue, W.: Monotone Matrix Function and Analytic Continuation, Grundlehren Math. Wiss. 207, Springer-Verlag, New York, 1974. Girko, V.: An Introduction to Statistical Analysis of Random Arrays, VSP, Utrecht, 1998. Hertz, J. A., Krogh, A. and Palmer, R. G.: Introduction to the Theory of Neural Computations, Santa Fe Inst. Stud. Sci. Complexity Lecture Notes, I, Addison-Wesley, Redwood City, CA, 1991. Hopfield, J. J.: Neural networks and physical systems with emergemt collective computational abilities, Proc. Natl. Acad. Sci. USA 79 (1982), 2554–2558. Khorunzhy, A.: Eigenvalue distribution of large random matrices with correpated entries, Mat. Fiz. Anal. Geom. 3(1–2) (1996), 80–101. Khorunzhy, A.: On dilute unitary random matrices, J. Phys. A: Math. Gen. 31 (1998), 4773– 4784. Khorunzhy, A., Khoruzhenko, B., Pastur, L. and Shcherbina, M.: Large-n limit in statistical mechanics and the spectral theory of disordered systems, In: C. Domb and J. L. Lebowitz (eds), Phase Transitions and Critical Phenomena 15, Academic Press, New York, 1992, pp. 67–237. Khorunzhy, A. and Pastur, L.: On the eigenvalue distribution of the deformed Wigner ensemble of random matrices, In: V. A. Marchenko (ed.), Spectral Operator Theory and Related Topics, Adv. Sov. Math. 19, Amer. Math. Soc., Providence, RI, 1994, pp. 97–127. Khorunzhy, A. and Rodgers, G. J.: Eignevalue distribution of large dilute random matrices, J. Math. Phys. 38 (1997), 3300–3320.

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

31

Khorunzhy, A. and Rodgers, G. J.: On the Wigner law in dilute random matrices, Rep. Math. Phys. 42 (1998), 297–319. Kolchin, V. F.: Random Graphs, Cambridge University Press, Cambridge, 1999. Lamperti, L.: Stochastic Processes: A Survey of the Mathematical Theory, Appl. Math. Sci. 23, Springer-Verlag, New York, 1977. Marchenko, V. A. and Pastur, L. A.: Eignevalue distribution of certain ensembleof random matrices, Mat. Sbornik 72, 507–536 [English translation: Math. USSR-Sb 1 (1967), 457–483]. Mézard, M., Parisi, G. and Virasoro, M. A.: Spin-Glass Theory and Beyond, World Scientific Lecture Notes in Phys. 9, World Scientific, Singapore, 1987. Mirlin, A. D. and Fyodorov, Ya. V.: Universality of level correlations of sparse random matrices, J. Phys. A: Math. Gen. 24(10) (1991), 2273–2286. Pastur, L. A. and Figotin, A.: Exactly solvable model of a spin glass, Soviet J. Low Temp. Phys. 3 (1977), 378–383. Porter, C. E. (ed.): Statistical Theories of Spectra: Fluctuations, Academic Press, New York, 1965. Rodgers, G. J. and Bray, A. J.: Density of states fo dilute random matrix, Phys. Rev. B 37(7) (1988), 3557–3562. Rodgers, G. J. and De Dominicis, C.: Density of states of sparse random matrices, J. Phys. A: Math. Gen. 23(9) (1990), 1567–1573. Stariolo, D. A., Curado, E. M. F. and Tamarit, F. A.: Eigenvalue distribution of dilute Hopfield model, J. Phys. A: Math. Gen. 29(15) (1996), 4733–4739. Talagrand, M.: Rigorous results for the Hopfield model with many patterns, Probab. Theory Related Fields 110 (1998), 177–276. Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math. 62 (1955), 548–564.

Mathematical Physics, Analysis and Geometry 3: 33–47, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

33

On Ground-Traveling Waves for the Generalized Kadomtsev–Petviashvili Equations A. PANKOV1 and K. PFLÜGER2

1 Department of Mathematics, Vinnitsa State Pedagogical University, Ukraine.

e-mail: [email protected] 2 Institut für Mathematik I, Freie Universität Berlin, Germany. e-mail: [email protected]

(Received: 18 March 1999; in final form: 18 October 1999) Abstract. As a continuation of our previous work, we improve some results on convergence of periodic KP traveling waves to solitary ones as the period goes to infinity. In addition, we present some qualitative properties of such waves, as well as nonexistence results, in the case of general nonlinearities. We suggest an approach which does not use any scaling argument. Mathematics Subject Classifications (2000): 35Q53, 35B10, 35A35, 35A15. Key words: generalized Kadomtsev–Petviashvili equation, traveling waves, variational methods.

1. Introduction Kadomtsev–Petviashvili (KP) equations, both original and generalized, appear in the theory of weakly nonlinear dispersive waves [7]. They read ut + uξ ξ ξ + f (u)ξ + εvy = 0,

vξ = uy

(1)

or, eliminating v, (ut + uξ ξ ξ + f (u)ξ )ξ + εuyy = 0.

(2)

More precisely, these are KP-I equations if ε = −1, and KP-II equations if ε = +1. The original KP equations correspond to the case f (u) = 12 u2 , form a completely integrable Hamiltonian system, and were studied extensively by means of algebrogeometrical methods (see, e.g., [8]). There are also a number of papers dealing with more general equations (1) or (2), mainly in the case of pure power nonlinearity: [1, 3 – 6, 10, 17, 20, 23], to mention a few. In particular, solitary traveling waves were studied [1, 4 – 6, 10, 17, 23]. Here we consider mainly the case of KP-I equations. Remark that KP-II equations do not possess traveling waves at all (see [5] for the case of solitary waves and pure power nonlinearity, and Section 4 for the general case). The present paper is a direct continuation of our previous work [17]. It concerns the existence of ground-traveling waves, both periodic and solitary, and the limit

34

A. PANKOV AND K. PFLÜGER

behavior of periodic waves, as the period goes to infinity. Corresponding equations for traveling waves read −cux + uxxx + f (u)x + εvy = 0,

vx = uy

(3)

and (−cux + uxxx + f (u)x )x + εuyy = 0,

(4)

respectively. Here x = ξ − ct, c > 0 is the wave speed. In [17], among other results we have proved that k-periodic in x ground waves converge to a solitary ground wave in a very strong sense (Theorem 5 of that paper). Unfortunately, that result does not cover the case of the original KP equation, but includes the case f (u) = u3 . The first aim of this paper is to extend the results of Theorems 4 and 5, [17], in order to include nonlinearities like f (u) = |u|p−1 , 2 < p < 6. This will be done in Section 2. Our second goal is to discuss, in Section 3, some qualitative properties of KP traveling waves: symmetry, continuity, and rate of decay. As for continuity and decay properties, we follow very closely the paper [6] and point out only the main differences. On the contrary, our proof of symmetry with respect to y-variable relies on a quite different variational characterization of ground waves and permits us to treat the case of nonhomogeneous nonlinearity. Finally, in Section 4, we discuss the nonexistence of traveling waves, both solitary and periodic, for general nonlinearities. All the assumptions we impose here are satisfied for the nonlinearities f (u) = c|u|p−1

and

f (u) = c|u|p−2 u +

k X

ci |u|pi −2 u,

i=1

with c, ci > 0,

2 < p < 6,

2 < pi < p.

Unfortunately, Assumptions (N) and (N1) below are not satisfied for a very interesting nonlinearity f (u) = u2 − u3 which appears in some physical models [18]. In addition, let us remark that it is natural to look at (k, l)-periodic traveling wave solutions with respect to (x, y), as well as l-periodic waves with respect to y which are decaying in x. All the results below, except of Theorem 2, have their straightforward counterparts in both these cases. The case of double periodic waves is even simpler, since the corresponding functional satisfies the Palais–Smale condition. However, as we already mentioned in our previous paper [17], our technique does not work, at least directly, when we try to study the behavior of such traveling waves as (k, l) → ∞ (respectively, l → ∞). 2. Ground Waves Denote by F (u) = assumptions:

Ru 0

f (t) dt the primitive function of f . We make the following

35

GROUND-TRAVELING WAVES AND KP EQUATIONS

(1) (2) (3) (4)

f ∈ C(R), f (0) = 0; |f (u)| 6 C(1 + |u|p−1 ), 2 < p < 6, andR f (u) = o(|u|) as u → 0; there exists ϕ ∈ C0∞ (R2 ) such that 1/λ2 R2 F (λϕx ) → +∞, as λ → +∞; there exists µ > 2 such that µF (u) 6 uf (u) for all u ∈ R.

Remark that if F (u) > 0 for all u 6= 0, then Assumption (3) follows from (4). Let Qk = (−k/2, k/2) × R, 0 < k 6 ∞. We set Z x −1 Dx,k u(x, y) = u(s, y) ds, k ∈ (0, ∞]. (5) −k/2

We shall simply write Dx−1 in the case k = ∞. Define the Hilbert space Xk as the completion of {ϕx : ϕ ∈ Ck∞ }, where Ck∞ is the space of smooth functions on R2 which are k-periodic in x and have finite support in y, with respect to the norm 1/2 kukk = (u, u)k , Z −1 −1 (u, v)k = ux vx + Dx,k uy · Dx,k vy + cuv. Qk

Similarly, X = X∞ is the completion of {ϕx : ϕ ∈ C0∞ (R2 )} with respect to the 1/2 −1 is well-defined on norm kuk = kuk∞ = (u, u)1/2 = (u, u)∞ . The operator Dx,k the space Xk , k ∈ (0, ∞]. For k-periodic traveling waves, k ∈ (0, ∞), Equation (4) may be written in the form [17] −2 uyy + cu − f (u))x = 0. (−uxx + Dx,k

(6)

Solitary waves are solutions of the same Equation (6), with k = ∞. The action functional associated with (6) reads [17] Z 1 2 Jk (u) = kukk − F (u); (7) 2 Qk Jk is of the class C 1 on Xk . We consider weak solutions of (6), i.e. critical points of Jk in Xk . Now let us consider the so-called Nehari functional Z 0 2 uf (u), (8) Ik (u) = hJk (u), ui = kukk − Qk

and the Nehari manifold Sk = {u ∈ Xk : Ik (u) = 0, u 6= 0}. All traveling wave solutions lie in the corresponding Nehari manifold and we will find ground waves, i.e. solutions with minimal action among all nontrivial solutions, solving the following minimization problem: mk = inf{Jk (u) : u ∈ Sk }.

(9)

36

A. PANKOV AND K. PFLÜGER

Remark that

Z

Jk (u) = Qk

1 uf (u) − F (u), 2

u ∈ Sk .

(10)

In what follows, we will omit the subscript k if k = ∞ and write simply J, I, . . . . Throughout this section, in addition to Assumptions (1)–(4), we impose the following one (N) For any u ∈ L2 (R2 ) such that Z uf (u) > 0, R2

the function of t Z t −1 uf (tu) R2

is strictly increasing on (0, +∞). In the proof of Theorem 1, [17], we have considered the Mountain Pass Values ck for Jk and proved that they are uniformly bounded from below and above by positive constants. More precisely, ck = inf max Jk (γ (t)), γ ∈0k t ∈[0,1]

where 0k = {γ ∈ C([0, 1], Xk ) : γ (0) = 0, Jk (γ (1)) < 0}. Here we have defined 0k in a slightly different way than in [17], but it does not effect on the value of ck . Consider also another minimax value ck0 = inf+ sup Jk (tv), v∈Xk t >0

where Xk+

Z = v ∈ Xk :

F (v) > 0 .

Qk

Due to Assumption (4), Xk+ 6= ∅. LEMMA 1. For every v ∈ Xk+ , there exists a unique tk = tk (v) such that tk v ∈ Sk , Jk (tk v) = max Jk (tv), t >0

and tk (v) depends continuously on v ∈ Xk+ .

GROUND-TRAVELING WAVES AND KP EQUATIONS

37

Proof. Assumption (4) implies that Z vf (v) > 0 Qk

for any v ∈ Xk+ . Therefore, due to Assumption (N), the function Z d 2 2 −1 vf (tv) Jk (tv) = Ik (tv) = t kvkk − t dt Qk vanishes at only one point tk = tk (v) > 0. Equation (10) and Assumption (4) imply that Jk is positive on Sk . Since Jk (0) = 0, we see that tk is a point of maximum for 2 Jk (tv). Continuity of tk (v) is easy to verify. LEMMA 2. ck = ck0 = mk . Proof. Since uf (u) is subquadratic at 0 and the quadratic part of Jk is positive defined, we see that Ik (v) > 0 in a neighborhood of the origin, except of 0. Hence, Ik (γ (t)) > 0, γ ∈ 0k , for small t > 0. Due to Assumption (4), for v ∈ Xk+ we have Z Z F (v) > kvk2k − µ F (v) 2Jk (v) = kvk2k − 2 Qk Qk Z 2 vf (v) = Ik (v). > kvkk − 2 Qk

Hence, Ik (γ (1)) < 0. Therefore, γ (t) crosses Sk and this implies that ck > mk . By Assumption (4), for any v ∈ Xk+ we have F (tv) > αt µ , α > 0, if t > 0 is large enough. This implies that Jk (tv) < 0 for every v ∈ Xk+ and sufficiently large t > 0. Hence, the half-axis {tv : t > 0} generates in a natural way an element of 0k . This implies the inequality ck 6 ck0 . R Now let v ∈ Sk . By the definition of Ik , σ = Qk vf (v) > 0, and (N) implies that Z Z Z d F (tv) = vf (tv) > t −1 vf (tv) > σ > 0 dt Qk Qk Qk provided t > 1. Hence, for t > 0 large enough Z F (tv) > 0. Qk

By definitions of ck0 and mk , we see that ck0 = mk .

2

THEOREM 1. Assume Assumptions (1)–(4) and (N) are fulfilled. Then, for any k ∈ (0, ∞), there exists a minimizer uk ∈ Sk of (9) which is a critical point of Jk . Moreover, Jk (uk ) = mk is bounded from above and below by positive constants independent on k.

38

A. PANKOV AND K. PFLÜGER

Proof. In the proof of Theorem 1, [17], it is shown that there exists a Palais– Smale sequence uk,n ∈ Xk at the level ck , i.e. Jk0 (uk,n ) → 0,

Jk (uk,n ) → ck p

as n → ∞. Moreover, uk,n → uk weakly in Xk and strongly in Lloc (R2 ), where uk ∈ Xk is a nontrivial solution of (6). Therefore, Ik (uk,n ) = hJk0 (uk,n ), uk,n i → 0 and

Z Jk (uk,n ) −

1 I (u ) 2 k k,n

= Qk

1 u f (uk,n ) 2 k,n

− F (uk,n ) → ck .

Due to Assumption (4), the integrand here is nonnegative and, since uk,n → uk in L2loc (R2 ), we have Z 1 u f (uk ) − F (uk ) 6 ck . 2 k Qk

However, uk is a nontrivial solution, hence, uk ∈ Sk . Therefore, we deduce from (10) that Z 1 u f (uk ) − F (uk ) > mk . Jk (uk ) = 2 k Qk

Now Lemma 2 implies that Jk (uk ) = mk and uk is a ground-wave solution. The last statement of the theorem follows immediately from Lemma 2 and 2 uniform estimates for ck . Remark 1. The Nehari variational principle suggested in [13] was used successfully in many papers (see, e.g., [2, 9, 14 – 17, 22]). In all these papers, except [16], the geometry of Nehari manifold is simple enough: it is a bounded surface without boundary around the origin, like a sphere. In the case we consider here, the picture is different: Sk may look like a sphere if, e.g., f (u) = |u|p−2 u, and may be unbounded if, e.g., f (u) = |u|p−1 . Nevertheless, in any case, Sk separates the origin and the domain of negative values of Jk , which is sufficient for our purpose. In [16], such a manifold is also unbounded in general, but there we have used different arguments. Now we are going to study the behavior of uk , as k → ∞. Recall the definition of cut-off operators Pk : Xk → X, [17]. Let χk ∈ C0∞ (R) be a nonnegative function such that χk (x) = 1 for x ∈ [−k/2, k/2], χk (x) = 0 for |x| > (k + 1)/2, and |χk0 |, |χk00 | 6 C0 , with some constant C0 > 0. We set −1 Pk u(x, y) = [χk (x)Dx,k u(x, y)]x .

39

GROUND-TRAVELING WAVES AND KP EQUATIONS

THEOREM 2. Assume that Assumptions (1)–(4) and (N) are satisfied. Let uk ∈ Xk be a sequence of ground-wave solutions. Then there exists a nontrivial ground wave u ∈ X and a sequence of vectors ζk ∈ R2 such that, along a subsequence, Pk uk (· + ζk ) → u weakly in X. If, in addition, |f (u + v) − f (u)| 6 C(1 + |u|p−2 + |v|p−2 )|v|,

v ∈ R,

(11)

then, along the same subsequence, lim kuk (· + ζk ) − ukk = 0.

k→∞

Proof. By Theorem 2, [17], there is a nontrivial solution u ∈ X such that Pk uk (· + ζk ) → u weakly in X for some ζk ∈ R2 (along a subsequence). Let us prove that u is a ground wave, i.e. J (u) = inf{J (v) : v ∈ S} = m. First of all, for any v ∈ S and any ε > 0, there exist kε and vk ∈ Sk such that Jk (vk ) 6 J (v) + ε,

k > kε .

Indeed, since J and I are continuous, we can find ϕk ∈ C0∞ (Qk ) such that ηk = Dx ϕk → v in X and, hence, J (ηk ) → J (v),

I (ηk ) → I (v) = 0.

Since I (v) = 0 and v 6= 0, we have Z vf (v) = kvk2 > 0. Qk

R Hence, Qk ηk f (ηk ) > 0 for k large enough. Due to (N), there exists τk > 0 such that I (τk ηk ) = 0 and τk → 1. Let vk be a unique k-periodic function which coincides with τk ηk on Qk . Then Jk (vk ) = J (τk ηk ) 6 J (v) + ε provided k is large enough. In particular, we have lim supk→∞ mk 6 m. Now, exactly as in the proof of Theorem 5, [17], we see that lim infk→∞ mk > J (u) > m. Hence, m = J (u) and u is a ground wave solution. The second part of the theorem follows from Theorem 3, [17], exactly as at the end of the proof of Theorem 5, [17]. 2

3. Qualitative Properties of Traveling Waves Now we are going to study such properties of KP traveling waves as symmetry, regularity, and decay. We start with the following:

40

A. PANKOV AND K. PFLÜGER

LEMMA 3. Suppose that Assumptions (1) and (2) are satisfied. In the case 2 < p 6 5, assume, in addition, that f ∈ C 2 (R) and |f (j ) (u)| 6 C(1 + |u|p−1−j ),

j = 1, 2,

u 6= 0.

(12)

Then any traveling wave is continuous. Moreover, any solitary (resp. periodic) wave tends to zero as (x, y) → ∞ (resp. y → ∞). Proof. For such a wave u ∈ Xk , we have −cvxx − vyy + vxxxx = f (u)xx = gxx . Let

Z (Fk,x h)(ξ ) =

(13)

k/2

h(x) exp(−iξ x) dx −k/2

be the Fourier transform if k = ∞ (then we simply write Fx ), and the sequence of Fourier coefficients if k < ∞. In the last case, ξ ∈ (2π/k)Z. Now we get from (13) Fk,x Fy u = p(ξ1 , ξ2 )(Fk,x Fy g),

(14)

where p(ξ ) = p(ξ1 , ξ2 ) =

ξ12 , cξ12 + ξ14 + ξ22

ξ1 and ξ2 are dual variables to x and y, respectively. If k = ∞, there is nothing to do. In the case 5 < p < 6, one needs only to repeat the proof of Theorem 1.1, [6], which does not use any particular property of power nonlinearity, except of its growth rate. In the case 2 < p 6 5 the arguments from the proof of Lemma 4.1, [5], work and just here assumption (12) is needed. Now we explain how to cover the case of periodic waves. Recall the following Lizorkin theorem [11]. Let p(ξ ), ξ ∈ Rn , be of the class C n for |ξj | > 0, j = 1, . . . , n. Assume that k1 ∂kp ξ · · · ξ kn 6 M, n 1 ∂ξ1k1 · · · ∂ξnkn with kj = 0 or 1, k = k1 +· · · +kn = 0, 1, . . . , n. Then p(ξ ) is a Fourier multiplier on Lr (Rn ), 1 < r < ∞. We rewrite now (14) as follows: Fk,x u = Fy−1 [p(ξ1 , ξ2 )Fy Fk,x g] = P (ξ1 )g, where P (ξ1 ) is the operator Fy−1 p(ξ1 , ·)Fy for any fixed ξ1 . It is easy to verify that P (ξ1 ) ∈ L(Lr (Ry )), the space of bounded linear operators in Lr (Ry ). Moreover, due to the Lizorkin theorem, p(ξ ) is a multiplier in Lr (R2 ). Hence, so is it for P (ξ1 ) in the space Lr (Rx , Lr (Ry )) = Lr (R2 ). It is not difficult to verify that

GROUND-TRAVELING WAVES AND KP EQUATIONS

41

P (ξ1 ) depends continuously on ξ1 with respect to the norm in L(Lr (Ry )) at any point ξ1 6= 0. Therefore, by Theorem 3.8 of Ch. 7, [21], we see that P (ξ1 ) is also a multiplier in the space Lr ((−k/2, k/2), Lr (Ry ))0 = Lr (Qk )0 considered as the space of k-periodic in x functions. The subscript 0 means that for functions from this space Fk,x u vanishes at ξ1 = 0. Since p(0, ξ2 ) = 0, the corresponding multiplier vanishes on {u ∈ Lr (Qk ) : Fk,x u = 0 if ξ1 6= 0} and, hence, is a bounded operator on the entire space Lr (Qk ). In fact, we need here an extension of that theorem for operator-valued multipliers which may be discontinuous at the point 0. However, in this case, the proof presented in [21] works without any change. To complete the proof in the case 5 < p < 6, we can now use the same reiteration argument as in [6]. In the case 2 < p 6 5, again one needs to invoke the arguments of the proof of Theorem 4.1, [5]. Here we have to apply the remark on operator-valued multipliers to p1 (ξ ) = ξ12 p(ξ ), p2 (ξ ) = ξ2 p(ξ ), as well as to p(ξ ) itself. 2 We also need the following additional assumption: R (N1) f ∈ C 1 (R) and, for any v ∈ L2 (R2 ) such that R2 f (v)v > 0, we have Z Z f (v)v < f 0 (v)v 2 R

R2

R2

f (tv)v > 0 ∀t > 0. R Calculating the derivative of t −1 R2 f (tv)v, we see that (N1) implies (N). Let us introduce the functional Z 1 f (v)v − F (v) , v ∈ Xk . Lk (v) = 2 and

R2

Qk

As we have seen, Lk = Jk on Sk and Lk (v) > 0, ∀v ∈ Xk . LEMMA 4. Under R Assumption (N1), Lk (tv) is a strictly increasing function of t > 0, provided Qk f (v)v > 0. Proof. It follows immediately from the following elementary identity Z Z 1 d Lk (tv) = f 0 (tv)t 2 v − f (tv)tv . 2 dt 2t Qk Qk We also need the following dual characterization of ground-traveling waves: LEMMA 5. Suppose Assumptions (1)–(4) and (N1) are satisfied. For nonzero u ∈ Xk , k ∈ (0, ∞], the following statements are equivalent:

42

A. PANKOV AND K. PFLÜGER

(i) u is a ground wave, (ii) Ik (u) = 0 and Lk (u) = mk = inf{Lk (v) : v ∈ Sk }, (iii) Ik (u) = 0 = sup{Ik (v) : v ∈ Xk , Lk (v) = mk }. Proof. Implication (i) ⇒ (ii) is proved in Section 2. To prove (ii) ⇒ (i) assume that u ∈ Xk satisfies (ii). Since Jk = Lk on Sk , there exists a Lagrange multiplier λ such that λIk0 (u) = Jk0 (u). Then λhIk0 (u), ui = hJk0 (u), ui = Ik (u) = 0. On the other hand, hIk0 (u), ui

=

Z −

0

Z

f (u)u − f (u)u Qk Z Z f (u)u − f 0 (u)u2 = 2Ik (u) + Qk Qk Z Z = f (u)u − f 0 (u)u2 . 2kuk2k

2

Qk

R

Qk

Qk

However, Qk f (v)v > 0 on Sk and, due to (N1), hIk0 (u), ui < 0. Therefore, λ = 0 and u is a ground wave. Now let us prove (ii) ⇒ (iii). For u as in (ii), Ik (u)R= 0. Assume that there is v ∈ Xk such that Lk (v) = mk and Ik (v) < 0. Then Qk f (v)v > 0 and there exists t0 ∈ (0, 1) such that Ik (t0 v) = 0. By Lemma 4, Lk (t0 v) < Lk (v) = mk , which is impossible. Finally, we prove (iii) ⇒ (ii). Let u ∈ XR k satisfies (iii). Then, Lk (u) > mk . Assume that Lk (u) > mk . Again we have Qk f (u)u > 0. By Lemma 4, there exists t0 ∈ (0, 1) such that Lk (t0 u) = mk . However, Ik (t0 u) > 0 and this contradicts (iii). 2 Now we are ready to prove the symmetry property for all kinds of ground waves we consider. As in [6], we use the approach suggested in [12] (see also [22]). THEOREM 3. In addition to Assumptions (1)–(4) and (N1), suppose that f ∈ C 2 (R). In the case 2 < p 6 5 assume also that inequality (12) is fulfilled. Then any ground wave u ∈ Xk , k ∈ (0, ∞], is symmetric with respect to some line 1 = {(x, y) ∈ R2 : y = b}. Proof. Choose b in such a way that Z Z 1 1 mk , f (v)v − F (v) = f (v)v − F (v) = 2 2 2 1+ ∩Qk 1− ∩Qk where 1+ and 1− are the corresponding upper and lower half-planes. Let u± be a symmetric (with respect to 1) function such that u± = u on 1± . Then u± ∈ Xk and Lk (u± ) = Lk (u) = mk .

GROUND-TRAVELING WAVES AND KP EQUATIONS

43

By Lemma 5, Ik (u± ) 6 0. On the other hand, Ik (u+ ) + Ik (u− ) = 2Ik (u) = 0. Using Lemma 5, we conclude that u± is a ground wave. To conclude that u± = u and, hence, complete the proof, it is sufficient to use the same unique continuation result as in [6], and just here we need the assumption f ∈ C 2 (R) and Lemma 3. Remark that a periodic version (with 5 = 1± ) of unique continuation Theorem A.1, [6], can be proved exactly as that theorem itself. 2 In addition, we formulate the following direct generalization of results of [6] for decay of solitary waves. THEOREM 4. Suppose Assumptions (1) and (2) to be satisfied. Let u ∈ Xk , k ∈ (0, ∞], be a traveling wave. If k = ∞, then r 2 u ∈ L∞ (R2 ),

r 2 = x2 + y2.

If 0 < k < ∞, then y 2 u ∈ L∞ (Qk ). The proof is essentially the same as in [6]. In the case k < ∞, one needs only to use the partially periodic Fourier transform as in Lemma 3. Let us remark that the classical rational KP-solitons decay exactly as r−2 . On the other hand, in [24] a family of traveling waves is constructed for the original KP-I equations which are periodic in x and decay exponentially fast with respect to the transverse variable. Thus, it seems that the statement of Theorem 4 is not exact in the case k < ∞, while it is so in the case k = ∞. Also unknown are Zaitsev’s ground-wave solutions. 4. On Nonexistence of Traveling Waves In this section we turn to general KP equations (3), with ε = ±1, and discuss the nonexistence problem. We use the same approach as in [5]. However, the case of periodic waves is more involved (see the proof of Lemma 6). Here we consider traveling waves belonging to the space −1 uyy ∈ L2 (Qk ), f (u)u ∈ L1 (Qk )} Yk = {u ∈ Xk : u ∈ H 1 (Qk ), uxx , Dx,k

if k < ∞, and −1 Y = Y∞ = {u ∈ X : u ∈ H 1 (R2 ), uxx , Dx,k uyy ∈ L2loc (R2 ), f (u)u ∈ L1 (R2 )}.

First, we collect some useful identities. LEMMA 6. Suppose that f satisfies Assumptions (1) and (2). Let u ∈ Yk , k ∈ (0, ∞], be a solution of Equations (3). Then

44

A. PANKOV AND K. PFLÜGER

Z

v2 c 2 3 2 u + ux + ε − uf (u) + F (u) = 0, 2 2 Qk 2 Z v2 c 2 1 2 u + ux + ε + F (u) = 0, 2 2 Qk 2 Z [cu2 + u2x − εv 2 − f (u)u] = 0.

(15) (16) (17)

Qk

Proof. First, we remark that, for any k, (17) is an extention of the case ε = ±1 of Ik (u) = 0 stated in Section 2. Therefore, we concentrate on (15) and (16) only. In the case of solitary waves (k = ∞), the calculations carried out in the proof of Theorem 1.1, [5], work equally well for general nonlinearities. Therefore, we look at periodic waves (k < ∞). Fixed κ ∈ (0, 1), let ϕT ∈ C0∞ (R) be a nonnegative function such that ϕT = 1 on [−T /2, T /2], ϕT (x) = 0 if |x| > (T + T κ )/2, and ϕ (j ) (x) 6 Cj /|x|j , j = 1, 2, . . ., if T /2 6 |x| 6 (T + T κ )/2 (the construction of such a function will be given later on). Multiplying the first equation (3) by xϕT u and integrating over R2 , after a number of integrations by parts, we get Z Z Z Z 3 c 2 ϕT u − ϕT uf (u) + ϕT F (u) + ϕT u2x + 2 2 Z Z Z Z 1 1 2 0 2 0 ϕT v + xϕT u − xϕT uf (u) + xϕT0 F (u)+ +ε 2 2 Z Z Z Z 3 1 0 00 0 2 xϕT ux + ε xϕT0 v 2 = 0. + 2 ϕT uux + xϕT uux + 2 2 Dividing the last identity by T , we are going to pass to the limit as T → ∞. First, we point out that here the integrals containing ϕT are taken over QT ∪ Q0T ∪ Q00T = QT ∪ {(T /2, (T + T κ )/2) × R} ∪ {(−(T + T κ )/2, −T /2) × R}, while those containing ϕT0 and ϕT00 are over Q0T ∪ Q00T . Moreover, ϕT = 1 on QT . Now, let g ∈ L1loc (R2 ) be a function which is k-periodic in x. Then, it is easy to verify that Z Z 1 g= g. lim T →∞ T Q Qk T Next, dueRto the properties of ϕT , all the integrals over Q0T can be estimated from above by Q0 g, with a nonnegative k-periodic in x function g ∈ L1loc (R2 ). Now T Z Z 1 Tκ +1 g6 g. T Q0T T Qk

GROUND-TRAVELING WAVES AND KP EQUATIONS

45

This justifies the passage to the limit and gives rise to (15). Identity (16) can be proved exactly as (2.8), [5], with the only change: take the cut-off functions χj depending on y only. Now we construct the function ϕT . Fix ε > 0 and let ( 1 if x 6 T /2, g(x) = 1 − log(x/T ) if T /2 < x 6 (T + T κ )/2 + ε, 0 if x > (T + T κ )/2 + ε. ∞ We R choose a nonnegative function h ∈ C0 (R) such that supp h ⊂ (0, ε) and h = 1, and set Z ϕT (x) = ϕ˜T (|x|). ϕ˜T (x) = h(x − t)g(t) dt,

For this function, it is easy to verify all the properties we need.

2

THEOREM 5. Suppose that f ∈ C(R) satisfies Assumption (4). Then there is no nontrivial traveling wave u ∈ Yk , k ∈ (0, ∞], provided ε = +1, or ε = −1 and µ > 6. Proof. Adding (15), (16) and subtracting (17), we get Z Z 2 ux = −2ε v2. Qk

Qk

This rules out the case ε = +1. In the case ε = −1 (KP-I equations) the last identity together with (15) and (17), respectively, implies Z c 2 5 2 u + v − f (u)u + F (u) = 0 2 Qk 2 and

Z [cu2 + 3v 2 − f (u)u] = 0. Qk

Eliminating v, we get Z Z 2c u2 = [6F (u) − f (u)u]. Qk

Qk

If µ > 6, we have Z Z 2 2c u 6 [µF (u) − f (u)u] 6 0. Qk

Qk

Hence, u = 0 and we conclude.

2

46

A. PANKOV AND K. PFLÜGER

Acknowledgements This work was carried out during the visit of the first author (A.P.) to the Institut für Mathematik, Humboldt Universität Berlin (September–December, 1998) under the support of Deutsche Forschungsgemeinschaft. A.P. is very grateful to K. Gröger for his kind hospitality and a lot of stimulating discussions. The authors thank the anonimous referee for information on paper [24]. References 1.

Ablowitz, M. J., Segur, H. and Wang, X. P.: Wave collapse and instability of solitary waves of a generalized Kadomtsev–Petviashvili equation, Physica D 78(3–4) (1994), 241–265. 2. Bartsch, Th. and Willem, M.: Infinitely many radial solutions of a semilinear elliptic problem on RN , Arch. Rat. Mech. Anal. 124 (1993), 261–276. 3. Bourgin, J.: On the Cauchy problem for the Kadomtsev–Petviashvili equation, Geom. Funct. Anal. 3(4) (1993), 315–341. 4. De Bouard, A. and Saut, J.-C.: Remarks on the stability of generalized KP solitary waves, Contemp. Math. 200 (1996), 75–84. 5. De Bouard, A. and Saut, J.-C.: Solitary waves of generalized Kadomtsev–Petviashvili equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 211–236. 6. De Bouard, A. and Saut, J.-C.: Symmetry and decay of the generalized Kadomtsev–Petviashvili solitary waves, SIAM J. Math. Anal. 28 (1997), 1064–1085. 7. Kadomtsev, B. B. and Petviashvili, V. I.: On stability of waves in weakly dispersive media, Soviet Phys. Dokl. 15 (1970), 539–541, transl. from Dokl. AN SSSR 192 (1970), 753–756. 8. Krichever, I. M. and Novikov, S. P.: Holomorphic bundles over algebraic curves and nonlinear equations, Russ. Math. Surv. 35(6) (1980), 53–79, transl. from Uspekhi Mat. Nauk 35(6) (1980), 47–68. 9. Lions, P.-L.: The concentration-compactness method in the calculus of variations. The locally compact case. I, II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145, 223–283. 10. Liu Yue and Wang, X. P.: Nonlinear stability of solitary waves of a generalized Kadomtsev– Petviashvili equation, Comm. Math. Phys. 183 (1997), 253–266. 11. Lizorkin, P. I.: Multipliers of Fourier integrals, Proc. Steklov Inst. Math. 89 (1967), 269–290. 12. Lopes, O.: A constrained minimization problem with integrals on the entier space, Bol. Soc. Brasil Mat. (N.S.) 25 (1994), 77–92. 13. Nehari, Z.: On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960), 101–123. 14. Pankov, A. A.: Semilinear elliptic equations in Rn with nonstabilizing coefficients, Ukrainian Math. J. 41(9) (1989), 1075–1078, transl. from Ukrain. Mat. Zh. 41(9) (1989), 1247–1251. 15. Pankov, A. A.: On positive solutions of nonlinear elliptic equations on whole space, Soviet Math. Dokl. 44 (1991), 337–341, transl. from Dokl. AN SSSR 319(6) (1991), 1318–1321. 16. Pankov, A. A. and Pflüger, K.: On a semilinear Schrödinger equation with periodic potential, Nonlinear Anal. 33 (1998), 593–609. 17. Pankov, A. A. and Pflüger, K.: Periodic and solitary traveling waves for the generalized Kadomtsev–Petviashvili equation, Math. Meth. Appl. Sci. 22 (1999), 733–752. 18. Pouget, J.: Stability of nonlinear structures in a lattice model for phase transformations in alloys, Phys. Rev. B 46 (1992), 10554–10562. 19. Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conf. Ser. Math. 65, Amer. Math. Soc., Providence, 1986. 20. Saut, J.-C.: Remarks on the generalized Kadomtsev–Petviashvili equation, Indiana Univ. Math. J. 42 (1993), 1011–1026.

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21. 22. 23. 24.

47

Stein, E. M. and Weiss, G.: Introduction to Fourier Analysis on Euclidian Spaces, Princeton Univ. Press, Princeton, 1971. Willem, M.: Minimax Methods, Birkhäuser, Boston, 1996. Willem, M.: On the generalized Kadomtsev–Petviashvili equation, Rapp. Semin. Math. Louvain Nov. Ser. 245–260 (1996), 213–222. Zaitsev, A. A.: On formation of nonlinear stationary waves by means of superposition of solitons, Dokl. Akad. Nauk SSSR 272 (1983), 583–587 (in Russian).

Mathematical Physics, Analysis and Geometry 3: 49–74, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

49

Pole Dynamics for Elliptic Solutions of the Korteweg–deVries Equation BERNARD DECONINCK1 and HARVEY SEGUR2

1 Department of Applied Mathematics, Box 352420, University of Washington, Seattle, Washington,

98195, U.S.A. 2 Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, U.S.A. (Received: 21 April 1999; in final form: 24 December 1999) Abstract. The real, nonsingular elliptic solutions of the Korteweg–de Vries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This constraint is solvable for any finite number of poles located in the fundamental domain of the elliptic function, often in many different ways. Special consideration is given to those elliptic solutions that have a real nonsingular soliton limit. Mathematics Subject Classifications (2000): 34M05, 35A20, 35Q53, 37K10, 37K20. Key words: KdV equation, elliptic, finite gap solutions, pole dynamics, Calogero–Moser.

1. Introduction In 1974, Kruskal [15] considered the interaction of solitons governed by the Korteweg–de Vries equation (KdV), ut = 6uux + uxxx .

(1.1)

Each KdV soliton is defined by a meromorphic function in the complex x-plane (i.e., sech2 k(x − x0 )), so Kruskal [15] suggested that the interaction of two or more solitons could be understood in terms of the dynamics of the poles of these meromorphic functions in the complex x-plane, where the poles move according to a force law deduced from (1.1). This was followed by the work of Thickstun [17] who considered the case of two solitons in great detail. Following a different line of thought, Airault, McKean and Moser [2] studied rational and elliptic solutions of the KdV equation. An elliptic solution of KdV is by definition a solution of the KdV equation that is doubly periodic and meromorphic in the complex x-plane, for all time. Note that the soliton case is an intermediary case between the elliptic and the rational case. It was treated as such in [2]. Airault, McKean and Moser [2] approached these elliptic solutions and their degenerate limits through the motion of their poles xi (t) in the complex x-plane. In particular, they looked for elliptic KdV solutions of the form u(x, t) = −2

N X i=1

℘ (x − xi (t)).

(1.2)

50

BERNARD DECONINCK AND HARVEY SEGUR

Here ℘ (z) denotes the Weierstrass elliptic function. It can be defined by its meromorphic expansion X 1 1 1 − , (1.3) ℘ (z) = 2 + 2 2 z (z + 2mω + 2nω ) (2mω + 2nω ) 1 2 1 2 (m,n)6=(0,0) with ω1 /ω2 not real.? More properties of the Weierstrass function will be given as they are needed. It is shown in [2] that the dynamics of the poles xi (t) is governed by the dynamical system x˙i = 12

N X

℘ (xi − xj ),

i = 1, 2, . . . , N,

(1.4a)

j =1,j 6=i

(the dot denotes differentiation with respect to time) with the invariant constraint N X

℘ 0 (xi − xj ) = 0,

i = 1, 2, . . . , N.

(1.4b)

j =1,j 6=i

Here the prime denotes differentiation with respect to the argument. The solutions (1.2) generalize an elliptic solution given earlier by Dubrovin and Novikov [7], corresponding to the case N = 3. These authors also recall the Lamé–Ince potentials [13] u(x) = −g(g + 1)℘ (x),

(1.5)

which are the simplest g-gap potentials of the stationary Schrödinger equation ∂ 2ψ + u(x)ψ = λψ. ∂x 2

(1.6)

The remarkable connection between the KdV equation and the stationary Schrödinger equation has been known since the work of Gardner, Greene, Kruskal and Miura [9]. Dubrovin and Novikov show [7] that the (N = 3)-solution discussed in [7] is a 2-gap solution of the KdV equation with a 2-gap Lamé–Ince potential as initial condition. If one considers the rational limit of the solution (1.2) (i.e., the limit in which the Weierstrass function ℘ (z) reduces to 1/z2 ), then the constraint (1.4b) is solvable only for a triangular number of poles, N=

n(n + 1) , 2

(1.7)

? In this paper it is always assumed that ω is real and ω is imaginary. This is necessary to 1 2

ensure reality of the KdV solution (1.2) when x is restricted to the real x-axis. Other considerations for reality of the elliptic KdV solutions will be discussed in Section 4.

POLE DYNAMICS FOR ELLIPTIC SOLUTIONS OF THE KdV EQUATION

51

for any positive integer n [2]. Notice that the Lamé–Ince potentials are given by g(g + 1)/2 times an N = 1 potential. Based on these observations, it was conjectured in [2] that also in the elliptic case given by (1.2), the constraint (1.4b) is solvable only for a triangular number N, ‘or very nearly so’. From the moment it appeared this conjecture was known not to hold, because it already fails in the soliton case, where the Weierstrass function degenerates to hyperbolic functions. This failure of the conjecture easily follows from the work of Thickstun [17]. A further understanding of the elliptic case had to wait until 1988, when Verdier provided more explicit examples of elliptic potentials of the Schrödinger equation [21]. Subsequently, Treibich and Verdier demonstrated that 4 X gi (gi + 1) ℘ (x − x0 − ωi ) u(x) = −2 2 i=1

(1.8)

(ω3 = ω1 + ω2 , ω4 = 0, the gi are positive integers) are finite-gap potentials of the stationary Schrödinger equation (1.6) and, hence, result in elliptic solutions of the KdV equation [19, 18, 20]. The potentials of Treibich and Verdier were generalized by Gesztesy and Weikard [10, 11]. They showed that any elliptic finite-gap potential of the stationary Schrödinger equation (1.6) can be represented in the form u(x) = −2

M X gi (gi + 1) ℘ (x − αi ), 2 i=1

(1.9)

for some M and positive integers gi . Notice that this formula coincides with (1.2) if all the gi are 1. The focus of this paper is the constrained dynamical system (1.4a–b). We return to the ideas put forth by Kruskal [15] and Thickstun [17]. This allows us to derive the system (1.4a–b) in a context which is more general than [2]: there it was obtained as a system describing a class of special solutions of the KdV equation. Here, it is shown that any meromorphic solution of the KdV equation which is doubly periodic in x is of the form (1.2). Hence the consideration of solutions of the form (1.2) and the system of equations (1.4a–b) leads to all elliptic solutions of the KdV equation. Simultaneously, some of the results of Gesztesy and Weikard [10, 11] are recovered here. Because of the connection between the KdV equation and the Schrödinger equation, any potential of Gesztesy and Weikard can be used as an initial condition for the KdV equation, which determines any time dependence of the parameters of the elliptic KdV solution with that initial condition. Our approach demonstrates which parameters in the solutions (1.9) are time independent and which are time dependent. The following conclusions are obtained in this paper:

52

BERNARD DECONINCK AND HARVEY SEGUR

• All finite-gap? elliptic solutions of the KdV equation are of the form (1.2), with all xi (t) distinct for almost all times (see below). In other words, if u(x, t) is a finite-gap KdV solution that is doubly periodic in the complex x-plane, then u necessarily has the form (1.2) except at isolated instants of time. • Any number N 6= 2 of xj is allowed in (1.2). This is trivially true, since it follows from the work of Thickstun [17] that it is true in the soliton limit, which is a special case of the elliptic case. As a consequence, the constraint (1.4b) is solvable for any positive integer?? N 6= 2. In our numerical method, the elliptic case is viewed as a deformation of the soliton limit of the system (1.4a–b). This viewpoint is useful because it provides good initial guesses for many of the numerical solutions of (1.4b) in Section 5. This deformation concept does not lead to all elliptic solutions of the KdV equation, but only to those that have nonsingular soliton limits. In particular, we are unable to find the solutions corresponding to the Treibich–Verdier potentials (1.8) in this way; for these solutions it is necessary to find an initial guess by some other means. In Section (5.2), a solution corresponding to one particular Treibich–Verdier potential is discussed. It is the only solution discussed in this paper that does not have a nonsingular soliton limit. Its inclusion allows us to point out some differences with the other examples in a very concrete way. • If |ω1 /ω2| 1, then for a given N > 4, nonequivalent configurations satisfying the constraint exist that do not flow into each other under the KdV flow and which cannot be translated into each other. To the best of our knowledge this is a new result. • The xi are allowed to coincide, but only in triangular numbers: if some of the xi coincide at a certain time tc , then gi (gi + 1)/2 of them coincide at that time tc . At this time tc , the solution can be represented in the form (1.9) with not all gi = 1. Such times tc are referred to as collision times and the poles are said to collide at the collision time. Before and after each collision time all xi are distinct, hence pole collisions are isolated events. At the collision times, the dynamical system (1.4a) is not valid. The dynamics of the poles at the collision times is easily determined directly from the KdV equation. Gesztesy and Weikard [10, 11] demonstrate that (1.9) are elliptic finite-gap potentials of the Schrödinger equation. These potentials generalize to solutions of the KdV equation, but this requires the gi to be nonsmooth functions of time. Only at the collision times tc are the gi not all one. Furthermore, at all times but the collision times, the number of parameters αi (which are time P g dependent) is N = M i=1 i (gi +1)/2. This shows that the generalization from [10, 11] to solutions of the KdV equation is nontrivial. ? See Section 3. ?? The constraint (1.4b) is solvable for N = 2 [2]. As noted there, the corresponding solution

reduces to a solution for N = 1, with smaller periods. This case is therefore trivial and is disregarded.

POLE DYNAMICS FOR ELLIPTIC SOLUTIONS OF THE KdV EQUATION

53

In the terminology of Chapter 7 of [3], the poles are in general position if all gi are equal to one. Otherwise, if not all gi = 1, the poles are said to be in special position. We conclude that at the collision times the poles are in special position. Otherwise they are in general position. • The solutions discussed here are finite-gap potentials of the stationary Schrödinger equation (1.6), with t treated as a parameter. In other words, each solution specifies a one-parameter family of finite-gap potentials of the stationary Schrödinger equation. It follows from our methods that to obtain a g-gap potential that corresponds to a nonsingular soliton potential, one needs at least N = g(g + 1)/2 poles xj . The Lamé–Ince potentials show that this lower bound is sharp. If we consider potentials that do not have soliton limits (such as the Treibich–Verdier potentials (1.8)) then it may be possible to violate this lower bound. • In Section 5, we present an explicit solution of the form (1.2) with N = 4. Notice that if in (1.8) all gi are one, the solution is reducible to a solution with N = 1 and smaller periods. To see this it is convenient to draw the pole configuration corresponding to (1.8) in the complex plane. This is actually true, even if gi is not one, but all gi are equal. In that case, (1.8) reduces to a Lamé–Ince potential (1.5). Unlike any of the Treibich–Verdier solutions, the (N = 4)-solution, presented in Section 5, has a nonsingular soliton limit. The first five conclusions are all discussed in Section 4. In Section 2, the results of Kruskal [15] and Thickstun [17] for the soliton solutions of the KdV equation are reviewed, but they are obtained from a point of view that is closer to the approach we present in Section 3 for the elliptic solutions of the KdV equation. Finally, in Section 5, some explicit examples are given, including illustrations of the motion of the poles in the complex x-plane. 2. The Soliton Case: Hyperbolic Functions In this section, the results of Kruskal [15] and Thickstun [17] for the dynamics of poles of soliton solutions are discussed from a point of view that will allow us to generalize to the periodic case. Consider the one-soliton solution of the KdV equation u(x, t) = 2k 2 sech2 k(x + 4k 2 t − ϕ).

(2.1)

Here k is a positive parameter (the wave number of the soliton) determining the speed and the amplitude of the one-soliton solution. Using the meromorphic expansion [12] ∞ X 1 1 2x cosech = 2 T T (x + inπ T )2 n=−∞

(2.2)

54

BERNARD DECONINCK AND HARVEY SEGUR

(uniformly convergent except at the points x = inπ T ), one easily obtains the following meromorphic expansion for the one-soliton solution of the KdV equation: u(x, t) = −2k 2

∞ X n=−∞

(k(x +

4k 2 t

1 . − ϕ) + i π2 + inπ )2

(2.3)

From this expression, one easily finds that the locations of the poles of the onesoliton solution of the KdV equation for all time are given by iπ 1 2 xn = ϕ − 4k t − n+ . (2.4) k 2 This motion is illustrated in Figure 1a. Notice that the locations of the poles are symmetric with respect to the real x-axis. This is a consequence of the reality of the solution (2.1). In order for a solution to be real it is necessary and sufficient that if xn (t) is a pole, then so is xn∗ (t), where ∗ denotes the complex conjugate.? The closest distance between any two poles is d = π/k and is constant both along the vertical line Re(x) = ϕ − 4k 2 t and in time. Note that the poles are moving to the left. This is a consequence of the form of the KdV equation (1.1), which has time reversed, compared to the version Kruskal [15] and Thickstun [17] used. Since two solitons of the KdV equation cannot move with the same speed, a two-soliton solution of the KdV equation asymtotically appears as the sum of two one-soliton solutions which are well-separated: the higher-amplitude soliton, which is faster, is to the right of the smaller-amplitude soliton as t → −∞. As t → ∞, the higher-amplitude soliton is to the left of the smaller-amplitude soliton. Hence as t → −∞, the pole configuration of a two-soliton solution with wave numbers k1 and k2 is as in Figure 1b. In this limit, the two-soliton solution is a sum of two one-soliton solutions. Each results in a vertical line of equispaced poles, with interpolar distance respectively d1 = π/k1 and d2 = π/k2 . As long as the solitons are well-separated, these poles move in approximately straight lines, parallel to the real axis, with respective velocities v1 = −4k12 and v2 = −4k22 . Since |v1 | > |v2 |, the solitons interact eventually. This interacting results in nonstraight line motion of the poles. After the interaction, the situation is as in Figure 1b, but with the two lines of poles interchanged. Thickstun [17] considered the case where k1 and k2 are rationally related, so k1 /k2 = p/q, where p and q are positive integers. In this case, one can define D = pd1 = qd2 . The complex x-plane is now divided into an infinite number of equal strips, parallel to the real x-axis, each of height D. The real x-axis is usually taken to be the base of such a strip. It is easy to show [17] that the motion of the poles in one strip is repeated in every strip. Hence, one is left studying the motion of a finite number N (= p + q) of poles in the fundamental strip, whose base is the ? The vertical line of poles can be rotated arbitrarily. The expression (2.3) still results in a solution

of the KdV equation, but it is no longer real. Again, we will only consider real, nonsingular solutions, when restricted to the real x-axis.

POLE DYNAMICS FOR ELLIPTIC SOLUTIONS OF THE KdV EQUATION

55

(a)

(b) Figure 1. (a) The motion of the poles of a one-soliton solution in the complex plane. (b) The asymptotic motion of the poles of a two-soliton solution.

56

BERNARD DECONINCK AND HARVEY SEGUR

real x-axis. Thickstun examined this motion by analyzing the exact expression for a two-soliton solution of the KdV equation. Any two-soliton solution is expressible as [1] u(x, t) = 2∂x2 ln τ (x, t).

(2.5)

It follows from this formula that the poles of u(x, t) are the zeros of τ (x, t) if τ (x, t) is entire in x. Then the Weierstrass Factorization Theorem [5] gives a factorization for τ (x, t): ∞ Y x x/xk 1− e . (2.6) τ (x, t) = C xk k=1 Since only the second logarithmic derivative of this function is relevant, the constant C is not important. If the solution is periodic in the imaginary x-direction, this is rewritten as N Y ∞ Y x 1− ex/(xn +ilD) , (2.7) τ (x, t) = C x + ilD n n=1 l=−∞ where the first product runs over the poles in the fundamental strip. The second product runs over all strips. Using the uniform convergence of (2.7), u(x, t) = −2

N X ∞ X

1 , (x − xn − ilD)2 n=1 l=−∞

(2.8)

which, using (2.2), is rewritten as N π(x − xn ) π2 X cosech2 u(x, t) = −2 2 D n=1 D

= −2

p+q k2 (x − xn ) k22 X cosech2 , 2 q n=1 q

(2.9)

where the pole locations xn depend on time: xn = xn (t). One recovers the onesoliton solution (2.1) easily, by equating k1 = 0, p = 0, q = 1. Equation (2.9) essentially expresses a two-soliton solution as a linear superposition of N onesoliton solutions with nonlinearly interacting phases. Note that the first equality in (2.9) is valid for arbitrary soliton solutions that are periodic in x with period iD. This is the case for a g-soliton solution if its wavenumbers ki , i = 1, 2, . . . , g are all commensurable: (k1 : k2 : . . . : kg ) = (p1 : p2 : . . . : pg ), for positive distinct integers pi , i = 1, 2, . . . , g which have no overall common integer factor. The total number of poles in a strip is then N = p1 + p2 + · · · + pg . In obtaining (2.8) and (2.9), we have deviated from Thickstun’s approach [17] to an approach that is generalized to the elliptic case of the next section in a straightforward way.

POLE DYNAMICS FOR ELLIPTIC SOLUTIONS OF THE KdV EQUATION

57

Next, we derive the dynamics imposed on the poles xn (t) by the KdV equation. This is conveniently done by substituting (2.8) into (1.1) and examining the behaviour near one of the poles: x = xn + . This results in several singular terms as → 0, corresponding to negative powers of . The dynamics of the poles is then determined by the vanishing of the coefficients of these negative powers and the zeroth power. This results in only two nontrivial equations, obtained at order −3 and −2 respectively: ∞ X 1 1 x˙n = 12 + 12 , 2 (xk − xn − ilD) (−ilD)2 k=1,k6=n l=−∞ l6=0,l=−∞ N X

0=

N X

∞ X

∞ X

1 , (xk − xn − ilD)3 k=1,k6=n l=−∞

(2.10a)

(2.10b)

for n = 1, 2, . . . , N. Using (2.2) and its derivative, x˙n = −4 0=

N π2 π2 X π(xk − xn ) , + 12 cosech2 2 2 D D k=1,k6=n D

N X k=1,k6=n

cosech2

π(xk − xn ) π(xk − xn ) coth , D D

(2.11a)

(2.11b)

for n = 1, 2, . . . , N. Hence the dynamics of the poles xn (t) is determined by (2.11a). This dynamics is constrained by the equations (2.11b). These constraint equations (2.11b) are invariant under the flow of (2.11a). This follows from a direct calculation. Remarks • Since the KdV equation has two-soliton solutions for any ratio of the wavenumbers k1 /k2 6= 1, the constraint (2.11b) is solvable for any value of N, excluding N = 2, which can only be obtained by p = q = 1, resulting in equal wavenumbers k1 and k2 . • In particular, it follows that for almost all times (i.e., all times except collision times) the minimum number of poles in a fundamental strip required to obtain a g-soliton solution is N = 1 + 2 + 3 + · · · + g = g(g + 1)/2, corresponding to a g-soliton solution with wavenumbers which are related as (k1 : k2 : . . . : kg−1 : kg ) = (g : g − 1 : . . . : 2 : 1). • Equating k1 = 0, p = 0, q = 1, one obtains from (2.11a) x˙1 = −4k22 , corresponding to the dynamics of the one-soliton solution. The asymptotic behavior of the poles of a two-soliton solution also follows from (2.11a): from the separation of the poles into distinct vertical lines, it follows from (1.4a) that the velocity of these vertical lines is given by the one-soliton velocity for

58

BERNARD DECONINCK AND HARVEY SEGUR

each line, as expected. This result follows from easy algebraic manipulation and the identity p−1 p2 − 1 X 2 nπ , (2.12) cosec = 3 p n=1 valid for any integer p > 1. • A full analysis of the interaction of the poles for the case of any two-soliton solution with k1 /k2 = p/q is given in [17].

3. The Elliptic Case Consider the quasiperiodic finite-gap solutions of the KdV equation with g phases [14] u(x, t) = 2∂x2 ln θg (kx + ωt + φ|B), where θg (z|B) =

X

exp

1 2

(3.1)

m · B · m + im · z ,

(3.2)

m∈Z g

a hyperelliptic Riemann theta function of genus g. The g × g real Riemann matrix (i.e., symmetric and negative definite) B originates from a hyperelliptic Riemann surface with only one point at infinity. Furthermore, k, ω and φ are g-dimensional vectors. The derivation of equations (2.9), (2.11a) and (2.11b) is easily generalized to the case where the solution is periodic not only in the imaginary x-direction, but also in the real x-direction: u(x + L1 , t) = u(x, t) = u(x + iL2 , t).

(3.3)

This divides the complex x-plane into an array of rectangular domains, each of size L1 × L2 . One of these domains, called the fundamental domain S, is conveniently placed in the lower left corner of the first quadrant of the x-plane. The theta function has the property [6] 1 θg (z + iB · M + 2π N|B) = θg (z|B) exp − M · B · M + iM · z , (3.4) 2

for any pair of g-component integer vectors M, N. This expression is useful to determine conditions on the wavevector k and on the Riemann matrix B in order for u(x, t), given by (3.1), to satisfy (3.3): ∃N0 , M0 ∈ Z g : kL1 = 2π N0 , kL2 = B · M0 .

(3.5)

POLE DYNAMICS FOR ELLIPTIC SOLUTIONS OF THE KdV EQUATION

59

These results are now used to determine the number of poles N of u(x, t) in the fundamental domain. The poles of (3.1) are given by the zeros of ϑ(x, t) = θg (kx+ ωt + φ|B), regarded as a function of x: I 1 N= d ln ϑ(x, t) 2π i ∂S Z L1 Z iL2 1 d ln ϑ(x, t) + d ln ϑ(x + L1 , t)− = 2π i 0 0 ! Z L1 Z iL2 − d ln ϑ(x + iL2 , t) − d ln ϑ(x, t) 0

Z

0 L1

1 ϑ(x, t) d ln 2π i 0 ϑ(x + iL2 , t) Z L1 using (3.4) 1 = d(−ixM0 · k) 2π i 0 L1 = −M0 · N0 = − M0 · B · M0 . 2π L2 using (3.4)

=

(3.6)

The first equality of (3.6) confirms that N is an integer. The second equality shows that N is positive, by the negative-definiteness of B. We now proceed to determine the dynamical system satisfied by the motion of the N poles of u(x, t) in the fundamental domain S. Again, the poles of u(x, t) are the zeros of ϑ(x, t). Furthermore, zeros of ϑ(x, t) result in double poles of u(x, t), as in the hyperbolic case. The Weierstrass Factorization theorem [5] gives the following form for ϑ(x, t): x x2 Y x x + 2 cx 2 /2 1− e k 2xk , (3.7) ϑ(x, t) = e xk k where the product runs over all poles xk . The additional exponential factors, as compared to (2.6), are required because the poles now appear in a bi-infinite sequence: both in the vertical and horizontal directions. These exponential factors ensure uniform convergence of the product. The parameter c is allowed to depend on time. It determines the behavior of ϑ(x, t) as x approaches infinity in the complex x-plane [6]. Using (3.3), this is rewritten as ∞ N ∞ Y Y Y x 2 1− × ϑ(x, t) = exp(cx /2) xn + mL1 + ilL2 n=1 m=−∞ l=−∞ x2 x × exp . (3.8) + xn + mL1 + ilL2 2(xn + mL1 + ilL2 )2 The first product runs over the number of poles (N) in the fundamental domain, the second and third products result in all translations of the fundamental domain.

60

BERNARD DECONINCK AND HARVEY SEGUR

From the uniform convergence of (3.8), u(x, t) = 2c − 2

∞ N ∞ X X X n=1 m=−∞ l=−∞

1 − (x − xn − mL1 − ilL2 )2 1 − . (xn + mL1 + ilL2 )2

(3.9)

Using the definition of the Weierstrass function (1.3), this is rewritten as u(x, t) = 2c − 2

N X

℘ (x − xn ) + 2

n=1

N X

℘ (xn ),

(3.10)

j =1

where the periods of the Weierstrass function are given by 2ω1 = L1 , 2ω2 = iL2 . Define c˜ = 2c + 2

N X

℘ (xn ).

(3.11)

n=1

The dynamics of the poles xn = xn (t) is determined by substitution of (3.10) or (3.9) into the KdV equation and expanding in powers of for x near a pole: x = xk + . Equating the coefficients of −3 , −2 and 0 to zero result in N X

x˙n = 12

℘ (xj − xn ),

(3.12a)

j =1,j 6=n N X

0=

℘ 0 (xj − xn ),

(3.12b)

j =1,j 6=n

,c˙˜ = 0 ⇐⇒ c(t) ˜ =α=0

(3.12c)

for n = 1, 2, . . . , N. (The constant α can always be removed by a Galilean shift, so it is equated to zero, without loss of generality.) The constraints (3.12b) are invariant under the flow, as can be checked by direct calculation. Notice that (3.12a–b) are identical to the equations obtained by Airault, McKean and Moser [2]. These equations are obtained here in greater generality: any solution (3.1) that is doubly periodic in the x-plane gives rise to a system (3.12a–b). This allows us to reach the conclusions stated in the next section. Remarks • In the limit L1 → ∞, Equations (3.12a–b) reduce to (2.11a–b). This limit is most conveniently obtained from the Poisson representation of the

POLE DYNAMICS FOR ELLIPTIC SOLUTIONS OF THE KdV EQUATION

61

Weierstrass function: ℘ (x) =

2

πx 1 + + cosech2 3 L2 ! ∞ X 2 π 2 nπ L1 cosech . (3.13) + (x + nL1 ) − cosech L L 2 2 n=−∞,n6=0 π L2

This representation is obtained from (1.3) by working out the summation in the vertical direction. It gives the Weierstrass function as a sum of exponentially localized terms, hence few terms have important contributions in the fundamental domain. A Poisson expansion for ℘ 0 (x) is obtained from differentiating (3.13) term by term with respect to x. • Define the one-phase theta function θ1 (z, q) [12]: ∞ X 2 θ1 (z, q) = 2 (−1)n q (n+1/2) sin(2n + 1)z,

(3.14)

n=0

with |q| < 1. If L2 < L1 , then the relationship ℘ (z) = a − ∂x2 ln θ1 (π z/L1, iL2 /L1 ), with a a constant [12], allows us to rewrite (1.2) as u(x, t) = aˆ +

2∂x2

N Y L2 π , ln θ1 (x − xj (t)), i L1 L1 j =1

(3.15)

with aˆ = −2aN. Hence, for the doubly-periodic solutions of the KdV equation of the form (3.1), it is possible to rewrite the g-phase theta function as a product of N 1-phase theta functions, with nonlinearly interacting phases. Note that this does not imply that the g-phase theta function appearing in (3.1) is reducible. Reducible theta functions do not give rise to solutions of the KdV equation [6]. • By taking another time derivative of (3.12a) and using (3.12b), one obtains x¨n = −(12)

2

N X

℘ 0 (xj − xn )℘ (xj − xn ).

(3.16)

j =1,j 6=n

It is known that this system of differential equations is Hamiltonian [4], with Hamiltonian

62

BERNARD DECONINCK AND HARVEY SEGUR N N N 1 X 2 (12)2 X X H= p + ℘ 2 (xk − xj ), 2 k=1 k 2 k=1 j =1,j 6=k

(3.17)

and canonical variables {xk , pk = x˙k }. A second Hamiltonian structure for the equations (3.16) is given in [4]. A Lax representation for the system (3.12a–c) is also given there. This Lax representation is a direct consequence of the law of addition of the Weierstrass function [12]. It is unknown to us whether a Hamiltonian structure exists for the constrained first-order dynamical system (3.12a–b). The Hamiltonian structure (3.17) shows that the system (3.12a–b) is a (constrained) member of the elliptic Calogero–Moser hierarchy [3].

4. Discussion of the Dynamics In this section, the constrained dynamical system (3.12a–c) is discussed. In particular, the assertions made in the introduction are validated here. For reality of the KdV solution (3.1) when x is restricted to the real line, it is necessary and sufficient that if xj (t) appears, then so does xj∗ (t). Because the Weierstrass function is a meromorphic function of its argument that is real valued on the real line, this reality constraint is invariant under the dynamics (3.12a). As a consequence, the distribution of the poles in the fundamental domain S is symmetric with respect to the horizontal centerline of S. Poles are allowed on the centerline. Most of what follows is valid for both real KdV solutions? and KdV solutions that are not real, but we restrict our attention to real KdV solutions. 4.1.

ALL FINITE - GAP ELLIPTIC SOLUTIONS OF THE KDV EQUATION ARE OF THE FORM (1.2) , UP TO A CONSTANT

A straightforward singularity analysis of the KdV equation [2] shows that any algebraic singularity of a solution of the KdV equation is of the type u(x, t) = −2/(x − α(t))2 + O(x − α), for almost all times t. At isolated times tc , the leading order coefficient is not necessarily −2. It can be of the form −g(g +1) (see below), but the exponent of the leading term is always −2. Hence, an elliptic function ansatz for u(x, t) can only have second-order poles and with the substitution u(x, t) = 2∂x2 ln ϑ(x, t) gives rise to a Weierstrass expansion of the form (3.8), with an arbitrary prefactor exp(c(x, t)), for an arbitrary function c(x, t), entire in x. Substitution of this ansatz in the KdV equation then determines that cxx is doubly periodic and meromorphic in x. The only such c is a constant. Hence, all finite-gap elliptic solutions of the KdV equation are of the form (1.2). ? ‘Real KdV solution’ refers to a solution of the KdV equation which is real when x is restricted to the real x-axis.

POLE DYNAMICS FOR ELLIPTIC SOLUTIONS OF THE KdV EQUATION

4.2.

63

IF L1 /L2 1, ANY NUMBER OF POLES N 6= 2 IN THE FUNDAMENTAL DOMAIN IS ALLOWED

This immediately follows from the soliton case, which is a special case of the elliptic case. Thickstun’s results [17] already implied that any value of N 6= 2 occurs. We do not provide a direct proof for finite values of L1 . On the other hand, the numerical evidence presented in Section 5 supports this statement. We have already argued that the equations (2.9) and (2.11a–b) are obtained from (1.2) and (3.12a–b) in the limit L1 → ∞. On the other hand, (3.13) allows us to rewrite each term in (1.2) as an infinite sum of solitons, each of which is localized in a different real-period interval of the solution. In the limit L1 → ∞, only two terms in each one of these series remain: the constant term and a onesoliton term. This gives rise to the soliton limit of the solution (2.9). The terms that vanish as L1 → ∞ are then regarded as a deformation of the soliton limit. Similar deformations are valid at the level of the dynamical system and the constraints. The success of the numerical method prompts us to formulate the following CONJECTURE. Every nonsingular soliton solution that is periodic in ix with period iL2 is obtained as the limit in which L1 → ∞ of an elliptic solution with periods iL2 and L1 . At this point, it is appropriate to remark that if one is interested in elliptic solutions of the Kadomtsev–Petviashvili (KP) equation, ∂x (−ut + 6uux + uxxx ) + 3σ 2 uyy = 0

(4.1)

(with parameter σ ), then the Equations (1.2), (3.12a–b) are replaced by [4] u(x, y, t) = −2

N X

℘ (x − xn (y, t)),

(4.2a)

n=1

2 N X ∂xn 2 ∂xn = 3σ + 12 ℘ (xj − xn ), ∂t ∂y j =1,j 6=n σ2

N X ∂ 2 xn = −16 ℘ 0 (xj − xn ). ∂y 2 j =1,j 6=n

(4.2b)

(4.2c)

This clarifies the appearance of the constraint (3.12b) on the motion of the poles of elliptic solutions of the KdV equation, where the poles are independent of y. For y-independent solutions, the KP equation reduces to the KdV equation, and Equations (4.2a–c) reduce to (1.2), (3.12a–b), forcing the poles to remain on the invariant manifold defined by (3.12b). For the KP equation, no such constraint exists and the number of poles N in the fundamental domain can be any integer, not equal to two.

64 4.3.

BERNARD DECONINCK AND HARVEY SEGUR

FOR ANY N > 4, NONEQUIVALENT CONFIGURATIONS EXIST, FOR L1 /L2 SUFFICIENTLY LARGE

Consider the asymptotic behavior for t → −∞ of limL1 →∞ u(x, t). In this soliton limit, as t → −∞, the poles are collected in groups corresponding to one-soliton solutions. In this section, two configurations are called nonequivalent if the above asymptotic behavior results in two different groupings of the poles. For N = 3, all configurations are equivalent to one configuration. In the limit L1 → ∞, this configuration corresponds to the two-soliton case with k1 : k2 = 2 : 1. This configuration is discussed in Section 5.1. For N = 4, all configurations are again equivalent to one configuration. This configuration corresponds to the two-soliton case with k1 : k2 = 3 : 1. Recall that k1 and k2 are not allowed to be equal, hence a configuration with two poles to the left and two poles to the right does not exist. Another way of expressing that only one configuration exists is that N = 4 can only be decomposed in one way as the sum of distinct positive integers without common factor (> 1), namely as N = 3 + 1. Again, all N = 4 configurations are equivalent. This configuration is discussed in Section 5.2. That section also discusses another example of an N = 4 potential which does not have a nonsingular soliton limit. This potential is a special case of one of the Treibich–Verdier potentials (1.8). Any integer N > 4 can be written as a sum of distinct positive integers without number of terms in the m-th overall common factor in more than one way.? Let theP m decomposition of N be denoted as Nm , then N = N i=1 ni , with the ni distinct and having no overall common factor. This configuration corresponds to the Nm soliton case with wavenumber ratios (k1 : k2 : . . . : kNm ) = (n1 : n2 : . . . : nNm ). A solution with these wave numbers has Nm phases and is an Nm -soliton solution. Hence for any N > 4 there exist at least as many different configurations as there are decompositions of N into distinct positive integers, without overall common factor. These configurations need not have the same number of phases. Two nonequivalent configurations corresponding to N = 5 are discussed in Section 5.3. 4.4.

THE POLES ONLY COLLIDE IN TRIANGULAR NUMBERS

A collision of poles is a local process in which only the colliding poles play a significant role. The analysis of the collisions is identical to that of the rational and the soliton cases because close to the collision point, the Weierstrass function reduces to 1/x 2 . Kruskal [15] already noticed that the poles do not collide in pairs, but triple collisions do occur. In fact, any triangular number of poles can participate in a collision, in which case the solution of the KdV equation at the collision time tc , nearby the collision point xc is given by u(x, tc ) = −g(g + 1)/(x − xc )2 . ? 5 = 1 + 4 = 2 + 3, N = 1 + (N − 1) = 1 + 2 + (N − 3), for N > 5.

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Asymtotically near the collision point xc , before the collision, i.e., t < tc , the poles lie on the vertices of a regular polygon [2] with g(g + 1)/2 vertices. For t > tc , the poles emanate from the collision points, again forming a regular polygon with g(g + 1)/2 vertices [2]. If g(g + 1)/2 is even, this polygon is identical to the polygon before the collision. If g(g + 1)/2 is odd, the polygon is rotated around the collision point by 2π/g(g + 1) radians. Of all these collision types, the one where three poles collide (corresponding to g = 2) is generic. It is the one observed in the examples illustrated in Section 5. Since the poles only collide in triangular numbers, it is possible that at any given (1.9), with not all gi = 1. time tc the solution of the KdV equation has the form P At almost every other time t, such a solution has N = M i=1 gi (gi + 1)/2 distinct poles. 4.5.

THE SOLUTIONS (1.2) ARE FINITE - GAP POTENTIALS OF THE STATIONARY SCHRÖDINGER EQUATION (1.6)

By construction the solutions (1.2) are periodic in x because they are obtained as a Weierstrass factorization of the theta function appearing in (3.1), upon which we have imposed the double periodicity. Hence the solutions (1.2) are finite-gap potentials of the Schrödinger operator. In [11], another proof of this can be found. For solutions that are elliptic deformations of the nonsingular solitons of Section 2, more can be said: an elliptic deformation of a g-soliton solution is a ggap potential of the Schrödinger equation. The reasoning is as follows: we already know that any elliptic deformation results in a finite-gap potential of the Schrödinger equation. On the other hand, any finite gap potential of the Schrödinger equation is of the form (3.1). The soliton limit of such a finite-gap solution with g-phases is a g-soliton solution [3]. Hence the number of phases of an elliptic deformation of a g-soliton solution is equal to g. This limit is the soliton limit of the periodic solutions, in which the fundamental domain reduces to the fundamental strip. In order to have a g-soliton solution of the KdV equation, we remarked in Section 2 that at least N = g(g + 1)/2 = 1+2+· · ·+g poles are required in the fundamental strip. Hence, this many poles are required in the fundamental domain to obtain a g-gap potential of the Schrödinger equation that is an elliptic deformation of a nonsingular soliton solution. 5. Examples In this section, some explicit examples of elliptic solutions of the KdV equation are discussed. These are illustrated with figures displaying the motion of the poles in the fundamental domain. Other figures display the solution of the KdV equation u(x, t) as a function of x and t. All these figures were obtained from numerical solutions of the corresponding constrained dynamical system. In all cases, the constrained dynamical system was solved using a projection method: the dynamical

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system (3.12a) is used to evolve the system for some time. Subsequently, the new solution is projected onto the constraints (3.12b) to correct numerical errors, after which the process repeats. In all examples given, L1 = 4 and L2 = π . This seems to indicate that one can wander far away from L1 /L2 1 and still obtain soliton-like elliptic solutions of the KdV equation. This is not surprising as (3.13) indicates that as perturbation parameter on the soliton case one should use = exp(−2π L1 /L2 ). For the values given above, this gives = 0.00034.

5.1.

THE SOLUTION OF DUBROVIN AND NOVIKOV [7]: N = 3

Dubrovin and Novikov [7] integrated the KdV equation with the Lamé–Ince potential u(x, 0) = −6℘ (x − xc ) as initial condition. They found the solution to be elliptic for all time, with N = 3. They gave explicit formulae for the solution, which they remarked was probably the simplest two-gap solution of the KdV equation. The dynamics of the poles in the fundamental domain is displayed in Figure 2a. Figure 2b displays the corresponding two-phase solution of the KdV equation. Animations of the behavior of the poles and of u(x, t) as t changes are also available at http://amath-www.colorado.edu/appm/other/kp/papers. Notice the solitonlike interactions of the two phases in the solution. In terms of the classification of Lax [16], these are interactions of type (c) (i.e., u(x, t) has only one maximum while the larger wave overtakes the smaller wave). From Figure 2a, it appears that the Dubrovin–Novikov solution is periodic in time. This was indeed proven by Ènol’skii [8]. For this specific solution only one of the three constraint equations is independent: since the derivative of the Weierstrass function is odd, the sum of the constraints is zero. Furthermore, labelling the three poles by x1 , x2 and x3 , for reality x2 = x1∗ + iL2 and x3 is on the centerline. Hence, the second constraint is the complex conjugate of the first constraint. The constraints (3.12b) reduce to the single equation ℘ 0 (x1 − x1∗ ) + ℘ 0 (x1 − x3 ) = 0.

(5.1)

This equation was solved numerically to provide the initial condition shown in Figure 2a. The initial guess required for the application of Newton’s method is based on the knowledge of the soliton limit. In that case two poles on the right represent a faster soliton, one pole on the left represents the slower soliton. The periodic case is not that different: the vertical line of poles with the smallest vertical distance between poles has poles closer to the real x-axis than the others and correspond to the wave crest with the highest amplitude, as seen in Figure 2b. We refer to the Dubrovin–Novikov solution as a (2 : 1)-solution because of the natural separation of the poles in a group of 2 poles (x1 , x2 ) and a single pole (x3 ).

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(a)

(b) Figure 2. The solution of Dubrovin and Novikov, with L1 = 4 and L2 = π. (a) The motion of the poles in the fundamental domain. The initial position of the poles is indicated by the black dots. The arrows denote the motion of the poles. (b) The KdV solution u(x, t).

Equating x1 = x3 + and only considering the singular terms of (5.1), it is possible to examine the location of the poles close to a collision points xc . With ℘ 0 (x) = −2/x 3 in this limit and = r + ii , one finds r3 − 3i2 r = 0,

9i3 − 3r2 i = 0.

(5.2)

This set of equations has three √ solutions, √ corresponding to the three distances between the poles: r /i ∈ {0, 3, − 3}. This allows for two triangular configurations of the poles: an equilateral triangle pointing left of the collision point and one pointing right.

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Using the dynamical system (3.12a) in the same way and only retaining singular terms results in ˙r = −

3 r2 − i2 + 3 2 , 4i2 r2 + i2

˙i =

6r i r2 + i2

2 .

(5.3)

Since the constraints (3.12b) are invariant under the flow (3.12a), the solutions to (5.2) give invariant directions of the system (5.3). Along these invariant directions, one obtains ordinary differential equations for the motion of the poles as they approach the collision point. It follows from these equations that the poles approach the collision point xc with infinite velocity. Integrating the equations with initial condition (tc ) = 0 gives =

35/6 √ ( 3 + i)(tc − t)1/3 . 2

(5.4)

Using the three branches of (tc − t)1/3 results in the dynamics of each edge of the equilateral triangle. If t < tc this triangle is pointing left, for t > tc it is pointing right. 5.2. N = 4:

AN ELLIPTIC DEFORMATION AND A TREIBICH – VERDIER SOLUTION

The next solution we discuss has 4 poles in the fundamental domain and is an elliptic deformation of a soliton solution. In the limit L1 → ∞, this solution corresponds to a two-soliton solution with wavenumber ratio k1 /k2 = 3/1, so this solution is refered to as a (3 : 1)-solution. The motion of the poles in the fundamental domain is displayed in Figure 3. Corresponding to the given wavenumber ratio, the amplitude ratio of the two phases present in the solution is roughly k12 /k22 = 9/1. As a consequence, the form of u(x, t) is not very illuminating and it has been omitted. Animations with the time dependence of both the positions of the poles and of u(x, t) are again available at http://amath-www.colorado.edu/appm/other/kp/papers. Note that the poles of the (3 : 1)-solution do not collide. This is in agreement with the results of Thickstun [17] who outlined which configurations lead to collisions and which do not, in the hyperbolic case. As mentioned before, the examination of collision behavior is essentially local and no differences appear among the rational, hyperbolic and elliptic cases. Another configuration with N = 4 exists. Consider the potential u(x, t = tc ) = −2℘ (x − x0 ) − 6℘ (x − x0 − ω1 ),

(5.5)

with x0 on the centerline. This is a Treibich–Verdier potential, obtained from (1.9) with M = 2, g1 = 1, g2 = 2, α1 = x0 and α2 = x0 + ω1 . It is referred to as a Treibich–Verdier potential because the position of the poles is given in terms of the

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Figure 3. N = 4: the pole dynamics of a (3 : 1)-solution. The initial positions of the four poles are indicated. The arrows on the centerline indicate that the poles there move in both directions.

periods of the Weierstrass function, as Pin (1.8). Also, it can be obtained from (1.8) as a degenerate case. As before N = 2i=1 gi (gi +1)/2 = 4, hence for all times that are not collision times, this solution has 4 distinct poles in the fundamental domain. The time t = tc is a collision time. Immediately after the collision time t = tc , the 3 poles located at x0 + ω1 separate, as in the Dubrovin–Novikov solution, along an equilateral triangle. The result appears to be a three-phase solution. However, it is known that the potential (5.5) is a two-gap potential of the Schrödinger equation and its hyperelliptic Riemann surface is given explicitly in [3]. This solution is not an elliptic deformation of a nonsingular soliton solution and the separation into different phases does not make sense. This is also seen from the following argument: if, for a fixed time which is not a collision time, we attempt to take the limit as L1 = 2ω1 → ∞, the poles seem to separate in three distinct solitons with respective wave numbers (k1 : k2 : k3 ) = (2 : 1 : 1). Such a nonsingular soliton solution does not exist for the KdV equation and the separation into different phases does not make sense. The dynamics of the poles is illustrated in Figure 4a. The corresponding KdV solution is shown in Figure 4b. The dynamics of the poles illustrated in Figure 4a exhibits behavior that appears qualitatively different from any other solution discussed here. The trajectories traced out by the motion of the poles in the fundamental domain appear to have singular points (cusps), away from the collision points. Upon closer investigation, these ‘cusps’ are only a figment of the resolution of the plot and the poles trace out a regular curve as a function of time, away from the collision times. Exactly why the global pole dynamics of the Treibich–Verdier potential (5.5) under the KdV flow appears so different from the pole dynamics of elliptic deformations of soliton solutions of the KdV equation is an open problem. Another question one may ask is whether similar behavior is observed for other solutions originating from Treibich–Verdier potentials.

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(a)

(b) Figure 4. An N = 4, M = 2 Treibich–Verdier solution, with L1 = 4 and L2 = π. (a) The motion of the poles in the fundamental domain. The initial position of the poles is indicated by the black dots. The initial time t = 0 was chosen different from the collision times tc . The arrows denote the motion of the poles. (b) The KdV solution u(x, t).

5.3. N = 5:

TWO DIFFERENT POSSIBILITIES

For N = 5, two soliton configurations are possible, and corresponding to each of these is an elliptic solution. The first solution is a (4 : 1)-solution. The second solution is a (3 : 2)-solution. The (4 : 1)-solution offers no new pole-dynamics: initially 1 pole is located on the centerline, at the left in the fundamental domain. The other poles are located at the right of the fundamental domain, symmetric with respect to the centerline. The three poles closest to the centerline interact as the (2 : 1)-solution. The two outer

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(a)

(b)

(c) Figure 5. N = 5: (a) The pole dynamics of a (4 : 1)-solution in the fundamental domain. (b) The pole dynamics of a (3 : 2)-solution in the fundamental domain. (c) The KdV solution u(x, t) corresponding to the pole dynamics in (b). In (a) and (b), the initial positions of the poles are indicated. Both solutions are quasiperiodic in time.

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Figure 6. The pole dynamics of a (3 : 2 : 1)-solution. The black dots mark the initial position of the poles; the grey dots mark the position of the poles at t = 0.4.

poles behave as the two outer poles of the (3 : 1)-solution. The pole dynamics of the (4 : 1)-solution is displayed in Figure 5a. The (3 : 2)-solution is more interesting. It is displayed in Figure 5c, together with the motion of the poles in the fundamental domain 5b. Again, the two crests of u(x, t) interact in a soliton-like manner. In Lax’s classification [16], this is an interaction of type (a), where at every time two maxima are observed. Figure 5b only displays the motion of the poles for a short time, in order not to clutter the picture. The motion of the poles is presumably quasiperiodic in time, as is the case for the (4 : 1)-solution. It appears that the two poles above (or below) the middle line of the fundamental domain share a common trajectory. It is an open problem to establish whether or not this is the case.

5.4. N = 6: TWO

DIFFERENT POSSIBILITIES . A THREE - PHASE SOLUTION

For N = 6, two distinct pole configurations are possible. The first one corresponds to a (5 : 1)-solution and results in a two-gap potential of the Schrödinger equation. It essentially behaves as the (3 : 1)-solution with two more poles added, which also behave as the outer poles of the (3 : 1)-solution. The second configuration is a (3 : 2 : 1)-solution, which limits to a three-soliton solution with wavenumber ratio (k1 : k2 : k3 ) = (3 : 2 : 1). This elliptic solution is a three-phase solution of the KdV equation. The amplitude ratio of the (3 : 2 : 1)-solution is (9 : 4 : 1), which explains why the third phase is hard to notice in Figure 7. Animations of the pole dynamics and of the time dependence of the (3 : 2 : 1)-solution are available at http://amath-www.colorado.edu/appm/other/kp/papers.

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Figure 7. A (3 : 2 : 1)-solution of the KdV equation, corresponding to the pole dynamics in Figure 6.

Acknowledgements The authors acknowledge useful discussions with B. A. Dubrovin, S. P. Novikov, C. Schober, A. Treibich and A. P. Veselov. This work was carried out at the University of Colorado and the Mathematical Sciences Research Institute. It was supported in part by NSF grants DMS 9731097 and DMS-9701755.

References 1. 2.

3.

4. 5.

Ablowitz, M. J. and Segur, H.: Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, PA, 1981. Airault, H., McKean, H. P., and Moser, J.: Rational and elliptic solutions of the Korteweg– de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30(1) (1977), 95–148. Belokolos, E. D., Bobenko, A. I., Enol’skii, V. Z., Its, A. R., and Matveev, V. B.: AlgebroGeometric Approach to Nonlinear Integrable Problems, Springer Ser. Nonlinear Dynam., Springer-Verlag, Berlin, 1994. Chudnovs’ki, D. V. and Chudnovs’ki, G. V.: Pole expansions of nonlinear partial differential equations, Nuovo Cimento B (11) 40(2) (1977), 339–353. Conway, J. B.: Functions of One Complex Variable, 2nd edn, Springer-Verlag, New York, 1978.

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Dubrovin, B. A.: Theta functions and nonlinear equations, Russian Math. Surveys 36(2) (1981), 11–80. 7. Dubrovin, B. A. and Novikov, S. P.: Periodic and conditionally periodic analogs of the manysoliton solutions of the Korteweg–de Vries equation, Soviet Phys. JETP 40 (1975), 1058–1063. 8. Ènols’kii, V. Z.: On solutions in elliptic functions of integrable nonlinear equations associated with two-zone Lamé potentials, Soviet Math. Dokl. 30 (1984), 394–397. 9. Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M.: Method for solving the Korteweg–de Vries equation, Phys. Rev. Lett. 19 (1967), 1095–1097. 10. Gesztesy, F. and Weikard, R.: On Picard potentials, Differential Integral Equations 8(6) (1995), 1453–1476. 11. Gesztesy, F. and Weikard, R.: Picard potentials and Hill’s equation on a torus, Acta Math. 176(1) (1996), 73–107. 12. Gradshteyn, I. S. and Ryzhik, I. M.: Table of Integrals, Series, and Products, 5th edn, Translation edited and with a preface by Alan Jeffrey, Academic Press, Boston, MA, 1994. 13. Ince, E. L.: Further investigations into the periodic Lamé functions, Proc. Roy. Soc. Edinburgh 60 (1940), 83–99. 14. Its, A. R. and Matveev, V. B.: Schrödinger operators with the finite-band spectrum and the Nsoliton solutions of the Korteweg–de Vries equation, Theoret. and Math. Phys. 23(1) (1976), 343–355. 15. Kruskal, M. D.: The Korteweg–de Vries equation and related evolution equations, Lectures in Appl. Math. 15, Amer. Math. Soc. Providence, 1974, pp. 61–83. 16. Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490. 17. Thickstun, W. R.: A system of particles equivalent to solitons, J. Math. Anal. Appl. 55(2) (1976), 335–346. 18. Treibich, A. and Verdier, J.-L.: Revêtements tangentiels et sommes de 4 nombres triangulaires, C.R. Acad. Sci. Paris Sér. I Math. 311(1) (1990), 51–54. 19. Treibich, A. and Verdier, J.-L.: Solitons elliptiques, In: The Grothendieck Festschrift, Vol. III, With an appendix by J. Oesterlé, Birkhäuser, Boston, 1990, pp. 437–480. 20. Treibich, A. and Verdier, J.-L.: Revêtements exceptionnels et sommes de 4 nombres triangulaires, Duke Math. J. 68(2) (1992), 217–236. 21. Verdier, J.-L.: New elliptic solitons, In: Algebraic Analysis, Vol. II, Academic Press, Boston, MA, 1988, pp. 901–910.

Mathematical Physics, Analysis and Geometry 3: 75–89, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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On Bianchi and Bäcklund Transformations of Two-Dimensional Surfaces in E 4 YU. AMINOV? and A. SYM Institute of Theoretical Physics of Warsaw University, Hoza street 69, 00-681 Warsaw, Poland (Received: 22 July 1999) Abstract. In this paper we discuss the existence and properties of the Bianchi transformations for pseudospherical surfaces in E 4 . The results of the paper show that the theory of Bianchi transformations in the discussed case is essentially different from the well-known case of pseudospherical surfaces in E 3 (in general n-manifolds of constant and negative curvature in E 2n−1 ). Mathematics Subject Classifications (2000): 53A05, 53A25, 37K35. Key words: Bianchi transformations, Bäcklund transformations, differential geometry, Gauss curvature, pseudospherical surface.

1. Historical Remarks These remarks serve as an introduction to the fascinating history of both Bianchi transformations and their generalization known as Bäcklund transformations. Simultaneously, we fix a terminology of the problem following the tradition of classical differential geometry. To begin, we recall the classical notion of an evolute of a surface S in E 3 [1]. Consider a family of all normals to S. In general, there exist two uniquely defined surfaces, say F and F¯ , such that each normal to S is tangent to both surfaces F and F¯ . Then F together with F¯ is an evolute of S. Luigi Bianchi in his 1879 habilitation thesis [2] introduced a notion of ‘complementary surfaces’. There are exactly two pieces F and F¯ of an evolute of some surface in E 3 . In this way, for Bianchi, the primary objects are F and F¯ , while S is a secondary one. Indeed, at the very beginning of his thesis he formulated the main problem as follows: “It is particularly interesting to consider complementary surfaces to a surface of constant negative curvatures”. In order to solve this problem, Bianchi made use of the following observation [1]: The tangents to curves of geodesic foliation of surface F are normal to an infinity of parallel surfaces. ? On leave of absence from B. I. Verkin Institute for Low Temperature of NAN of the Ukraine, 47 Lenin Ave, 310164, Kharkov, Ukraine.

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Let F be a pseudospherical surface of Gaussian curvature K = −b2 = const. F can be always equipped with coordinates (x, y) such that the metric form of F is equal to ds 2 = e2by dx 2 + dy 2 .

(1)

The system (x, y) is called an ‘horocyclic system’ since the curves y = const are horocycles. Simultaneously, the lines x = const are geodesics. Consider now a set of all tangents to the curves of the geodesic foliation of F defined by geodesics x = const. Relying on the earlier work of Weingarten [3], Bianchi pointed out (without proof) that the complementary surface F¯ is also pseudospherical of the same Gaussian curvature K¯ = −b2 . The corresponding analytical expression is very simple. Let r = r(x, y) be a position vector to F . A position vector r¯ = r¯ (x, y) to F¯ is given by 1 r¯ = r − ry . b

(2)

In fact, (2) is the Example 4 in [1], p. 290. To summarize, one can speak of the well-defined transformation of the pseudospherical (K = −b2 ) surface F into the pseudospherical surface F¯ of the same curvature. After Darboux [4], we call F → F¯ the ‘Bianchi transformation’. In 1883, A. V. Bäcklund published a paper, [5], which contains a generalization of the Bianchi transformation. This is still a transformation between two pseudospherical surfaces of the same Gaussian curvature. This paper is an essentially analytical one. Bäcklund represented F (F¯ ) in the Monge form z = f (x, y)(z0 = f 0 (x 0 , y 0 )) and replaced the original Bianchi transformation by a system of four equations relating (x, y, z, p, q) and (x 0 , y 0 , z0 , p 0 , q 0 ), where p = zx , q = zy , etc. Next, he formulated a generalization of this system containing a real parameter which defines an angle between normals to the corresponding surfaces at the corresponding points. Within the theory of pseudospherical surfaces in E 3 , this generalization is called the ‘Bäcklund transformation’. Bäcklund paper [5] is based on his earlier paper [6]. In any case, the Bäcklund transformation of [5] gave rise to further developments which were finally completed by the French school of J. Clairin and E. Goursat as a theory of Bäcklund transformations [7]. In a remarkable paper, Bianchi [8] reformulated the Bäcklund transformation in a purely geometric setting. In the same paper, he pointed out, for the first time the possibility, of interpreting Bäcklund’s results within Kummer’s theory of congruences. We recall that according to the classical terminology, a congruence in E 3 is a 2-parameter family of straight lines. Indeed, the Bianchi transformation, defines, for instance, a congruence: this is a 2-parametric family of tangent lines defined by ry . The important point is that F and F¯ can be now interpreted as the so-called focal surfaces of the congruence. This idea is fully developed in [9] published in 1887. For some other historical and modern aspects of this topic, see [10].

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2. Modern Developments Surprisingly enough, the topic of Bäcklund transformations has been out fashion for many years. A strong resurgence in interest in them has been evident since the creation of the theory of solitons (the theory of completely integrable systems of partial differential equations) [11]. In certain respects, a series of papers by G. L. Lamb [12, 13] and H. Wahlquist and F. Estabrook [14, 15] is a direct continuation of the work produced by the school of Clairin and Goursat. We have already pointed out that the Bianchi transformation (‘complementary transformation’ in Bianchi’s terminology) is a particular case of the Bäcklund transformation for pseudospherical surfaces corresponding to the special choice of Bäcklund parameters. This is why – despite its geometrical beauty – not much of attention was paid to this transformation in the past. There is, for instance, a discussion of Bianchi transformations by G. Darboux [4]. A generalization of the Bianchi transformation (with replacements: ‘pseudospherical surface’ by ‘an isometric immersion of the domain of the n-dimensional Lobachevski space’ and E 3 by E 2n−1 ) was formulated and proved in 1978 [16], almost one hundred years after Bianchi [2]. On the other hand, a similar generalization of the Bäcklund transformation for pseudospherical surfaces was presented by Tenenblat and Terng in 1979 [18], see also [19]. Their approach is based on Bianchi’s formulation of the Bäcklund transformations in terms of congruences. Paper [16] initiated further investigations concerning the generalized Bianchi transformations [17]. Similarly, papers [18, 19] are a point of departure for further studies [20]. 3. The Questions To the best of our knowledge, investigation of the Bäcklund (Bianchi) transformations for isometric immersions of n-dimensional Lobachevski space into E N with N > 2n − 1 has never been undertaken. In this paper we attempt to answer some questions related to the Bianchi transformations for isometric immersions of domains of Lobachevski plane L2 into E 4 . We start with L2 equipped with the horocyclic system of coordinates (x, y). The metric of L2 in these coordinates is given by ds 2 = e2y dx 2 + dy 2 .

(3)

Note that (3) is in fact (1) with b = 1, (K = −1). Coordinate curves x = const are geodesics while curves y = const are of the geodesic curvature equal to 1. F 2 denotes some isometric immersion of the domain of L2 into E 4 . r = r(x, y) stands for a position vector to F 2 . Guided by formula (2), we put r¯ = r − ry . A surface in E 4 defined by its position vector r¯ is called a Bianchi transform of F 2 and is denoted by F¯ 2 while a map F 2 → F¯ 2 is called a Bianchi transformation in ¯ E 4 . The Gauss curvature of F¯ 2 is denoted by K.

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A few questions can be asked here. QUESTION 1. When, in the definition above, E 4 is replaced by E 3 , we are assured that F¯ 2 is a pseudospherical surface as well. Does the same hold in the discussed case (K¯ = −1?)? QUESTION 2. Is the vector ry tangent to F¯ 2 as well? Soon we shall see that the answers to the above-mentioned questions are negative. Hence, one can weakes these two questions. QUESTION 3. Does a pseudospherical surface F 2 in E 4 exist (the case F 2 ⊂ E 3 is excluded) for which F¯ 2 is pseudospherical as well? QUESTION 4. Does a pseudospherical surface F 2 in E 4 exist such that the vector ry is tangent to F¯ 2 as well? In this case, we say that the vector field ry on F 2 enjoys a bi-tangency property. For simplicity, we also say that F 2 enjoys a bi-tangency property. QUESTION 5. exist in E 4 ?

How many pseudospherical surfaces with bi-tangency property

This question in turn inspires the following general question. QUESTION 6. Consider now an arbitrary surface F 2 in E 4 . Let a be a field of unit tangent vectors to F 2 . Let F¯ 2 be given by its position vector r¯ = r +ha, where h = h(u1 , u2 ) is a function on F 2 . The question is: under which conditions does the vector field ha exist on F 2 with the bi-tangency property? It is evident that this transformation r → r¯ is a generalization of the Bäcklund transformation. 4. The Answers The answer to Questions 1 and 2 is given by the following theorem: THEOREM 1. There exist pseudospherical surfaces F 2 in E 4 such that its Bianchi transformation has K¯ 6= −1. For surfaces F 2 ⊂ E 4 in general position, the vector ry is not tangent to F¯ 2 . If the bi-tangency property is fulfilled, then the Bianchi transformation has its remarkable property as in E 3 : THEOREM 2. If F 2 enjoys the bi-tangency property, then K¯ = −1 (F¯ 2 is a pseudospherical surface as well).

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Theorem 2 answers Question 3. The answer to Questions 4 and 5 is given by the following theorem: THEOREM 3. The set of all pseudospherical surfaces in E 4 with a bi-tangency property is arbitrarily of four functions, each with a single variable. Consider now Question 6. A partial answer (a necessary condition) to Question 6 follows below (Theorem 4). First of all, we recall that the set of all m-dimensional subspaces of n-dimensional Euclidean space through fixed point O (called a Grassmann manifold and denoted by Gm,n ) is a Riemannian space [25, 26]. The corresponding sectional curvature for the element of the area tangent to the Grassmann image of F 2 is denoted by KG . We recall that the Grassmann image of F 2 is elliptic (parabolic, hyperbolic) if and only if KG > 1, (KG = 0, KG < 1). THEOREM 4. (1) If the Grassmann image of F 2 is elliptic, then the vector field a tangent to both F 2 and F¯ 2 never exists. (2) If the Grassmann image is hyperbolic, then at every point x ∈ F 2 there exist at most two directions of the vector field tangent to both F 2 and F¯ 2 . One can complete Theorem 4 by the following theorem: THEOREM 5. Suppose F 2 ⊂ E 4 is of a flat normal connection and its Gaussian curvature K 6= 0. If, additionally, there exists a vector field ha with constant lenght h and tangent to regular surface F¯ 2 , then necessarily F 2 ⊂ E 3 . If F 2 6⊂ E 3 , then for every field of principal directions τi , it is possible to define the function h – by putting it equal to the minus radius of the geodesic curvature of the orthogonal curvature line – such that the vector field hτi will be tangent to F 2 and F¯ 2 .

5. The Proofs PROOF OF THEOREM 1

We put x = y1 , y = y2 . Select any field of the normalized normal vectors nσ , σ = 1, 2. Correspondingly, Lσij are the coefficients of the second fundamental forms. We denote the first and second derivatives of r by ri and rij . The Gauss decompositions for F 2 imply j

r¯i = ri − r2i = ri − 02i rj − Lσ2i nσ . 1 2 Since 021 = 1 and 022 = 0, we have

r¯1 = −Lσ21 nσ ,

r¯2 = r2 − Lσ22 nσ .

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Then we obtain the formula for the metric of F¯2 . d s¯2 = (¯r1 dy1 + r¯2 dy2 )2 = (−Lσ21 nσ dy1 + (r2 − Lσ22 nσ )dy2 )2 2 X = (Lσ12 dy1 + Lσ22 dy2 )2 + dy22 .

(4)

σ =1

Now we make an additional assumption: the normal connection of F 2 is flat. Hence, one can choose the field n1 , n2 with the torsion coefficients µασ |i = 0. Fortunately, this implies an existence of two functions φσ = φσ (y1 , y2 ) such that Lσ12 dy1 + Lσ22 dy2 = e−y2 dφσ .

(5)

In other words, (5) is equivalent to ∂Lσ12 ey2 ∂Lσ22 ey2 − = 0. ∂y2 ∂y1

(6)

But the Codazzi equations for F 2 have the form Lσ12,2 − Lσ22,1 = 0 or, in detail, ∂Lσ12 ∂Lσ22 − + Lσ12 = 0. ∂y2 ∂y1 Therefore, Equations (5), (6) are true. The metric d¯s 2 of F¯ 2 can be written as d¯s 2 = e−2y2 (dφ12 + dφ22 ) + dy22 .

(7)

If φ1 , φ2 , y2 were independent coordinates, (6) could serve as a metric of the threedimensional Lobachevski space of the curvature equal to −1. This means that, apart from F¯2 ⊂ E 4 , we can consider some immersion of its metric into L3 . This resulting surface is denoted by M 2 ⊂ L3 . The Gauss equation for F 2 reads L111 L122 − (L112 )2 + L211 L222 − (L212 )2 = −e2y2 .

(8)

The assumption of the flat normal connection is expressed by e2y2 (L111 L112 − L112 L211 ) + (L112 L222 − L122 L221 ) = 0.

(9)

We can solve (8) and (9) with respect to L111 and L211 and the resulting expressions can be substituted into the Codazzi equations: Lσ11,2 − Lσ12,1 = 0, σ = 1, 2. Let P = (e2y − e−2y (u2x + vx2 ))ux + vy (ux vy − uy vx ), Q = (e2y − e−2y (u2x + vx2 ))vx + uy (uy vx − ux vy ), where we have put φ1 = u, φ2 = v. Finally, we obtain the following system: ∂ P + uxx e−2y = −uy , (10) ∂y ux uy + vx vy ∂ Q + vxx e−2y = −vy . (11) ∂y ux uy + vx vy

ON BIANCHI AND BÄCKLUND TRANSFORMATIONS

81

The important point is that the whole system of Gauss–Codazzi–Ricci equations in the discussed case reduces to system (10)–(11). This system is of the form uxx = A(y, ux , uy , vx , vy , uxy , uyy , vxy , vyy ),

(12)

vxx = B(y, ux , uy , vx , vy , uxy , uyy , vxy , vyy ),

(13)

where A and B are analytical functions of their variables. As initial data, we can select the following functions: u(0, y), ux (0, y), v(0, y), vx (0, y). We can certainly, use these functions as parameters labelling isometric immersions of L2 into E 4 . The arbitrariness is four functions of a single argument. After these preliminary remarks, we go over the proof of Theorem 1. In fact, we present two proofs (one indirect and one direct). The indirect proof. Let us consider the surface M 2 ⊂ L3 . For surfaces in L3 , ¯ eL ) is the Gauss (extrinsic) curvature of M 2 . we have K¯ = KeL − 1, where K(K L Obviously K¯ = −1 iff Ke = 0. As a convenient model of L3 we take an open ball D ⊂ E 3 of the unit radius equipped with the metric dsl2 =

(ldl)2 + (1 − l 2 )(dl)2 , (1 − l 2 )2

(14)

where l is a position vector starting at the center of the ball and ending at the point of D. In this way, any surface M 2 ⊂ D can be interpreted either as a Lobachevskian surface or as a Euclidean one. Sidorov [21] in particular, derived the following formula relating KeL (extrinsic curvature of M 2 ⊂ L3 ) to K E (Gauss curvature of M 2 ⊂ E3) 1 − l2 2 L E Ke = K , 1 − (ln)2 where n is a unit normal to M 2 ⊂ E 3 . This formula implies that KeL = 0 iff K E = 0. Now the set of all developable (K E = 0) surfaces is parametrized by two functions of a single argument. In other words, of the set of all surfaces F¯ 2 of the Gaussian curvature = −1 the arbitrariness is smaller than the arbitrariness of all isometric immersions of L2 into E 4 with a flat normal connection. The direct proof. Consider again metric (4) rewritten as follows: d¯s 2 = e−2y (du2 + dv 2 ) + dy 2 . The two-dimensional metric ds02 = e−2y du2 + dy 2

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is of the Gaussian curvature equal to −1 as well. The equations of the immersion read u = u(x, y),

v = v(x, y).

We want to compute the value of K¯ at a given point P0 , say x = 0, y = 0. Let us assume that vx (0, 0) = vy (0, 0) = 0 at P0 . According to the Frobenius formula for the Gaussian curvature [22], one can write Eyy − 2Fxy + Gxx + 9(E, F, . . . , Ex , . . . , Gy ), K¯ = − 2(EG − F 2 ) where we have E = e−2y (u2x + vx2 ),

F = e−2y (ux uy + vx vy ),

G = e−2y (u2y + vy2 ) + 1. We point out that all expressions for the first derivatives of E, F and G at point P0 do not contain the derivatives of the function v. At point P0 we have 2 Eyy − Fxy + Gxx = −2(vxx vyy − vxy ) + 3(ux , uy , . . . , uyy ).

EG − F 2 = u2x . Moreover, the expression of 9 at P0 does not involve the derivatives of v. The Gaussian curvature at P0 is given by K¯ = −1 +

2 vxx vyy − vxy

u2x

.

As initial data we select v(0, y) = 0,

vx (0, y) = y,

ux (0, 0) 6= 0,

Thus, at point P0 vyy = 0,

vxy = 1,

vx = vy = 0.

So, finally, we obtain at P0 1 K¯ = −1 − 2 = 6 −1. ux Theorem 1 is proved in a direct way.

uy (0, 0) 6= 0.

ON BIANCHI AND BÄCKLUND TRANSFORMATIONS

83

PROOF OF THEOREM 2

Let ξ1 and ξ2 denote two fields of normalized normals to F¯ 2 . We select ξ1 and ξ2 as follows. It is not difficult to show that e−y2 ry1 is orthonormal to F¯ 2 . Hence, we put ξ1 = e−y2 ry1 , ξ2 = λ[¯ry1 r¯y2 ry1 ] (skew product in E 4 ), and λ = |[· · ·]|−1 is a normalized factor. Trilinearity of the skew product implies ξ2 = λ(L112[n1 ry2 ry1 ] + L212 [n2 ry2 ry1 ] + (L122 L212 − L112 L222 )[n1 n2 ry1 ]).

(15)

This in turn implies (ry2 ξ2 ) = (L122 L212 − L112 L222 )(ry2 n1 n2 ry1 ) = 0, where the second factor of the the RHS is a four-dimensional ‘mixed product’. For regular surfaces the ‘mixed product’ is nonzero everywhere. The vector ry2 is tangent to F¯ 2 iff (ry2 , ξ2 ) = 0, or iff L122 L212 − L112 L222 = 0.

(16)

If so, then (15) reduces to −L2 n1 + L112 n2 . ξ2 = q 12 (L112 )2 + (L212 )2

(17)

(17) implies that normal planes to F 2 and F¯ 2 at the corresponding point intersect along a line generated by ξ2 . It is natural to choose ξ2 = n1 (L112 = 0), while (16) implies L122 L212 = 0.

(18)

The case L212 = 0 is excluded (F¯ 2 is not regular). So we are left with L112 = L122 = 0. The Codazzi equation reduces to ∂L212 ∂L222 − + L212 = 0. ∂y2 ∂y1 Thus L212 dy1 + L222 dy2 = e−y2 dφ, and the metric of F¯ 2 assumes the form d¯s 2 = e−2y2 dφ 2 + dy22 .

(19)

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Certainly, the Gauss curvature of (19) is equal to −1. Theorem 2 has been proved. PROOF OF THEOREM 3

We recall (see the previous proof) that in the discussed case L112 = L122 = 0, L212 = e−y2 φy1 ,

L222

−y2

=e

φy2 .

(20) (21)

Thanks to (20) and (21) one of the Codazzi equations is satisfied. Certainly, the other Codazzi equation is to be satisfied L112,2 − L122,1 = µ21|2 L212 − µ21|1 L222 ,

(22)

where the µ coefficients are called ‘torsion coefficients’ [23]. From (20) it follows that the right-hand side of (22) vanishes. We can put µ21|2 = λL222,

µ21|1 = λL212 ,

(23)

where λ = λ(y1 , y2 ) is an unknown factor. The Gauss equation L211 L222 − (L212 )2 = −e2y2

(24)

implies L211 =

e−2y2 φy21 − e2y2 e−y2 φy2

.

Combining the Codazzi equation L211,2 − L212,1 = µ12|2 L111 − µ12|1 L212 = µ12|2 L111 and (23) we obtain ∂ e−2y2 φy21 − e2y2 − e−2y2 φy1 y1 = −λφy2 L111 e−2y2 . ∂y2 φy2

(25)

By using (23) and (24) we can rewrite the last Codazzi equation L111,2 − L112,1 = µ21|2 L211 − µ21|1 L212 as ∂L111 − L111 = λ[L211 L222 − (L212 )2 ] = −λe2y2 . ∂y2

(26)

(25) is equivalent to ∂ (L1 e−y2 ) = −λey2 . ∂y2 11

(27)

ON BIANCHI AND BÄCKLUND TRANSFORMATIONS

85

Finally we consider the (single) Ricci equation µ21|2,1 − µ21|1,2 + g ll (L2l2 L1l1 − L2l1 L1l2 ) = 0, where g ll stands for contravariant components of the metric tensor. By using (23) we obtain ∂ ∂ (λe−y2 φy2 ) − (λe−y2 φy1 ) + e−3y2 L111 φy1 = 0. (28) ∂y1 ∂y2 (27) gives λ which we insert into (25) and (28). Let θ stand for L111 . We conclude the calculation of the current proof with the following statement: the system of Gauss–Codazzi–Ricci equations in the discussed case reduces to the system of two equations for two functions φ(y1 , y2 ) and θ(y1 , y2 ) of the form ∂ e−2y2 φy21 − e2y2 ∂θ − e−2y2 φy1 y1 = e−3y2 φy2 θ , ∂y2 φy2 ∂y2 ∂ ∂e−y2 θ ∂e−y2 θ ∂ − e−2y2 φy2 + e−2y2 φy1 + e−3y2 φy1 θ = 0. ∂y1 ∂y2 ∂y2 ∂y2 Rewrite these equations in the brief form φy2 y2 = A(y2 , θ, φy1 , φy2 , θy2 , φy1 y1 , φy1 y2 ), θy2 y2 = B(y2 , θ, φy1 , φy2 θy1 , θy2 , φy1 y2 , θy1 y2 ). As usual φ(y1 , 0), φy2 (y1 , 0), θ(y1 , 0) and θy2 (y1 , 0) denote the initial conditions. Thus, the arbitrariness of isometric immersions of L2 into E 4 (no flat normal condition is assumed) with the bi-tangency property is of four functions of a single argument. Moreover, we can select the initial functions in such a way that the Gauss torsion κ0 := g ll (L1l1 L2l2 − L1l2 L2l1 ) 6= 0 and hence the discussed immersion of L2 essentially lies in E 4 not in E 3 . PROOF OF THEOREM 4

Let F 2 ⊂ E 4 be any regular surface and (u1 , u2 ) any orthogonal coordinates on it. The metric of F 2 reads ds 2 = Edu21 + Gdu22 .

(29)

We write r¯ = r + ha,

(30)

where r = r(u1 , u2 ) is a point vector to F 2 , while a is a field tangent to F 2 of unit lenght: |a| = 1, r¯ is a position vector to a surface F¯ 2 . Instead of (29) we can write r¯ = r + h(cos γ τ1 + sin γ τ2 ),

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where τi (i = 1, 2) are normalized coordinate vectors. Let b = − sin γ τ1 +cos γ τ2 . We recall that our aim is to state necessary conditions for ha to be tangent to F¯ 2 as well. To this end we compute cos γ Lσi1 sin γ Lσi2 + √ nσ , √ (31) r¯ui = rui + hui a + hbAi + h E G where A1 =

∂γ 1 ∂E − √ , ∂u1 2 EG ∂u2

A2 =

∂γ 1 ∂G + √ . ∂u2 2 EG ∂u1

(32)

We simplify the right-hand side of (31) as r¯u1 = a1 τ1 + a2 τ2 + c1 n1 + c2 n2 , r¯u2 = b1 τ1 + b2 τ2 + d1 n1 + d2 n2 ,

(33) (34)

where ai , . . . , di are some coefficients. The vector a is a linear combination of the vectors r¯u1 , r¯u2 iff the following matrix   a2 c1 c2 a1 b2 d1 d2  D =  b1 (35) cos γ sin γ 0 0 has the rank D 6 2. Hence c1 d2 − c2 d1 = 0.

(36)

In the left-hand side of (36) we replace ci and di by their explicit forms given in terms of the coefficients of I I 1 and I I 2 . We arrive at cos2 γ 1 2 sin γ cos γ 1 2 (L11 L12 − L112 L211 ) + √ (L11 L22 − L122 L211 ) + E EG sin2 γ 1 2 (L12 L22 − L122 L212 ) = 0. + (37) G So far we have not specified orthogonal coordinates u1 , u2 . Now we select them as those defined by the indicatrix of the normal curvature of F 2 ⊂ E 4 [24]. We recall it is an ellipse in our case. If n1 and n2 are parallel to its axes then L111 = E(α + a),

L211 = Eβ, √ = 0, L212 = EGb, = G(α − a), L222 = Gβ,

(38)

L112 L122

(39) (40)

where a and b are lengths of semi-axes while α and β are coordinates of the ellipse center. Substituting the expressions (38)–(40) into (37) gives cos2 γ b(α + a) + 2 cos γ sin γ aβ − sin2 γ b(α − a) = 0.

(41)

ON BIANCHI AND BÄCKLUND TRANSFORMATIONS

87

This quadratic equation has two roots iff a 2 β 2 + α 2 b2 − a 2 b2 > 0.

(42)

To conclude our proof, let us recall the formula for the Gaussian curvature KG of the Grassmann image of F 2 in G2,4 expressed in terms of the Gaussian curvature K of F 2 [25, 26] KG =

K 2 + 4a 2 b2 . K 2 + 4(a 2 β 2 + b2 α 2 )

(43)

It is evident that (41) implies KG < 1. According to the definition given in [25, 26] the Grassmann image of F 2 is hyperbolic. This ends the proof of Theorem 4. PROOF OF THEOREM 5

In this proof we employ all the notations of the previous proof. The assumption of a flat normal connection implies b = 0. Thus the ellipse of normal curvature is degenerated and Equation (40) reduces to sin γ cos γ aβ = 0.

(44)

One should consider three cases: (1) sin γ = 0 (cos γ = 0 is treated similarly), (2) a = 0, and (3) β = 0. In case (1) a = τ1 and we have r¯ = r + hτ1 , r¯u1 = a1 τ1 + a2 τ2 + c1 n1 + c2 n2 , √ r¯u2 = hu2 τ1 + ( G + hA2 )τ2 .

(45)

¯2 Let ξi (i = 1, 2) be normalized √ normal vectors to F . Consider at first the case h = const and denote b2 = G + hA2 . But (¯ru2 ξi ) = 0 implies b2 = 0 or (τ2 ξi ) = 0. If b2 = 0, hu2 = 0, then F¯ 2 is necessarily degenerated. So, let b2 6= 0. Now (τ2 ξi ) = 0 when combined with (τ1 ξi ) = (aξi ) = 0 gives an identity of normal spaces to F 2 and F¯ 2 at the corresponding points. Thus c1 = c2 = 0 and therefore c2 = L211 = β = 0. By the result of [27] we know that the identities β = 0, b = 0 and inequality K 6= 0 imply F 2 ⊂ E 3 . In case (2) or a = 0 and b = 0 implies F 2 is a standard sphere S 2 ⊂ E 3 . Case (3) has been already discussed above. The consideration above and (44) imply: if F 2 ⊂ E 4 is of a flat normal connection, F 2 6⊂ E 3 , K 6= 0 and if the field ha enjoys the bi-tangency property, then a must be necessarily a field of principal directions. As we have proved above, h must be some nonconstant function on F 2 . Now we shall prove that in general it is possible to select the function h in such a way that the field hτi will enjoy the bi-tangency property.

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Consider, for example, the transformation r¯ = r + hτ1 . We recall formula (44). To determine the function h, we put √ √ 1 ∂G G + hA2 = G + h √ = 0. 2 EG ∂u1 This equation can be rewritten in the following simple form 1+h

1 = 0, ρgv

where 1/ρgv denotes the geodesic curvature of the curvature line u = const. So, if the geodesic curvature of the orthogonal curvature line is not equal to zero, then we can take −h equal to the radius of the geodesic curvature of this line. If this geodesic curvature is not constant, then hu2 6= 0, r¯u2 6= 0 and F¯ 2 is a regular surface. In this case, matrix D has rank = 2. Therefore field ha is tangent to both F 2 and F¯ 2 . Acknowledgement Work partially supported by the Polish Committee of Scientific Researches (KBN grant 2 PO3B 185 09). References 1.

Eisenhart, L. P.: A Treatise on the Differential Geometry of Curves and Surfaces, Ginn, New York, 1909. 2. Bianchi, L.: Ricerche sulle superficie a curvatura costante esulle elicoidi, Ann. Sci. Norm. Sup. Pisa (1) 2 (1879), 285–340. 3. Weingarten, J.: Ueber eine Klasse auf einander abwickelbarer Flachen, Crelle J. 59 (1861), 382–390. 4. Darboux, G.: Leçons sur la théorie generale des surfaces, vol. III, Gauthier-Villars, Paris, 1894, ch. XII. 5. Bäcklund, A. V.: Om ytor med konstant negativ krokning, Lunds Univ. Arsskr. 19 (1883), 1–41. 6. Bäcklund, A. V.: Zur Theorie der partiellen Differentialgleichung erster Ordnung, Mat. Ann. 17 (1880), 285–328. 7. Forsyth, A. R.: Theory of Differential Equations, Vol. VI, Dover Publications, New York, 1959. 8. Bianchi, L.: Sopra i sistemi tripli ortogonali di Weingarten, Ann. di Mat. (2) 13 (1885), 177– 234. 9. Bianchi, L.: Sui sistemi doppiamente infiniti di raggi, Ann. di Mat. (2) 15 (1887–1888), 161– 172. 10. Prus, R. and Sym, A.: Rectilinear congruences and Bäcklund transformations: roots of the soliton theory, In: D. Wojcik and J. Cie´sli´nski (eds), Nonlinearity and Geometry, PWN (Polish Scientific Publishers), Warsaw, 1998. 11. Ablowitz, M. J. and Clarkson, P. A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. 12. Lamb, G., Jr.: Bäcklund transformations at the turn of the century, In: R. M. Miura (ed.), Bäcklund Transformations, Lecture Notes in Math. 515, Springer, New York, 1976.

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13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

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Lamb, G. L., Jr.: Bäcklund transformations for certain nonlinear evolution equations, J. Math. Phys. 15 (1974), 2157–2165. Wahlquist, H. D. and Estabrook, F. B.: Bäcklund transformations for solutions of the Korteweg– de Vries equation, Phys. Rev. Lett. 31 (1973), 1386–1390. Wahlquist, H. D. and Estabrook, F. B.: Prolongation structures of nonlinear evolution equations, J. Math. Phys. 16 (1975), 1–7. Aminov, Yu.: Bianchi transform for the domain of multidimensional Lobachevsky space, Ukrain. Geom. Sb. 21 (1978), 3–5 (in Russian). Masal’tzev, L.: Pseudospherical Bianchi congruencies in E 2n−1 , Math. Phys. Anal. Geom. 1(3/4) (1994), 505–512. Tenenblat, K. and Terng, C.-L.: A higher dimension generalization of the sine-Gordon equation and its Bäcklund transformation, Bull. Amer. Math. Soc. (N. S.) 1 (1979), 589–599. Tenenblat, K. and Terng, C.-L.: Bäcklund theorem for n-dimensional submanifolds of R 2n−1 , Ann. of Math. 111 (1980), 477–490. Tenenblat, K.: Bäcklund’s theorem for submanifolds of space form and a generalized wave equation, Bol. Soc. Brasil Mat. 16 (1985), 67–92. Sidorov, L.: Some properties of surfaces of negative extrinsic curvature in the Lobachevsky space, Mat. Zametki 4 (1968), 165–169. Blaschke, W.: Einfuhrung in die Differentialgeometrie, Springer-Verlag, Berlin, 1950. Eisenhart, L. P.: Riemannian Geometry, Princeton University Press, Princeton. Cartan, E.: Léçons sur la géométrie des espaces de Riemann, 2nd edn, Gauthier-Villars, Paris, 1946. Aminov, Yu.: Grassmann transform of a two-dimensional surface in four-dimensional Euclidean space, Ukrain. Geom. Sb. 23 (1980), 3–16. Aminov, Yu.: Determination of a surface in 4-dimensional Euclidean space by means of its Grassmann image, Mat. USSR Sb. 45 (1983), 155–167. Aminov, Yu.: Torsion of two-dimensional surfaces in Euclidean spaces, Ukrain. Geom. Sb. 17 (1975), 3–14.

Mathematical Physics, Analysis and Geometry 3: 91–100, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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The Ground State of Certain Coulomb Systems and Feynman–Kac Exponentials AMÉDÉE DEBIARD and BERNARD GAVEAU Université Pierre et Marie Curie (Paris 6), Mathématiques, T-46, 5ème étage, Boîte 172, 4 place Jussieu, 75252 Paris Cedex 05, France (Received: 10 January 2000) Abstract. We give a lower bound for the ground state energy of certain Coulomb Hamiltonians using the Feynman–Kac formula. We show that this bound is very precise for two electron atoms. Mathematics Subject Classifications (2000): 58J70, 60J65, 81S40. Key words: Coulomb systems, Brownian motion, ground state.

1. Introduction In this short note, we show that it is possible to use the Feynman–Kac formula to obtain a very sharp (indeed, almost exact) lower bound for the ground state of a two-electron atom, using only the basic facts of Brownian motion theory and elementary calculus. The starting point is the Feynman–Kac formula which provides a way of writing the solution of the heat equation with a potential, using a Wiener expectation on the Brownian motion. A consequence is the formula for the upper bound of the spectrum of 12 1−V in terms of the asymptotic estimate of such Wiener expectations (see [1] for the original reference, [2, 3] for a recent review of the subject and, more recently, [4] for another method of derivation which can be generalized to other situations). The estimation of such Wiener expectations have been the subject of many works (see [5 – 7], and their references). We thus obtain an upper estimate of the spectrum of 12 1−V , or a lower bound for the spectrum of the Hamiltonian H . The usual variational method (Rayleigh–Ritz) would provide an upper bound of the ground state of H . We obtain, in the special case of a Coulomb potential, an upper bound of the Wiener expectation, which uses the fact that the Coulomb potential is homogeneous of degree −1. That fact allows us to reduce the problem to an ergodic theorem for the spherical Brownian motion. For the twoelectron system, the absolute ground state is also the physical ground state. The estimation that we obtain in this case can be compared to the experimental value and is surprisingly sharp (1.2% of error), indicating the power of the Feynman–Kac formula and path integrals.

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In Section 2, we recall the basic facts about the Feynman–Kac formula and in Section 3, we specialize to the case of the Coulomb system. Section 4 gives the main estimate, Section 5 gives the final result. The Appendix contains the calculations of various integrals on a sphere. 2. The Feynman–Kac Formula for the Ground State We consider the operator H = − 12 1 + V where 1 is the standard Laplace operator in an N-dimensional space RN and V is a real function, such that H is essentially a self-adjoint operator. We consider the Cauchy problem ∂ψ (2.1) = −H ψ, ψ|t =0 = ψ0 . ∂t This can be solved, using the Feynman–Kac formula as follows. We consider the N-dimensional standard Brownian motion b(t) = (b1 (t), . . . , bN (t)), where bj (t) are independent one-dimensional standard Brownian motions. Then, if x ∈ RN , the solution of Equation (2.1) is Z t ψ(x, t) = E exp − V (b(s)) ds ψ0 (b(t)) b(0) = x , (2.2) 0

where E{. . . | b(0) = x} is the conditional expectation on Brownian paths, starting from x at time t = 0. We refer to [1 – 3] and also to [4] for a different proof. Because H is essentially a self-adjoint operator, it has a spectral decomposition and its spectrum is bounded from below. Call 3 an upper bound of the spectrum of −H . If ψ0 is not orthogonal to the generalized eigenfunction of eigenvalue 3, then ψ(x, t) ∼ C exp(3t)

(2.3)

and comparing Equations (2.2) and (2.3), we deduce Kac’s formula for the upper bound 3: Z t 1 (2.4) V (b(s)) ds ψ0 (b(t)) b(0) = x . 3 = lim log E exp − t →∞ t 0 We shall apply this formula to the situation where 3 is an eigenvalue of −H , the corresponding eigenfunction is integrable (and, as usual, square integrable), and it does not change sign, so that we can use any ψ0 > 0, bounded. This last hypothesis is normally fulfilled for standard Schrödinger Hamiltonians because the ground state does not change sign. 3. N Electrons Atom Hamiltonian → → rN ) in the Coulomb We consider the situation of N electrons (positions − r1 , . . . , − field of a nucleus of charge Ze. In atomic units, the Hamiltonian is 1X → → H =− 1i + V (− r1 , . . . , − rN ), 2 i=1 N

(3.1)

THE GROUND STATE OF CERTAIN COULOMB SYSTEMS

93

→ ri and where 1i is the three-dimensional Laplace operator for the variables − → → V (− r1 , . . . , − rN ) = −

N X Z i=1

ri

X

+

16i<j 6N

1 . → → |− ri − − rj |

(3.2)

The Hamiltonian H satisfies all the hypotheses of the previous section. We shall apply Equation (2.4) with V given as the Coulomb potential Equation (3.2) and with ψ0 = exp(−ϕ), where ϕ is a real function such that ψ0 is bounded. Our problem is, then, to find the large time behavior of the quantity Q(N, Z, t) Z t → → → V (− r1 (s), . . . , − rN (s)) ds − ϕ(− r1 (t), . . . = E exp − 0 − (0) → → − − → . . . , rN (t)) ri (0) = ri ,

(3.3)

→ → where − r1 (s), . . . , − rN (s) are N independent three-dimensional Brownian motions. 4. Transformation of Q(N, Z, t) To obtain a bound of Q, we shall use a method related to the one used in [7] for compact manifolds. Here, the fact that the Coulomb potential is homogeneous of degree −1 replaces the compactness. We define → → ϕ(− r1 , . . . , − rN ) =

N X j =1

→ Z− rj −

1 X − → |→ ri − − rj |. 2 16i<j 6N

(4.1)

It is easy to see that, using 12 1r = 1r , X N 1 1i ϕ = −V . 2 i=1

(4.2)

Now, we use Itô’s formula for the Brownian motion (see, e.g., [8]), under the form → → ϕ(− r1 (t), . . . , − rN (t)) → − → = ϕ( r1 (0), . . . , − rN (0)) + Z tX N − → → → − (∇j ϕ)(− r1 (s), . . . , − rN (s)) · d→ rj (s) + + 0 j =1

Z +

0

t

! N 1 X → → r1 (s), . . . , − 1j ϕ(− rN (s)) ds. 2 j =1

(4.3)

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Equations (4.2) and (4.3) allow us to rewrite the Feynman–Kac exponential inside Equation (3.3) as Z t → − − → → − − → V r1 (s), . . . , rN (s) ds − ϕ r1 (t), . . . , rN (t) exp − 0

Z tX N − → → → − = exp −ϕ(− r1 (0), . . . , − rN (0)) exp − rj (s) . ∇j ϕ · d →

(4.4)

0 j =1

We use the exponential martingale (see [8]). Given a N-dimensional Brownian mo→ − → − tion b (s) and a vector-valued random function f (s, ω) (ω being the simple path − → of the Brownian motion, s the time) such that f is nonanticipating, the quantity Z t Z 1 t − − → → − →2 Mt ≡ exp f (s, ω) d b (s) − | f | ds (4.5) 2 0 0 is a martingale and, in particular, its expectation is equal to 1. We shall rewrite the exponential of the stochastic integral in Equation (4.4) as Z tX N − → − → ∇j ϕ · d rj (s) = F · G exp − 0 j =1

with F and G given as Z tX N N pX − − → − → 2 → − → ∇j ϕ( r (s)) · d rj (s) − |∇j ϕ| ds , F = exp − 2 j =1 0 j =1

p G = exp 2

Z tX N − → 2 |∇j ϕ| ds 0 j =1

and p is a positive number. Then Q can be rewritten Q(N, Z, t) = exp −ϕ(rEj (0)) E(F G)

(4.6)

and we apply the Hölder–Young inequality with (1/p) + (1/q) = 1 1/p

E(F G) 6 E(F p )

1/q

E(Gq )

to the expectation in Equation (4.6). Thus, E(F p ) is of the type E(Mt ) for a certain Mt , as in Equation (4.5), with → − → − − → − → → f = ( fj ), fj = −p (∇j ϕ)(− r (s))

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THE GROUND STATE OF CERTAIN COULOMB SYSTEMS

so that E(Mt ) = 1 and, thus, Q(N, Z, t)

1/q Z N −→ − 2 pq t X − → → (0) → − 6 exp −ϕ(r ) E exp ∇j ϕ r (s) ds rj (0) = rj . (4.7) 2 0 j =1 (0)

P − → 2 Now, N j =1 |∇j ϕ| is a function homogeneous of degree 0 so that it is a function on the (3N −1)-dimensional unit sphere in R3N and we can use the ergodic theorem on the sphere [9] to get * N + Z N X − pq pq t X − → − 2 → ∇j ϕ → (4.8) r (s) ds ∼ |∇j ϕ|2 t, 2 0 j =1 2 j =1

where h · i is the average on the (3N − 1)-dimensional sphere. Then, from Equations (4.7)–(4.8), * N + 1 p X − → 2 lim log Q(N, Z, t) 6 |∇j ϕ| . t →∞ t 2 j =1 This upper bound is valid for any p > 1, noticing that q has disappeared and so, finally, we have the upper bound: + * N 1 1 X − → 3 = lim log Q(N, Z, t) 6 |∇j ϕ|2 ≡ 8. (4.9) t →∞ t 2 j =1 This estimate gives an upper bound of the spectrum of −H = 12 1 − V , and so a lower bound of the ground state of the Coulomb Hamiltonian. 5. Estimation of 3 We need to calculate the second member of Equation (4.9), namely the quantity 8 defined there. We have → → − → rj − − rj 1X − ri − → − ∇j ϕ = Z → − − → rj 2 i6=j | rj − ri | and so 2 N N − → → − X − X X → r r 1 − r − → 2 j j i Z |∇j ϕ| = → − → − r −2 | r − r | j j i i6=j j =1 j =1 → ri N(N − 1) N X − = NZ 2 + + 4 2 i,j 6=1 ri i6=j

→ − rj + r j

→ − → − → X → ri − − ri rj − rj − − , + NZ → → → |− ri − − rj | |− rj | |→ ri | i<j

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where, in the last summation, each couple (i, j ) is counted once. Moreover, calling → → ri and − rj , one has θij the angle between − − → − → → → rj ri ri + rj ri − − rj − − = − (1 − cos θij ) → − − → → − → − → → | ri − rj | | rj | | ri | | ri − − rj | so that, finally, using hcos θij i = 0, * N + 1 X − → 2 8 = |∇j ϕ| 2 j =1 N N −1 N(N − 1) r1 + r2 2 = Z + − Z − (1 − cos θ12 ) . → 2 4 4 |→ r1 − − r2 |

(5.1)

We obtain a trivial upper bound, saying that r1 + r2 > 1, − → → | r1 − − r2 |

hcos θ12 i = 0,

namely 86

N − 1 (N − 1)Z N Z2 + − . 2 4 2

In the Appendix, it is proved that 4 r1 + r2 (1 − cos θ12 ) = , → − → − | r1 − r2 | π

(5.2)

(5.3)

so that

N − 1 2(N − 1)Z N 2 Z + − . 8= 2 4 π

(5.4)

For N = 2, the Pauli principle states that the physical ground state is indeed the absolute ground state, while this is no more correct for N > 3. For N = 2, we can restore the units (see Appendix), and obtain − from Equation (5.2) 8 6 88 eV; − from the more exact Equation (5.4) 8 ' 80 eV to be compared with the actual value of the physical ground state 79 eV. See, e.g., [10] or [11].

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Appendix: Calculation of Spherical Integrals In this Appendix we calculate the spherical integral r1 + r2 I= − (1 − cos θ12 ) → |→ r1 − − r2 |

(A.1)

and prove Equation (5.3). In particular, we prove that this is independent of N. (I)

COORDINATES ON THE SPHERE

Call

q 2 2 → − → r1 + − r2 , q 2 2 → → r3 + . . . + − r 00 = − rN , p R = r 02 + r 002 . r0 =

→ In the space − r1 , we use the traditional polar angles (θ1 , ϕ1 ) and write → d− r1 = r12 dr1 sin θ1 dθ1 dϕ1 . → → r1 , namely angles In the space − r2 , we use polar angles around the axis defined by − (θ12 , ϕ12 ) → d− r2 = r22 dr2 sin θ12 dθ12 dϕ12 . Call r1 = r 0 cos γ , r2 = r 0 sin γ (0 6 γ 6 π/2), dr1 dr2 = r 0 dr 0 dγ . → → rN ), we use any polar coordinate system and write Moreover, in the space (− r3 , . . . , − − → → → rN = r 003N−7 dr 00 dσ (− r3 , . . . , − rN ), d− r3 . . . d→ where dσ is the spherical volume element of the unit sphere in R3N−6 . Finally, we write r 0 = R cos ρ,

r 00 = R sin ρ

(0 6 ρ 6 π/2).

In this way, we obtain → → → r2 . . . d− rN d− r1 d− 3N−1 =R dR cos5 ρ(sin ρ)3N−7 dρ cos2 γ sin2 γ dγ × → → r3 , . . . , − rN ) × sin θ1 dθ1 dϕ1 sin θ12 dθ12 dϕ12 dσ (−

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and thus the spherical volume of the unit sphere in R3N → → dσ (− r1 , . . . , − rN ) 2 = cos γ sin2 γ dγ sin θ1 dθ1 dϕ1 sin θ12 dθ12 dϕ12 × → → × cos5 ρ(sin ρ)3N−7 dρ dσ (− r3 , . . . , − rN ).

(A.2)

The function to be integrated is r1 + r2 cos γ + sin γ (1 − cos θ12 ) = √ (1 − cos θ12 ). − → → − | r1 − r2 | 1 − sin 2γ cos θ12

(A.3)

→ → It is independent of − r3 , . . . , − rN by definition, but also on ρ, θ1 , ϕ1 , ϕ12 , so that the average value is the average value with respect to the measure dσ = cos2 γ sin2 γ dγ · sin θ12 dθ12 , π 06γ 6 , 0 6 θ12 6 π. 2

( II )

(A.4)

CALCULATION OF THE INTEGRAL

First of all we calculate Z Z π/2 Z π π (sin 2γ )2 dσ = dγ sin θ12 dθ12 = . 4 8 0 0

(A.5)

Then we need to calculate the integral of Equation (A.3), namely abbreviating θ = θ12 , Z Z π 1 π/2 cos γ + sin γ 2 J ≡ (sin 2γ ) dγ sin θ dθ √ (1 − cos θ) (A.6) 4 0 1 − sin 2γ cos θ 0 so that I=

8J . π

For convenience, we give details of the calculation: Z Z 1 1 π/4 cos γ + sin γ 2 J = (1 − u). (sin 2γ ) dγ du √ 2 0 1 − (sin 2γ )u −1 Thus Z 1 1−u du √ 1 − (sin 2γ )u −1 p p 2 1 − sin 2γ − 1 + sin 2γ + =− sin 2γ

(A.7)

(A.8)

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Z 1 1 −(sin 2γ )u + √ du sin 2γ −1 1 − (sin 2γ )u 2 =− (1 − sin 2γ )1/2 − (1 + sin 2γ )1/2 + sin 2γ 2 (1 − sin 2γ )1/2 − (1 + sin 2γ )1/2 − + 2 sin 2γ 2 − (1 − sin 2γ )3/2 − (1 + sin 2γ )3/2 . 2 3 sin 2γ But

π , 4 (1 + sin 2γ )3/2 − (1 − sin 2γ )3/2 = (2 sin γ )(1 + 2 cos2 γ ), Z 1 2 sin γ 1−u 2 − . du √ = cos γ 3 cos2 γ 1 − (sin 2γ )u −1 (1 + sin 2γ )1/2 − (1 − sin 2γ )1/2 = 2 sin γ

06γ 6

From Equations (A.8)–(A.9), we deduce Z 4 π/4 2 J = sin γ (4 cos2 γ + 2 sin γ cos γ − 1) dγ = 3 0

(A.9)

1 2

because Z π/4 π 4 sin2 γ cos2 γ dγ = , 8 0 Z π/4 1 2 sin3 γ cos γ dγ = , 8 0 Z π/4 π 1 sin2 γ dγ = − , 8 4 0 so that I = 4/π . ( III )

RESTORING THE UNITS

In the unit of this article, the exact ground state of an hydrogenoïd atom of charge Z would be Z 2 /2. For Z 2 = 1 (hydrogen), we obtain 13.5 eV, so 1 unit corresponds to 27 eV, from which we deduce the numerical values given in this article. Acknowledgement We thank the referee for comments.

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References 1.

Kac, M.: Probability and Related Topics in the Physical Sciences, Interscience, New York, 1989. 2. Kac, M.: Integration in Function Spaces and Some of its Applications, Lezioni Fermiane, Pisa, 1980. 3. Schulman, L. S.: Techniques and Applications of Path Integration, Wiley, New York, 1981. 4. Gaveau, B. and Schulman, L. S.: Grassmann-valued processes for the Weyl and the Dirac equations, Phys. Rev. D 36 (1987), 1135–1140. 5. Berthier, A.-M. and Gaveau, B.: Convergence des exponentielles de Kac et applications en physique et en géométrie, J. Funct. Anal. 29 (1978), 416–424. 6. Gaveau, B. and Mazet, E.: Divergence des fonctionnelles de Kac et diffusion quantique, Publ. RIMS, Kyoto 18 (1982), 365–377. 7. Gaveau, B.: Estimation des fonctionnelles de Kac sur une variété compacte et première valeur propre de 1 + f , Proc. Japan Acad. Sci. 60 (1985), 361–364. 8. McKean, H. Jr.: Stochastic Integrals, Academic Press, New York, 1969. 9. Itô, K. and McKean, H. Jr.: Diffusion Processes and their Sample Paths, Academic Press, New York, 1964. 10. Levine, I. N.: Quantum Chemistry, Prentice-Hall, Englewood, 1991. 11. Karplus, M. and Porter, R. N.: Atoms and Molecules, Benjamin Cummings, Reading, 1970.

Mathematical Physics, Analysis and Geometry 3: 101–115, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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Periodic Ground State Configurations in a One-Dimensional Hubbard Model of Statistical Mechanics M. M. KIPNIS Chelyabinsk Pedagogical University, 69 Lenin Ave, Chelyabinsk, 454080, Russia e-mail: [email protected] (Received: 1 July 1999; in final form: 17 April 2000) Abstract. This paper considers an averaging procedure for the description of a particles arrangement in a Hubbard model with antiferromagnetic interactions. The arrangements are described by the devil’s staircase. Completeness of the staircase is proved. Mathematics Subject Classification (2000): 82B20. Key words: statistical mechanics, Hubbard model, periodic ground state configurations, symbolic dynamics, phenomenon of even two-colouring, averaging procedure, devil’s staircase, completeness.

1. Introduction In the Hubbard model [1, 3, 8, 13, 14] of statistical mechanics, the ground states are described by bilateral sequences (un ) ((un ): Z → {−1; 1}). The values of variable un may be interpreted either as an electron (un = 1) and a hole (un = −1) or as a particle with an up (un = 1) or down (un = −1) spin. The ground states provide the minimum of the formally defined Hamiltonian H : H = −ψ

X i∈Z

ui +

X

γi−j ui uj ,

(1)

i>j ; i,j ∈Z

where γi ((γi ): N → R) may be interpreted as the interaction energy of two particles at the distance of i units; ψ stands for a chemical potential. We shall give a euristical procedure for calculating the minimal arrangement for the Hamiltonian (1) in the spirit of dynamic programming. Let us suppose that the values um (m < n) are constructed and we have no information about um for m > n. Excluding um for m > n from (1) and defining the value of un , we get the energy increment ∞ X 1Hn = un −ψ + γi un−i . i=1

(2)

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To decrease energy according to (2), it is quite natural to provide 1Hn 6 0 by putting ∞ X (3) γi un−i . un = sgn ψ − i=0

Equation (3) will be called a Boolean averaging system. It will be the main object of our investigations. Another method of minimization of Hamiltonian (1) is given in [3]. We may consider that in (3) one selects either un = 1 or un = −1 in an attempt to provide the equality of ψ and a weighted average of the sequence un , un−1 , . . . with weights γ1 , γ2 , . . . That is why we refer to (3) as an averaging system. Equation (3) was introduced by the author [13, 14]. The averaging system (3) describes a variety of systems: relay periodic processes in a sampled-data control system [6, 11]; periodic output in analog-to-digital converter with sigma-delta modulation and leaky integration [4, 5, 14, 18]; the point itineraries in the iterates of some maps on an interval [10, 14, 15]; the rotation of the circle through a rational angle [17]. The common property of these systems is the even mixture of two kinds of objects on a circle in a given proportion, i.e. the even 2-colouring [16, 17, 21]. We treat of two equivalent formal descriptions of the periodic configurations in the above-mentioned systems (all as words in the −+ alphabet): Hubbard configurations [3] and the set J [11]. The frequency of pluses in the periodic configurations is called its rotation number. The rotation numbers are described in a parameter space by the devil’s staircase. The completeness of the devil’s staircase is the main result in the article. It is stated in Section 3 (Theorem 2) and proved in Section 8. Some results of this paper were given earlier without proofs by the author [13, 14]. 2. Periodic Ground State Configurations It is convenient to consider the periodic ground state configurations in Boolean averaging system (3) as words in the −+ alphabet. DEFINITION 1. Let for each i (i ∈ N, 1 6 i 6 p, p ∈ N) εi = − or εi = +. The word ε1 . . . εp is called a periodic ground state configuration in Boolean averaging system (3) for the given ψ, if there exists a bilateral sequence (un ) (n ∈ Z), satisfying Equation (3), such that (1) for each i (1 6 i 6 p) if εi = − then ui = −1; if εi = + then ui = 1; (2) for each integer n un+p = un . Let A be a word in the −+ alphabet. The set of points ψ, such that A is a periodic ground state configuration in (3) for a given ψ, is called the domain of the periodic configuration A.

PERIODIC GROUND STATE CONFIGURATIONS

103

EXAMPLE 1. Let γ1 = 10, γ2 = 9, γ3 = 6, γ4 = 4, and γn = 0 when n > 4. The complete list of periodic ground state configurations (up to shifts and degrees) is −+ for ψ ∈ (0, 3); − − ++ for ψ ∈ (0, 1); − + − + + for ψ ∈ (3, 11); − + + for ψ ∈ (11, 17); − + ++ for ψ ∈ (17, 21); − + + + + for ψ ∈ (21, 29); + for ψ > 29, and symmetrically for ψ < 0 (interchanging plusses and minuses). We have here non-uniqueness of periodic configurations: there exist simultaneously two configurations −+ and − − ++ for ψ ∈ (−1, 1). DEFINITION 2. For the word A the ratio of the number of plusses to the total number of letters is called the rotation number of A. The rotation number of A is denoted as ω(A). For example, ω(− + − + +) = 3/5. In the following definition [a] stands for the integer part of a. For each word A in the −+ alphabet the symbol A0 denotes the empty word. For the nonnegative integer n An+1 = An A. The word B is called the shift of A if there exist words C, D such that A = CD and B = DC. DEFINITION 3 [3]. The word A in −+ alphabet is called a Hubbard configuration, if in each of its shifts B there are either [i/ω(A)] or [i/ω(A)] − 1 letters between every plus and ith plus on the right-hand side of it. Besides, the words −n (n > 0), −n + (n > 0) and their shifts are also called Hubbard configurations. EXAMPLE 2. The words −+, −+−++, −−+−+ are Hubbard configurations, while the word − − ++ is not.

3. Statement of the Main Result Now we are ready to connect the periodic ground state configurations in averaging procedure (3) and the Hubbard configurations. THEOREM 1 [13]. Let for each i (1 6 i < ∞) γi > γi+1 > 0, let the series P∞ i=1 γi be convergent and (γi ) be convex (i.e. γi+1 < (γi + γi+2 )/2 for i > 1). Then the word A is a periodic configuration in Boolean averaging system (3) if and only if it is a Hubbard configuration. We avoid giving the proof of Theorem 1 because in Theorem 2 which comes later, we’ll deal with the wider class of interaction functions, free from the condition of convexity. In addition, results, similar to Theorem 1, were stated earlier [3, 8] for systems, which are not described by Equation (3), but also originate from a Hamiltonian minimization problem. The dependence of rotation numbers of the periodic ground state configurations on the values of ψ is described by the devil’s staircase [3, 14]: it is an increasing function, whose derivative equals to zero almost everywhere. It is a discontinuous function, but it may be extended to a continuous function. The case of nonconvex (γi ) had not been investigated until the author’s paper [13]. It is considered in the following theorem:

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THEOREM 2. If for each i (1 6 i < ∞) γi > γi+1 > 0, and the series is convergent, then:

P∞ i=1

γi

(1) Every Hubbard configuration is the periodic ground state configuration in Boolean averaging system (3) with some value of ψ. (2) Let periodic ground state configuration A in system (3) not be a Hubbard configuration. Let B be a Hubbard configuration with the same rotation number as A. Then the domain of A is a proper subset of the B domain. (3) Let P be the set of values of ψ such that there exist no periodic ground state configurations in Boolean averaging system (3) with a given ψ. Then the set P is a Cantor perfect set of the Lebesgue zero measure. Theorem 2 is our main result. It is proved in Section 8. When (γi ) is nonconvex, some ψ values may correspond to more than one periodic ground state configuration (see Example 1). However, Theorem 2 states the preservation of the devil’s staircase in this case. Part (3) of Theorem 2 means the completeness of the staircase. 4. The Point Itineraries in the Maps of Interval in Itself Consider the variant of Boolean averaging procedure (3) with the exponential weight sequence (γn ): γn = e−nα (eα − 1) (α > 0). If we replace (3) by two equations un = sgn σn ; σn = ψ − from (4): −α e σ + (ψ − 1)(1 − e−α ), if σn > 0; σn+1 = −α n e σn + (ψ + 1)(1 − e−α ), if σn < 0.

P∞ i=1

(4) γi un−i , we get

(5)

Map (5) is a piecewise linear discontinuous transformation of the line in itself. The orbits of transformation (5) were investigated earlier [15]. Write + (plus) when σn > 0 and − (minus) when σn < 0. Then the periodic trajectories in map (5) are described by words in the −+ alphabet. By Theorem 1, the periodic ground state configurations in map (5) are Hubbard configurations and vice versa. Besides, by Theorem 2 the devil’s staircase associated with map (5) is complete. Hence, Theorem 2 gives a new proof of the result obtained earlier [10, 18]. Transformation (5) is associated with selfsimilar structures [14] and deterministic chaos [11]. Changing the variables in Equation (5) by means of equations σn = 2(δn + ω − 1); ψ = 2ω − 1, and letting α go to zero (5) gives the map δ + ω − 1, if δn + ω > 1; δn+1 = n (6) if δn + ω < 1, δn + ω, where ω is a rotation number, 0 6 ω 6 1. Map (6) is the classical model of circle rotation through the angle ω. Write + (plus) if δn ∈ [0, ω) and − (minus)

PERIODIC GROUND STATE CONFIGURATIONS

105

otherwise. If ω is rational, then the sequence of letters −, + is periodic and it forms the so-called Sturmian chain [17]. The latter are equivalent to the Hubbard configurations. 5. Even 2-Colouring and the Set J The even 2-colouring problem has been widely discussed [2, 3, 16, 19, 21]. The problem is how to distribute a objects of the first kind (say, pluses) and b objects of the second kind (say, minuses) on a circle as evenly as possible. In the definition of the Hubbard configuration (Section 2), we found the requirement of the even distribution of plusses in words. The second description of the words in the −+ alphabet with the even distribution of plusses is given by the linear ordered set J [11]. To define J , we define the sequence of finite linear ordered sets Jn , each having words in the −+ alphabet as components. DEFINITION 4. (1) J0 = (−, +). The order in J0 : − < +. (2) Assume Jn be defined. Let the words A and B be arbitrary components of Jn , let A < B and let there be no C in Jn , such that A < C < B. Then the words A, B and AB are the components of Jn+1 and A < AB < B in Jn+1 . There are no other components in Jn+1 . DEFINITION 5. J is the linear ordered set consisting of all components of every set Jn (n > 0) in the order induced by the order in the sets Jn . The sets Jn are constructed by the operations of concatenation and insertion. J0 consists of two words: − (minus) and + (plus). We get J1 by inserting the concatenation of the − (minus) and + (plus) between the aforesaid words: J1 = (−, −+, +). Further, we insert the concatenation of the neighboring words between the neighbors: J2 = (−, − − +, −+, − + +, +), J3 = (−, − − −+, − − +, − − + − +, −+, − + − + +, − + +, − + ++, +), and so on. The sets Jn are the redefinitions of the Farey tree [7, 19]. The two definitions mentioned above are equivalent in a certain sense. THEOREM 3 [12]. For each word A in −+ alphabet the following two assertions are equivalent: (1) A is a Hubbard configuration (2) A is a shift of a nonzero degree of some component of J . EXAMPLE 3. The word − + − + + − + − + + − + + is simultaneously a Hubbard configuration and a component of J (namely, J5 ).

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6. Properties of the Words in J The number of the minuses and plusses in the word A we denote by q− (A) and q+ (A) accordingly. We call the word L the left neighbor of the word R in Jn , if L and R are components of Jn , L < R and there exist no M in Jn , such that L < M < R. For example, the word − + − + + is the left neighbour of − + + in J3 . If L is the left neighbor of R in Jn , then L and R are called the left and the right predecessors of the word LR accordingly. If L is the left neighbour of R in Jn (n > 0), then either R is the right predecessor of L, or L is the left predecessor of R. (For example, the word − + + is the right predecessor of − + − + +.) This dichotomy will be used in the forthcoming lemmas. LEMMA 1. If L is the left neighbor of R in Jn , then q− (L)q+ (R) − q+ (L)q− (R) = 1.

(7)

Proof (by induction on n). The case n = 0 is evident. Induction Step. Let the assertion be valid for n. Let L be the left neighbor of R in Jn+1 . Case 1: L is the left predecessor of R. Then L is a component of Jn and there exists a word R1 , such that L is the left neighbor of R1 in Jn and LR1 = R. By induction hypothesis q− (L)q+ (R1 ) − q+ (L)q− (R1 ) = 1. From the latter equality and equalities q− (R) = q− (R1 ) + q− (L) and q+ (R) = q+ (R1 ) + q+ (L), we get (7). Case 2: R is the right predecessor of L. The proof is similar. Lemma 1 is proved. 2 LEMMA 2. For each two pairs of nonnegative integers (a, b) and (c, d) if ad − bc = 1, then there exist the words L, R and the integer n, such that L is the left neighbor of R in Jn and q− (L) = a, q+ (L) = b, q− (R) = c, q+ (R) = d. Proof (by induction on b + d). The case b + d = 1: since ad − bc = 1, we get b = 0, d = 1. Put L = −, R = +, n = 0. Induction Step. Suppose b + d > 1 and for each quadruple (a1 , b1 , c1 , d1 ) if b1 + d1 < b + d, then the assertion is true. Case 1: b = d. Since ad − bc = 1, we get b(a − c) = 1, hence b = d = 1, a = c + 1. Put L = −a +; R = −c +; then L is the left neighbor of R in Ja . Case 2: b > d. Since ad−bc = 1, we have a > c. Then a−c > 0 and b−d > 0. Take two pairs (a − c, b − d) and (c, d). We have (a − c)d − (b − d)c = 1. The sum of the second components of the pairs in question is less than b + d, hence by the induction hypothesis, there exist two words L1 , R1 and a natural number n, such that L1 is the left neighbor of R1 in Jn , and q− (L1 ) = a − c, q+ (L1 ) = b − d, q− (R1 ) = c, q+ (R1 ) = d. Then the word L1 R1 is the left neighbor of R1 in Jn+1 . Put L = L1 R1 , R = R1 and we are done. Case 3: b < d. The proof is similar to that of case 2. Lemma 2 is proved. 2

107

PERIODIC GROUND STATE CONFIGURATIONS

THEOREM 4. (1) If A is a word in J , then q− (A) and q+ (A) are coprime numbers. (2) If A < B in J , then ω(A) < ω(B) (recall that ω(A) is the rotation number, ω(A) = q+ (A)/(q− (A) + q+ (A))). (3) For each pair of coprime natural numbers (a, b) there exists the only word A in J , such that q− (A) = a, q+ (A) = b. Proof. (1) Part (1) is an evident consequence of Lemma 1. (2) Let A < B in J , then there exists n, such that A and B are in Jn . If A is the left neighbor of B, then the proof is attained by Lemma 1; otherwise there exist the words Ai (1 6 i 6 m) in J , such that A is the left neighbor of A1 , Ai is the left neighbor of Ai+1 (1 6 i 6 m − 1), Am is the left neighbor of B. By Lemma 1 for every i (1 6 i 6 m − 1) ω(A) < ω(Ai ) < ω(Ai+1 ) < ω(B), as required. (3) Let a, b be coprime numbers. Construct numbers c, d, such that ad−bc = 1, and by Lemma 2 we get the word A, such that A is in J and q− (A) = a, q+ (A) = b. The uniqueness of the word A is followed from part (2) of Theorem 4 by the linear ordering of J . Theorem 4 is proved. 2 EXAMPLE 4 (to Theorem 4). (1) The word A = − + − + − + + is a component of J . We have q− (A) = 3, q+ (A) = 4, and 3, 4 are coprime numbers. (2) A = − + − + − + + < − + − + + = B in J , and ω(A) = 4/7 < ω(B) = 3/5. (3) There exists a unique word A in J , such that q− (A) = 11, q+ (A) = 71. The word A is (−+6 )2 − +7 (− +6 −+7 )4 .

7. Domain of the Periodic Ground State Configurations in Boolean Averaging System Let us introduce certain notations. Define the function sg: {−, +} → {−1, 1} by the equations sg(−) = −1, sg(+) = 1. Given a sequence γ = (γn ) and integers P j j j, p (1 6 j 6 p) we define Kp (γ ) by the equation Kp (γ ) = ∞ m=0 γj +mp . j +p j For each integer j we put Kp (γ ) = Kp (γ ). Let A = ε1 . . . εp , where εi = − or εi = + (1 6 i 6 p). Define F (γ , A) =

p X

Kpp−i (γ ) sg εi

=

p X

i=1

Kpi (γ ) sg εp−i .

i=1

Let Shiftp(A) (respectively, Shiftn(A)) stand for the set of the shifts of A, which end with the letter + (plus) (respectively, with − (minus)). THEOREM 5 (Domain Theorem). Let Shiftp(A) 6= φ and Shiftn(A) 6= φ. Then the word A is a periodic ground state configuration in Boolean averaging system (3) with the given ψ, if and only if min

B∈Shiftn(A)

F (γ , B) > ψ >

max

B∈Shiftp(A)

F (γ , B).

(8)

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M. M. KIPNIS

Proof. (1) Let A = ε1 . . . εp , where εi = − or εi = + (1 6 i 6 p), and A be a periodic ground state configuration in Boolean averaging system (3) with the given ψ. Let (un ) be a bilateral sequence (n ∈ Z) satisfying Equation (3) and the conditions (1), (2) of Definition 1. Then un+mp = sg εn for all integers m and = A, An = 1 6 n 6 p. Denote the various shifts of A: A1 = ε2 . . . εp ε1 , Ap P εn+1 . . . εp ε1 . . . εn (2 6 n 6 p − 1). Introduce the notation σn = ψ − ∞ i=1 γi un−i (1 6 n 6 p). With 1 6 n 6 p we get σn = ψ −

p X

sg εi Kpn−i (γ ) = ψ − F (γ , An ).

i=1

If εn = + (1 6 n 6 p) then by part (1) of Definition 1 σn = ψ − F (γ , An ) > 0; if εn = − (1 6 n 6 p) then σn = ψ − F (γ , An ) < 0. Hence, we get (8). condition (8) be hold. Then put for each integer n σn = ψ − Pp(2) Let the n−i sg ε K i p (γ ) and un = sgn σn . If 1 6 n 6 p, then σn = ψ − F (γ , An ). i=1 By (8) with εn = + (1 6 n 6 p), we have σn = ψ − F (γ , An ) > 0 and un = 1, with εn = − (1 6 n 6 p) we have σn = ψ − F (γ , An ) < 0 and un = −1. The sequence (σn ) is p-periodic by the p-periodicity of Kpi (γ ) as to the superscript, hence un+mp = un (1 6 n 6 p, m ∈ Z). The sequence (un ) satisfies Equation (3) because ψ−

∞ X

sgn σn−i γi = ψ −

p X

i=1

sg εi Kpn−i (γ ) = σn .

i=1

2

Theorem 5 is proved.

COROLLARY 1. The word A in the −+ alphabet is a periodic configuration in Boolean averaging system (3) if and only if min

B∈Shiftn(A)

F (γ , B) >

max

B∈Shiftp(A)

F (γ , B).

8. Proof of Theorem 2 8.1.

SUBSIDIARY ASSERTIONS TO THE PROOF OF THEOREM 2

We need some definitions and lemmas. Let A be a component of J , A 6= −, A 6= +. By + A (resp. by A− ) let us denote the word A where the first (resp., last) letter is replaced by the letter + (plus) (resp., − (minus)). By + A− denote the result of the two operations mentioned above. Define the functions λ and ρ over Jn by recursion on n. DEFINITION 6. (1) For words in J0 : λ(−) = −, λ(+) = 3 (3 is an empty word), ρ(−) = 3, ρ(+) = +.

PERIODIC GROUND STATE CONFIGURATIONS

109

(2) Let λ and ρ be defined in Jn . Let A be a component of Jn+1 and L, R be components of Jn , L and R be the left and the right predecessors of A respectively. Case 1: L is the left predecessor of R. Take λ(A) = λ(LR) = Lλ(R), ρ(A) = ρ(LR) = ρ(R). Case 2: R is the right predecessor of L. Take λ(A) = λ(LR) = λ(L), ρ(A) = ρ(LR) = ρ(L)R. EXAMPLE 5. Let A = − + + − + + − + + + − + + − + + +. Then λ(A) = − + + − + + −, ρ(A) = + + + − + + − + ++. LEMMA 3. If the word A is a component of Jn (n > 1) and L, R are the left and the right predecessors of A respectively, then ρ(A) = + L, λ(A) = R − , ρ(A)λ(A) = + A− . Proof (by induction on n). The case n = 1 is obvious. Induction Step. Suppose the assertion be valid for n. Let A be a component of Jn+1 (n > 1), let L and R be the left and the right predecessors of A, respectively. Case 1: R is the right predecessor of L. Then L = L1 R, where L1 , R are the components of Jn−1 . In this case λ(LR) = λ(L), ρ(LR) = ρ(L)R. Hence, ρ(A) = ρ(L)R = + L1 R = + L; λ(A) = λ(LR) = λ(L) = R − . We’ve used the equalities ρ(L) = + L1 and λ(L) = R − which are valid by the induction hypothesis. Thus, ρ(A)λ(A) = + LR − = + A− . Case 2: L is the left predecessor of R. The proof is similar. Lemma 3 is proved. 2 LEMMA 4. If the word A is a component of J and A 6= −, A 6= +, then + A− ∈ Shiftn(A). Proof follows immediately from Lemma 3 and the evident equality λ(A)ρ(A) = A. Lemma 4 is proved. 2 DEFINITION 7. Define the function g(i, A) for the words A in the −+ alphabet and the integers i (1 6 i 6 q+ (A)). Let us number the positions of the letters in the word A. Let the position of the extreme right letter of A be 0; the other positions of the letters from right to left are the integers from 1 to q− (A) + q+ (A) − 1. If A = B− for some B and 1 6 i 6 q+ (A), then g(i, A) is the number of the position of ith plus from the right edge of A. If A = B+ for some B and 1 6 i < q+ (A), then g(i, A) is the number of the position of (i + 1)th plus from the right edge of A. Besides, if A = B+ and i = q+ (A), then g(i, A) = q− (A) + q+ (A). EXAMPLE 6. If A = + + − + − + −, then g(1, A) = 1, g(2, A) = 3, g(3, A) = 5, g(4, A) = 6. If A = − + − + − + +, then g(1, A) = 1, g(2, A) = 3, g(3, A) = 5, g(4, A) = 7. LEMMA 5. For each word A in Jn (n > 0, A 6= −) and for each natural number m g(i, Am ) = [i + iq− (A)/q+ (A)] (1 6 i 6 q+ (Am )).

110

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Proof (by induction on n) is being demonstrated for m = 1. The expansion on arbitrary m is obvious. The case n = 1 is evident. Induction Step. Suppose the lemma is valid for n. Let L be the left neighbor of R in Jn . Denote q− (L) = a, q+ (L) = b, q− (R) = c, q+ (R) = d. We have g(i, L) = [i + ia/b] (1 6 i 6 b), g(i, R) = [i + ic/d] (1 6 i 6 d). We aim to show that g(i, LR) = [i + i(a + c)/(b + d)] (1 6 i 6 b + d). In view of ad − bc = 1 (Lemma 1), if 1 6 i 6 d, then 0 < i(a + c)/(b + d) − ia/b = i/(d(b + d)) 6 1/(b + d). Hence, [ic/d] = [i(a + c)/(b + d)] and g(i, LR) = g(i, R) = [i + i(a + c)/(b + d)], as required. If d < i 6 b + d, then g(i, LR) = c + d + g(i − d, L) = c + d + [i − d + (i − d)a/b]. Hence, by 0 < i(a + c)/(b + d) − c − (i − d)a/b = (b + d − i)/(b(b + d)) 6 1/(b + d) (the latter equality is the concequence of ad − bc = 1 as well) we have g(i, LR) = [i + i(a + c)/(b + d)], as required. Lemma 5 is proved. 2 LEMMA 6. For any word A (A 6= −n , A 6= +n , n > 0) in the −+ alphabet there exist the words B ∈ Shiftp(A) and C ∈ Shiftn(A) such that for each i (1 6 i 6 q+ (A)) g(i, B) 6 i + [iq− (A)/q+ (A)]; g(i, C) > i − 1 + iq− (A)/q+ (A). Proof. Given the word A we define the number j by max (g(i, A) − i − iq− (A)/q+ (A)) = g(j, A) − j − j q− (A)/q+ (A). (9)

16i6q+ (A)

Let the word B be a shift of A, such that the j th plus of A is the rightmost letter of B, i.e. ( g(j + k, A) − g(j, A), if 1 6 k 6 q+ (A) − j ; (10) g(k, B) = g(k − q+ (A) + j, A) − g(j, A) + q− (A) + q+ (A), if q+ (A) − j < k 6 q+ (A). Then B ∈ Shiftp(A). When 1 6 k 6 q+ (A) − j , we get by (9) g(j, A) − j − j q− (A)/q+ (A) > g(j + k, A) − (j + k) − (j + k)q− (A)/q+ (A). Hence, by (10) g(k, B) = g(j +k, A)−g(j, A) 6 k+kq− (A)/q+ (A), as required. When q+ (A)− j < k 6 q+ (A), we get by (9) and (10) similarly g(k, B) 6 k + kq− (A)/q+ (A). The word C we construct analogously. Lemma 6 is proved. 2 LEMMA 7. If A is a component of J , A 6= −, A 6= +, and B ∈ Shiftp(A), C ∈ Shiftn(A), then for each i (1 6 i 6 q+ (A)) g(i, B) > g(i, A);

g(i, C) 6 g(i, + A− ).

Proof. By Lemma 5 for each i (1 6 i 6 2q+ (A)) g(i, A2 ) = i + [iq− (A)/q+ (A)].

(11)

PERIODIC GROUND STATE CONFIGURATIONS

111

Let us assume that B ∈ Shiftp(A). Then B is the subword of A2 . Let the last letter of B be in the position g(j, A2 ) in the word A2 . Then for any i (1 6 i 6 q+ (A)) by (11) we get g(i, B) = g(i + j, A2 ) − g(j, A2 ) = i + [(i + j )q− (A)/q+ (A)] − [j q− (A)/q+ (A)] > i + [iq− (A)/q+ (A)] = g(i, A). Let us assume that C ∈ Shiftn(A). If q+ (A) = 1, then the conclusion of the Lemma follows from the inequality g(q+ (A), C) 6 q+ (A) + q− (A) − 1 = g(q+ (A), + A− ). Now, consider the variant q+ (A) > 1. The word C is a subword of A2 . We may assume, without loss of generality, that the first letter to the right of the subword C in the word A2 is the letter + (plus). Let the position of this letter in A2 be g(j, A2 ). Then for any i (1 6 i 6 q+ (A) − 1) by (11) and Definition 7 g(i, C) = g(i + j, A2 ) − g(j, A2 ) − 1 = i − 1 + [(i + j )q− (A)/q+ (A)] − [j q− (A)/q+ (A)] 6 i + [iq− (A)/q+ (A)] = g(i, + A− ). Besides, if i = q+ (A), we have g(i, C) 6 q+ (A) + q− (A) − 1 = g(i, + A− ). Lemma 7 is proved.

2

LEMMA 8. If A, B are components of J and A < B in J , then under the conditions of Theorem 2 F (γ , + A− ) < F (γ , B). Proof. Let the conditions of the lemma be valid. Without loss of generality, we may assume that A is the left neighbor of B in some set Jk (k > 0). Hence, by Lemma 1 q− (A)q+ (B) − q− (B)q+ (A) = 1.

(12)

Then there exist integers m, n (m > 0, n > 0), such that q− (Am ) + q+ (Am ) = q− (B n ) + q+ (B n ) = p. By (12) q+ (Am ) < q+ (B n ). By Lemma 5 if 1 6 i 6 q+ (B n ) then g(i, B n ) = i + [iq− (B)/q+ (B)]. Besides, if 1 6 i 6 q+ (Am ) and i/q+ (A) 6∈ N then by Lemma 5 g(i, (+ A− )m ) > g(i, B n ). Consider the case i/q+ (A) = r ∈ N and 1 6 i 6 q+ (B n ). By Lemma 5 and by (12) g(i, B n ) = i + [iq− (B)/q+ (B)] = rq+ (A) + [rq− (A) − r/q+ (B)] 6 rq+ (A) − 1 + rq− (A) = g(i, (+ A− )m ). So, for each i (1 6 i 6 q+ (Am ) < q+ (B n )) g(i, (+ A− )m ) > g(i, B n ). Then by the equalities F (γ , (+ A− )m ) = F (γ , +A− ) and F (γ , B n ) =

112

M. M. KIPNIS j

F (γ , B) and by the monotony of Kp (γ ) as to superscribe with 1 6 j 6 p (j ∈ Z) we get F (γ , + A− ) = F (γ , (+ A− )m ) q+ (Am )

X

= 2

Kpg(i,(

+ A− )m )

(γ ) − K(γ )

i=1 q+ (B n )

X

< 2

Kpg(i,(

+ A− )m )

(γ ) − K(γ )

i=1 q+ (B n )

6 2

X

n

Kpg(i,B ) (γ ) − K(γ ) = F (γ , B n ) = F (γ , B).

i=1

2

Lemma 8 is proved.

8.2.

PROOF OF PART (1) OF THEOREM 2

Evidently, the word + (plus) is a periodic ground state configuration in Boolean P∞ averaging system (3) with the given ψ if and only if ψ > i=1 γi = K(γ ). For − (minus) we have analogously ψ < −K(γ ). Now, consider the components of J of length more than 1. Let A = ε1 . . . εp , where εi = − or εi = + (1 6 i 6 p). We have F (γ , A) =

p X

X

q+ (A)

Kpp−i (γ ) sg εi

=2

i=1

Kpg(i,A) (γ )

−

i=1

p X

Kpi (γ ).

(13)

i=1 j

Because of the monotonicity of (γi ) we get Kpi (γ ) > Kp (γ ) when 1 6 i < j 6 p. Hence, for the word A of length greater than 1 in J by Lemma 7 F (γ , A) =

max

B∈Shiftp(A)

F (γ , B);

F (γ , + A− ) =

min

B∈Shiftn(A)

F (γ , B);

(14)

(Recall that + A− ∈ Shiftn(A)). By (13) we get F (γ , + A− ) = F (γ , A) + 2(Kpp−1 (γ ) − Kpp (γ ) > F (γ , A).

(15)

Hence by (14), (15) and the corollary of Theorem 5 the domain of A is nonempty. Part (1) of Theorem 2 is proved. 8.3.

PROOF OF PART (2) OF THEOREM 2

Let the periodic ground state configuration A in Boolean averaging system (3) not be a Hubbard configuration. By part (3) of Theorem 4 there exists a word B, such that B is a component of J and q+ (A)q− (B) = q+ (B)q− (A). Then q+ (B n ) =

113

PERIODIC GROUND STATE CONFIGURATIONS

q+ (A), q− (B n ) = q− (A), where n = q+ (A)/q+ (B). By Lemma 6 we’ll find C ∈ Shiftp(A), such that for each i (1 6 i 6 q+ (A)) g(i, C) 6 i + [iq+ (A)/q− (A)] = g(i, B n ) (the latter equality is valid by Lemma 5). Besides, there exists i (1 6 i 6 q+ (A)), such that we have a strict inequality in the latter inequality; otherwise the words C and A would be Hubbard configurations. Hence, X

q+ (A)

F (γ , C) = 2

Kpg(i,C) (γ ) − K(γ ) > F (γ , B n )

i=1

X

q+ (A)

= 2

n

Kpg(i,B ) (γ ) − K(γ ).

i=1

Similarly we’ll find D ∈ Shiftn(A), such that F (γ , D) < F (γ , (+ B − )n ). By Theorem 5 the domain of A is defined by the inequalities F (γ , (+ B − )n ) > F (γ , D) >

min

E∈Shiftn(A) n

F (γ , E) > ψ >

> F (γ , C) > F (γ , B ).

max

E∈Shiftp(A)

F (γ , E) (16)

The domain of B n coincides with that of B and is defined by the inequalities (see Theorem 5 and Equation (14)) F (γ , (+ B − )n ) > ψ > F (γ , B n ).

(17)

In (16) and (17), we proceed from the fact that for any word E in the −+ alphabet and for any integer n (n > 0) F (γ , E n ) = F (γ , E). Comparing the domains of ψ, defined by inequalities (16) and (17), we get what is required. Part (2) of Theorem 2 is proved. 8.4.

PROOF OF PART (3) OF THEOREM 2

By Theorem 5, Lemma 8 and part (1) of Theorem 2 the domain of the periodic ground state configurations in J is a disjoint system of intervals on the ψ axis. By part (2) of Theorem 2 the set P is a complement to the system mentioned above on the ψ axis. The intervals (−∞; −K(γ )) and (K(γ ); ∞) in the ψ axis are the domains of the periodic ground state configurations − (minus) and + (plus) respectively, which are the components of J0 . The interval [−K(γ ), K(γ )] in the ψ axis is being divided in three parts: [−K(γ ); K22 (γ ) − K21 (γ )], (K22 (γ ) − K21 (γ ); K21 (γ ) − K22 (γ )), [K21 (γ ) − K22 (γ ); K(γ )]. The middle part is the domain of the periodic ground state configuration −+ in J1 . Furthermore, the domains of − − + and − + + are being deleted respectively from the left-handed and right-handed intervals mentioned above. Thus we have deleted the domains of the components of J2 . This procedure goes on according to the definition of Jn . So we have constructed the Cantor perfect set P . Let us prove that the total length of the deleted intervals is equal to 2K(γ ). True, any periodic configuration in

114

M. M. KIPNIS

J , consisting of p letters, is located in the ψ axis in the interval of the length p−1 p 2(Kp (γ ) − Kp (γ )) (Theorem 5 and Equations (14), (15)). By Theorem 4 there are ϕ(p) components of J of the length p (ϕ is a number-theoretic Euler function, ϕ(p) is the number of integers k with 1 6 k 6 p and k relatively prime to p). So, P p−1 p we must prove that ∞ p=2 ϕ(p)(Kp (γ ) − Kp (γ )) = K(γ ). Indeed, ∞ X

ϕ(p)(Kpp−1 (γ ) − Kpp (γ ))

p=2

=

∞ X

ϕ(p)

p=2

= γ1 +

∞ X

(γmp+p−1 − γmp+p )

m=0 ∞ X n=2

γn

X

p|(n+1),p>1

ϕ(p) −

X p|n,p>1

X ∞ ϕ(p) = γn = K(γ ). n=1

In this chain P of equalities, we have used the Gauss theorem about the Euler function [7]: p|n,p>1 ϕ(p) = n − 1. Part (3) of Theorem 2 is proved. Acknowledgement The author would like to thank B. Slepchenko and M. Zelikin for useful discussions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Bak, P. and Bruinsma, R.: One-dimensional Ising model and the complete devil’s staircase, Phys. Rev. Lett. 49 (1982), 249–251. Bernoulli, J., III.: Sur une nouvelle espèce de calcul, Recueil pour les astronomes (Berlin), Vol. 1, 1772, pp. 255–284. Burkov, S. and Sinay, Ya.: Phase diagrams of one-dimensional lattice models with long-range antiferromagnetic interactions, Russian Math. Surveys 38(4) (1983), 235–257. Delchamps, D.: Nonlinear dynamics of oversampling A-to-D converters, Proc. 32nd IEEE CDC, San-Antonio, 1993. Feely, O. and Chua, L.: The effect of Integrator leak in Sigma-delta modulation, IEEE Trans. Circuits Systems 38 (1991), 1293–1305. Gelig, A. and Churilov, A.: Stability and Oscillations in Nonlinear Pulse-modulated Systems, Birkhäuser, Basel, 1998. Hardy, G. and Wright, E.: Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1976. Hubbard, J.: Generalized Wigner lattices in one dimension and some applications to tetracianoquinodimethane (TCNQ) salts, Phys. Rev. B. 17 (1978), 494–505. Jury, E.: Sampled-data Control Systems, Wiley, New York, 1958, 2nd edn, Krieger, 1977. Kieffer, J. C.: Analysis of dc input response for a class of one-bit feedback encoders, IEEE Trans. Comm. 38(3) (1990), 337–340. Kipnis, M. M.: Symbolic and chaotic dynamics of a pulse-width control system, Soviet Phys. Dokl. 324(2) (1992), 273–276.

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12. 13.

14. 15. 16. 17. 18. 19. 20. 21.

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Kipnis, M. M.: On the formalizations of the even 2-colouring, Proc. Chelyabinsk Pedagogical Univ. Series 4. Natural Sciences 1 (1996), 96–104. Kipnis, M. M.: One-dimensional model of statistical mechanics with the Hubbard Hamiltonian and the interaction function, free from the convexity condition, Phys. Dokl. 336(3) (1994), 316–319. Kipnis, M. M.: Boolean Averaging in a statistical mechanics model and in an analog-to-digital converter, Russian J. Math. Phys. 14(3) (1996), 397–402. Leonov, N. N.: On the pointwise transformation of the line in itself, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 2(6) (1959), 942–956 (in Russian). Markoff, A.: Sur une question de Jean Bernoulli, Mat. Ann. 19 (1882), 27–36. Morse, M. and Hedlund, G.: Symbolic dynamics II: Sturmian trajectories, Amer. J. Math. 62 (1940), 1–42. Park, S. and Gray, R.: Sigma-delta modulation with leaky integration and constant input, IEEE Trans. Inf. Theory 38 (1992), 1512–1533. Rockmore, D., Siegel, R., Tongring, N. and Tresser, C.: An approach to renormalization on n-torus, Chaos 1(1) (1991), 25–30. Siegel, R., Tresser, C. and Zettler, G.: A decoding problem in dynamics and in number theory, Chaos 2(4) (1992), 473–493. Smith, H. J. S.: Note on continued fraction, Messenger Math. VI (1877), 1–14.

Mathematical Physics, Analysis and Geometry 3: 117–138, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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Polynomial Asymptotic Representation of Subharmonic Functions in a Half-Plane P. AGRANOVICH Institute for Low Temperature Physics, Mathematical Division, Lenin Ave. 47, 61164 Kharkov, Ukraine (Received: 5 August 1999; in final form: 1 May 2000) Abstract. Let u(z) be a subharmonic function in a half-plane such that its Riesz measure is concentrated on the finite system of rays. In the paper the connection between the behavior of u(z) and the distribution of its measure (including boundary measure) is investigated in terms of polynomial asymptotic representations. Mathematics Subject Classifications (2000): 30E15, 31A05. Key words: half-plane, subharmonic function, measure, asymptotic representation.

1. Introduction The relation between the asymptotic distribution of zeros of a holomorphic function and the growth of this function at infinity is one of the most important questions of function theory. We will say that a function f (t), t > 0, has polynomial asymptotics if it can be represented in the following way: f (t) = 11 t ρ1 + 12 t ρ2 + · · · + 1n t ρn + κ(t),

t → ∞,

(1)

where 1j , j = 1, . . . , n, are real constants, 0 < [ρ1 ] < ρn < ρn−1 < · · · < ρ1 , and the last term on the right, i.e., the function κ(t), is small in a certain sense in comparison with the previous term. Similarly, we will understand the expression ‘polynomial asymptotics of a function f (z), z → ∞’. In this case, the coefficients 1j are functions of 0 = arg z and t = |z|. The theory of functions of completely regular growth establishes a close connection between the growth of an entire function and the distribution of its zeros by means of one-term asymptotic representations. This theory was constructed by B. Levin and A. Pflüger simultaneously and independently in the thirties and immediately found intensive use in different parts of mathematics and physics. Later, this theory?? was extended to other classes of functions, including holomorphic functions in a half-plane (N. Govorov [5]). Note that, in the latter case, new ?

? [a] is the integral part of a. ?? An extensive bibliography is in [9].

118

P. AGRANOVICH

effects appeared. They are connected with possible singularity of the holomorphic function on the boundary of the half-plane. In addition to such generalizations of this theory, a number of authors have considered problems relating to the connection between the behavior of a subharmonic function and the growth of its Riesz measure in terms of polynomial asymptotic representation. They have shown that it is not possible to use the methods of the function theory of completely regular growth to answer these questions ( see, for example, [4]). The relation between the existence of polynomial asymptotic representations of a subharmonic function in the plane and its Riesz measure was investigated in [1–3,6,7]. In this paper, we consider this problem for a subharmonic function in a halfplane, where the Riesz masses are concentrated on a finite system of rays. For the formulation of the main results, we need the following notations and definitions. Let Uρ be the class of subharmonic functions in the upper half-plane C+ = {z : Imz > 0}, which are bounded from above on every semi-circle CR = {z : z ∈ C+ , |z| < R}, 0 < R < ∞, and have noninteger order ρ in C+ , and where the Riesz measures are concentrated on the positive ray of the imaginary axis. Recall that the order ρu of a subharmonic function u(z) in C+ is the following value (see, for example [9]): ρu = lim sup r→∞

ln+ max{81 (r), 82 (r)} , ln r

where 81 (r) = sup u(reiθ ) 0<θ<π

and

Z

π

82 (r) =

|u(reiθ )| sin θ dθ. 0

If u(z) ∈ Up , then the distribution of its Riesz measure µu can be characterized by the function τu (r) = µu (Ir ), where Ir = {iy : 0 < y 6 r}. In addition, we will use the so-called “boundary measure” µ∂,u defined as Z µ∂,u (φ) = lim u(x + ih)φ(x) dx, h→+0

where φ(x) is a test function on R and the limit is considered in the distribution space D 0 (R). It is known [9] that this limit exists and that it is a real measure (charge). As was shown in [5], the function u(z) ∈ Uρ1 can be represented in the form (p = [ρ1 ]) Z 1 1 1 u(z) = ImPp (z) − Im dµ∂,u + π −1 t − z

POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS

119

z z − ln 1 + dτu (t) + + Re ln 1 − it it −1 Z ∞ z z + Re ln 1 − − ln 1 + + it it 1 j p X 1 z z j dτu (t) + − + j it −it j =1 Z 1 1 zp + Im − − · · · − p+1 dµ∂,u (t) t −z t t 1<|t |<∞ = U1 + U˜1 Z

1

for p > 1 or Z 1 1 1 dµ∂,u + Im u(z) = C − π −1 t − z Z 1 z z + − ln 1 + dτu (t) + Re ln 1 − it it 1 Z ∞ z z Re ln 1 − + − ln 1 + dτu (t) + it it 1 Z z + dµ∂,u (t) = U2 + U˜2 Im t (t − z) 1<|t |<∞ for p = 0, where Pp (z) is a polynomial of degree p. Since ρ1 is noninteger and the functions U1 , U2 are harmonic for |z| > 1, lim r −ρ1 |Uj (reiθ )| = 0,

j = 1, 2,

r→∞

uniformly for θ ∈ [0, π ]. Without loss of generality, we further consider only the case p > 1 and assume that τu (t) = 0, t ∈ (0, 1], so Z ∞ z z Re ln 1 − − ln 1 + + u(z) = it it 1 j p X z 1 z j + dτu (t) + − j it −it j =1 Z 1 zp 1 − − · · · − p+1 t dτ˜u (t), Im (2) + t −z t t 1<|t |<∞ where τ˜u (t) := τu+ (t) = −

1 2π

Z 1

t

1 dµ∂,u (x) x

120

P. AGRANOVICH

for t ∈ (1, ∞) and τ˜u (t) :=

τu− (t)

1 = 2π

Z

−1

−r

1 dµ∂,u (x) x

for t ∈ (−∞, −1). All results below will be formulated, without these additional restrictions. The smallness of the function κ(t) in (1) will be expressed in terms of the following definition: DEFINITION. We will say that a function f (t) satisfies the condition (q, ρ) if Z 2T |f (t)|q dt = o(T ρq+1 ), T → ∞, T

where ρ > 0 is any fixed number. Theorem 1, which will be formulated below, shows the existence of polynomial asymptotics of a subharmonic function u(z) if there are polynomial asymptotic representations of the functions τu , τu+ and τu− . THEOREM 1. Let u(z), z ∈ C+ , be a subharmonic function of the class Uρ1 , / Z, and let the support of its Riesz measure be contained in the ray {z : Re z = ρ1 ∈ 0, Im z > 0}. Suppose that τu (r) = τu+ (r) = τu− (r) =

n X

1j r ρj + 8(r),

j =1 n X

δj+ r ρj + γ + (r),

j =1 n X

(3)

δj− r ρj + γ − (r),

j =1

where p = [ρ1 ] < ρn < ρn−1 < · · · < ρ1 and the functions 8(r), γ ± (r) satisfy the condition (q, ρn ) for some fixed q > 1. Then there exists a C0,1 -set E ? such that, in any sector η 6 θ 6 π − η, 0 < η < π/2, u(z) has the representation n X π π r ρj 1j π iθ cos %j θ − − π − cos %j θ − − u(re ) = sin πρ 2 2 j j =1 δj− πρj r ρj δj+ πρj r ρj sin ρj (θ − π ) + sin ρj θ + − sin πρj sin πρj + κ(reiθ ),

(4)

? The set E is called a C 0,1 set if it can be covered by a system of circles Kj = {|z − zj | < rj } P such that limj →∞ R1 j :|zj |
POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS

121

where lim

z→∞,z∈ /E

|z|ρn κ(z) = 0.

Moreover, if in the conditions (q, ρn ) for the functions 8(r), γ ± (r), the number q is strictly greater than 1, then the functions Z π κ(reiθ ) sin θ dθ sup |κ(reiθ )| and η6θ6π−η

0

satisfy the condition (q, ρn ). Remark 1. The assumption ρn > [ρ1 ] is a natural condition. Indeed, let ρn < [ρ1 ]. It is not difficult to show that one can slightly modify the Riesz measure of the subharmonic function u(z) without changing the asymptotics (3) and the conditions (q, ρn ) for the functions 8(r), γ ± (r) and change the asymptotics of u(z). In a certain sense, the following theorem is the converse to Theorem 1. THEOREM 2. Let u(z), z ∈ C+ , be a function of class Uρ1 , ρ1 ∈ / Z, such that its Riesz measure is concentrated on the positive ray of the imaginary axis. Assume equality (4) holds on the union 0 of the positive ray of the imaginary axis and sectors {0 < arg z < θ1 }, {θ2 < arg z < π }, 0 < θ1 < π/2 < θ2 < π , i.e. n X π π π r ρj 1j iθ cos %j θ − − π − cos %j θ − − u(re ) = sin πρj 2 2 j =1 δj+ πρj r ρj δj− πρj r ρj − sin ρj (θ − π ) + sin ρj θ + κ(reiθ ), sin πρj sin πρj where [ρ1 ] < ρn < ρn−1 < · · · < ρ1 ; 11 > 0 and for some fixed q ∈ (1, ∞) and η > 0 the functions iθ iθ κ˜ 1 (r) := sup |κ(re )|, sup |κ(re )|, |κ(ir)| , θ2 6θ6π−η

η6θ6θ1

and

Z κ˜ 2 (r) :=

θ1

Z

π

κ(re ) sin θ dθ + iθ

κ(reiθ ) sin θ dθ θ2

0

satisfy the condition (q, ρn ). Then the functions τu (r), τu+ and τu− in (2) have the asymptotic representations τu (r) =

n X j =1

1j r ρj + 8(r),

122

P. AGRANOVICH

τu+ (r) =

n X

δj+ r ρj + γ + (r),

j =1

τu− (r)

=

− X

δj− r ρj + γ − (r),

j =1

where the remainder terms 8(r) and γ ± (r) satisfy the condition (q, ρn ). Remark 2. The main difficulties in the proof of this theorem are connected with the presence of singularities of the subharmonic function u(z) on the boundary of the half-plane (see Theorem 3 in Section 2).

2. Proof of Theorem 1 Proof. ? As was noted above we may suppose the function u(z) to have representation (2). Integrating by parts, we obtain p n X 1 − itz 1 z j X1 z j τu (t)|R1 − u(z) = lim Re ln − z + R→∞ 1 + it j it j −it j =1 j =1 Z R 2izp+1 (z sin p π2 − t cos p π2 ) τu (t) dt + − t p+1 (t 2 + z2 ) 1 p+1 Z τ˜u (t) z R p+1 τ˜u (t)|1 + pz dt + + Im p p+1 t (t − z) (t − z) 1<|t |
z it z it

+

p X 1 z j

− j it R |t |=R p X zp+1 1 z j τu (t) +Im (p+1) τ˜u (t) = 0. − j −it t j =1 j =1

(5)

Since τu (1) = 0,

τu+ (1) = 0 and

τu− (−1) = 0,

? Only for simplicity in this paper we will consider the case of two-term asymptotic representa-

tion.

POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS

123

then the expression in (5) is zero when |t| = 1. Thus Z ∞ z sin p π2 − t cos p π2 ) p+1 τu (t) dt + u(z) = −Re 2iz t p+1 (t 2 + z2 ) 1 Z Z τ˜u (t) τ˜u (t) p+1 p dt + Im z dt + p+1 (t − z) p 2 1<|t |<∞ t 1<|t |<∞ t (t − z) (6) = I1 + I2 . Let us consider the integral I1 . Substituting in I1 the expressions of τu (t) from (3) and integrating the main members with the help of residues, we obtain 2 X π r ρj 1j π π iθ cos ρj θ − − π − cos ρj θ − + I1 (re ) = sin πρj 2 2 j =1 + ψ1 (reiθ ),

(7)

where

Z ψ1 (re ) = Re 2i(reiθ )p+1 iθ

1

∞

t cos p π2 − reiθ sin p π2 8(t) dt . t p+1 (t 2 + r 2 e2iθ )

(8)

To estimate the function ψ1 (reiθ ), let us divide the ray [1, ∞) into intervals [2k , 2k+1 ), k = 0, 1, . . . . Let |z| ∈ [2k , 2k+1 ). We can represent the function ψ1 (z) as the sum of three terms: Z 2k−1 Z 2k+2 Z ∞ (k) (k) (k) +Re +Re = I1,1 + I1,2 + I1,3 , ψ1 (z) = Re 2k−1

1

2k+2

(k) ls taken as the principal value when Re z = 0. where I1,2 (k) (k) Estimates of I1,1 and I1,3 can be obtained with the help of the following lemma:

LEMMA A [7]. Let γ (t) be from the space Lq , q > 1, on any finite R the function −p−1 dt < ∞, where p > 0 is an integer. Suppose also interval of R and 0 |γ (t)|t that, for any T > 0 the function γ (t) satisfies the condition (q, ρ), 0 < p < ρ < p + 1. Then, for any fixed positive numbers k1 < 1 and k2 > 1 the asymptotic estimates Z A γ (t) = o(r ρ−p−1 ), r → ∞, sup dt sup p+1 iθ (t − re ) A∈[0,k1 r] 06θ62π 0 t and

sup

Z sup

B∈[k2 r,∞] 06θ62π

are valid.

∞

B

γ (t) dt = o(r ρ−p−1 ), p+1 iθ t (t − re )

r → ∞,

124

P. AGRANOVICH

From our hypothesis on 8, Lemma A and the fact that ρ2 ∈ ([ρ1 ], ρ1 ), we obtain (k) (k) + I1,3 ) → 0, |z|−ρ2 (I1,1

(9)

when |z| → ∞, uniformly for θ = arg z ∈ [0, π ]. (k) into two terms: Let us divide the expression I1,2 Z 2k+2 t cos p π2 − z sin p π2 (k) p+1 I1,2 = Re z 8(t) dt − t p+2 (z − it) 2k−1 Z 2k+2 t cos p π2 − z sin p π2 8(t) dt −Re zp+1 t p+2 (z + it) 2k−1 (k) (k) + I1,2,2 . = I1,2,1

Since z ∈ C+ , it is easy to see that (k) | = o(|z|ρ2 ), |I1,2,2

|z| → ∞.

(10)

(k) For the estimate of the integral I1,2,1 , we will use the following theorem about the ? Cauchy-type integrals:

THEOREM A. (a) For each q ∈ (1, ∞) there exists a constant Cq < ∞ such that, for each function g(t) ∈ Lq (R) and s ∈ R+ the function Z ∞ g(t) g(s) ˆ = sup dt {z:|Im(z−is)|>|Re(z−is)|} −∞ z − it satisfies the estimate kg(t)k ˆ Lq 6 Cq kgkLq . (b) There also exists a constant C < ∞ such that, for any function g(t) ∈ L1 (R) and any h > 0 mes{s : g(s) ˆ > h} 6 CkgkL1 h−1 . (k) consists of two terms: The integral I1,2,1 Z 2k+2 cos p π2 8(t) dt i1 = zp+1 p+1 (z − it) 2k−1 t

and

Z i2 = zp+2

2k+2

2k−1

sin p π2 8(t) dt. t p+2 (z − it)

To estimate the integral i1 , set cos pπ/28(t)t −(p+1), gk (t) = 0, t ∈ / [2k−1 , 2k+2 ).

t ∈ [2k−1 , 2k+2 ),

? In [2] this theorem was formulated for the half-plane {z : Re z > 0}.

125

POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS

Since the function 8(t) satisfies the condition (q, ρ2 ), where q > 1 is fixed, we have that q k(ρ −p− q10 ) . ), k → ∞, q 0 = kgk kLq = o(2 2 q −1 From this and part (a) of Theorem A, for q > 1 we obtain Z 2k+2 sup |i1 |q ds 2k−1

6 =

Z

{z:|Im(z−is)|>|Re(z−is)|} ∞

sup

Z (k+1)(p+1) 2

∞

gk (t) q ds dt z − it

−∞ {z:|Im(z−is)|>|Re(z−is)|} −∞ q q (k+1)(p+1)q k(p+1)q 2 kgˆk kLq 6 Cq 2 kgk kLq = o(2k(ρ2 q+1) ),

k → ∞.

Now let {εk } be a sequence of positive numbers. Then the measure of the set k k+1 kρ2 ˜ Ek = s ∈ [2 , 2 ) : sup |i1 | > εk 2 {z:|Im(z−is)|>|Re(z−is)|}

satisfies −q mes E˜ k = εk o(2k ),

k → ∞.

If q = 1, then using part (b) of Theorem A with h = εk 2kρ2 , we obtain that mes E˜ k = εk−1 o(2k ),

k → ∞.

Thus, for any fixed q > 1 −q

mes E˜ k = εk o(2k ),

k → ∞.

Since the integral i2 has similar estimates, we conclude that if q > 1, then q Z 2k+1 (k) sup |I1,2,1 | ds = o(2k(ρ2 q+1) ), k → ∞. 2k−1

{z:|Im(z−is)|>|Re(z−is)|}

and for q > 1 the measure of the set sup Ek = s : [2k , 2k+1 ) :

{z:|Im(z−is)|>|Re(z−is)|}

(11)

(k) |I1,2,1 |

kρ2

> εk 2

−q

is εk o(2k ), k → ∞. If the sequence {εk } tends to zero S sufficiently slowly, then it is easy to see that the relative measure? of the set e = k Ek is zero. By comparing this with estimates (9) and (10) we conclude that if [ z∈ / E := {z : |Im(z − is)| > |Re(z − is)|} s∈e ? The relative measure of a set G ⊂ (0, ∞) is defined to be the limit lim −1 t →∞ t mes(G ∩ (0, t)).

126

P. AGRANOVICH

then r −ρ2 |ψ1 (reiθ )| → 0,

r→∞

uniformly for θ ∈ [0, π ], where ψ1 (z) is defined by (8). The relative S measure of E is zero. Evidently we can assume that the set E is open, so E = Ij where Ij is an interval. Let us consider squares with diagonals Ij , j = 1, . . . , and circumscribe circles around each such square. The union of these circles covers E and it is easy to see that their radii satisfy the condition from the definition of C0,1 -set. So the set E is C0,1 -set. If q > 1, then direct calculation shows that Z 2k+1 (k) iθ q sup |I1,2,1(re )| dr = o(2k(ρ2 q+1) ), k → ∞. 2k−1

θ∈[0, π4 ]∪[ 3π 4 ,π]

From this and (9), (10), (11) we obtain that Z 2k+1 sup |ψ1 (reiθ )|q dr = o(2k(ρ2 q+1) ), 2k−1

k → ∞,

06θ6π

and, hence, the function sup06θ6π |ψ1 (reiθ )| satisfies the condition (q, ρ2 ). Now let us consider the integral I2 (see (6)). From (3), integrating the main terms with the help of residues, we conclude that I2 =

2 X −π δ + ρj r ρj

sin πρj

j =1

+

2 X π δj− ρj r ρj j =1

where

sin πρj

sin ρj (θ − π ) + sin ρj θ + ψ2 (reiθ ),

(12)

Z ∞ Z ∞ γ + (t) γ − (t) p ψ2 (re ) = Im z p p dt + (−1) dt + t p+1 (t − z) t p+1 (t + z) 1 1 Z Z ∞ ∞ γ + (t) γ − (t) p dt + (−1) dt + t p (t − z)2 t p (t + z)2 1 1 4 X p+1 = Im z I2,j (z) . (13) iθ

p+1

j =1

Let η, 0√< η < π/2, be fixed. If z = reiθ , 0 < η 6 θ 6 π − η, then |t − z| > 2 tr sin η/2 and, hence, it is easy to see that |ψ2 (reiθ )|r −ρ2 → 0,

r → ∞.

POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS

127

From here taking into account (7), (12) and that ψ1 satisfies condition (q, ρ2 ), we obtain (4) with the function κ(reiθ ) = ψ1 (reiθ ) + ψ2 (reiθ ) such that supη6θ6π−η |κ(reiθ )| satisfies condition (q, ρ2 ). For the R πcompletion of the proof of Theorem 1 we have to establish that the function 0 κ(reiθ ) sin θ dθ also satisfies the condition (q, ρ2 ). In fact, q Z 2T Z π κ(reiθ ) sin θ dθ dr T 0 q q Z 2T Z π Z 2T Z π iθ iθ 6 Bq ψ (re ) sin θ dθ dr + ψ (re ) sin θ dθ dr 1 2 T 0 T 0 with some constant Bq > 0. By virtue of the condition (q, ρ2 ) for the function sup06θ6π |ψ1 (reiθ )| it follows that q Z 2T Z π iθ ψ1 (re ) sin θ dθ dr = o(T ρ2 q+1 ), T → ∞. (14) T 0 Also, from Theorem A (the case of the half-plane {z : Re z > 0}), we have (see (13)) q Z 2T Z π dr r p+1 ei(p+1)θ )I2,j (reiθ ) dθ = o(T ρ2 q+1 ), T → ∞, j = 1, 2. 0 T The integrals I2,3 and I2,4 can be estimated in the same way, so we will consider only one of them, for example I2,3 . Let us introduce the function + γ (r) sin θei(p+1)θ , r ∈ [1, ∞), θ ∈ [0, π ], + iθ γ˜ (re ) = 0, otherwise. It is easy to see that γ˜ + (reiθ ) ∈ Lq (C) and kγ˜ + (reiθ )kLq (C) = o(r ρ2 q+1 ), q

Then Z π

r → ∞. Z

sin θr

p+1 i(p+1)θ

e

I2,3 (re ) dθ = r iθ

p+1 C

0

γ˜ + (teiθ ) dλ, t p+1 (t − reiθ )2

where dλ is the Lebesgue measure on C. It is clear that this integral is the Beurling transformation [10] of the function γ˜ + (teiθ )t −(p+1) and, hence, for q > 1 q

kr p+1 I2,3 (reiθ )kLq {T 6r62T } = o(T ρ2 q+2 ),

T → ∞.

128

P. AGRANOVICH

So in view of (13), we have q Z 2T Z π iθ ψ2 (re ) sin θ dθ dr = o(T ρ2 q+1 ), T 0

T → ∞, q > 1. 2

Hence, by virtue of (14), we have finished the proof of Theorem 1. 3. Proof of Theorem 2 For the proof of this statement, we need the following theorem.

/ THEOREM 3. Let u(z), z ∈ C+ , be a subharmonic function of the class Uρ1 , ρ1 ∈ Z, and let its Riesz measure be concentrated on the ray {z : Re z = 0, Im z > 0}. In some sector let Y (θ1 , θ2 ) = {z : θi 6 arg z 6 θ2 , 0 < θ1 < π/2 < θ2 < π } and the function u(z) have the asymptotic representation (4) with the remainder term κ(reiθ ) such that supθ1 6θ6θ2 |κ(reiθ )| satisfies the condition (q, ρn ). Then τu may be represented as τu (t) =

n X

1j t ρj + 8(t),

j =1

where the function 8(t) satisfies the condition (q, ρn ). The proof of this theorem is analogous to the proof of Theorem 6 of [2]. Proof of Theorem 2. First we consider the sector Y1 = {z : θ1 6 arg z 6 π/2}. Under the conformal mapping z → (zeiθ1 )π/(π/2−θ1) (so that θ → ψ ≡ π(θ − θ1 )/((π/2) − θ1 )) the sector Y1 is transformed into the upper half-plane and the function u(z) turns into a function v(z) of order ρ1 /π((π/2) − θ1 ) which is harmonic in the upper half-plane. Without loss of generality we can assume that ρ2 /π((π/2) − θ1 ) is noninteger. Otherwise we will take ρ2 π 0 0 0 /Z θ1 , 0 < θ1 < θ1 , such that − θ1 ∈ π 2 and will do all further reasonings for this θ10 . It is easy to see that the hypotheses of Theorem 2 yield that corresponding to θ = θ1 , θ = π/2 ρj π 2 X π π π 1j t π ( 2 −θ1 ) v(t) = cos ρj θ1 + − cos ρj θ1 − + sin πρ 2 2 j j =1 +

δj+ πρj t

ρj π

( π2 −θ1 )

sin πρj 1 π

+ κ(t π ( 2 −θ1 ) eiθ1 ),

sin ρj (π − θ1 ) + t > 0;

δj− πρj t

ρj π

( π2 −θ1 )

sin πρj

sin ρj θ1 +

(151 )

POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS

129

and v(−t) =

ρj π 2 X π 1j t π ( 2 −θ1 )

j =1

sin πρj

(cos ρj π − 1) +

ρj π πρj t π ( 2 −θ1 ) π + − + sin ρj (δj + δj ) + sin πρj 2

+ κ(t ( 2 −θ1 ) π ei 2 ), π

π

1

t > 0;

(152 )

Nevanlinna’s formula ([8], p. 34) implies that s X

1 v(z) = ak r sin kθ + π k=1 k

zs+1 v(t)Im s+1 dt + O(1), t (t − z) |t |>1

Z

where

ρ1 π s= − θ1 , π 2

z = reiθ1 .

Substituting expressions (151 ) and (152 ) into this, we obtain v(reiθ1 ) =

s X k=1

ak0 r k sin kθ +

ρj π 2 X π 1j r π ( 2 −θ1 )

π 1 π cos ρj +θ − θ1 + θ1 − + sin πρj 2 2 π j =1 π π 1 − cos ρj θ − θ1 + θ1 − − 2 π 2 ρj π δj+ πρj r π ( 2 −θ1 ) π 1 − sin ρj θ − θ1 + θ1 − π + sin πρj 2 π ρj π δj− πρj r π ( 2 −θ1 ) π 1 + I + O(1), + − θ1 + θ1 sin ρj θ sin πρj 2 π where the remainder I is Z 1 ∞ zs+1 ( π2 −θ1 ) π1 iθ κ(t e )Im s+1 dt+ π 1 t (t − z) Z 1 −1 zs+1 π 1 π dt. + κ(|t|( 2 −θ1 ) π ei 2 )Im s+1 π −∞ t (t − z)

130

P. AGRANOVICH π

Define the function κ(t) ˆ as κ(t ( 2 −θ1 ) π eiθ1 )t −(s+1) when t > 1 and π 1 π κ(|t|( 2 −θ1 ) π ei 2 )t −(s+1) when t 6 −1. Then, by virtue of the condition (q, ρ2 ) for κ we have Z π 1 q |κ(t)| ˆ dt = o(T ρ2 ( 2 −θ1 ) π −s−1)q+1), T → ∞, (16) 1

T 6|t |62T

Let us estimate the remainder term I using (16). For this we represent I as the sum of three integrals: Z s+1 Z Z 1 z dt κ(t)Im ˆ + + I = |z| π 16|t |6 |z|2 t − z 6|t |62|z| 2|z|6|t |6∞ 2 = I1 + I2 + I3 . From Theorem A (for the case of the half-plane {z : Re z > 0}) it follows that Z 2T ρ2 π sup |I2 (reiθ )|q dr = o(T π ( 2 −θ1 )q+1 ), T → ∞, (17) T

06θ6π

Since s < ρ2 /π( π2 − θ1 ), then by virtue of Lemma A ρ2 π

|z|− π ( 2 −θ1 ) (|I1 | + |I3 |) → 0, when |z| → ∞. From this and (17) we conclude that Z 2T ρ2 π sup |I (reiθ )|q dr = o(T π ( 2 −θ1 )q+1 ), T → ∞, T

(18)

06θ6π

P 0 k Moreover, it is clear that we can adjoin the sum m k=1 ak r sin kθ to the remainder term I without changing its estimate. Returning to the original function u(z), we deduce that for θ1 6 θ 6 π/2 2 X π π π r ρj 1j − cos ρj θ − − cos ρj θ + u(re ) = sin πρj 2 2 j =1 iθ

−

δj+ πρj r ρj sin πρj

sin ρj (θ − π ) +

δj− πρj r ρj sin πρj

sin ρj θ + κ1 (reiθ ), (191 )

and by virtue of (18) the function supθ1 6θ6 π |κ1 (reiθ )| satisfies the condition 2 (q, ρ2 ). Now let us consider the sector Y2 = {z : π/2 6 arg z 6 θ2 } and repeat the reasonings which we have made for the sector Y1 . Then we obtain that, in Y2 , 2 X 3π π π r ρj 1j − cos ρj θ − − cos ρj θ − u(re ) = sin πρj 2 2 j =1 iθ

POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS

−

δj+ πρj r ρj sin πρj

sin ρj (θ − π ) +

δj− πρj r ρj sin πρj

131

sin ρj θ + κ2 (reiθ ), (192 )

and the function sup π 6θ6θ2 |κ2 (reiθ )| satisfies the condition (q, ρ2 ). 2 So in the whole half-plane C+ , in view of (191 ), (192 ) and the conditions of Theorem 2, the function u(z) can be represented in the following form: 2 X π π r ρj 1j π iθ u(re ) = cos ρj θ − − π − cos ρj θ − − sin πρj 2 2 j =1 −

δj+ πρj r ρj sin πρj

sin ρj (θ − π ) +

δj− πρj r ρj sin πρj

sin ρj θ + κ(re ˜ iθ ), (190 )

where for q ∈ (1, ∞) the functions Z π iθ ˜ )| and κ(re ˜ iθ ) sin θ dθ sup |κ(re η6θ6π−η

0

satisfy the condition (q, ρ2 ). Using Theorem 3, we now conclude that τu (t) = 11 t ρ1 + 12 t ρ2 + 8(t)

(20)

and the function 8(t) satisfies the condition (q, ρ2 ). Let us now consider in the half-plane C+ the subharmonic function u1 (z) of the class Uρ1 , which has the Riesz masses the same as u(z) and with the distribution of the boundary measure τu±1 (t) ≡ 0. Then in view of Theorem 1 and (20), we have 2 X π r ρj 1j π π u1 (re ) = cos ρj θ − − π − cos ρj θ − + sin πρj 2 2 j =1 iθ

+ w1 (reiθ ), where the function sup06θ6π |w1 (reiθ )| satisfies the condition (q, ρ2 ). From (190 ), it is easy to see, in C+ u2 (reiθ ) := u(reiθ ) − u1 (reiθ ) 2 X δj+ πρj r ρj δj− πρj r ρj − sin ρj (θ − π ) + sin ρj θ + = sin πρj sin πρj j =1 + w2 (reiθ ), where the function supη6θ6π−η |w2 (reiθ )|, η > 0, satisfies the condition (q, ρ2 ). From this we obtain that πρ1 u2 (reiθ ) = [δ + sin ρ1 (π − θ1 ) + δ1− sin ρ1 θ1 ] ρ r→∞ r 1 sin πρ1 j lim

132

P. AGRANOVICH

and π

πρ1 sin ρ1 π2 + u2 (rei 2 ) lim = [δ1 + δ1− ]. r→∞ r ρ1 sin πρ1 So we can calculate δ1+ and δ1− . Now let us consider the function (reiθ ∈ C+ ) δ1+ πρ1 r ρ1 δ1− πρ1 r ρ1 sin ρ1 (θ − π ) − sin ρ1 θ. u3 (re ) := u2 (re ) + sin πρ1 sin πρ1 iθ

iθ

Evidently, the condition (q, ρ2 ) for the function supη6θ6π−η |w2 (reiθ )| guarantees the existence of a sequence {rj } such that limj →∞ rj = ∞ and π

w2 (rj ei 2 ) w2 (rj eiθ1 ) lim = lim = 0. ρ ρ j →∞ j →∞ rj 2 rj 2 Going over to the limit with respect to the sequence {rj }, we have π

πρ2 u3 (rj ei 2 ) π = [δ2+ + δ2− ] sin ρ2 ρ2 j →∞ sin πρ2 2 rj lim

and u3 (rj eiθ1 ) πρ2 = [δ + sin ρ2 (π − θ1 ) + δ2− sin ρ2 θ1 ]. ρ2 j →∞ sin πρ2 2 rj lim

So we can calculate δ2+ and δ2− , too. Thus, taking into account (20), we obtain that τu (r) = 11 t ρ1 + 12 t ρ2 + 8(t), τu+ (r) = δ1+ r ρ1 + δ2+ r ρ2 + γ + (r), τu− (r) = δ1− r ρ1 + δ1− r ρ2 + γ − (r), where the function 8(t) satisfies the condition (q, ρ2 ). It remains to estimate the terms γ +(r) and γ − (r). First we assume that sup u(z) < 0.

(21)

Im z>0

By virtue of the generalized Carleman formula [5, 9] we have the equality Z r Z π 1 1 1 dσ (x) = − u(reiθ ) sin θ dθ + A(u, r), 2 2 x r π r 1 0 where

Z

σ (r) = 1

r

Z t dτu (t) −

16|t |6r

t dτ˜u (t);

(22)

POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS

133

A(u, r) = C1 r12 + C2 with the constants C1 and C2 independent of r. Integrating (22) by parts and taking into account that σ (1) = 0, we conclude that Z π Z r σ (x) 1 1 dx = u(reiθ ) sin θ dθ + A(u, r). 3 x 2π r 2 1 0 If we replace r by kr, 0 < k < 1, and subtract the second equation from the first, we obtain Z π Z π Z r 1 1 σ (x) iθ dx = u(re ) sin θ dθ − u(kreiθ ) sin θ dθ + 3 2π r 0 2π kr 0 kr x d(1 − k 2 ) + . (23) 4π r 2 k 2 By virtue of the condition (21), the measure t dτ˜u (t) is nonpositive on R, and that implies monotonic growth of the function σ (x). Hence, Z r 2 2 σ (r) −2 1 − k −2 1 − k 6 . dx 6 σ (r)(rk) σ (kr)(kr) 3 2 2 kr x From this and (23) we have Z π Z π kr iθ k σ (kr) 6 u(re ) sin θ dθ − u(kreiθ ) sin θ dθ + (1 − k 2 )π 0 0 d(1 − k 2 ) 6 σ (r). + 2kr Rπ Let us calculate 0 u(reiθ ) sin dθ. According to (190 ), Z

π

u(re ) sin θ dθ = iθ

0

2 X πρj r ρj j =1

ρj2 − 1

(1j −

δj+

−

δj− )

Z +

π

κ(re ˜ iθ ) sin θ dθ.

0

Therefore X 2 πρj r ρj kr k (1j − δj+ − δj− ) + σ (kr) 6 (1 − k 2 )π j =1 ρj2 − 1 Z

π

2 X πρj r ρj

(1 − δj+ − δj− ) − 2 ρ − 1 j =1 j Z π d(1 − k 2 ) iθ 6 σ (r). κ(kre ˜ ) sin θ dθ + − 2kr 0 +k

κ(re ˜ ) sin θ dθ − iθ

0

It is easy to show that τu (r) − τu+ (r) − τu− (r) =

σ (r) + r

Z 1

r

σ (x) dx. x2

(24)

134

P. AGRANOVICH

From this and (24), we obtain τu (r) − τu+ (r) − τu− (r) " 2 X πρj r ρj k > (1j − δj+ − δj− )(k − k ρj )+ (1 − k 2 )π j =1 ρj2 − 1 Z π Z π d(1 − k 2 ) iθ iθ +k κ(re ˜ ) sin θ dθ − κ(kre ˜ ) sin θ dθ + + 2kr 0 0 2 X π(r ρj − 1) + (1j − δj+ − δj− )(k − k ρj )+ 2 ρ − 1 j j =1 Z Z Z r Z r dt π dt π iθ iθ κ(te ˜ ) sin θ dθ − κ(kte ˜ ) sin θ dθ+ +k t t 1 0 1 0 # d(1 − k 2 ) 1 + 1− . 2k r

(25)

On the other hand, from (24) it follows that (t = kr) τu (t) − τu+ (t) − τu− (t) X 2 1 πρj t ρj 6 (1j − δj+ − δj− )(k 1−ρj − 1)+ (1 − k 2 )π j =1 ρj2 − 1 Z π Z π d(1 − k 2 ) t iθ sin θ dθ − e + κ˜ κ(te ˜ iθ ) sin θ dθ + + k 2t 0 0 2 X

π (1j − δj+ − δj− )(k 1−ρj − 1)(t ρj − 1)+ −1 Z dx π x iθ +k κ˜ sin θ dθ− e k 1 x 0 # Z t Z 1 d(1 − k 2 ) dx π iθ 1− . − κ(xe ˜ ) sin θ dθ + 2 t 1 x 0

+

ρ2 j =1 j Z t

Since k 1−ρj − 1 k(k − k ρj ) 1 − ρj = lim =− , 2 2 k→1 1 − k k→1 1−k 2 from (25) and (26) we have that lim

τu (r) − τu+ (r) − τu− (r) − Z > lim sup k k→1

0

2 X (1j − δj+ − δj− )r ρ j =1

π

Z

κ(re ˜ ) sin θ dθ − iθ

0

π

iθ κ(kre ˜ ) sin θ dθ+

(26)

135

POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS

Z +k

r

1

dt t

Z

π

Z κ(te ˜ iθ ) sin θ dθ −

0

1

r

dt t

Z

π

iθ κ(kte ˜ ) sin θ dθ

0

k , 1 − k2

and τu (r) − τu+ (r) − τu− (r) − Z

2 X (1j − δj+ − δj− )r ρj j =1

r iθ sin θ dθ− 6 lim inf κ˜ e k→1 k 0 Z Z π Z r dt π t iθ iθ − κ(re ˜ ) sin θ dθ + k κ˜ sin θ dθ− e k 0 1 t 0 Z Z r dt π 1 − κ(te ˜ iθ ) sin θ dθ . 1 − k2 1 t 0 Rπ By virtue of the condition (q, ρ2 ) for the function 0 κ(te ˜ iθ ) sin θ dθ, it follows that q Z 2T Z r Z dt π iθ κ(te ˜ ) sin θ dθ dr 6 o(T ρ2 q+1 ), T → ∞, T 1 t 0 π

hence τu (r) − τu+ (r) − τu− (r) −

2 X (1j − δj+ − δj− )r ρj = γ (r), j =1

where the function γ (r) satisfies the condition (q, ρ2 ). So Theorem 2 is proved for this special case. For the complete proof of Theorem 2, we have to remove restriction (21). To this end, it suffices to consider the function U (z) = u(z) − KRe zρ1 e−iρ1 2 − ln M, π

where K and M are sufficiently large constants. Theorem 2 is proved.

2

4. General Case The statements of Theorems 1 and 2 can be extended to the case when the Riesz measure of a subharmonic function is concentrated on a finite system of rays. Let us formulate the corresponding results. / THEOREM 4. Let u(z), z ∈ C+ , be a subharmonic function of the class Uρ1 , ρ1 ∈ Z, and let the support of its Riesz measure µu be contained on a finite system of rays {arg z = θj },

j = 1, . . . , m, 0 = θ0 < θ1 < · · · < θm < θm+1 = π.

136

P. AGRANOVICH

Suppose that τu,j (r) =

n X

(j )

1k r ρk + 8j (r),

j = 1, . . . , m,

k=1

where τu,j (r) = µu ({teiθj : 0 < t 6 r}), and τu+ (r) = τu− (r) =

n X k=1 n X

δk r ρk + γ + (r), δk− r ρk + γ − (r).

k=1

Here [ρ1 ] < ρn < ρn−1 < · · · < ρ1 and the functions 8j (r), j = 1, . . . , m, γ ± (r) satisfy the condition (q, ρn ), q > 1. Then m n X π r ρk X (j ) 1 [cos ρk (|θ − θj | − π ) − cos ρk (θ − θj )] − u(z) = sin πρk j =1 k k=1 δk− πρk r ρk δk+ πρk r ρk sin ρk (θ − π ) + sin ρk θ + κ(reiθ ), − sin πρk sin πρk where the function κ(reiθ ) = o(r ρn ) when r → ∞ uniformly for θ ∈ [η, π − η], 0 < η < π/2, if the point z = reiθ does not belong to any C0,1 -set. If q > 1, then the functions π sup |κ(reiθ )|, 0 < η < , 2 η6θ6π−η and

Z

π

κ(reiθ ) sin θ dθ 0

satisfy the condition (q, ρn ). / THEOREM 5. Let u(z), z ∈ C+ , be a subharmonic function of the class Uρ1 , ρ1 ∈ Z, let its Riesz measure be concentrated on a finite system of rays {arg z = θj }, j = 1, . . . , m, 0 = θ0 < · · · < θm < θm+1 = π . Suppose, further, that m n X π r ρk X (j ) iθ 1 [cos ρk (|θ − θj | − π ) − cos ρk (θ − θj )] − u(re ) = sin πρk j =1 k k=1 δk− πρk r ρn δk+ πρk r ρn ρk (θ − π ) + sin ρk θ + κ(reiθ ), − sin πρk sin πρk

POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS

θ ∈ 0 = (0, α1 ) ∪ (α2 , π ) ∪

[ m

137

{θj } ,

j =1

where ρn /Z (θj +1 − θj ) ∈ π

[ρ1 ] < ρn < · · · < ρ1 , for any

j, j = 1, 2, . . . , m − 1, 0 < α1 < θ1 , θm < α2 < π and the functions n [ o sup |κ(reiθ )| : {η 6 θ 6 α1 } ∪ {α2 6 θ 6 π − η} ∪ θj , η > 0 and

Z

α1

Z κ(re ) sin θ dθ +

π

iθ

κ(reiθ ) sin θ dθ α2

0

satisfy the condition (q, ρn ). Then τu,j (r) =

n X

(j )

1k r ρk + 8j (r),

k=1

τu+ (r) = τu− (r) =

n X k=1 n X

δk r ρk + γ + (r), δk− r ρk + γ − (r),

k=1

where the functions 8(r), j = 1, . . . , m, and γ ± satisfy the condition (q, ρn ).

Acknowledgements With deep sorrow I say my last thankful words to Professor L. Ronkin for some very useful conversations related to this problem. I am grateful to the referee for constructive criticism directed at improving the quality of my exposition.

References 1.

Agranovich, P. Z.: Polynomial asymptotic representations of subharmonic functions with masses on the finite system of rays, MAG 3(3/4) (1996), 219–230 (Russian).

138 2.

3. 4. 5. 6. 7. 8. 9. 10.

P. AGRANOVICH

Agranovich, P. Z. and Logvinenko, V. N.: The analogue of the Valiron–Titchmarsh theorem for two-term asymptotics of the subharmonic function with masses on a finite set of rays, Sibirsk. Mat. Z. 24(5) (1985), 3–19 (Russian). Agranovich, P. Z. and Logvinenko, V. N.: Polynomial asymptotic representation of subharmonic function in the plane, Sibirsk. Math. Z. 32(1) (1991), 3–21 (Russian). Anderson, J. M.: Integral functions and Tauberian theorems, Duke Math. J. 32(4) (1965), 145– 163. Govorov, N. V.: Riemann Boundary Value Problems with Infinite Index, Nauka, Moscow, 1986 (Russian). Logvinenko, V. N.: About entire functions with zeros on a half-line. I, Theory of functions, Funct. Anal. Appl. 16 (1972), 154–158. Logvinenko, V. N.: About entire functions with zeros on a half-line. II, Theory of functions, Funct. Anal. Appl. 17 (1973), 84–99. Nevanlinna, R.: Uber die Eigenschaften Meromorpher Funktionen in einem Winkebraum, Acta Soc. Sci. Fenn. 50(12) (1925), 1–45. Ronkin, L. I: Functions of Completely Regular Growth of Several Variables, Kluwer Acad. Publ., Dordrecht, 1992. Zygmund, A.: Integrales Singulieres, Lecture Notes in Math. 204, Springer-Verlag, New York, 1971.

Mathematical Physics, Analysis and Geometry 3: 139–177, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

139

Differential Equations Compatible with KZ Equations G. FELDER1, Y. MARKOV2, V. TARASOV3 and A. VARCHENKO2?

1 Departement Mathematik, ETH-Zentrum, 8092 Zürich, Switzerland. e-mail: [email protected] 2 Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599 – 3250, U.S.A.

e-mail: {markov, av}@math.unc.edu 3 St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011, Russia. e-mail: [email protected] (Received: 3 April 2000) Abstract. We define a system of ‘dynamical’ differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra g. These are equations on a function of n complex variables zi taking values in the tensor product of n finite dimensional g-modules. The KZ equations depend on the ‘dual’ variable in the Cartan subalgebra of g. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions. Mathematics Subject Classifications (2000): Primary 35Q40; secondary 17B10. Key words: hypergeometric solutions, Kac–Moody Lie algebras, KZ equations.

1. Introduction In the theory of the bispectral problem [5, 12], one considers a commutative algebra A of differential operators L(z, ∂/∂z) acting on functions of one complex variable z. Such an algebra is called bispectral if there exists a non-trivial family u(z, µ) of common eigenfunctions depending on a spectral parameter µ Lu(z, µ) = fL (µ)u(z, µ),

L ∈ A,

(1)

which is also a family of common eigenfunctions of a commutative algebra B of differential operators 3(µ, ∂/∂µ) with respect to µ: 3u(z, µ) = θ3 (z)u(z, µ),

3 ∈ B.

(2)

J. Duistermaat and A. Grünbaum [5] studied the case where A is the algebra of differential operators that commute with a Schrödinger operator (d2 /dz2 ) − V (z) with meromorphic potential V (z). They give a complete classification of bispectral ? The last author is supported in part by NSF grant DMS-9801582.

140

G. FELDER ET AL.

algebras arising in this way. In particular they show that A is bispectral if V (z) is a rational KdV potential (a rational function which stays rational under the flow of the Korteweg–de Vries equation). G. Wilson [12] classified bispectral algebras of rank one, i.e., such that the greatest common divisor of the orders of the differential operators in A is one. He showed that the maximal bispectral algebras of rank one are in one to one correspondence with conjugacy classes of pairs (Z, M) of square matrices so that ZM − MZ + I has rank one. The bispectrality then follows from the existence of the involution (Z, M) 7→ (M T , Z T ), which corresponds to exchanging z and µ. The higher-dimensional version of the bispectral problem, in which A consists of partial differential operators in z ∈ Cn is open. However, O. Chalykh, M. Feigin and A. Veselov [3, 2] constructed examples of algebras in higher dimensions which have the bispectral property (see Veselov’s contribution to [7]). In these examples, A consists of differential operators commuting with an n-particle Schrödinger operator with certain special rational potentials, including those of Calogero–Moser. These potentials are in many respects the natural generalization of rational KdV potential associated to rank one algebras. In these examples, the Baker–Akhiezer function u(z, µ) is symmetric in the two arguments, thus B = A. A good source of material on the bispectral problem is the volume [7]. In this paper we study a class of examples of commutative algebras of partial differential operators acting on vector-valued functions with the bispectral property. This means that in (1), (2), u takes values in a vector space and fL (µ), θ3 (z) are endomorphisms of the vector space. In our class of examples, the algebra A is generated by Knizhnik–Zamolodchikov differential operators. They are commuting first-order differential operators associated to a complex simple Lie algebra g with a fixed non-degenerate invariant bilinear form and a non-zero complex parameter κ. They act on functions of n complex variables zi taking values in the tensor product of n finite-dimensional g-modules. The ‘dual’ variable µ is in a Cartan subalgebra of g. The first set of Equations (1) is then the set of generalized Knizhnik–Zamolodchikov equations X (ij ) ∂ − u(z, µ) = µ(i) u(z, µ), i = 1, . . . , n. κ ∂zi j :j 6=i zi − zj Here ∈ g ⊗ g is dual to the invariant bilinear form and (ij ) acts as on the ith and j th factors of the tensor product and as the identity on the other factors. Similarly µ(i) is µ acting on the ith factor. It is well-known that these equations form a compatible system, i.e., they are the equations defining horizontal sections for a flat connection. For µ = 0 they reduce to the classical Knizhnik–Zamolodchikov equations. The algebra B is generated by rank(g) first order partial differential operators in µ with rational coefficients. We call the corresponding Equations (2) dynamical differential equations, and show that they form, together with the generalized Knizhnik–Zamolodchikov equations, a compatible system. We also give simultaneous solutions of both systems of equations in terms of hypergeometric

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

141

integrals for a more general class of Lie algebras, which includes in particular all Kac–Moody Lie algebras. In the case of g = sl2 , the algebra B is generated by one ordinary differential operator. In this case, the corresponding equations where first written and solved by H. Babujian and A. Kitaev [1], who also related the equations to the Maxwell– Bloch system. Our paper is organized as follows. In Section 2 we introduce the systems of Knizhnik–Zamolodchikov and dynamical differential equations for arbitrary simple Lie algebra and prove their compatibility. We then give formulae for hypergeometric solutions in Section 3 and give as an application a determinant formula. The fact that the hypergeometric integrals provide solutions is a consequence of a general theorem valid for a class of Lie algebras with generic Cartan matrix, introduced in [10]. We introduce in Section 4 the Knizhnik–Zamolodchikov and dynamical differential equations in this more general context and explain in the next Section the results on complexes of hypergeometric differential forms from [10]. In Section 6 we prove that the hypergeometric integrals for generic Lie algebras satisfies the dynamical differential equations. Finally in Section 7 we prove that our hypergeometric integrals are solutions of both systems of equations for any Kac–Moody Lie algebra. We also find a determinant formula, which implies a completeness result for solutions in the case of generic parameters. 2. Dynamical Differential Equations 2.1. Let g be a simple complex Lie algebraL with an invariant bilinear form ( , ) and a root space decomposition g = h ⊕ ( α∈1 Ceα ). The root vectors eα are = 1. Then the quadratic Casimir element of g ⊗ g has normalized so P that (eα , e−α ) P the form = s hs ⊗ hs + α∈1 eα ⊗ e−α , for any orthonormal basis (hs ) of the Cartan subalgebra h. We also fix a system of simple roots α1 , . . . , αr . Consider the Knizhnik–Zamolodchikov (KZ) equations with an additional parameter µ ∈ h, for a function u on n variables taking values in a tensor product V = V1 ⊗ · · · ⊗ Vn of highest weight modules of g with corresponding highest weights 31 , . . . , 3n , κ

X (ij ) ∂u = µ(i) u + u, ∂zi z − z i j i6=j

i = 1, . . . , n,

(3)

where κ is a complex parameter. We are interested in a differential equation for u with respect to µ which are compatible with KZ equations. If µ0 ∈ h, denote by ∂µ0 the partial derivative with respect to µ in the direction of µ0 . THEOREM 2.1. The equations κ∂µ0 u =

n X i=1

zi (µ0 )(i) u +

X hα, µ0 i α>0

hα, µi

e−α eα u,

µ0 ∈ h,

(4)

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form together with the KZ equations (3), a compatible system of equations for a function u(z, µ) taking values in V = V1 ⊗ · · · ⊗ Vn . Equations (4) will be called dynamical differential equations. EXAMPLE. Let g = slN = glN /C. View slN -modules as glN -modules by letting the center of glN act trivially. Denote by Ea,b ∈ glN the matrix whose entries are zero except for a one at the intersection of the bth column. The P ath row with the P fundamental coweights $a = (1 − a/N) b6a Eb,b − (a/N) b>a Eb,b , a = 1, . . . , N − 1 form a basis of the standard Cartan subalgebra of slN . Write µ = P N−1 a=1 µa $a . Then our equations may be written as

2.2.

κ

(i) (j ) N−1 X X Ea,b X Eb,a ∂u = µa $a(i) u + u, ∂zi z − z i j a=1 j :j 6=i a,b

κ

(i) (j ) n X X X Eb,c Ec,b ∂u = zi $a(i) u + u. ∂µa µb + µb+1 + · · · + µc−1 i=1 b,c:b6a
PROOF OF THEOREM

2.1

It is rather easy to verify that most terms of the compatibility equations vanish. P The only non-trivial thing to check is that the operators α>0 (hα, λi/hα, µi)e−α eα commute for different values of λ. The operators obtained by extending the sum to all roots differ from the sum over positive roots by an element of the Cartan subalgebra. Since the operators commute with the Cartan subalgebra, it is sufficient to prove the following proposition. PROPOSITION 2.2. Let for λ, µ ∈ h, T (λ, µ) = Then for all λ, µ ∈ h,

P

α∈1 (hα, λi/hα, µi)e−α eα .

T (λ, µ)T (ν, µ) = T (ν, µ)T (λ, µ). The proof is based on the following fact. LEMMA 2.3. Let α, β ∈ 1 with α 6= ±β,P and let S = S(α, β) be the set of integers j such that β + j α ∈ 1. Then j ∈S [eα , eβ+j α e−β−j α ] = 0 and P [e , e e ] = 0. j ∈S −α β+j α −β−j α Proof. For roots γ , δ such that γ +δ is a root, let Nγ ,δ = ([eγ , eδ ], e−γ −δ ), so that [eγ , eδ ] = Nγ ,δ eγ +δ . By considering the adjoint action on g of the sl2 sub-algebra generated by e±α , we see that for β 6= ±α, S is a finite sequence of subsequent integers. We may thus assume that S = {0, . . . , k} by replacing β by β − j α for some j if necessary.

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

143

We then have k X [eα , eβ+j α e−β−j α ] j =0

=

k−1 X

Nα,β+j α eβ+(j +1)α e−β−j α +

j =0

=

k X

Nα,−β−j α eβ+j α e−β−(j −1)α

j =1

k−1 X (Nα,β+j α + Nα,−β−(j +1)α )eβ+(j +1)α e−β−j α . j =0

By the invariance of the bilinear form, Nα,β+j α = ([eα , eβ+j α ], e−β−(j +1)α ) = −(eβ+j α , [eα , e−β−(j +1)α ]) = −Nα,−β−(j +1)α . P Therefore j ∈S [eα , eβ+j α e−β−j α ] vanishes. The other statement is proved by replacing α by −α and noticing that S(−α, β) = S(α, β). 2 Proof of Proposition 2.2. Consider X hα, λihβ, νi T (λ, µ)T (ν, µ) = [eα e−α , eβ e−β ]. hα, µihα, µi α,β

(5)

Let us show that this expression is a regular function of µ ∈ h. Since for nontrivial λ, µ it converges to zero at infinity, it then vanishes identically. We compute the residue of (5) at hα, µi = 0: X hα, λihγ , νi − hγ , λihα, νi (6) [eα e−α , eγ e−γ ], hγ , µi γ :γ 6=±α a function on the hyperplane hα, µi = 0. The sum over γ of the form β + j α, j ∈ S(α, β) gives for hα, µi = 0, X hα, λihβ, νi − hβ, λihα, νi [eα e−α , eβ+j α e−β−j α ] = 0, (7) hβ, µi j ∈S by the previous lemma. Since the sum over γ in (6) can be written as a sum of such terms, it vanishes.

3. Hypergeometric Solutions Let g be a simple complex Lie algebra. Choose a set f1 , . . . , fr , e1 , . . . , er of Chevalley generators of the Lie algebra g associated with simple roots α1 , . . . , αr .

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P Let λ = (m1 , . . . , mr ) ∈ Nr . Let Q+ = P Nαi be the positive root lattice for mi αi . Let V be a tensor product g. Define a map α: Nr → Q+ by α(λ) = of highest weight modules V of g with respective highest weights 3j , where Pn j j = 1, . . . , n. Set 3 = j =1 3j . Denote Vλ the weight space of V with weight 3 − α(λ). The hypergeometric solutions of the KZ equations in Vλ , see [10], have R the form u(z) = γ (z) 8(z, t)1/k ω(z, t). We will describe the explicit construction. P The number of integration variables (tk )m mi . Let c be the unique k=1 is m = non-decreasing function from {1, . . . , m} to {1, . . . , r} (i = 1, . . . , r), such that #c−1 ({i}) = mi . Define Y Y Y 8(z, t) = (zi − zj )(3i ,3j ) (tk − zj )−(αc(k) ,3j ) (tk − tl )(αc(k) ,αc(l) ) . i<j

k,j

k
The m-form ω(z, t) is a closed logarithmic differential form on Cn × Cm with values in Vλ . It has the following combinatorial description. Let P (λ, n) be the set of sequences I = (i11 , . . . , is11 ; . . . ; i1n , . . . , isnn ) of integers in {1, . . . , r} with sj > 0, j = 1, . . . , n and such that, for all 1 6 j 6 r, j appears precisely |c−1 (j )| times in I . For I ∈ P (λ, n), and a permutation σ ∈ 6m , set σ1 (l) = σ (l) and σj (l) = σ (s1 + · · · + sj −1 + l), j = 2, . . . , n, 1 6 l 6 sj . Define 6(I ) = {σ ∈ 6m | c(σj (l)) = ijl for all j and l}. Fix a highest-weight vector vj for each representation Vj , j = 1, . . . , n. To every I ∈ P (λ, n) we associate a vector fI v = fi11 · · · fis1 v1 ⊗ · · · ⊗ fi1n · · · fisnn vn 1 in Vλ , and meromorphic differential m-forms ωI,σ = ωσ1 (1),...,σ1 (s1 ) (z1 ) ∧ · · · ∧ ωσn (1),...,σn (sn ) (zn ), labeled by σ ∈ 6(I ), where ωi1 ,...,is (z) = d log(ti1 − ti2 ) ∧ · · · ∧ d log(tis−1 − tis ) ∧ d log(tis − z) is a meromorphic one form on C × Cs . Finally X X (−1)|σ | ωI,σ fI v. ω(z, t) = I ∈P (λ,n) σ ∈6(I )

P dz −dz It obeys the equation 8−1 d8 ∧ ω = K ∧ ω, where K = i<j (ij ) zii −zj j . As a consequence, for each horizontal family of twisted cycles γ (z) in {z}×Cm, u obeys the KZ equations (3) with µ = 0 (see [10]). This construction can be modified to give solutions for general µ: THEOREM 3.1. The integrals ! Z m n X X 1 8µk ω, 8µ = exp − hαc(i) , µiti + h3j , µizj 8, γ (z)

i=1

are solutions of the KZ equations (3). Proof. The proof follows from the identity 8−1 µ d8µ ∧ ω =

n X i=1

µ(i) dzi ∧ ω + K ∧ ω.

j =1

(8)

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DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

To prove this identity notice that ω is a sum of several terms ωs , which can be grouped according to how the integration variables t1 , . . . , tm are distributed among the points z1 , . . . , zn . If the variable tk is associated to the point zi in one term ωs , then (dtk − dzi ) ∧ ωs = 0. Moreover, if the term ωs obeys h(i) ωs = (h3i , hi − P m0j (i)hαj , hi)ωs , h ∈ h, where 3i is the highest weight of the ith representation, then the number of variables tk associated to zi such that c(k) = j is m0j (i) (j = 1, . . . , r). Thus we have n X X −1 0 h3i , µi − mj (i)hαj , µi dzi ∧ ω + K ∧ ω. 8µ d8µ ∧ ω = i=1

j

2

The proof is complete.

THEOREM 3.2. The hypergeometric integrals of Theorem 3.1 obey the Dynamical differential equations (4). The proof of the Theorem is given in Section 7. An application of the above two theorems is a determinant formula. COROLLARY Pd3.3. Fix a basis v1 , . . . , vd of a weight space Vλ . Suppose that ui (µ, z) = j =1 ui,j vj , i = 1, . . . , d is a basis of the space of solutions in a neighbourhood of a generic point (µ, z) ∈ h × Cn . Let δα = trVλ (e−α eα ) (α ∈ 1, α > 0), ij = trVλ (ij ). Then there is a constant C = C(V1 , . . . , Vn , λ, κ) 6= 0 such that ! n X Y Y zi det(uij ) = C exp hα, µiδα /κ (zi − zj )ij /κ . trVλ (µ(i) ) κ i=1 α>0 i<j

4. Free Lie Algebras. Dynamical Differential Equations 4.1.

THE DEFINITION OF KZ AND DYNAMICAL DIFFERENTIAL EQUATIONS

Following [10] let us fix the following data: (1) A finite-dimensional complex vector space h; (2) A non-degenerate symmetric bilinear form ( , ) on h; (3) Linearly independent covectors (‘simple roots’) α1 , . . . , αr ∈ h∗ . We denote by b: h → h∗ the isomorphism induced by ( , ), and we transfer the form ( , ) to h∗ via b. Set bij = (αi , αj ), hi = b−1 (αi ) ∈ h. Denote by g the Lie algebra generated by ei , fi for i = 1, . . . , r and h subject to the relations: [h, ei ] = hαi , hiei ,

[h, fi ] = −hαi , hifi ,

[ei , fj ] = δij hi ,

[h, h0 ] = 0,

for all i, j = 1, . . . , r and h, h0 ∈ h. Thus we have constructed a Kac–Moody Lie algebra without Serre’s relations. We denote by n− (resp. by n+ ) the subalgebra

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of g generated by fi (resp. ei ) for i = 1, . . . , r. We have g = n− ⊕ h ⊕ n+ . Set b± = n± ⊕ h. These are subalgebras of g. Let 3 ∈ h∗ . Denote by M(3) the Verma module over g generated by a vector v subject to the relations n+ v = 0 and hv = h3, hiv for all h ∈ h. Let us fix weights 31 , . . . , 3n ∈ h∗ , and let M = M(31 ) ⊗ · · · ⊗ M(3n ). For λ = (m1 , . . . , mr ) ∈ Nr , set D X E (n± )λ = x ∈ n± | [h, x] = ± mi αi , h x, for all h ∈ h , D E X mi αi , h x, for all h ∈ h , M(3)λ = x ∈ M(3) | hx = 3 − DX E X Mλ = x ∈ M | hx = 3j − mi αi , h x, for all h ∈ h . L L L We λ (n± )λ , b± = h ⊕ ( λ (n± )λ ), M(3) = λ M(3)λ , M = L have n± = λ Mλ . Let τ : g → g be the Lie algebra automorphism such that τ (eL i ) = −fi , τ (fi ) = L −ei , τ (h) = −h, for h ∈ h. Set n∗± = λ (n± )∗λ . Set M(3)∗ = λ M(3)∗λ . Define a structure of a g-module on M(3)∗ by the rule hgφ, xi = hφ, −τ (g)xi

for φ ∈ M(3)∗ , g ∈ g, x ∈ M(3).

(9)

There is a unique bilinear form K( , ) on g such that: K coincides with ( , ) on h; K is zero on n+ and n− ; h and n− ⊕ n+ are orthogonal; K(fi , ej ) = K(ej , fi ) = δij for i, j = 1, . . . , r; K is g-invariant, that is K([x, y], z) = K(x, [y, z]) for all x, y, z ∈ g. A bilinear form S on g is defined by the rule S(x, y) = −K(τ (x), y). The form S is symmetric, τ -invariant, and S([x, y], z) = S(x, [τ (y), z]). The subspaces n+ , h, n− are pairwise orthogonal with respect to S. For a Verma module M(3) with highest weight 3 and generating vector v there is a unique bilinear form S on M(3) such that: S(v, v) = 1; S(ei x, y) = S(x, fi y); S(fi x, y) = S(x, ei y), for all x, y ∈ M(3) and i = 1, . . . , r. S is symmetric. The subspaces M(3)λ are pairwise orthogonal with respect to S. The form S induces a homomorphism of g-modules S: M(3) → M(3)∗ . The module M(3)/ ker S V is the irreducible g-module with highest weight 3. More generally, on the space p n− ⊗ M(31 ) ⊗ · · · ⊗ M(3n ) for 3j ∈ h∗ , a bilinear form S is defined by the rule S(g1 ∧ · · · ∧ gp ⊗ x1 ⊗ · · · ⊗ xn , g10 ∧ · · · ∧ gp0 ⊗ x10 ⊗ · · · ⊗ xn0 ) = det S(gi , gj0 ) ·

n Y

S(xi , xi0 ).

i=1

This form induces a map ^p n− ⊗ M(31 ) ⊗ · · · ⊗ M(3n ) MS:

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

→

^p

∗ n− ⊗ M(31 ) ⊗ · · · ⊗ M(3n ) ,

147 (10)

S is called the contravariant form. The linear map (10) depends analytically on (bij ) and 31 , . . . , 3n ∈ h∗ for a fixed r. It is non-degenerate for general values of the parameters, see Theorem 3.7 in [10]. A Lie Bialgebra Structure on b± [10]. A Lie bialgebra is a vector space g with a Lie algebra structure and a Lie coalgebra structure, such that the cocommutator map ν: g → g ∧ g is a one-cocycle: xν(y) − yν(x) = ν([x, y]), for all x, y ∈ g. Here the action of g on g ∧ g is the adjoint one: a(b ∧ c) = [a, b] ∧ c + b ∧ [a, c]. The dual map to the cocommutator map, ν ∗ : (g ∧ g)∗ → g∗ , defines a Lie algebra structure on g∗ . Let g be a Lie bialgebra. The double of g is the Lie algebra equal to g ⊕ g∗ as a vector space with the bracket on g and g∗ defined by the Lie algebra structure on g and g∗ , and for x ∈ g and l ∈ g∗ , [l, x] = l¯ + x, ¯ where x¯ ∈ g and l¯ ∈ g∗ are defined ¯ by the rules l(y) = l([x, y]), m(x) ¯ = [m, l](x), for all y ∈ g, m ∈ g∗ . The double is denoted by D(g). Let g be the Kac–Moody algebra we defined at the beginning of the section. g is a Lie bialgebra with respect to the following cobracket. There exists a unique map ν: g → g ∧ g such that xν(y) − yν(x) = ν([x, y]), and ν(h) = 0, and ν(fi ) = 1 f ∧ hi , and ν(ei ) = 12 ei ∧ hi . In the previous four equalities, h ∈ h, i = 1, . . . , r, 2 i the action of g on g ∧ g is the adjoint one, see [4], Example 3.2 and [10]. b− and b+ are subbialgebras. The map ν has the property τ ν + ντ = 0. Thus, if ρ: b− ∗ → End(V ) is a representation of the Lie algebra (ν|b− )∗ : 32 b− ∗ → b− ∗ , then −ρ ◦ τ : b+ ∗ → End(V ) is a representation of the Lie algebra (ν|b+ )∗ : 32 b+ ∗ → b+ ∗ . Note that the coalgebra map ν defines a Lie algebra structure on b∗± . Comultiplication. Let M = M(31 ) ⊗ · · · ⊗ M(3P n ). Let v = v1 ⊗ · · · ⊗ vn ∈ M be the product of the generating vectors. Set 3 = 3j . Let b− act on b− ⊗ M by the rule a(b ⊗ m) = [a, b] ⊗ m + b ⊗ am. For 1 6 i 6 n, and a, b ∈ g, m = x1 ⊗ · · · ⊗ xn ∈ M, set a (i) m = x1 ⊗ · · · ⊗ xi−1 ⊗ axi ⊗ xi+1 ⊗ · · · ⊗ xn and a (i) (b ⊗ m) = [a, b] ⊗ m + b ⊗ a (i) m. There is a unique linear map νM : M → b− ⊗ M such that νM (h · x) = h · νM (x) for any h ∈ h and x ∈ M; νM (x) = 12 (b−1 (3 − α(λ))) ⊗ x + νM− (x) for x ∈ Mλ . P P Recall that b−1 (α(λ)) = mi b−1 (αi ) = mi hi , and b−1 : h∗ → h is defined n− ⊗ M is defined via an at the beginning of Section 4.1. The map νM− P:n M → (k) inductive definition. νM− (v) = 0, νM− (x) = k=1 νM− (x) where (j )

(j )

(k) (k) (fi x) = fi νM− (x) νM− (k) (fi(k) x) νM−

= fi ⊗

h(k) i x

for k 6= j, and

(k) + fi(k) νM− (x).

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In all formulae we have 1 6 i 6 r, and 1 6 j, k 6 n. Remark. The corresponding definition of νM− (x) in [10] should be corrected as above. Note that the two definitions coincide if we have one tensor factor, i.e. n = 1. We have the following lemma (cf. Lemma 6.15.2 in [10]). LEMMA 4.1. For any x, y ∈ M, a ∈ b− , 1 S(x, ay) if a ∈ h; S(νM (x), a ⊗ y) = 2 S(x, ay) if a ∈ n− .

(11)

Here S is defined on b− ⊗ M by the rule S(a ⊗ x, b ⊗ y) = S(a, b)S(x, y), cf. (10). The proof of Lemma 4.1 is given in Section 4.5. Note that the above lemma renders the following diagram commutative. n− ⊗ M

standard

S

n− ∗ ⊗ M ∗

M S

∗ νM−

M∗

S is an isomorphism for general values of parameters (bij ), (3k )nk=1 , see [10], ∗ : b− ∗ ⊗ M ∗ → M ∗ is a b− ∗ -module structure sections (3.7) and (6.6). Hence, νM with respect to the Lie algebra structure ν ∗ : 32 b− ∗ → b− ∗ for any values of the above parameters. COROLLARY 4.2. If y ∈ ker S: n− → n− then yM ⊂ ker S: M → M.

ACTIONS OF THE DOUBLES OF

b±

ON

m AND m∗

RESPECTIVELY

Consider the standard action of b− on M ∗ , i.e. ∀a ∈ b− ∀φ ∈ M ∗ , ha · φ, · i = hφ, −a · i, where the action on the right-hand side is the standard action of g on M. ∗ defines an action of b− ⊕ b− ∗ on M ∗ . Lemma 6.17.1 This map together with νM− ∗ [10] asserts that M is a D(b−)-module under this action, where D(b−) is the double of b−. ∗ ∗ defines a map νM (a, ·): M ∗ → M ∗ . Set For any a ∈ b− ∗ , the action of νM ∗ ∗ ∗ ρ(a) = −(νM (a, ·)) : M → M. The map ρ: b− → End(M) gives an action of b− ∗ on M. An action of b+ ∗ on M is defined by ω = −ρ ◦ τ : b+ ∗ → End(M). The rule a ⊗ x → τ (a)x, for a ∈ b+ , x ∈ M, defines an action of b+ on M. This action and the map ω define an action of b+ ⊗ b+∗ on M. Lemma 11.3.28 [11] implies that M is a D(b+) module. Note that for any Kac–Moody Lie algebra g without Serre’s relations, and a g-module V we can define a g-module structure on V ∗ by the rule hgφ, · i =

149

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

hφ, −g · i for all g ∈ g and φ ∈ M ∗ . With respect to this module structure, M becomes a D(b−)-module and M ∗ becomes a D(b+)-module. KZ Equations and Dynamical Differential Equations in M and M ∗ . For a vector space V denote by (V ) ∈ V ⊗V ∗ the canonical element. For λ ∈ Nr , set − λ,± := + ∗ ∗ ∗ ((n± )λ ) ∈ (n± )λ ⊗ (n± )λ , set λ,± := ((n± )λ ) ∈ (n± )λ ⊗ (n± )λ , 0 := ((h) + (h∗ )) ∈ h ⊗ h∗ + h∗ ⊗ h. Set X X 0 − + ± = λ,± + + λ,± ∈ D(b± ) ⊗ D(b± ). λ

λ

Let +,ij be the operator on M (or M ∗ ) acting as + on M(3i )⊗M(3j ), (M(3i )∗ ⊗M(3j )∗ respectively) and as the identity on the other factors. The action of D(b±) on M(3j ) is the action decribed in the previous paragraph when n = 1. The KZ equations with additional parameter µ ∈ h, for a function u(µ, z) on n variables z = (z1 , . . . , zn ) taking values in M (or M ∗ ) are κ

X +,ij ∂u = µ(i) u + u, ∂zi z − zj i6=j i

i = 1, . . . , n, κ ∈ C.

(12)

Let α be a positive root for g and (yi(α) ) a basis of (n+ )α . Set xi(α) = τ (yi(α) ). Then ((yi(α) )∗ ), (xi(α) ) and ((xi(α) )∗ ) are bases of (n+∗ )α , (n− )α , (n− ∗ )α , respectively. P (α) (α) ∗ Define operators 1±,α on M (or M ∗ ) via the formulae 1+,α = i yi (yi ) , P (α) ∗ (α) 1−,α = i (xi ) xi . The dynamical differential equations for the function u(µ, z) with values in M (M ∗ respectively) are κ∂µ0 u =

n X

zi (µ0 )(i) u +

i=1

4.2.

X hα, µ0 i α>0

hα, µi

PROPERTIES OF THE OPERATORS

1+,α u,

µ0 ∈ h, κ ∈ C.

(13)

1+,α

The properties of the operator +,ij are thoroughly described in [11]. Now we are going to study the operators 1+,α . LEMMA 4.3. The following diagram is commutative: M

1+,α

S

M∗

M S

−1−,α

M∗

In particular, the operators 1+,α preserve the kernel of the map S: M → M ∗ .

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Proof. We fix α throughout this proof and will drop it from the notation of the bases. Fix a basis (uk ) of ker S: (n+ )α → (n+ ∗ )α . Complete it to a basis of (n+ )α by vectors (vl ). Let (u∗k ), (vl∗ ) be the dual basis of (n+ ∗ )α . Moreover, (vl∗ ) =

X (A−1 )lp S(vp , ·)

and

(τ (vl )∗ ) =

X (A−1 )lp S(τ (vp ), ·),

p

(14)

p

where A = (alp ) is a nondegenerate matrix with entries alp = S(vl , vp ). For y ∈ (n+ )α , consider the map y: M → M via the action of D(b+). Let y · p denote the D(b+) action, and yp denote the standard action of b+ . Then we have S(y · p, q) = S(τ (y)p, q) = S(p, −yq)

for any p, q ∈ M.

(15)

Consider the map (vl )∗ : M → M via the action of D(b+). For any p, q ∈ M we have ∗ (τ (vl )∗ , ·))∗ p, S(q, · )i S((vl )∗ p, q) = h(vl )∗ p, S(q, · )i = h(νM ∗ = hp, νM (τ (vl )∗ , S(q, · ))i X ∗ −1 (A )lj S(τ (vj ), · , S(q, · )) = p, νM j

X (A−1 )lj hp, S(τ (vj )q, ·)i = j

X = (A−1 )lj S(p, τ (vj )q).

(16)

j

The first three equalities come from the definition of the action of D(b+) on M, the last two from formula (14). We combine (15), (16), and Corollary 4.2 to obtain: X X ∗ ∗ uk (uk ) + vl (vl ) p, q S(1+,α p, q) = S k

= S

X

vl (vl∗ )p, q

l

= S((vl∗ )p, (−vl )q)

l

X (A−1 )lj S(p, τ (vj )vl q). = −

(17)

j,l

Now we trace the arrows in the alternative direction. For x ∈ (n− )α , consider the map x: M ∗ → M ∗ via the action of D(b−). Denote this action by ‘·’. Let xq denote the standard action of b− on M. We have hx · S(p, ·), qi = hS(p, −τ (τ (x)) · ), qi = S(p, −xq) for all x ∈ b− ; p, q ∈ M.

(18)

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

151

Consider the map (τ (vl ))∗ : M ∗ → M ∗ via the action of D(b− ). h(τ (vl ))∗ S(p, · ), qi =

X (A−1 )lj S(τ (vj ), ·) S(p, · ), q j

X (A−1 )lj hS(τ (vj )p, · ), qi = j

X = (A−1 )lj S(τ (vj )p, q).

(19)

j

Finally combine (18), (19), and Corollary 4.2 with uk ∈ ker S to get h−1−,α S(p, · ), qi X X ∗ ∗ τ (uk ) τ (uk ) + τ (vl ) τ (vl ) S(p, · ), q = − k

l

X X ∗ ∗ τ (uk ) S(p, −τ (uk ) · ), q − τ (vl ) S(p, −τ (vl ) · ), q =− k

= −

X

k

+ −

l

τ (uk )∗ S(−τ (−τ (uk ))p, · ), q +

X

∗

τ (vl ) S(−τ (−τ (vl ))p, · ), q

l

X −1 (A )lj S(τ (vj )(vl )p, · ), q =0− j,l

X (A−1 )lj S(τ (vj )(vl )p, q) =− j,l

X =− (A−1 )lj S(p, τ (vl )vj q).

(20)

j,l

Since the matrix A is symmetric (17) and (20) prove that the diagram is commutative. 2 As a corollary of the lemma we have that 1+,α naturally acts on L = M/ ker(S: M → M ∗ ). We describe this action. Consider the Kac–Moody algebra g¯ = g/ ker(S: g → g∗ ). Let x 7→ x¯ denote the canonical projections M → L, g → g¯ . ker S is an ideal and the form S induces a non-degenerate Killing form on g¯ via the formula K(x, y) = −S(τ (x), y), see [11]. K induces a non-degenerate pairing between root spaces g¯ α and g¯ −α . Let (el(α) ) be a basis of g¯ α , and let (fl(α) ) be the P ¯ α = l fl(α)el(α) . dual basis of g¯ −α with respect to K. Let 1

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COROLLARY 4.4. The following diagram is commutative M

L

1+,α D(b+ )-act ion ¯α 1 st andard

M

L.

∗ x¯ 7→ S(x, · ). We keep the notation Proof. L(3) ∼ = Im{S: M → M P } via from the lemma above. Set wl = − j (A−1 )lj τ (vj ). From the computation in the lemma we have X X −1 − 1+,α p = S(1+,α p, · ) = S (A )lj τ (vj ) vl p, ·

= S

X l

l

wl vl p, ·

=

j

X l

wl vl p =

X

w¯ l v¯l p. ¯

l

Finally notice that the set (v¯l ) forms a basis of g¯ α , and the set (w¯ l ) forms the dual 2 basis of g¯ −α with respect to K. COROLLARY 4.5. Fix λ ∈ Nr . Let m ∈ Mλ , and let (mj ) be a basis of Mλ , and let (m∗j ) be the dual basis of Mλ∗ . Then the following decomposition holds X h−1−,α m∗j , mimj . 1+,α m = j

Proof. Let y ∈ b+ and x = τ (y) ∈ b− . Let p ∈ M, and φ ∈ M ∗ . As in the proof of Lemma 4.3 hy ∗ p, φi = hp, x ∗ φi, where D(b+) acts on M and D(b−) acts on M ∗ . Moreover hy · p, φi = hp, −xP · φi, where D(b+) acts on P M and D(b−) acts on M ∗ . Finally noting that 1+,α = i yi(α) (yi(α) )∗ and 1−,α = i τ (yi(α) )∗ τ (yi(α) ) we have X X hm∗j , 1+,α mimj = h−1−,α m∗j , mimj . 2 1+,α m = j

4.3.

j

AN INTEGRAL FORM OF THE DYNAMICAL DIFFERENTIAL EQUATIONS

Our aim now is to rewrite the Dynamical equations in a form related to the hypergeometric solutions. Fix λ = (m1 , . . . , mr ) ∈ Nr . Let M be a tensor product of highest weight P modules of the Kac–Moody Lie algebra without Serre’s relations g. Set 3 = 3j , the sum of the respective highest weights. P Consider the weight mi αi . Fix a highestspace Mλ of M with weight 3 − α(λ), where α(λ) = weight vector vj for each module M(3j ), j = 1, . . . , n. To every I ∈ P (λ, n) we associate a vector fI v = fi11 · · · fis1 v1 ⊗ · · · ⊗ fi1n · · · fisnn vn in Mλ , cf. Section 3. 1 Note that the vectors (fI v)I ∈P (λ,n) form a basis of Mλ .

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

153

n− acts on Mλ∗ via the D(b−) action. Therefore U (n−) acts on Mλ∗ . Explicitly, x · φ( · ) = φ(−x · ) for x ∈ n−, φ ∈ M ∗ (cf. Section 4.1), where the action on the left-hand side is the D(b−) one and the action on the right-hand side is the standard one. Let V be a vector space freely generated by f1 , . . . , fr . Therefore we have an inclusion of tensor algebras T (V ) ⊂ T (n− ). Moreover T (V ) is an associative enveloping algebra of the Lie algebra n− . Since T (V ) is a free associative algebra, T (V ) is isomorphic to the universal enveloping algebra U (n−). From now on we will refer to the monomial basis of T (V ) as to the monomial basis of U (n− ), and to the dual of the monomial basis of T (V ) as to the monomial basis of UP (n−)∗ . aj xj Rewrite a commutator x ∈ n− as an element of U (n−) in the form x = where aj ∈ Z, and xj ’s are elements of the monomial basis of U (n−). Thus E D X (21) hx · φ, · i = φ, − aj xj · . Denote i: n− → U (n− ) the inclusion monomorphism. Let σj ∈ {1, . . . , r} P for j = 1, . . . , k. Let the positive root of g, nj=1 ασj , correspond to the r-tuple P λ0 ∈ Nr , i.e. α(λ0 ) = nj=1 ασj . Define an element 1σ1 ,...,σk of (n− )∗λ0 via the rule h1σ1 ,...,σk , xi = h(fσ1 · · · fσk )∗ , i(x)i, (fσ1 · · · fσk )∗ ∈ U (n−)∗ .

where x ∈ n− , and (22)

Thus h1σ1 ,...,σk , xi is the coefficient of fσ1 · · · fσk in the decomposition of i(x) into a sum of monomials. P LEMMA 4.6. Let α = rk=1 m0i αi be a positive root of g. Let I ∈ P (λ, n). Set λ0 = (m01 , . . . , m0r ) ∈ Nr . Then we have X ∗ 1i1 ,...,im0 (eim0 · · · ei1 ) (fI v)∗ , (23) −1−,α (fI v) = (i1 ,...,im0 )∈P (λ0 ,1)

where 1i1 ,...,im0 acts according to the D(b−) action on Mλ∗ , the product of e’s acts P (j ) (j ) on one tensor factor at a time (eim0 · · · ei1 ) = nj=1 eim0 · · · ei1 , and ej acts via the standard action (9) on each tensor factor. Proof. Note that α = α(λ0 ). Let x ∈ (n− )λ0 . If there exists i such that m0i > mi , then formula (21) implies x · (fI v)∗ = 0 for every I ∈ P (λ, n) because each than fI v. monomial xj in the U (n−) expansion of x has more fi ’s P m0i . First consider the Now let m0i 6 mi for any 1 6 i 6 r. Set m0 = ∗ case n = 1, I = (i1 , . . . , im ) ∈ P (λ, 1). Let (xj ) be a basis of (n− )∗λ0 such that x1∗ = 1i1 ,...,im0 , and let (xj ) be the dual basis of (n−)λ0 . Formula (22) implies that the coefficient of the monomial fi1 · · · fim0 in the U (n−) expansion of x1 is 1, and the coefficient of fi1 · · · fim0 in the U (n−) expansion of xj is 0 for j > 1. Now use (21) to obtain −x1 · (fI v)∗ = (fim0 +1 · · · fim v)∗ = eim0 · · · ei1 (fI v)∗ , −xj · (fI v)∗ = 0 for j > 1.

(24)

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G. FELDER ET AL.

For any element I 0 = (i10 , . . . , im0 0 ) ∈ P (λ0 , 1), such that I 0 6= (i1 , . . . , im0 ) we have eim0 0 · · · ei10 (fI v)∗ = 0. The proof for n = 1 is finished. Let n be arbitrary natural number. (fI v)∗ = (fi11 · · · fis1 v1 )∗ ⊗ · · · ⊗ (fi1n · · · fisnn vn )∗ . 1

Formula (21) implies h−x · (fI v)∗ , · i = h(fI v)∗ , x · i = h(fI v)∗ , This and the computation for n = 1 give X

∗

−1−,α (fI v) =

n X

(i1 ,...,im0 )∈P (λ0 ,1) j =1

Pn

j =1

x (j ) · i.

1i1 ,...,im0 eim0 · · · ei1 (fI v)∗ . (j )

(j )

2

P α For every positive root α = mi αi of g, set λα = (mα1 , . . . , mαr ) ∈ Nr . Now we combine Corollary 4.5 and Lemma 4.6 to obtain the following form of the Dynamical KZ equation. P 0 LEMMA 4.7. Let u(µ, z) = I ∈P (λ,n) uI fI v, and let µ ∈ h be a direction of differentiation. The Dynamical differential equation (13) is equivalent to the equation κ∂µ0 u =

n X

zi (µ0 )(i) u +

i=1

×

X J ∈P (λ,n)

4.4.

X

X X hα, µ0 i × hα, µi I ∈P (λ,n) α>0 1i1 ,...,im0 (eim0

∗ · · · ei1 ) (fJ v) , fI v uI fJ v. 2

(25)

(i1 ,...,im0 )∈P (λα ,1)

A SYMMETRIZATION PROCEDURE

The definition of the hypergeometric differential form involves a symmetrization procedure, see Section 3. Now we will study the behavior of the operator 1i1 ,...,im0 eim0 · · · ei1 for (ii , . . . , im0 ) ∈ P (λ0 , 1), where α = α(λ0 ), under the same type of symmetrization procedure. Complexes [10]. For a Lie algebra g and a g-module M, denote by C• (g, M) the standard chain complex of g with coefficients in M. Cp (g, M) = 3p g ⊗ M and d: gp ∧ · · · ∧ g1 ⊗ x p X = (−1)i−1 gp ∧ · · · ∧ gbi ∧ · · · ∧ g1 ⊗ gi x+ i=1

+

X

16i<j 6p

(−1)i+j gp ∧ · · · ∧ gbj ∧ · · · ∧ gbi ∧ · · · ∧ g1 ∧ [gj , gi ] ⊗ x.

155

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

the complex Let 31 , . . . , 3n ∈ h∗ . Set M = M(31 ) ⊗ · · · ⊗ M(3n ). Consider L C• (n− , M). We have the weight decomposition C• (n− , M) = λ∈Nr C• (n− , M)λ . In Section 4.1 we recalled a Lie algebra structure on n− ∗ and a n− ∗ -module Let C• (n− ∗ , M ∗ ) be the corresponding standard chain complex: structure on M ∗ . L ∗ ∗ ∗ ∗ C• (n− , M ) = λ∈Nr C• (n− , M )λ . The covariant form induces a graded homomorphism of complexes, S: C• (n− , M) →PC• (n−∗ , M ∗ ), see [10]. mi . Define a subgroup 6λ of the Let λ = (m1 , . . . , mr ) ∈ Nr , and m = symmetric group 6m via the direct product 6λ = 6m1 × · · · × 6mr , where 6mj Pj −1 Pj −1 permutes the set of indices { p=1 mp + 1, . . . , p=1 mp + mj }. Introduce a free f˜1 , . . . , f˜m . Define a map of Lie algebras n− → n f Lie algebra n f − on − Pgenerators mi ˜ by setting fi 7→ j =1 fm(i)+j , where m(i) = m1 + · · · + mi−1 . It induces a map of n− , U (f n− )⊗n )6λ . Set e λ = (1, 1, . . . , 1). Let complexes C• (n− , U (n− )⊗n ) → C• (f | {z } m ⊗n

s: C• (n−, U (n− ) )λ →

C• (f n− , U (f n− )⊗n )eλ6λ

(26)

be the previous map composed with the projection on the e λ-component. e On the other hand there is a map of Lie algebras n f − → n− defined by fj 7→ ⊗n n− , U (f n− ) )eλ → fi , for m(i) < j 6 m(i + 1). It induces the map πλ : C• (f C• (n− , U (n− )⊗n )λ . Note that s(y) equals the sum over the preimages of y under πλ , for any y ∈ C• (n− , U (n−)⊗n )λ . Each such preimage is uniquely described by an element σ ∈ 6λ . . . . , im ) ∈ P (λ, 1). Consider fI as an element EXAMPLE. Let n = 1 and I = (i1 ,P of C0 (n− , U (n−))λ . Then s(fI ) = σ feσ1 . . . feσm , where the sum is over the set {σ ∈ 6(I )} ∼ = 6λ . LEMMA 4.8. Let n = 1, I ∈ P (λ, 1), and m0 6 m. Then the map s ∗ : n− , U (f n−))eλ6λ )∗ → (C1 (n− , U (n−))λ )∗ has the following property (C1 (f 1 X ∗ ∗ e e e s 1σ ,...,σ 0 ⊗ (fσm0 +1 . . . fσm ) |6λ | σ ∈6(I ) 1 m = 1i1 ,...,im0 ⊗ (fim0 +1 . . . fim )∗ .

(27)

Proof. Choose a basis (xj∗ ) of n− ∗ such that x1∗ = 1i1 ,...,im0 and let (xj ) be the dual basis of n− . Take a tensor product of the basis (xj ) with the monomial basis in U (n− ) to get a basis in n− ⊗ U (n− ). Let us compare the two sides of (27) on that basis. 1i1 ,...,im0 ⊗ (fim0 +1 . . . fim )∗ (x1 ⊗ fim0 +1 . . . fim ) = 1. The right-hand side is zero on any other element of the basis. i: n f n− ) → U (f n−)⊗U (f n− ) Let i: n− ⊗U (n−) → U (n− )⊗U (n−), ande − ⊗U (f e e be the natural inclusion maps. Clearly i ◦ s = s ◦ i. Moreover, h1σi1 ,...,σi 0 , s(x)i = m i ◦ s(x))i by definition for x ∈ n− . For a fixed σ ∈ 6λ we have h(feσi1 · · · feσi 0 )∗ , (e m

eσ1 ,...,σ 0 ⊗ (feσ 0 . . . feσm )∗ ), x1 ⊗ fi 0 . . . fim i hs (1 m m +1 m +1 ∗

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G. FELDER ET AL.

eσi ,...,σi ⊗ (feσi = h1 0 1

m0 +1

m

. . . feσim )∗ , s(x1 ⊗ fim0 +1 . . . fim )i

= h(feσ1 . . . feσm0 )∗ ◦e i ⊗ (feσm0 +1 . . . feσm )∗ , s(x1 ⊗ feim0 +1 . . . feim )i = h(feσ1 . . . feσ 0 )∗ ⊗ (feσ 0 . . . feσm )∗ , s(i(x1 ) ⊗ fei 0 . . . feim )i. m

m +1

m +1

The duality of the bases implies i(x1 ⊗ fim0 +1 . . . fim ) = (fi1 . . . fim0 + other monomials) ⊗ fim0 +1 · · · fim , and s ◦ i(x1 ⊗ fim0 +1 . . . fim ) = (feσ1 . . . feσm0 + other eσ1 ,...,σ 0 ⊗ (feσ 0 . . . feσm )∗ ), x1 ⊗ monomials) ⊗ feσm0 +1 · · · feσm . Therefore, hs ∗ (1 m m +1 fim0 +1 . . . fim i = 1. The same way we check that the left-hand side is zero on the other basis elements. 2 COROLLARY 4.9. Let π be the restriction of the projection πλ to the subspace n− , U (f n− )⊗n )eλ6λ of C. (f n−, U (f n− )⊗n )eλ . Let J = (j1 , . . . , jm ) ∈ P (λ, n) and C. (f I = (i1 , . . . , im0 ) ∈ P (λα , 1) for a positive root α. Then we have X X ∗ e e eτ1 ,...,τ 0 (e e . . .e e ) ( f . . . f ) 1 τm0 τ1 σ1 σm m τ ∈6(I ) ∗

σ ∈6(J )

= π ((1i1 ,...,im0 (eim0 . . . ei1 ))(fj1 · · · fjm )∗ ).

(28)

Proof. Assume n = 1. The general case follows from this one because the eτ1 ) and (eim0 . . . ei1 ) act on one tensor factor at a time. Since operators (e eτm0 . . .e s ◦ π = |6λ |id we have π ∗ ◦ s ∗ = |6λ |id. s ∗ and π ∗ are maps of complexes, i.e. they commute with the corresponding differentials. Thus, Lemma 4.8 implies after applying differentials and taking π ∗ from both sides X eσ1 ,...,σ 0 (feσ 0 · · · feσm )∗ . π ∗ (1j1 ,...,jm0 (fjm0 +1 · · · fjm )∗ ) = 1 m m +1 σ ∈6(J )

The right-hand side of our formula is non-zero if and only if m0 6 m and ik = jk for 1 6 k 6 m0 . Thus we compute π ∗ (1i1 ,...,im0 eim0 . . . ei1 (fj1 . . . fjm )∗ ) = π ∗ (1j1 ,...,jm0 (fjm0 +1 . . . fjm )∗ ) X eσ1 ,...,σ 0 (feσ 0 . . . feσm )∗ = 1 m m +1 σ ∈6(J )

=

X

eσ1 ,...,σ 0e e . . .e eσ1 (feσ1 . . . feσm )∗ . 1 m σm0

σ ∈6(J )

Note that (σ1 , . . . , σm0 ) ∈ 6(I ). For any other τ ∈ 6(I ), we have e eτm0 . . .e eτ1 (feσ1 . . . feσm )∗ = 0. Therefore we rewrite the last equality in the form (28).

2

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

157

Our last step is to show that the Dynamical equations for any Lie algebra in any weight space follow from the Dynamical equations for any Lie algebra in a weight space with weight e λ = (1, 1, . . . , 1). Fix a finite-dimensional complex vector space h, a non-degenerate symmetric bilinear form ( · , · ) on h, and a set of linearly independent ‘simple roots’ α1 , . . . , αr ∈ h∗ . Consider the corresponding Kac–Moody Lie algebra without Serre’s relations, g, defined at the beginning of Section 4.1. Recall that b: h → h∗ denotes the isomorphism induced by the bilinear form, hi = b−1 (αi ), and and the form ( · , · ) is transfered to h∗ via the map b. r Ppα(λ) = PrFix λ = (m1 , . . . , mr ) ∈ N . Consider the corresponding positive root m α of g. Up to reordering of the α’s we can assume that α(λ) = i=1 mi αi i=1 i i where mi > 0 for 1 6 i 6 P p and p 6 r is fixed. The corresponding coloring mi } → {1, . . . , p}. We use the following linear function is cλ : {1, . . . , m = algebraic fact when symmetrizing. PROPOSITION 4.10. Let h be a finite-dimensional vector space with a non-degenerate symmetric bilinear form ( · , · ), and a set of linearly independent vectors h with a non(hi )ri=1 ⊂ h. Then there exists a finite-dimensional vector space e degenerate symmetric bilinear form ( · , · )1 , a set of linearly independent vectors e e (e hj )m j =1 ⊂ h, and a monomorphism sh : h → h such that P i e (a) sh (hi ) = m1i m j =1 hm(i)+j , where m(i) = m1 + · · · + mi−1 , i = 1, . . . , p; 0 (b) (e hj , sh (h ))1 = (hc(j ) , h0 ) and (h0 , h00 ) = (sh (h0 ), sh (h00 ))1 for any h0 , h00 ∈ h, j = 1, . . . , m. Proof. Let q = dim h. Complete the set h1 , . . . , hr to a basis h1 , . . . , hr , hr+1 , h0 = C{e h1 , . . . , e hm , e hm+1 , . . . , hq of h. Consider a complex linear space e . . . ,e hm+q−p }. Extend the coloring function c: {1, . . . , m + q − p} → {1, . . . , q} setting c(m + j ) = r + j for j = 1, . . . , q − p. Define a symmetric degenerate hj , e hk )1 = (hc(j ) , hc(k) ) for 1 6 j, k 6 m + q − p. bilinear form on e h0 by the rules (e The rank of the form is q and the dimension of its kernel is m − p. There exists an extension e h of the vector space e h0 and an extension of ( · , · )1 to a non-degenerate symmetric bilinear form on e h. P i e Define a monomorphism sh by sh (hi ) = m1i m j =1 hm(i)+j , where m(i) = m1 + hm+j for j = 1, . . . , q − p. · · · + mi−1 , i = 1, . . . , q, and note that sh (hp+j ) = e Now checking (b) on a basis is straightforward. 2 hj , · )1 ∈ e h∗ for j = 1, . . . , m. Consider e g, a Kac–Moody Lie algeSet e αj = (e αj )m bra without Serre’s relations corresponding to the data e h, ( · , · )1 , and (e j =1 . Note 0 0 0 e that 1 6 j 6 m implies he αj , sh (h )i = (hj , sh (h ))1 = (hc(j ) , h ) = hαc(j ) , h0 i for 0 any h ∈ h. Let M = M(31 )⊗· · ·⊗M(3n ) be a tensor product of Verma modules for g with corresponding highest weights 31 , . . . , 3n ∈ h∗ . Since sh is a monomorphism, e1 , . . . , 3 en ∈ e h∗ → h∗ is a linear epimorphism. Choose highest weights 3 h∗ sh∗ : e

158

G. FELDER ET AL.

ej ) = 3j for 1 6 j 6 n, and consider the corresponding tensor such that sh∗ (3 e 3 e = M( e 3 en ). e1 ) ⊗ · · · ⊗ M( product of Verma modules for e g, M P LEMMA 4.11. Let e λ = (1, 1, . . . , 1). Let e u(e µ, z) = K∈P (eλ,n) e uK feK be a hyper| {z } m

geometric solution of the Dynamical equations with values in the e λ weight space e∼ u)(sh (µ), z) is a hypergeometof a e g-module M n−)⊗n . Then u(µ, z) = π(e = U (f ric solution of the Dynamical equations with values in the λ weight space of a P ⊗n ∼ g-module M = U (n−) , i.e. u = I ∈P (λ,n) uI fI . S Proof. Note that by definition P (e λ, n) = I ∈P (λ,n) {K ∈ 6(I )}. From the definition P of the hypergeometric differential form, see Section 3, it follows that uK . Therefore uI = k∈6(I ) e X X e uk feK π(e u) = π I ∈P (λ,n) K∈6(I )

X X

=

I ∈P (λ,n)

e uk fI =

X

uI fI .

(29)

I ∈P (λ,n)

K∈6(I )

µ = sh (µ) and Fix a point µ ∈ h and a direction of differentiation µ0 ∈ h. Denote e e µ0 = sh (µ0 ). Since ! m n X X hαc(j ) , µitj + h3l , µi ∂µ0 exp − j =1

=

−

l=1

m X

0

hαc(j ) , µ itj +

j =1

=

−

0

he αj , e µ itj +

= ∂e µ0 exp

! m n X X − hαc(j ) , µitj + h3l , µi

0

h3l , µ i exp

l=1

m X j =1

!

n X

j =1

!

n X

el , e h3 µ i exp 0

−

l=1

! m n X X el , e − he αj , e µitj + h3 µi , j =1

l=1

m X

n X

j =1

l=1

he αj , e µitj +

!

el , e h3 µi

(30)

l=1

we have π(∂e u(e µ, z)) = ∂µ0 π(e u)(µ, z). If I = (I1 , . . . , In ) ∈ P (λ, n) and K = µ0e j j j j (K1 , . . . , Kn ) ∈ 6(I ), where Kj = (k1 , . . . , ksj ) and Ij = (i1 , . . . , isj ), then + + * * sj sj X X 0(j ) e 0 0 ej − e α j,e µ feK = 3j − α j , µ feK , e µ fK = 3 kl

il

l=1

l=1

+ * sj X X X 0(j ) 0 e e π e µ e uK fK = 3j − αi j , µ e uK π(fK ) l

K∈6(I )

* = 3j −

l=1 sj

X

0

αi j , µ

+

K∈6(I )

X

l

l=1

K∈6(I )

e uK fI

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

* = 3j −

sj X

159

+ 0

αi j , µ uI fI = µ0(j ) fI .

(31)

l

l=1

Combine formulae (29) and (31) to obtain ! n n X X 0(j ) zj e µ e u = zj µ0(j ) u. π j =1

(32)

j =1

P Let α = ri=1 m0i αj be a positive root for g. Lemma 4.6 gives a necessary conMλ∗ . Namely m0i 6 mi for all i = 1, . . . , r. dition for a non-zero action of 1−,α on P αj of e g a necessary condition for Analogously, for a positive root e α = m j =1 pj e ee∗ is pj = 0, 1 for j = 1, . . . , m. Call all such a non-zero action of 1−,eα on M λ α’s (e α ’s) λ-admissible (e λ-admissible). Since sh∗ (e αj ) = αc(j ) , sh∗ maps the set of e λ-admissibleProots of e g onto the set of λ-admissible roots ofP g. 0 0 m0i . For any e α , such Let α = mi αi be a λ-admissible root for g and m = ∗ 0 0 α ) = α, we have he α, e µ i/he α, e µi = hα, µ i/hα, µi. Consider that sh (e X he α, e µ0 i 1+,eαe π u , he α, e µi ∗ e α , sh (e α)=α

where the sum is over e λ-admissible roots. Corollary 4.5 applied to the basis (feK )K∈P (eλ,n) of Mλ gives X hα, µi he α, e µ0 i u π 1+,eαe hα, µ0 i he α, e µi e α, sh∗ (e α)=α X X ∗ − =π 1−,eα (feK ) ,e u feK e α , sh∗ (e α)=α

K∈P (e λ,n)

=

X I ∈P (λ,n)

X

−

X u fI . 1−,eα (feK )∗ ,e

e α, sh∗ (e α)=α

Lemma 4.6 asserts that X − 1−,eα =

X

e α , sh∗ (e α)=α

e α , sh∗ (e α)=α

(33)

K∈6(I )

X

e elm0 . . .e el1 ) . 1l1 ,...,lm0 (e

(l1 ,...,lm0 )∈P (e λe α ,1)

Rearrange the summation using that sum over (l1 , . . . , lm0 ) ∈ P (e λeα , 1) such that α ) = α equals the sum over (p1 , . . . , pm0 ) ∈ 6(J ) such that J = (j1 , . . . , lm0 ) ∈ sh∗ (e P (λα , 1). Combine such rearrangement with Lemma 4.6 and Corollary 4.9 to simplify formula (33). X he α, e µ0 i hα, µi π 1 e u +,e α hα, µ0 i he α, e µi ∗ e α, sh (e α)=α

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G. FELDER ET AL.

=

X I ∈P (λ,n)

J ∈P (λα ,1)

× =

X I ∈P (λ,n)

=

X

I ∈P (λ,n)

=

X

X

X

ep1 ,...,p 0 (e e . . .e e ) × 1 pm0 p1 m

X (p1 ,...,pm0 )∈6(J )

∗ e u fI (fK ) ,e

K∈6(I )

π ∗ (1j1 ,...,jm0 (ejm0 . . . ej1 )(fI )∗ ),e u fI

X J ∈P (λα ,1)

X

. . . ej1 )(fI ) , π(e u) fI ∗

1j1 ,...,jm0 (ejm0

J ∈P (λα ,1)

h−1−,α (fI )∗ , uifI = 1+,α u.

(34)

I ∈P (λ,n)

Finally (32) and (34) imply n X

∂µ0 u = π(∂e u) = π µ0e

0(j )

zj e µ

+

j =1

=

n X

zj µ0(j ) +

j =1

4.5.

X he α, e µ0 i e α >0

he α, e µi

X hα, µ0 i α>0

hα, µi

! ! 1+,eα e u !

1+,α u.

2

(35)

THE PROOF OF LEMMA 4.1

Recall that the linear map νM : M → b− ⊗ M has the following property νM (x) = 1 −1 (b (3−α(λ)))⊗x +νM− (x), where x ∈ Mλ , νM− (x) ∈ n− ⊗M, and b−1 : h∗ → 2 h is defined at the beginning of Section 4.1. Let a ∈ h, x ∈ Mλ . Since S(b ⊗ x, a ⊗ y) = S(a, b)S(x, y) for any b ∈ g, y ∈ M, and h is orthogonal to n− with respect to S, and ( · , · ) coincides with S on h we have (36) S(νM− (x), a ⊗ y) = S 12 (b−1 (3 − α(λ))), a S(x, y) = 12 S(x, ay). This proves the first equality in the lemma. To prove the second part for a monomial x, we use double induction by the number of tensor factors and the number of f 0 s in x. We use νM− (x) instead of νM (x) because of the orthogonality mentioned above. Let M = M(31 ) be a highest weight module of g with a highest vector v, and a ∈ n− , y ∈ M. Since S(νM− (v)) = 0, we have S(νM− (v), a ⊗y) = 0 = S(v, ay). The inductive step is as follows. Assume S(νM− (x), a ⊗ y) = S(x, ay). Then S(νM− (fi x), a ⊗ y) = S(fi ⊗ hi x, a ⊗ y) + S(fi νM− (x), a ⊗ y) = S(fi , a)S(hi x, y) + S(νM− (x), ei (a ⊗ y))

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

161

= S(fi , a)S(hi x, y) + S(νM− (x), [ei , a] ⊗ y) + S(νM− (x), a ⊗ ei y) = S(fi , a)S(hi x, y) + S(νM− (x), [ei , a] ⊗ y) + S(x, aei y) = S(fi , a)S(hi x, y) + S(νM− (x), [ei , a] ⊗ y)+ + S(x, ei ay) − S(x, [ei , a]y) = S(fi , a)S(hi x, y) + S(νM− (x), [ei , a] ⊗ y)+ + S(fi x, ay) − S(x, [ei , a]y). (37) If a = fi , then (37) and the properties S(x, hy) = S(hx, y), S(νM− (x), h ⊗ y) = 0 for h ∈ h imply S(νM− (fi x), a ⊗ y) = S(hi x, y) + S(νM− (x), hi ⊗ y) + S(fi x, ay) − S(x, hi y) = S(fi x, y). If a is orthogonal to fi with respect to S, then (37) and the inductive hypothesis give S(νM− (fi x), a ⊗ y) = 0 + S(x, [ei , a]y) + S(fi x, ay) − S(x, [ei , a]y) = S(fi x, a). Thus the statement is proved for one tensor factor. Assume that S(νM− (fi x), a ⊗ y) = S(x, ay) for a module M, which is a tensor product of up to n − 1 tensor factors (n > 2). Pn (k) Let M = M(31 ) ⊗ · · · ⊗ M(3n ). Recall that νM− (x) = k=1 νM (x)− , (j ) (j ) (k) (k) (k) (fi x)− = fi νM (x)− for k 6= j , and νM (fi(k) x)− = fi ⊗ h(k) where νM i x + (j ) (k) (k) (k) fi νM (x)− . The following commutation relations will be useful. S(fi νM− (x), (j ) (j ) (j ) (j ) (k) (x), a ⊗ei y), for j 6= k, and S(fi νM (x)− , a ⊗y) = S(νM− (x), a ⊗y) = S(νM− (j ) (j ) [ei , a]⊗y)+S(νM− (x), a⊗ei y). Both equalities are corollaries of the Lemma 4.1 for one tensor factor, and the definition of S, e.g. S(x1 ⊗ · · · ⊗ xj −1 Q⊗ νM− (xj ) ⊗ xj +1 ⊗· · · ⊗xn , a ⊗y1 ⊗· · · ⊗yj ⊗· · ·⊗yn ) = S(νM− (xj ), a ⊗yj ) k6=j S(xk , yk ). In all formulae 1 6 i 6 r, and 1 6 j, k 6 n, and the upper script indicates the tensor factor where the action is applied. Let a ∈ n− . The base for the induction is exactly as for n = 1. The inductive step is as follows. (j )

S(νM− (fi x), a ⊗ y) X (j ) (j ) (j ) (k) = S(νM− (fi x), a ⊗ y) + S(νM− (fi x), a ⊗ y) k6=j

=

X

(j )

(k) S(fi νM− (x), a ⊗ y)+

k6=j (j )

(j ) (j )

+ S(fi ⊗ hi x, a ⊗ y) + S(fi νM (x)− , a ⊗ y) (j )

(j )

= S(fi , a)S(hi x, y) + S(νM− (x), [ei , a] ⊗ y)+ n X (j ) (k) + S(νM− (x), a ⊗ ei y). k=1

(38)

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G. FELDER ET AL.

The result for one tensor factor gives X X (j ) (j ) (k) S(νM− (x), a ⊗ ei y) = S(x, a (k) ei y) k

k

=

X

(j )

S(x, ei a (k) y) − S(x, [ei , a](j ) y)

k (j )

= S(fi x, ay) − S(x, [ei , a](j ) y).

(39)

(k) If a = fi , then (38), (39) and the properties S(x, h(k) y) = S(h(k) x, y), S(νM− (x), h ⊗ y) = 0 for h ∈ h, k = 1, . . . , n imply (j )

(j )

(j )

(j )

S(νM− (fi x), a ⊗ y) = S(hi x, y) + 0 + S(fi x, ay) − S(x, hi y) (j )

= S(fi x, ay). If a is orthogonal to fi with respect to S, then (38), (39) and the result for one tensor factor give (j )

S(νM− (fi x), a ⊗ y) (j )

= 0 + S(x, [ei , a](j ) y) + S(fi x, ay) − S(x, [ei , a](j ) y) (j )

= S(fi x, ay). This finishes the inductive argument. The lemma is proved.

2

5. Flags, Orlik–Solomon Algebra, Hypergeomertic Differential Forms In this section we will formulate results from [10] which define a map between the complex of hypergeometric differential forms and the complex C• (n− ∗ , M ∗ ) for a suitable Lie algebra n− and a n− -module M. 5.1.

COMPLEXES

Let W be an affine complex m-dimensional space and let C be a configuration of hyperplanes in W . Define Abelian groups Ak (C, Z), 0 6 k 6 m, as follows. A0 (C, Z) = Z. For k > 1, Ak (C) is generated by k-tuples (H1 , . . . , Hk ), Hi ∈ C, subject to the relations: (H1 , . . . , Hk ) = 0 if H1 , . . . , Hk are not in general position (codim H1 ∩ · · · ∩ Hk 6= k); (Hσ (1), . . . , Hσ (k) ) = (−1)|σ | (H1 , . . . , Hk ) for any permutation σ ∈ 6k ; Pk+1 i ˆ i=1 (−1) (H1 , . . . , Hi , . . . , Hk+1 ) = 0 for any (k + 1)-tuple H1 , . . . , Hk+1 which is not in general position L and such that H1 ∩ · · · ∩ Hk 6= 0. k The direct sum A• (C, Z) = m k=0 A (C, Z) is a graded skew commutative algebra with respect to the multiplication (H1 , . . . , Hk ) · (H10 , . . . , Hl0 ) =

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

163

(H1 , . . . , Hk , H10 , . . . , Hl0 ). A• (C, Z) is called the Orlik–Solomon algebra of the configuration C. Flags. For 0 6 k 6 m, denote by Flagk (C) the set of all flags L0 ⊃ L1 ⊃ k · · · ⊃ Lk , where Li is an edge of C of codimension i. Denote by Flag (C) the k free Abelian group on Flagk (C) and by F l k (C, Z) the quotient of Flag (C) by the following relations. For every i, 0 < i < k, and a flag with a gap, Fˆ = P (L0 ⊃ · · · ⊃ Li−1 ⊃ Li+1 ⊃ k j L ), where L is an edge of codimension j , we set F ⊃Fˆ F = 0 in F l k (C, Z), where the summation is over all flags F = (L˜ 0 ⊃ L˜ k ) ∈ Flagk (C) such that L˜ j = Lj for all j 6= i. To define the relation between Ak (C, Z) and F l k (C, Z) we define the following map. For (H1 , . . . , Hk ) in the general position, Hi ∈ C, define F (H1 , . . . , Hk ) = (H1 ⊃ H12 ⊃ · · · ⊃ H12...k ) ∈ Flagk (C), where H12...i = H1 ∩ H2 ∩ · · · ∩ Hi . For a flag F ∈ Flagk (C), define a functional δF ∈ F l k (C, Z)∗ as δF (F 0 ) = 1 if F 0 = F and δF (F 0 ) = 0 otherwise. For (H1 , . . . , Hk ) in general position, define a map X (−1)|σ | δF (Hσ1 ,...,Hσk ) . (40) ϕ k (H1 , . . . , Hk ) = σ ∈6k

Thus we have a homomorphism ϕ k : Ap (C, Z) → F l p (C, Z)∗ . The following statements are from [10]. All groups F l p (C, Z) are free over Z. Ap (C, Z) and F l p (C, Z) are dual and the map ϕ k is an isomorphism. Set Ak (C) = Ak (C, Z) ⊗Z C and F l k (C) = F l k (C, Z) ⊗Z C for all k. From now on we assume that the configuration C is weighted, that is, to any hyperplane H ∈ C its weight, a number a(H ) ∈ C, is assigned. Define the quasiclassical weight of any edge L of C as the sum of the weights of all hyperplanes that contain the edge. Say that a k-tuple H¯ = (H1 , . . . , Hk ), Hi ∈ C, is adjacent to a flag F if there exists σ ∈ 6k such that F = F (Hσ1 , . . . , Hσk ). This permutation σ is unique. Denote it by σ (H¯ , F ). Define a symmetric bilinear form S k on F l k (C). For F, F 0 ∈ Flagk (C), set S k (F, F 0 ) =

1X ¯ ¯ 0 (−1)σ (H ,F )σ (H ,F ) a(H1 ) . . . a(Hk ), k

(41)

where the summation is over all H¯ = (H1 , . . . , Hk ) adjacent to both F and F 0 . The form S k is called the quasiclassical contravariant form of the configuration C. It defines a bilinear symmetric form on F l k (C). See [10]. Flag Complex. Define a differential d: F l k → F l k+1 by d(L0 ⊃ · · · ⊃ Lk ) = P 0 k k+1 ), where the sum is taken over all edges Lk+1 of Lk+1 (L ⊃ · · · ⊃ L ⊃ L k codimension k + 1 such that L ⊃ Lk+1 . From the definition of the groups F l k it follows that d2 = 0.

164

G. FELDER ET AL.

P A Complex (A• , d(a)). Set ω = ω(a) = H ∈C a(H )H, ω(a) ∈ A1 . Define a differential d = d(a): Ak → Ak+1 by the rule dx = ω(a) · x. It is clear that d2 = 0. For any k, the quasiclassical bilinear form on C defines a homomorphism S k : F l k → (F l k )∗ ' Ak ,

(42)

where S k (F ) = (−1)k(k−1)/2S(F, ·). LEMMA 5.1. S • defines a map of complexes S • = S • (a): (F l • (C), d) → (A• (C), d(a)). Note. There is a misprint in [10] in the definition of S k where the factor (−1)k(k−1)/2 is missing. P 1 Proof. For any edge L, set S(L) = H ∈C, L⊂H a(H )H , S(L) ∈ A . It is k k 0 easy to see that the homomorphism S is defined by S (L ⊃ · · · ⊃ Lk ) = (−1)k(k−1)/2S(L1 ) · S(L2 ) · · · S(Lk ). In other words X a(H1 ) . . . a(Hk )(H1 , . . . , Hk ), S k (L0 ⊃ · · · ⊃ Lk ) = (−1)k(k−1)/2 where the sum is over all k-tuples (H1 , . . . , Hk ) such that Hi ⊃ Li for all i. Therefore, we have S k+1 d(L0 ⊃ · · · ⊃ Lk ) X = S k+1 (L0 ⊃ · · · ⊃ Lk ⊃ Lk+1 ) Lk+1 ,Lk+1 ⊂Lk

(k+1)k X = (−1) 2 a(H1 ) . . . a(Hk )a(Hk+1 )(H1 , . . . , Hk , Hk+1 ) X k(k−1) X = (−1) 2 a(H1 ) . . . a(Hk )(H1 , . . . , Hk ) · (−1)k a(Hk+1 )Hk+1 k(k−1) X = (−1) 2 a(H1 ) . . . a(Hk )(H1 , . . . , Hk ) · (−1)k ω(a)

= S k (L0 ⊃ · · · ⊃ Lk ) · (−1)k ω(a) = ω(a) · S k (L0 ⊃ · · · ⊃ Lk ) = d(a)S k (L0 ⊃ · · · ⊃ Lk ). The second, third and fourth sum are over all Hi , such that Hi ⊃ Li , for 1 6 i 6 k and Hk+1 ∩ Lk 6= 0. Note that Hk+1 ∩ Lk = 0 implies (H1 , . . . , Hk , Hk+1 ) = 0, and thus the fourth equality is justified. The sixth one comes from the skew symmetry 2 in A• (C). Recall that we have a weighted configuration of hyperplanes in a complex mdimensional space W , and a = {a(H ) | H ∈ C} are the weights. Fix an affine S equation lH = 0 for each hyperplane H ∈ C. Set Y = W − H ∈C H . Consider the trivial line bundle L(a) over Y with an integrable connection d(a): O → 1

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

165

P given by d + (a) = d + H ∈C a(H )d log lH , where d is the de Rham differential. Denote by • (L(a)) the complex of Y -sections of the homomorphic de Rham complex of L(a). To any H ∈ C assign the one-form i(H ) = d log lH ∈ 1 (L(a)). This construction defines a monomorphism i(a): (A• (C), d(a)) → (• (L(a)), d(a)). The image of this monomorphism is called the complex of the hypergeometric differential forms of weight a. It is denoted by (A• (C, a), d(a)). The image of the homomorphism i(a)S: (F l • (C), d) → (• (L(a)), d(a)) is called the complex of the flag hypergeometric differential forms of weight a. It is denoted by (F l • (C, a), d(a)). For further details see [10]. 5.2.

DISCRIMINANTAL CONFIGURATIONS

Let W be an affine complex space of dimension m. Let z1 , . . . , zn be pairwise distinct complex numbers. Denote by Cm a configuration in W consisting of hyperplanes Hkl : tk − tl = 0; 1 6 k < l 6 m. So C1 = ∅, and Y (Cm ) is the space of m-tuples of ordered distinct points in C. Denote by Cn;m (z) a configuration in j W consisting of hyperplanes Hk : tk − zj = 0, 1 6 k 6 m, 1 6 j 6 n, and Hkl , 1 6 k < l 6 m. Thus, Y (Cn;m (z)) = p −1 (z) where p: Y (Cn+m ) → Y (Cn ) is the projection on the first n coordinates. Define C0;m = Cm . Edges and Flags of Cn,m . For every non-empty subset J = {j1 , . . . , jk } ⊂ [m] k−1 . L is an edge of codimension k −1. In set LJ = Hj1 j2 ∩Hj2 j3 ∩· · · ∩Hjk−1 jk ∈ Cn;m particular set LJ = W , for k = 1. For i ∈ [n] define LiJ = Hji1 ∩ Hji2 ∩ · · · ∩ Hjik ∈ k Cn;m . LiJ is an edge of codimension k. Set Li∅ = W . Given non-intersecting subsets T T J1 , . . . , Jk ; I1 , . . . , In ⊂ [m], define LJ1 ,...,Jk ;I1 ,...,In = ( kj =1 LJj ) ∩ ( ni=1 LiIi ). Multiplication of Flags. Given two subsets J ⊂ [m] and I ⊂ [n], denote by CJ ;I ⊂ Cm;n the subset consisting of all hyperplanes Hj1 j2 with j1 , j2 ∈ J and Hji with j ∈ J, i ∈ I . Given subsets J, J 0 ⊂ [m]; I, I 0 ⊂ [n] such that J ∩ J 0 = ∅; I ∩ I 0 = ∅, define maps ◦: Flagk (CI ;J ) × Flagl (CI 0 ;J 0 ) → Flagk+l (CI ∪I 0 ,J ∪J 0 ) as follows. For F = F (H1 , . . . , Hk ) ∈ Flagk (CI ;J ), F = F (H10 , . . . , Hl0 ) ∈ Flagl (CI 0 ;J 0 ), set F ◦ F 0 = (H1 , . . . , Hk , H10 , . . . , Hl0 ). The following lemma, [10], Lemma 5.7.2, takes place. LEMMA 5.2. The above map correctly defines the map F l k (CI ;J )⊗F l l (CI 0 ;J 0 ) → F l k+l (CI ∪I 0 ,J ∪J 0 ). Moreover, for all x ∈ F l k (CI ;J ), y ∈ F l l (CI 0 ;J 0 ) we have x◦y = (−1)kl y ◦ x.

166 5.3.

G. FELDER ET AL.

TWO MAPS OF COMPLEXES

Let g be a Kac–Moody Lie algebra without Serre’s relations. Let M = M(31 ) ⊗ · · ·⊗M(3n ) be a tensor product of Verma modules with weights 31 , . . . , 3n ∈ h∗ . Set λ = (1, 1, . . . , 1). In this case the number of generators (fj )r1 of n− equals m, | {z } m

i.e. r = m. Two maps of complexes ψ• and η• are described in [10]: ψp : Cp (n− , M)λ → F l m−p , ηp = ϕ −1 ◦ (ψp∗ )−1 : Cp (n− ∗ , M ∗ )λ → Am−p ,

(43)

where ϕ is the map (40). Note. The maps ψp define isomorphism of complexes. Theorem 6.6 [10] implies that the maps (−1)p (−1)(m−p)(m−p−1)/2ηp define isomorphism of complexes. The sign is due to Lemma 5.1 and the fact that the contravariant form S in this paper is minus the contravariant form in [10], see formula (10) and [10], formula (6.2.3). We will recall the explicit description of ψ• under the above assumption on λ. Let g ∈ n− . A length l = l(g) of a commutator g is given via an inductive definition. Set l(fj ) = 1 for j = 1, . . . , m. If g = [g1 , g2 ] and l1 = l(g1 ), l2 = l(g2 ), then set l(g) = l1 + l2 . So l(g) = ‘the number of f ’s in g’. To every commutator g assign a bracket sign b(g) ∈ Z/2Z as follows. Set b(fj ) = 0; b([g1 , g2 ]) = b(g1 ) + b(g2 ) + l(g1 ) mod 2. To every commutator g assign a flag F l(g) ∈ F l l(g)−1(C0;|g|) as follows. Set F l(fj ) = . If g = [g2 , g1 ], set F l(g) equal to F l(g1 ) ◦ F l(g2 ) completed by the edge L|g| . Finally, for a commutator g set F (g) = (−1)b(g) F l(g) ∈ F l l(g)−1(C0;|g| ). For I = (i1 , . . . , il ) ⊂ {1, . . . , m} and 1 6 i 6 n, set fI = fil . . . fi1 ∈ U (n− ) and F i (fI ) = F (Hii1 , . . . , Hiil ) ∈ F l l (C{i};I ). Let z ∈ Cp (n− , U (n− )⊗n )λ and z = gp ∧ gp−1 ∧ · · · g1 ⊗ fIn ⊗ fIn−1 ⊗ · · · ⊗ fI1 , where all gi are commutators, li = l(gi ). Let {fi1 , . . . , fim } be the list of fi ’s in z read from right to left. Define σ (z) ∈ 6m by σ (z)(j ) = ij . Set ψp (z)

Pp

= (−1)|σ (z)|+

i=1 (i−1)(li −1)

F 1 (fI1 ) ◦ · · · ◦ F n (fIn ) ◦ F (g1 ) ◦ · · · ◦ F (gp ).

(44)

Note. There is a correction of the sign in the definition of ψ compared with [10]. EXAMPLES. Let n = 1. ψ(fm . . . f1 ) = F (H11 , . . . , Hm1 ), and η((fσ1 . . . fσm )∗ ) = (−1)|σ | Hσ1 ,σ2 ◦ · · · ◦ Hσm−1 ,σm ◦ Hσ1m . Compose the inclusion map i(a): (A• (C), d(a)) → (• (L(a)), d(a)) (see Section 5.1) with the map η to get i(a) ◦ η((fσ1 . . . fσm )∗ ) = (−1)|σ | d ln(tσ1 − tσ2 ) ∧ · · · ∧ d ln(tσm−1 − tσm ) ∧ d ln(tσm − z1 ).

167

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

Let I ∈ P (λ, n) and I = (i11 , . . . , is11 , . . . , i1n , . . . , isnn ). Since λ = (1, 1, . . . , 1), j j ∗ I ∈ 6m . Let Ij = (isj , . . . , i1 ) for 1 6 j 6 P n. We have∗ i(a) ◦ η((fIn ) ⊗ ∗ |I | · · · ⊗ (fI1 ) ) = (−1) ωI . Therefore i(a) ◦ η( I ∈P (λ,n)(fI ) fI ) = ω(z, t), see Section 3. Let I = (i11 , . . . , is11 , . . . , i1n , . . . , isnn ) ∈ P (λ, n), and 1 6 k 6 sj . Define fI ;i j = fIn ⊗ · · · ⊗ fIj+1 ⊗ fi j . . . fisj ⊗ fIj−1 ⊗ · · · ⊗ fI1 , k

k+1

j

θI ;i j = ωi 1 ,...,is1 ∧ · · · ∧ ωi j−1 ,...,isj−1 ∧ ωi j 1

k

∧ω

1

1

i1n ,...,isnn

1

j −1

n

◦Hj

ik+1 ,...,i

j k−1 ,ik

1 2

p

j+1

− ti j )],

j sj

(45)

k

j +1

j

j−1

,...,isj−1

◦Hinn ,...,isn ◦ [Hi j ,i j ◦ · · · ◦ Hi j 1

1

k−1

2

i1

s1

1

∧ ωi j+1 ,...,isj+1 ∧ · · ·

∧ [d ln(ti j − ti j ) ∧ · · · ∧ d ln(ti j

HI ;i j = Hi11 ,...,i 1 ◦ · · · ◦ H j−1 k

k+1 ,...,is j j

j−1

◦ H j+1 i1

j+1

,...,isj+1

◦ ···

],

p

where Hi1 ,...,il = Hi1 ,i2 ◦ · · · ◦ Hil−1 ,il ◦ Hil . j

LEMMA 5.3. Let I ∈ P (λ, n). Let k = k((sj − k) + sj +1 + · · · + sn ). Then j

i(a) ◦ η(1σ j ,...,σ j ⊗ (fI ;i j )∗ ) = (−1)|I |+k θI ;i j . i1

ik

k

(46)

k

Proof. The statement of the lemma is equivalent to the equation: j

1σ j ,...,σ j ⊗ (fI ;i j )∗ = (−1)|I |+k η−1 (HI,i j ). i1

ik

k

(47)

k

It is sufficient to compute the two sides on elements of type g ⊗ fI,i j where g is a k commutator of length k on fi j , . . . , fi j . Let σ ∈ 6k and fiσj , . . . , fiσj be the list k 1 1 k of fi j ’s entering g from right to left. The left hand side and the right-hand side of (47) evaluated on g ⊗ fI,i j give k

∗

1σ j ,...,σ j ⊗ (fI ;i j ) (g ⊗ fI ;i j ) = 1σ j ,...,σ j (g), i1

ik j |I |+k

(−1)

k

i1

k

ik

j

η−1 (HI,i j )(g ⊗ fI ;i j ) = (−1)|I |+k ϕ(HI,i j )(ψ(g ⊗ fI ;i j )), (48) k

k

k

k

respectively. See formula (43). Use the definition of ψ to obtain ψ(g ⊗ fI ;i j ) = (−1)|τ | F (Hi11 , . . . , Hi11 ) ◦ · · · ◦ F (Hi 1 , . . . , H j ) ◦ · · · s1

k

1

j

j

sj

ik+1

◦ F (Hinsn , . . . , Hinn ) ◦ F (g), n

1

1 . . . . . . . . . . . j j is11 · · · i11 · · · isj · · · ik+1 · · · isnn · · · i1n iσj1 I as an element of 6m has the form 1 . . . . . . . . . m . I= 1 j j i1 · · · is11 · · · i1 · · · isj · · · i1n · · · isnn where τ =

. ···

!

m . iσjk

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G. FELDER ET AL.

Thus j

|I | = (|τ | + Sk + k((sj − k) + sj +1 + · · · + sn ) + |σ |) mod 2,

(49)

where j

Sk =

n X sl (sl − 1) (sj − k)(sj − k − 1) + . 2 2 l=1, l6=j p

p

Note that Hi1 ,...,il = (−1)l(l−1)/2Hil ◦ Hil−1 ,il ◦ · · · ◦ Hi1 ,i2 . Use the definition of ϕ, (40), to compute p

p

ϕ(Hi1 ,...,il ) = (−1)l(l−1)/2ϕ(Hil ◦ Hil−1 ,il ◦ · · · ◦ Hi1 ,i2 ) = (−1)l(l−1)/2δF (Hip ,...,Hip ) + other δ-summands. l

j

ϕ(HI,i j ) = (−1)Sk ϕ(Hi11 ◦ Hi 1

1 s1 −1 ,is1

s1

k

j

◦ H j ◦ Hi j

j sj −1 ,isj

isj

◦ · · · ◦ Hi 1 ,i 1 ◦ · · · 1 2

◦ · · · ◦ Hi j

j k+1 ,ik+2

◦ Hi1n ,i2n ◦ [Hi j ,i j ◦ · · · ◦ Hi j

j k−1 ,ik

1 2

(50)

1

◦ · · · ◦ Hinsn ◦ Hisnn −1 ,isnn ◦ · · · n

]).

(51)

We use formulae (49), (50), (51) to simplify (48). j

(−1)|I |+k η−1 (HI,i j )(g ⊗ fI ;i j ) k

k

= (−1)|σ | ϕ(Hi j ,i j ◦ · · · ◦ Hi j 1 2

j k−1 ,ik

)(F (g)).

The proof of Lemma 5.3 is finished modulo the following result.

(52) 2

LEMMA 5.4. Let ηI , ψI be the combinatorial maps (43), (44) defined on the set of distinct indices I, I = {i1 , . . . , ik } ⊂ {1, . . . , m}. Then ηI (1i1 ,...,ik ) = Hi1 ,i2 ◦ · · · ◦ Hik−1 ,ik , for any k = 2, . . . , m. Proof. Induction by k. For k = 2, g = [fi1 , fi2 ] forms a base of the commutators of length 2 on fi1 and fi2 . 1i1 ,i2 (g) = 1. Since b(g) = 1 and F (g) = (−1)b(g)F (Hi1 ,i2 ) and σ = 12 21 , we have η−1 (Hi1 ,i2 )([fi1 , fi2 ]) = ϕ(Hi1 ,i2 )(ψ(g)) = δF (Hi1 ,i2 ) ((−1)|σ |+b(g)F (Hi1 ,i2 )) = 1. Let 2 < k 6 m. Assume that for any j , 2 6 j < k, and 1 6 s1 < · · · < sj 6 k we have η(1is1 ,...,isj ) = His1 ,is2 ◦ · · · ◦ Hisj−1 ,isj . Let g be a commutator of length k on fi1 , . . . , fik . Then g = [g1 , g2 ] with l(g1 ) = l1 , and l(g2 ) = l2 , and l1 + l2 = k. Let σ ∈ 6k be such that fiσ1 , . . . , fiσk is the list of fi ’s in g read from right to left. In order to evaluate η−1 (Hi1 ,i2 ◦ · · · ◦ Hik−1 ,ik )(g) = ϕ(Hi1 ,i2 ◦ · · · ◦ Hik−1 ,ik )(ψ(g)) remark that ψ(g) = (−1)|σ |+b(g) (F l(g2 ) ◦ F l(g1 ), L|g| )

169

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

= (−1)|σ |+l(g1 ) (F (g2 ) ◦ F (g1 ), Li1 ,...,ik ), η−1 (Hi1 ,i2 ◦ · · · ◦ Hik−1 ,ik )(ψ(g)) X = (−1)|τ | δF (Hiτ ,iτ +1 ,...,Hiτ ,iτ +1 ) (ψ(g)). 1

τ ∈6k−1

k−1

1

(53)

k−1

Since a link corresponding to a hyperplane Hj,j +1 connects only neighbouring indices in a flag of a type F (Hiτ1 ,iτ1 +1 , . . . , Hiτk−1 ,iτk−1+1 ) we have F (Hiτ1 ,iτ1 +1 , . . . , Hiτk−1 ,iτk−1 +1 ) = (, . . . , (ti1 = · · · = tiτk−1 ; tiτk−1 +1 = · · · = tik ), Li1 ,...,ik = (ti1 = · · · = tik )).

(54)

|σ |+b(g)

Let δF (Hiτ ,iτ +1 ,...,Hiτ ,iτ +1 ) (ψ(g)) 6= 0. Since (−1) ψ(g) = (F l(g2 ) ◦ 1 1 k− k−1 F l(g1 ), L|g| ) = (, . . . , L|g2 | ∩ L|g1 | , L|g| ) formula (53) implies either L|g1 | = (ti1 = · · · = tiτk−1 ); L|g2 | = (tiτk−1 +1 = · · · = tik ), or L|g2 | = (ti1 = · · · = tiτk−1 ); L|g1 | = (tiτk−1 +1 = · · · = tik ). Without loss of generality we will assume that the second case takes place, i.e. L|g2 | = (ti1 = ti2 = · · · = tiτk−1 ); L|g2 | = (tiτk−1 +1 = · · · = tik−1 = tik ). Compare the lengths of the flags to conclude that τk−1 = l2 . In order to have non-zero multiples in the product δF (Hiτ

1 ,iτ1 +1

,...,Hiτ

k−1 ,iτk−1 +1

= δF (Hiτ

1 ,iτ1 +1

,...,Hiτ

◦ F l(g1 ), L|g| )

) (F l(g2 )

l2 −1 ,iτl2 −1 +1

) (F l(g2 ))δF (Hiτ

l2 ,iτl2 +1

,...,Hiτ

k−2 ,iτk−2 +1

) (F l(g1 ))

(55)

we need (τ1 , . . . , τl2 −1 ) to be a permutation of the set (1, . . . , l2 − 1) and (τl2 , . . ., τk−2 ) to be a permutation of the set (l2 + 1, . . . , k − 1). Set τ 0 = τ1 ·· ·· ·· lτ2 − 1

and τ 00 = τ 1− l ·· ·· ·· τ l1 −−1l . Then (−1)|τ | = (−1)|τ |+|τ l2 2 k−2 2 b(g1 ) + b(g2 ) mod 2, and 0

00 |+l

η−1 (Hi1 ,i2 ◦ · · · ◦ Hik−1 ,ik )(ψ(g)) X 0 (−1)|τ | δF (Hi = (−1)|σ |+b(g)+l1 −1 ×

X τ 00 ∈6l1 −1

1 −1

,...,Hi 0 ,i 0 ) ,i τ10 τ10 +1 τl −1 τl −1 +1 2 2

τ 0 ∈6l2 −1 00

(−1)|τ | δF (Hi

,...,Hi 00 ) ,i ,i τ100 +l2 τ100 +l2 +1 τl −1 +l2 τl00 −1 +l2 +1 1 1

1

l2 −1

, b(g) + l1 =

(F l(g2 ))×

(F l(g1 ))

= (−1)|σ |−1 η−1 (Hi1 ,i2 ◦ · · · ◦ Hil2 −1 ,il2 )(F (g2 ))× ×η−1 (Hil2 +1 ,il2 +2 ◦ · · · ◦ Hik−1 ,ik )(F (g1 )) = (−1)η−1 (Hi1 ,i2 ◦ · · · ◦ Hil2 −1 ,il2 )(ψ(g2 ))× ×η−1 (Hil2 +1 ,il2 +2 ◦ · · · ◦ Hik−1 ,ik )(ψ(g1 )). 1 σ1

··· k · · · σk

The last equality holds because σ = Using the inductive hypothesis rewrite (56) as

=

(56) 1 σ1

· · · l2 l2 + 1 · · · lk · · · σl2 × σl2 +1 · · · σlk .

η−1 (Hi1 ,i2 ◦ · · · ◦ Hik−1 ,ik )(ψ(g)) = (−1)1i1 ,...,il2 (g2 )1il2 +1 ,...,ik (g1 ) 2 = 1i1 ,...,ik ([g1 , g2 ]).

(57)

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6. Derivation of the Dynamical Differential Equation In this section g will be a Kac–Mody Lie algebra without Serre’s relations, λ = (1, 1, . . . , 1), r = m. We will work in a weight space Mλ of the module M = M(31 ) ⊗ · · · ⊗ M(3n ). We will differentiate the hypergeometric form ω(z, t), express the result in terms of the complex C• (n− ∗ , M ∗ ), and derive the Dynamical differential equation in the form (25). The integrand of a hypergeometric solution have the following form, see Section 3. !! m n X X 1 − 81/κ , hαc(i) , µiti + h3j , µizj 81/κ µ ω = exp κ i=1 j =1 where 8(z, t) =

Y

(zi − zj )(3i ,3j )

Y Y (tk − zj )−(αc(k) ,3j ) (tk − tl )(αc(k) ,αc(l) ) .

i<j

k,j

k
Fix µ0 ∈ h and let ∂µ0 be the partial derivative with respect to the parameter µ in the direction of µ0 . Then κ∂µ0 (81/κ µ ω) =

! m n X X X hαc(i) , µ0 iti + h3j , µ0 izj 81/κ (−1)|I | ωI fI v. (58) − µ j =1

i=1

I ∈P (λ,n)

Let I = (i11 , . . . , is11 ; . . . ; i1n , . . . , isnn ) ∈ P (λ, n). Moreover I ∈ 6m because of the form of λ. Since r = m we have c(i) = i. Set ti j = zj and αik;j = αi j + αi j + · · · + αi j . Rearrange the following expression:

sj +1

k

k−1

1

! m n X X 0 0 − hαi , µ iti + h3j , µ izj ωI j =1

i=1

=

−

! n X 0 hαi j , µ iti j + h3j , µ izj ωI

sj n X X

k

j =1 k=1

=

−

0

k

j =1

sj n X X

hαi j , µ0 i(ti j − ti j

j =1 k=1

k

k

k+1

+ ti j

k+1

− ti j

!

n X h3j , µ0 izj ωI + j =1

=

! sj n X X − hαik;j , µ0 i(ti j − ti j ) ωI + j =1 k=1

k

k+1

k+2

+ · · · + tisj − zj + zj )+ j

171

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

!

n X

+

0

zj h3j − αi j − · · · − αisj , µ i ωI . 1

j =1

LEMMA 6.1. Let u(µ, z) = κ∂µ0 u −

n X

=

R

1/κ γ (z) 8µ ω.

Then

zj µ0 u (j )

j =1

X

(59)

j

(−1)|I |

! sj n X X − hαik;j , µ0 i(ti j − ti j ) ωI fI v. (60)

Z 81/κ µ γ (z)

I ∈P (λ,n)

k

j =1 k=1

k+1

Proof. Combine formulae (58), (59) with the fact µ0 (j ) fI v = h3j − αi j − · · · − 1 2 αisj , µ0 ifI v to obtain the result. j

LEMMA 6.2. Define an operator L by Lu = κ∂µ0 u − X hα, µ0 i Lu = 1+,α u. hα, µi α>0

Pn

0 (j ) u. j =1 zj µ

Then

Proof. Let I ∈ P (λ, n) and 1 6 k 6 sj . (ti j − ti j )ωI k

k+1

= ωi 1 ,...,is1 ∧ · · · ∧ (ti j − ti j )ωi j ,...,isj ∧ · · · ∧ ωi1n ,...,isnn 1

k

1

k−1

1

j

= ωi 1 ,...,is1 ∧ · · · ∧ [d ln(ti j − ti j ) ∧ · · · ∧ d ln(ti j 1

1

1

∧ d ln(ti j

k+1

k−1

2

k+2

k

k+1

j

j

= (−1) ωi 1 ,...,is1 ∧ · · · ∧ ωi j

j k+1 ,...,isj

1

∧ [d ln(ti j − ti j ) ∧ · · · ∧ d ln(ti j 1

k

− ti j ) ∧ · · · ∧ d ln(tisj − zj )] ∧ · · · ∧ ωi1n ,...,isnn

k

1

− ti j ) ∧ d(ti j − ti j ) ∧

k−1

2

∧ · · · ∧ ωi1n ,...,isnn ∧ − ti j )] ∧ d(ti j − ti j ) k

j k

k

k+1

j

= (−1) θI ;i j ∧ d(ti j − ti j ) = (−1)k +(m−1) d(ti j − ti j ) ∧ θI ;i j , k

k

k+1

k

k+1

(61)

k

j

where k = k(sj +1 + · · · + sn + sj − k). It is clear that dθI ;i j = 0. Thus k ! m X κdt (81/κ − hαi , µidti + 8−1 (dt 8) 81/κ µ θI ;i j ) = µ θI ;i j . k

k

(62)

i=1

Rearrange as in formula (59) and simplify to get ! sj n X X hαil;j , µid(ti j − ti j ) ∧ θI ;i j l

j =1 k=1 −1

l+1

k

= (8 (dt 8)) ∧ θI ;i j − κ8−1/κ dt (81/κ µ µ θI ;ikj ), k hαik;j , µid(ti j − ti j )) ∧ θI ;i j k

k

k+1

= (8−1 (dt 8)) ∧ θI ;i j − κ8−1/κ dt (81/κ µ µ θI ;i j ). k

k

(63)

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Since γ (z) is a cycle, formulae (61), (63) and Lemma 5.3 allow us to rewrite Lu as Z X j (−1)|I | (−1)k × Lu = γ (z)

I ∈P (λ,n)

n X X hαik;j , µ0 i

!

sj

× −

hαik;j , µi

j =1 k=1

X Z

=

−

81/κ µ

I ∈P (λ,n) γ (z)

−1 (81/κ µ 8 (dt 8))

sj n X X hαik;j , µ0 i j =1 k=1

hαik;j , µi !

∧ θI ;i j − k

(−1)m−1

κdt (81/κ µ θI ;ikj )

dt 8 ∧ 8

∧ (i(a) ◦ η(1i j ,...,i j ⊗ (fI,i j )∗ ) . k

1

(64)

k

Since θI,i j is closed, its differential in the complex of the hypergeometric differenk tial forms reduces to multiplication by i(a)((a) = dt 8/8. Taking into account formula (43) and the note after it we have sj Z n X X X hαik;j , µ0 i 1/κ (−1)m−1 d(a) × − fI 8µ Lu = hα , µi i γ (z) k;j I ∈P (λ,n) j =1 k=1 ! ×(i(a) ◦ η(1i j ,...,i j ⊗ (fI,i j )∗ ) k

1

=

X

Z

=

γ (z)

=

I ∈P (λ,n)

j =1 k=1

Z γ (z)

j =1 k=1

Z

hαik;j , µi

sj n X X hαik;j , µ0 i

81/κ µ

fI

hαik;j , µi

sj n X X hαik;j , µ0 i

81/κ µ

fI

I ∈P (λ,n)

X

sj n X X hαik;j , µ0 i

81/κ µ

fI

I ∈P (λ,n)

X

k

γ (z)

j =1 k=1

hαik;j , µi !

! (i(a) ◦ η(d1i j ,...,i j ⊗ (fI,i j )∗ ) k

1

k

! (i(a) ◦ η(1i j ,...,i j (fI,i j )∗ ) 1

k

k

×

×(i(a) ◦ η(1i j ,...,i j ei j . . . ei j (fI )∗ ) . 1

k

k

(65)

1

Note that αik;j , k = 1, . . . , sj describe all λ-admissible roots such that 1+,α (fI )∗ 6= 0, see Section 4.4. Therefore Z X X hα, µ0 i i(a) ◦ η × fI 81/κ Lu = µ hα, µi γ (z) I ∈P (λ,n) α>0 !! X ∗ × 1i1 ,...,im0 eim0 . . . ei1 (fI ) (j )

(i1 ,...,im0 )∈P (λα ,1)

173

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

X

=

Z fI γ (z)

I ∈P (λ,n)

X

=

γ (z)

Lu =

hα, µi

X hα, µ0 i α>0

hα, µi

X

hα, µi

!! ∗

− 1−,α (fI )

X hα, µ0 i

i(a) ◦ η 81/κ µ

fI

X hα, µ0 i α>0

X hα, µ0 i α>0

Z

K,I ∈P (λ,n)

=

i(a) ◦ η

81/κ µ

α>0

hα, µi

!

!

h−1−,α (fI )∗ , fK i(fK )∗

h−1−,α (fI )∗ , fK iuK fI ,

(66)

K,I ∈P (λ,n)

h−1−,α (fI )∗ , uifI =

X hα, µ0 i α>0

hα, µi

1+,α u.

2

(67)

The statement of Lemma 6.2 is equivalent to the Dynamical differential equation in the direction of µ0 for a function u with values in the (1, 1, . . . , 1) weight space of a g module M. The Symmetrization Lemma 4.11 deduces the general case from this one.

7. Main Theorems In this section we conclude the proofs of the theorems from Section 3 in the setting of Kac–Moody Lie algebras without Serre’s relations. Then we deduce the corresponding results for any simple Lie algebra. Let g be a Kac–Moody Lie algebra without Serre’s relations. Let λ ∈ Nr . Let of Verma modules for g with M = M(31 ) ⊗ · · · ⊗ M(3n ) be a tensor productP highest weights 31 , . . . , 3n ∈ h∗ . Let u(µ, z) = I ∈P (λ,n) uI fI be a hypergeometricR integral with weight space Mλ as described in Section 3, i.e. P values in |σthe 1/κ | uI = γ (z) 8µ ( σ ∈6(I )(−1) ωI,σ ). THEOREM 7.1. The function u(µ, z) solves the KZ equations (12) in Mλ . Proof. The proof given in Section 3 holds, because all relations we used are proved in [10] in the general setting described above. 2 THEOREM 7.2. The function u(µ, z) solves the dynamical differential equations (13) in Mλ . Proof. Lemma 4.11 reduces the case of a general weight space Mλ to the case λ = (1, 1, . . . , 1). Lemma 6.2 derives the theorem in of a weight space Meλ , where e | {z } m

that case.

2

Proof of Theorem 3.2. Combine Corollary 4.4 and Theorem 7.2 to derive the dynamical differential equations for any Kac–Moody Lie algebra. In particular we have it for a simple Lie algebra. 2

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Finally we will prove a determinant formula which establishes a basis of solutions for the system of KZ and dynamical differential equations in a weight space Mλ . From that formula we will derive the compatibility of the system of KZ and Dynamical differential equations. Fix λ ∈ Nr . Fix a basis (fI v)I ∈P (λ,n) of the weight space Mλ . Assume that a set families of twisted cycles in {z} × Cm is given. (γI (z))I ∈P (λ,n) of horizontal R P 1/κ Denote uI J = γI (z) 8µ ( σ ∈S(J )(−1)|σ | ωJ,σ ). PROPOSITION 7.3. Let δα = trMλ (1+,α ) for a positive root α of g. Denote ij = trMλ (ij,+ ). Then we have (a) For any horizontal families of twisted cycles (γI (z))I ∈P (λ,n) in {z} × Cm , there exists a constant C = C(31 , . . . , 3n , λ, κ) such that det(uI J ) = C exp

n X zi i=1

κ

! trMλ (µ(i) )

Y α>0

hα, µiδα /κ

Y (zi − zj )ij /κ .

(68)

i<j

In the first product only finite number of factors are different from 1, i.e. δα 6= 0 if and only if 0 < α 6 λ. r (b) For generic values of the parameters (3j )m j =1 , (αi )i=1 , κ in a neighbourhood n of a generic point (µ, z) ∈ h × C we can choose cycles (γI (z))I ∈P (λ,n) such that the constant C from (a) is non-zero. Moreover the set of functions {uI = P J ∈P(λ,n) uI J fJ }I ∈P (λ,n) form a fundamental system of solutions for the system of KZ and dynamical differential equations. Proof. Part (a) is a corollary of Theorems 7.1 and 7.2. We will prove part (b) for values of the parameters such that all numbers (αi , αj )/κ, −(αi , 3k )/κ have positive real parts for 1 6 i, j 6 r, 1 6 k 6 n, and for a point (µ, z) such z ∈ Rn , z1 < z2 < · · · < zn and hαi , µi/κ > 0 for any i = 1, . . . , r. For generic values of r continuation. (3j )m j =1 , (αi )i=1 , z, µ, κ (b) holds by analytic Pm The case λ = (1, 1, . . . , 1). Set f0 (t) = j =1 hαcλ (j ) , µitj . Let I = (i11 , . . . , is11 ; | {z } m

. . . ; i1n , . . . , isnn ) ∈ P (λ, n). Set γI (z) = {t ∈ Rn : zj < ti j < · · · < tisj < 1

j

zj +1 for all j = 1, . . . , n}, where zn+1 = ∞. Note that {γI (z)}I ∈P (λ,n) is the set of all domains for the configuration of hyperplanes Hij : ti − tj = 0, Hik : ti − zk = 0, 1 6 i < j 6 m, 1 6 k 6 n which are either bounded, or the limit of f0 on them is +∞ when ktk → ∞. In [13] a linearly independent set of hypergeometric differential n-forms, called βnbc differential n-forms, associated to those domains is defined. An explicit non-vanishing formula for the corresponding determinant is given in [8], Theorem 6.2, see also [6]. Since λ = (1, 1, . . . , 1) the space of hypergeometric n-forms is isomorphic to the space C0 (n− ∗ , M ∗ )λ , see Section 5.3. The latter has basis (fI v ∗ )I ∈P (λ,n) which gives the basis (ωI = i(a) ◦ η(fI v ∗ ))I ∈P (λ,n) of the space of hypergeometric n-forms. Since this basis and the βnbc set have the

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

175

same cardinality the non-zero determinant formula for the integrals of βnbc forms over the domains {γI (z)}I ∈P (λ,n) implies a non-zero determinant formula for the integrals of (ωI )I ∈P (λ,n) over the same domains. Since the determinant is non-zero at one point (µ, z), it is non-zero at any point (µ, z) under the above conditions on the parameters. e∗ )(1,1,...,1) as n− , M The case of generic λ ∈ Nr . Consider C0 (n− ∗ , M ∗ )λ and C0 (f hypergeometric differential forms is in Section 4.4. A basis for the 6λ -symmetric P ωJ and e ωJ = i(a) ◦ η(feJ∗ ). given by (ωI )I ∈P (λ,n) , where ωI = J ∈6(I ) e [ P ((1, 1, . . . , 1), n) = S(I ) | {z } m

I ∈P (λ,n)

a disjoint union. Thus the set (ωI )I ∈P (λ,n) consists of linearly independent forms in the space of all hypergeometric forms. The integral pairing described in the previous case is non-degenerate. Therefore there there exists a subset of the set (γJ (z))J ∈P ((1,1,...,1),n) indexed by the set P (λ, n) such that the corresponding determinant is non-zero. 2 COROLLARY 7.4. The system consisting of the union of KZ and Dynamic differential equations for any Kac–Moody Lie algebra with (or without) Serre’s relations is a compatible system of differential equations. Remark. An algebraic proof of the compatibility of the system of KZ equations is given in [10]. Proof. Let us write the differential operators which determine the KZ equations (12) and the dynamical equations (13) in the form ∂ + Bj , Dynamical: ∂zj where j = 1, . . . , n and µ0 ∈ h.

KZ:

∂ + Cµ0 , ∂µ0 (69)

The operators Bj and Cµ0 are linear for any j = 1, . . . , n, µ0 ∈ h. In order to prove the compatibility of the system of KZ and Dynamical differential equations we need to check [

∂ ∂ + Bj , + Bk ] = 0, ∂zj ∂zk

[

∂ ∂ + Bj , 0 + Cµ0 ] = 0, ∂zj ∂µ

[

∂ ∂ + Cµ0 , 00 + Cµ00 ] = 0. 0 ∂µ ∂µ

and

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First consider the case of a Kac–Moody Lie algebra without Serre’s relations, g, acting on a tensor product of highest weight modules M. We have ∂ ∂ ∂ ∂ 0 0 + Bj , 0 + Cµ = Cµ − Bj + [Bj , Cµ0 ]. (70) ∂zj ∂µ ∂zj ∂µ0 The result is a linear operator with meromorphic coefficients depending on parameters {z, µ, (αi )ri=1 , (3j )nj=1 , κ}. Analogously, the commutators ∂ ∂ ∂ ∂ and + Bj , + Bk + Cµ0 , 00 + Cµ00 ∂zj ∂zk ∂µ0 ∂µ are linear operators with meromorphic coefficients depending on the above set of parameters, where j, k = 1, . . . , n, µ0 , µ00 ∈ h. It is enough to show the commutativity of the above operators for generic values of the parameters. Then the commutators will be zero for any values of the parameters by analytic continuation. Take such parameters {z, µ, (αi )ri=1 , (3j )nj=1 , κ} that the set of hypergeometric solutions of the system of KZ and dynamical differential equations forms a basis of M. According to Proposition 7.3(b) this is a generic choice of parameters. Since the KZ and the Dynamical differential operators act as zero on the set of hypergeometric solutions, their commutators also act as zero on the same set. Therefore the commutators act as zero on the g-module M. Finally, consider a Kac–Moody Lie algebra with Serre’s relations g¯ = g/ ker(S: g → g∗ ) which acts on L = M/ ker(S: M → M ∗ ). Corollary 4.4 and [10], Corollary 7.2.11 show that the Dynamical and the KZ operators for g correspond to the the the Dynamical and the KZ operators for g¯ under this factorization. Then the commutativity of the operators on g implies that they are commutative on g¯ as well. 2

References 1.

2. 3. 4. 5. 6.

Babujian, H. and Kitaev, A.: Generalized Knizhnik–Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell– Bloch system with pumping, J. Math. Phys. 39 (1988), 2499–2506. Chalykh, O. A., Feigin, M. V. and Veselov, A. P.: New integrable generalizations of Calogero– Moser quantum problem, J. Math. Phys. 39 (1998), 695–703. Chalykh, O. A. and Veselov, A. P.: Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys. 126 (1990), 597–611. Drinfeld, V.: Quantum groups, in: Proc. ICM (Berkeley, 1986), Vol. 1, Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. Duistermaat, J. J. and Grünbaum, F. A.: Differential operators in the spectral parameter, Comm. Math. Phys. 103 (1986), 177–240. Douai, A. and Terao, H.: The determinant of a hypergeometric period matrix, Invent. Math. 128 (1997), 417–436.

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS

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

177

Harnad, J. and Kasman A. (eds.): The Bispectral Problem (Montréal, 1997), CRM Proc. Lecture Notes 14, Amer. Math. Soc., Providence, RI, 1998. Markov, Y., Tarasov, V. and Varchenko, A.: The determinant of a hypergeometric period matrix, Houston J. Math. 24(2) (1998), 197–219. Orlik, P. and Solomon, L.: Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189. Schechtman, V. and Varchenko, A.: Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139–194. Varchenko, A.: Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum groups, Adv. Ser. Math. Phys. 21, World Scientific, Singapore, 1995. Wilson, G.: Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442 (1993), 177–204. Ziegler, G.: Matroid shellability, β-systems, and affine arrangements, J. Algebraic Combin. 1 (1992), 283–300.

Mathematical Physics, Analysis and Geometry 3: 179–193, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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A Riemann–Hilbert Problem for Propagation of Electromagnetic Waves in an Inhomogeneous, Dispersive Waveguide DMITRY SHEPELSKY Mathematical Division, Institute for Low Temperature Physics, 47 Lenin Avenue, 310164, Kharkov, Ukraine. e-mail: [email protected] (Received: 10 January 2000; accepted: 30 May 2000) Abstract. We consider the inverse scattering problem for a model of electromagnetic wave propagation in a rectangular waveguide filled with dispersive material. The waveguide is inhomogeneous in the longitudinal direction but homogeneous in the transverse directions. Dispersive properties of the material are described by a single-resonance Lorentz model. By reformulating the scattering problem in the frequency domain as a Riemann–Hilbert problem, we prove that the constitutive parameters of the inhomogeneous waveguide are reconstructed uniquely from the scattering data. Mathematics Subject Classifications (2000): 34L25, 34A55, 78A50. Key words: inverse problem, scattering, waveguide.

1. Introduction Recently, increasing attention has been paid to wave propagation, scattering, and guidance in complex media, such as bi-isotropic materials [7, 8, 15], uniaxial bianisotropic chiral materials [11, 23], chiral-omega materials [12, 16, 22], etc. (for the classification of bi-anisotropic media, see, e.g., [21]). Due to the advancement of material sciences, it is possible now to manufacture these materials by putting small metal elements in a host dielectric medium. These materials possess additional degrees of freedom (i.e. in the form of additional parameters in the constitutive relations), which may be used to provide solutions to current engineering problems. A class of new complex materials, called (omega) media, was introduced in [16]. The microstructure of these materials consists of small metal elements in the shape of an (a half-loop with two extended arms) embedded in a dielectric. Possible applications for materials are microwave phase-shifters, scanning antennae, perfectly matched layers, antireflection coatings (see, e.g., [12, 14, 16, 22, 20]). Transient electromagnetic wave propagation in waveguides filled with homogeneous as well as inhomogeneous materials has been studied extensively in the past few years, see, e.g., [2, 6, 10, 19]. The inverse problem of determining the

180

DMITRY SHEPELSKY

constitutive parameters of waveguides filled with homogeneous materials was addressed in [9] and [13]. In the present paper, the propagation of transient electromagnetic waves in a metallic rectangular waveguide filled with inhomogeneous material is considered. The waveguide is placed along the z direction, its inner dimensions are 0 6 x 6 a and 0 6 y 6 b, and the material inhomogeneous in the longitudinal direction z occupies the region 0 6 z 6 L. It is assumed that all the loops of the inclusions have their extended arms parallel to each other (along the direction y) and all the normals to the plane of the loops are also parallel (to the direction x). The constitutive relations for a general bi-anisotropic medium are (cf. [21]) D = εˆ E + ξˆ H,

B = ζˆ E + µH, ˆ

(1)

where εˆ and µˆ are the permittivity and the permeability tensors, respectively, and ξˆ , ζˆ are tensors describing the crosscoupling between the electric and magnetic fields. In the case of the material considered in the present paper, these tensors have the following forms (cf. [13, 9]): ! ! µd 0 0 εh 0 0 µˆ = 0 µh 0 , εˆ = 0 εd 0 , 0 0 εh 0 0 µh ! ! 0 0 0 0 i 0 ξˆ = −i 0 0 , ζˆ = 0 0 0 . (2) 0 0 0 0 0 0 Since materials are highly dispersive [9] (the resonance character of the parameters is due to that of a single particle), the frequency dependence of material parameters must be modelled whenever the analysis is expected to be performed over a broad frequency band. It is assumed that the resonant components, εd , µd , and , are inhomogeneous, whereas the non-resonant components, εh and µh , are constants (and known when considering the inverse problem). The dispersion in the frequency domain (the time dependence of all fields is exp iωt) is described by a single-resonance Lorentz model [18] with a fixed resonance frequency, ω0 , but varying (in z) amplitudes: εa (z) , ω02 − ω2 µa (z)ω2 , µd (z, ω) = µh + 2 ω0 − ω 2 a (z)ω (z, ω) = 2 . ω0 − ω 2

εd (z, ω) = εh +

(3)

The homogeneous parts of the waveguide, z < 0 and z > L, are assumed to be vacuum regions, with εˆ = ε0 I , µˆ = µ0 I , and = 0, where I is the unitary matrix, and ε0 and µ0 are the vacuum permittivity and permeability, respectively.

A RIEMANN–HILBERT PROBLEM FOR PROPAGATION OF ELECTROMAGNETIC WAVES

181

We consider the scattering matrix consisting of the reflection and transmission coefficients for T Em0 modes propagating in the waveguide. We are particularly interested in the inverse problem, when the scattering matrix (as a function of the frequency ω) is given, and the material parameters εa , µa , and a (as functions of z, z ∈ [0, L]) are to be reconstructed. Our approach to the scattering problem is based on its reformulation as a Riemann–Hilbert problem relative to a contour in the ω plane, where the jump matrix across the contour is constructed from the scattering matrix. Such an approach for studying inverse scattering problems is applied in [3] and [4] for stratified bi-isotropic non-dispersive materials. The inverse problem for a dispersive chiral slab is studied in [5]. In the present paper, this approach allows us to prove that the dispersive parameters of the inhomogeneous waveguide, εa (z), µa (z), and a (z), are determined uniquely by the scattering matrix relative to the basic mode in the waveguide, T E10 . 2. The Scattering Problem We assume that εa (z), µa (z), and a (z) are positive absolutely continuous functions on the interval z ∈ [0, L], µh > µa (z), and εa (z)µa (z) > 2a (z). Since the principal axes of the medium coincide with the principal directions of the rectangular waveguide, Maxwell’s equations ∇ × H = iωD,

∇ × E = −iωB,

considered together with the constitutive relations (1) allow the propagation of T Em0 and T E0n modes, m, n = 0, 1, . . .. Particularly, for the T Em0 modes, with ∂/∂y = 0, one obtains the following equations for E2 , H1 , and H3 : ∂E2 = −iωµh H3 , ∂x ∂E2 = iωµd H1 − ωE2 , ∂z ∂H1 ∂H3 (4) − = iωεd E2 + ωH1. ∂z ∂x The boundary conditions for the T Em0 modes are ∂H 3 = 0. (5) E2 x=0,a = H1 x=0,a = ∂x x=0,a In the vacuum regions of the waveguide, z < 0 and z > L, the fields of the T Em0 modes are E2 (x, z, ω) = Cml sin λm xe±γm z , γm H1 (x, z, ω) = ±Cml sin λm xe±γm z , iωµ0 λm H3 (x, z, ω) = −Cml cos λm xe±γm z , iωµ0

(6)

182

DMITRY SHEPELSKY

where l = 0, L, πm , λm = a

q

γm (ω) = i ω2 ε0 µ0 − λ2m ,

m = 0, 1, 2, . . . .

The continuity of the tangential fields E2 and H1 at z = 0 and L implies, for −∞ < z < ∞, E2 (x, z, ω) = sin λm x · e(z, ω), H1 (x, z, ω) = sin λm x · h(z, ω), where e(z, ω) and h(z, ω) satisfy the system of differential equations dU = W (z, ω)U, −∞ < z < ∞. dz e(z, ω) Here U (z, ω) = , h(z, ω) ! −ω iωµd 2 λ . W= iωεd + m ω iωµh

(7)

(8)

In the vacuum regions, z 6∈ [0, L], W (z, ω) ≡ Wb (ω) =

0 iωε0 +

λ2m iωµ0

iωµ0 0

! .

Introduce the matrix Tb (ω) diagonalizing Wb (ω):  iωµ0  1 − 0 −γm (ω)  γm  −1 . Tb (ω) =  iωµ0  , Tb (ω)Wb (ω)Tb (ω) = 0 γm (ω) 1 γm The propagating solutions of (7), U + (z, ω), and U − (z, ω), −∞ < z < ∞, are determined by the boundary conditions U −(0, ω) = U + (L, ω) = I . In the empty parts of the waveguide, they are written as follows: −γ (ω)z 0 e m −1 − Tb (ω), for z 6 0, U (z, ω) = Tb (ω) 0 eγm (ω)z −γ (ω)(z−L) 0 e m −1 + Tb (ω), for z > L. (9) U (z, ω) = Tb (ω) 0 eγm (ω)(z−L) The scattering matrix S(ω) relates U − (z, ω) and U + (z, ω) through U − (z, ω) · S(ω) = U + (z, ω),

−∞ < z < ∞.

(10)

A RIEMANN–HILBERT PROBLEM FOR PROPAGATION OF ELECTROMAGNETIC WAVES

183

The scattering matrix is expressed in terms of the reflection and transmission coefficients for a particular mode. Namely, the reflection coefficient for the wave propagating in the positive direction of z, rp , can be expressed as rp (ω) = S˜21 (ω)/S˜ 11(ω), ˜ where S(ω) = Tb (ω)S(ω)Tb−1 (ω). For the wave propagating in the opposite direction, rn (ω) = −S˜12 (ω)/S˜ 11(ω). The transmission coefficients are tp (ω) = tn (ω) = 1/S˜11 (ω). In the inverse problem, the 2 × 2 scattering matrix S as a function of ω is supposed to be known and is used to reconstruct the unknown material parameters (as functions of z). THEOREM 1. The 2 × 2 scattering matrix S(ω) given in a finite frequency band and corresponding to the T E10 mode uniquely determines the inhomogeneous parameters εa (z), µa (z), and a (z), z ∈ [0, L], of the dispersive waveguide. Our approach to the inverse problem is based on the reformulation of the scattering problem (10) as a Riemann–Hilbert problem. We seek for an invertible matrix-valued function piecewice holomorphic in the ω-plane (z being considered as a parameter) which, on the one hand, has good behavior near the poles (finite or infinite) of W (z, ω), and, on the other hand, is constructed from the properly chosen solutions of (10). Near ω = ∞, W (z, ω) is written as W (z, ω) = iωW1∞ (z) + W0∞ (z) + O(1/ω), where W1∞ (z)

=

0 εh

µ1 (z) , 0

W0∞ (z)

(11)

1 0 = a (z) 0 −1

,

µ1 (z) = µh − µa (z). The transformation matrix T∞ (z) is defined as T∞ (z) = Tˆ 1∞ (z), where 0 (εh /µ1 )1/4 −1 1 ˆ (z), T = , 1∞ (z) = 0 (εh /µ1 )−1/4 1 1 −1 hence T∞ (z)W1∞ (z)T∞ (z) = D∞ (z) with p −1 0 D∞ (z) = εh µ1 (z) . 0 1

The transformation F∞ (z, ω) = T∞ (z)U (z, ω), considered for z ∈ [0, L], gives the differential equation dF∞ (z, ω) dz = {iωD∞ (z) + A∞ (z) + R∞ (z, ω)} F∞ (z, ω),

0 6 z 6 L,

(12)

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DMITRY SHEPELSKY

where

A∞ (z) =

εh 1 d 0 −1 ln + a (z) , −1 0 4 dz µ1 (z)

(13)

R∞ (z, ω) = O(1/ω) as ω → ∞. Notice that A∞ (z) is off-diagonal, therefore A∞ (z) ∈ ran ad D∞ (z) for each z ∈ [0, L], where (ad A)B ≡ [A, B] = AB − BA. ± (z, ω), for |ω| > R, where R is big enough, and We seek solutions of (12), F∞ such that: ± (z, ω) are holomorphic in the half-neighborhoods of ω = ∞: {|ω| > (a) F∞ R, ± Im ω > 0}; ± −1 (b) F∞ (z, ω)E∞ (z, ω) → I2

as ω → ∞, RL where E∞ (z, ω) = exp{−iω z D∞ (t) dt}; ± ± (c) F∞ (z, ω) z=0,L are triangular matrices, (F∞ (z, ω))jj (L, ω) = 1, j = 1, 2. ± ± (z, ω) = Fˆ∞ (z, ω)E∞ (z, ω) where Such solutions are determined by F∞ ± ˆ F∞ (z, ω) are the solutions of the system of Fredholm equations m )j k (z, ω) (Fˆ∞

Z z exp iω [(D∞ )jj (s) − (D∞ )kk (s)] ds ×

Zz

= δj k +

t Q(j,k,m)

m (t, ω) j k dt, × (A∞ (t) + R∞ (t, ω)) Fˆ∞

where j, k ∈ {1, 2}, Q(j, k, m) =

(14)

0, m(j − k) > 0, L, j = k or m(j − k) < 0.

PROPOSITION 1. The system of integral equations (14), m = +, −, has a unique m satisfying conditions (a) and (b) for a sufficiently large R. solution Fˆ∞ Proof of the proposition is similar to the case with a constant main term D∞ , see, e.g., [25]. Note that the off-diagonal structure of A∞ is of importance here. On the other hand, the construction of (14) yields triangular boundary values of m (z, ω) at z = 0 and z = L: F∞ − −,l − −,u F∞ (0, ω) = V∞ (ω), F∞ (L, ω) = V∞ (ω), + +,u + +,l F∞ (0, ω) = V∞ (ω), F∞ (L, ω) = V∞ (ω),

(15)

±,l ±,u (ω) are lower-triangular, V∞ (ω) are upper-triangular, and the diagonal where V∞ +,l −,u elements of V∞ (ω) and V∞ (ω) are equal to 1.

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A RIEMANN–HILBERT PROBLEM FOR PROPAGATION OF ELECTROMAGNETIC WAVES

m,l m,u (ω) and V∞ (ω), m = +, −, for PROPOSITION 2. Triangular matrices V∞ |ω| = R, ω ∈ (−∞, −R), and ω ∈ (R, ∞) are uniquely determined by a scattering matrix S(ω) given in any (real) frequency band (outside ω = ±ω0 ). ± (z, ω) and T∞ (z)U +(z, ω), one gets Proof. Relating F∞ ± −1 ± (z, ω) = T∞ (z)U +(z, ω)T∞ (L)F∞ (L, ω). F∞

(16)

Setting z = 0 in (16) and using the scattering relation (10), one obtains ± −1 ± F∞ (0, ω) = T∞ (0)U − (0, ω)S(ω)T∞ (L)F∞ (L, ω) −1 ± = T∞ (0)S(ω)T∞ (L)F∞ (L, ω),

or, in view of (15), −,l −,u (ω) = S∞ (ω)V∞ (ω), V∞ +,u +,l V∞ (ω) = S∞ (ω)V∞ (ω),

(17)

−1 where S∞ (ω) = T∞ (0)S(ω)T∞ (L).

Relations (17) may be viewed as triangular factorizations of the matrix S∞ (ω) ±,l ±,u (ω) and V∞ (ω) (for each fixed ω), which allows a unique reconstruction of V∞ provided S∞ (ω) is known. The scattering matrix S(ω) is analytically continuable −1 (L), are in C from the given frequency band. The constant factors, T∞ (0) and T∞ uniquely determined by the asymptotics of S(ω) as ω → ∞. Indeed, relations (17) ± ± (L, ω) and F∞ (0, ω) imply and the large-ω behavior of the triangular factors F∞ that, as ω → ∞, S∞ (ω) = E∞ (0, ω)(I + O(1/ω)) iωκ 0 e (I + O(1/ω)), = 0 e−iωκ RL√ where κ = 0 εh µ1 (z) dz. Therefore, −1 ˆ −1 S∞ (ω)Tˆ 1∞ (L) S(ω) = T∞ (0)S∞ (ω)T∞(L) = 1−1 ∞ (0)T −1 1 cos ωκ −i sin ωκ σL 0 0 σ0 I +O , = −1 −i sin ωκ cos ωκ 0 σ0 0 σL ω

where σ0 = (µ1 (0)/εh )1/4 and σL = (µ1 (L)/εh )1/4 , that allows us to determine µ1 (0), µ1 (L), and κ. For an ω close to ω0 , W (z, ω) can be written as W (z, ω) =

1 W 1 (z) + W01 (z) + O(ω − ω0 ), ω − ω0 1

where 1 W11 (z) = 2

ω0 a (z) −iω02 µa (z)

! ,

−iεa (z)

−ω0 a (z)

W01 (z) =

3 (z) 4 a

B(z)

C(z)

− 34 a (z)

! ,

186

DMITRY SHEPELSKY

B(z) = iω0

5 µh − µa (z) , 4

C(z) = iω0 εh −

i λ2m εa (z) + . 4ω0 iω0 µh

The diagonalization of W11 (z) is performed by using T1 (z) = E(z)T˜1 (z), where γ2 (z) −1 ˜ , E(z) = diag{e1 (z), e2 (z)}, T1 (z) = −γ1 (z) 1 q a + iQ a − iQ , γ2 = , Q = εa µa − 2a , γ1 = iω0 µa iω0 µa Z L dγ2 1 3 e1 (z) = exp a (γ2 + γ1 ) + Bγ2 γ1 − C + dt , 4 dt z γ2 − γ1 Z L 3 dγ1 1 − a (γ2 + γ1 ) − Bγ2 γ1 + C − dt . e2 (z) = exp 4 dt z γ2 − γ1 Then T1 (z)W11 (z)T1−1 (z) = D1 (z), where i −1 0 D1 (z) = ω0 Q(z) . 0 1 2 The transformed differential equation for F1 (z, ω) = T1 (z)U (z, ω) is dF1 1 D1 (z) + A1 (z) + R1 (z, ω) F1 (z, ω), (z, ω) = dz ω − ω0 0 6 z 6 L, where

(18)

0 β1 (z) A1 (z) = , R1 (z, ω) = O(ω − ω0 ) as ω → ω0 , 0 β2 (z) 1 dγ2 3 2 β1 = a γ2 + Bγ2 − C + e1 e2−1 , γ2 − γ1 2 dt 3 1 dγ1 −1 2 β2 = − a γ1 − Bγ1 − C e1 e2 . γ2 − γ1 2 dt

The solutions of Equation (18) for |ω − ω0 | < δ, F1+ (z, ω) and F1− (z, ω), are constructed via the solutions Fˆ1± (z, ω) of the Fredholm integral equations similar to (14): (Fˆ1m )j k (z, ω)

Zz

= δj k + Q(j,k,m)

1 exp ω − ω0

Z

z

[(D1 )jj (s) − (D1 )kk (s)] ds × t

i h × (A1 (t) + R1 (t, ω)) Fˆ1m (t, ω)

jk

dt.

(19)

A RIEMANN–HILBERT PROBLEM FOR PROPAGATION OF ELECTROMAGNETIC WAVES

187

One has F1± (z, ω) = Fˆ1± (z, ω)E1 (z, ω), where E1 (z, ω) = exp −

1 ω − ω0

Z

L

D1 (t) dt .

z

Functions Fˆ1± (z, ω) are holomorphic in the half-disks {ω : |ω−ω0| < δ, ± Im ω > 0} for sufficiently small δ > 0, and Fˆ1± (z, ω) → I as ω → ω0 , ± Im ω > 0. Arguing as in the case of the infinite pole, we obtain the relations between F1± (z, ω) and U + (z, ω) F1± (z, ω) = T1 (z)U +(z, ω)T1−1 (L)F1± (L, ω)

(20)

and another set of triangular factorizations for the scattering matrix: V1−,u (ω) = S1 (ω)V1−,l (ω), V1+,l (ω) = S1 (ω)V1+,u (ω),

(21)

where S1 (ω) = T1 (0)S(ω)T1−1 (L). The triangular factors in (21) are the boundary values of F1± (z, ω) at z = 0 and z = L: −,u (ω), F1− (0, ω) = V∞ + +,l F1 (0, ω) = V∞ (ω),

−,l F1− (L, ω) = V∞ (ω), + +,u F1 (L, ω) = V∞ (ω),

(22)

where V1±,l (ω) are lower-triangular, V1±,u (ω) are upper-triangular, and the diagonal elements of V1−,l (ω) and V1+,u (ω) are equal to 1. The constant factors T1 (0) and RL T1−1 (L) (or, in other words, the constants γ1,2 (0), γ1,2 (L), e1,2 (0), and 0 Q(z) dz) are determined by the asymptotics of S(ω) as ω → ω0 . The considerations near −ω0 are literally the same, giving i −1 0 D2 (z) = − ω0 Q(z) , T2 (z) = E(z)T˜2 (z), 0 1 2 T˜2 (z) =

γ2 (z) −1 −γ1 (z) 1

,

E2 (z, ω) = exp −

1 ω + ω0

Z

L

D2 (t) dt ,

z

V2−,l (ω) = S2 (ω)V2−,u (ω), V2+,u (ω) = S2 (ω)V2+,l (ω), where S2 (ω) = T2 (0)S(ω)T2−1 (L).

(23)

188

DMITRY SHEPELSKY

Considering a neighborhood of ω = 0, we notice that that the function P (z, ω) =

KU + (z, ω)K −1 , where K =

1 iω

0

0

1

, is holomorphic near ω = 0, det P (z, ω) ≡

1, and P (z, ω) = P (z, 0)(I + O(ω)) as ω → 0, where µh sinh λm (z − L) cosh λm (z − L) λm . P (z, 0) = λm sinh λm (z − L) cosh λm (z − L) µh Now we are able to construct a piecewise holomorphic function m(z, ω) (z is a parameter) relative to a contour in the ω-plane, such that its jumps across the contour are essentially determined by the scattering matrix. The contour 0 consists of the circles |ω| = R, |ω − ω0 | = δ, |ω + ω0 | = δ, and the real axis Im ω = 0. Set  |ω| > R, ± Im ω > 0, Fˆ ± (z, ω),    ∞ −1 ± ˆ (24) m(z, ω) = T∞ (z)T1−1 (z)F1±(z, ω), |ω − ω0 | < δ, ± Im ω > 0,  ˆ ω), |ω + ω0 | < δ, ± Im ω > 0,   T∞ (z)T2+ (z)F2 (z, |ω| < R, |ω ± ω0 | > δ. T∞ (z)U (z, ω)K −1 , The contour 0 divides the complex plane into two open sets, Q+ and Q− , being the positively oriented boundary of Q+ . Denote by m± (z, η), η ∈ 0, the boundary values of m(z, ω) as ω → η, ω ∈ Q± . Then (24) together with (16) and (20) yields m+ (z, η) = m− (z, η) · V (z, η),

η ∈ 0,

(25)

where V (z, ω)  −1 +,l −1 (L)V∞ (ω)E∞ (z, ω), KT∞     E (z, ω) V −,u (ω) −1 T (L)K −1 ,   ∞ ∞ ∞  −1 +,l   −,u −1  E∞ (z, ω) V∞ (ω) V∞ (ω)E∞ (z, ω),    +,u −1 −1   KT1 (L)V1 (ω)E1 (z, ω),   −1   −,l  (z, ω) V (ω) T1 (L)K −1 , E  1  1 −1 = −,l (z, ω) V (ω) V1+,u (ω)E1−1 (z, ω), E  1 1      KT2−1 (L)V2+,l (ω)E2−1 (z, ω),   −1    E2 (z, ω) V2−,u (ω) T2 (L)K −1 ,   −1    E2 (z, ω) V2−,u (ω) V2+,l (ω)E2−1 (z, ω),     I,   From the definition of m(z, ω) it follows that 1 as ω → ∞, m(z, ω) = I + O ω

|ω| = R, Im ω > 0, |ω| = R, Im ω < 0, Im ω = 0, |ω| > R, |ω − ω0 | = δ, Im ω > 0, |ω − ω0 | = δ, Im ω < 0, Im ω = 0, |ω − ω0 | < δ,

(26)

|ω + ω0 | = δ, Im ω > 0, |ω + ω0 | = δ, Im ω < 0, Im ω = 0, |ω + ω0 | < δ, Im ω = 0, |ω| < R, |ω ± ω0 | > δ.

(27)

A RIEMANN–HILBERT PROBLEM FOR PROPAGATION OF ELECTROMAGNETIC WAVES

m(z, ω) = T∞ (z)K −1 P (z, ω) = Tˆ iω 0 ˜ ˆ = T P (z, ω), 0 1

iω 0

0 1

189

1∞ (z)P (z, ω) (28)

where P˜ (z, ω) is holomorphic and invertible near ω = 0. Note that Ek (z, ω), k = ∞, 1, 2, are expressed in terms of two combinations RL√ RL of the (unknown) material parameters, z εh µ1 (t) dt and z Q(t) dt. In order to construct a family of the Riemann–Hilbert problems with jumps independent of the unknown functions, two auxiliary real parameters J1 and J2 are introduced. If one defines Eˆ ∞ (J1 , J2 ; ω) = diag e−iωJ1 , eiωJ1 , iω0 − iω0 J J Eˆ 1 (J1 , J2 ; ω) = diag e 2(ω−ω0) 2 , e 2(ω−ω0 ) 2 , iω0 J − iω0 J Eˆ 2 (J1 , J2 ; ω) = diag e 2(ω+ω0 ) 2 , e 2(ω+ω0 ) 2 , then

, Ek (z, ω) = Eˆ k (J1 , J2 ; ω) J1 =J1 (z)

k = 1, 2, ∞,

(29)

J2 =J2 (z)

where

Z

L

J1 (z) = −

p

εh µ1 (t) dt,

z

Z J2 (z) = −

L

Q(t) dt.

(30)

z

A family of the Riemann–Hilbert problems parametrized by J1 and J2 is given as follows: find a 2 × 2 matrix function G(J1 , J2 ; ω) that satisfies the following conditions: G(·, ·; ω) is piecewise holomorphic relative to the contour 0; 1 as ω → ∞; G(·, ·; ω) = I + O ω

(31)

G(·, ·; ω) is invertible for ω 6= 0;

(33)

−i/ω 0

0 1

−1 1 1 1

(32)

G(·, ·; ω) is holomorphic and invertible in a neighborhood of ω = 0;

(34)

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G+ (J1 , J2 ; η) = G− (J1 , J2 ; η)Vˆ (J1 , J2 ; η),

η ∈ 0,

(35)

where G(·, ·; ω), G± (·, ·; η) = lim ω→η ω∈Q±

and Vˆ (J1 , J2 ; η) is constructed via (26) with Ek (z, ω) replaced by Eˆ k (J1 , J2 ; ω), k = 1, 2, ∞. By the Liouville theorem, the solution of the Riemann–Hilbert problem (31)– (35) is unique for each J1 and J2 (if G1 and G2 are two solutions, then G1 G−1 2 is an −1 −1 entire function, G1 G2 (ω) → I as ω → ∞, therefore G1 G2 ≡ I ). This solution is related to m(z, ω) as follows: (36) G(J1 , J2 ; ω) J1 =J1 (z) = m(z, ω). J2 =J2 (z)

3. Reconstruction of Material Parameters The fact that the solution of the Riemann–Hilbert problem (31)–(35) is unique together with relation (36) allows us to develop the procedure of unique simultaneous reconstruction of the material parameters and, hence, to prove Theorem 1. First, the jump matrix Vˆ (J1 , J2 ; η), η ∈ 0, is constructed from the scattering matrix S(ω) related to some T Em0 mode, for instance, the basic mode T E10 . The construction involves analytic continuation of S(ω) on the related parts of 0, evaluation of S(ω) as ω → ∞ and ω → ±ω0 , and triangular factorizations (17), (21), and (23). Second, the Riemann–Hilbert problems (31)–(35) are solved for each J1 and J2 , giving G(J1 , J2 ; ω). Evaluating the solution of the Riemann–Hilbert problem at ω = 0 and ω = ω0 gives, in view of (36) and (24), 1 −i/ω 0 −1 1 lim = 1∞ (z)P (z, 0), (37) G(J1 , J2 ; ω) 0 1 1 1 J1 =J1 (z) ω→0 2 J2 =J2 (z)

Tˆ −1 G(J1 , J2 ; ω0 ) J1 =J1 (z) = 1∞ (z)T1−1 (z).

(38)

J2 =J2 (z)

Let us denote 1 G (J1 , J2 ) = lim ω→0 2 (0)

M1 (J1 , J2 ) = M3 (J1 , J2 ) =

−i/ω 0

G(0) 22 (J1 , J2 ) G(0) 11 (J1 , J2 )

,

G22 (J1 , J2 ; ω0 ) . G12 (J1 , J2 ; ω0 )

0 1

−1 1 1 1

M2 (J1 , J2 ) =

G(J1 , J2 ; ω), G21 (J1 , J2 ; ω0 ) , G11 (J1 , J2 ; ω0 )

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Then, in view of (37) and (38), p εh µ1 (z) = εh M1 (J1 , J2 ) J1 =J1 (z) , J2 =J2 (z)

ω0 (M2 (J1 , J2 ) − M3 (J1 , J2 ))(µh M1−1 (J1 , J2 ) − εh M1 (J1 , J2 )) J1 =J1 (z) , Q(z) = J2 =J2 (z) 2 2 µa (z) = µh − εh M1 (J1 , J2 ) J1 =J1 (z) , J2 =J2 (z)

iω µa (z)M1−1 (J1 , J2 )(M2 (J1 , J2 ) + M3 (J1 , J2 )) J1 =J1 (z) , J2 =J2 (z) 2 −2 2 εa (z) = −ω0 µa (z)M1 (J1 , J2 )M2 (J1 , J2 )M3 (J1 , J2 ) J1 =J1 (z) . a (z) =

J2 =J2 (z)

(39)

√ Since, by the definition of J1 (z) and J2 (z), dJ1 /dz = εh µ1 (z) and dJ2 /dz = Q(z), we arrive at the system of differential equations for determining J1 (z) and J2 (z): dJ1 = εh M1 (J1 , J2 ), J1 (L) = 0, dz dJ2 ω0 = (M2 (J1 , J2 ) − M3 (J1 , J2 ))(µh M1−1 (J1 , J2 ) − εh M1 (J1 , J2 )), dz 2 J2 (L) = 0. Finally, substituting J1 (z) and J2 (z) into (39) gives µa (z), a (z), and εa (z). 4. Conclusion When implementing any inversion method, one faces the uniqueness problem, i.e., the question about the amount of information which is necessary and sufficient to achieve, in principle, a unique reconstruction. Ill-posed nature of most inverse problems requires searching for a minimum information which determines the problem uniquely (overdetermining an inverse problem may increase its illposedness). The connection of the inverse scattering problems and the Riemann–Hilbert problems was first established by Shabat [17] and rigorously developed further in [1, 24, 25]. The present paper illustrates usefulness of the method of Riemann– Hilbert problem for studying scattering problems relevant to wave propagation in complex inhomogeneous dispersive media. We believe that the Riemann–Hilbert approach to the inverse scattering problem is an efficient tool for achieving a better understanding of the relations between the scattering data and the material parameters to be reconstructed.

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Acknowledgements The author is grateful for the hospitality at the Laboratory of Mathematical Physics and Geometry, University Paris-7 (where the work was finalized) and to the Embassy of France in Ukraine (MAE) for financial support.

References 1. 2.

3. 4. 5. 6. 7. 8.

9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19.

Beals, R. and Coifman, R. R.: Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math. 37 (1984), 39–90. Bernekorn, P., Karlsson, A. and Kristensson, G.: Propagation of transient electromagnetic waves in inhomogeneous and dispersive waveguides, J. Electro. Waves Appl. 10 (1996), 1263–1286. Boutet de Monvel, A. and Shepelsky, D.: Direct and inverse scattering problem for a stratified nonreciprocal medium, Inverse Problems 13 (1997), 239–251. Boutet de Monvel, A. and Shepelsky, D.: Inverse scattering problem for a stratified bi-isotropic medium at oblique incidence, Inverse Problems 14 (1998), 29–40. Boutet de Monvel, A. and Shepelsky, D., 1999: A frequency-domain inverse problem for a dispersive stratified chiral medium, Preprint BIBOS 3/6/99. Dvorak, S. L. and Duldy, D. G.: Propagation of ultra-wide-band electromagnetic pulses through dispersive media, IEEE Trans. Electromagn. Compatibility 37(2) (1995), 192–200. He, S.: A time-harmonic Green’s function technique and wave propagation in a stratified nonreciprocal chiral slab with multiple discontinuities, J. Math. Phys. 33 (1992), 4103–4110. Jaggard, D. L. and Engheta, N.: Chirality in electrodynamics: Modelling and apllications, in H. L. Bertoni and L. B. Felsen (eds), Directions in Electromagnetic Modelling, Plenum Publishing Co., New York, 1993. Kharina, T. G., Tretyakov, S. A., Sochava, A. A., Simovski, C. R. and Bolioli, S.: Experimental studies of artificial omega media, Electromagnetics 18 (1998), 423–437. Kristensson, G.: Transient electromagnetic wave propagation in waveguides, J. Electro. Waves Appl. 9 (1995), 645–671. Lindell, I. V. and Sihvola, A. H.: Plane-wave reflection from uniaxial chiral interface and its application to polarization transformer, IEEE Trans. Antennas and Propagation 43 (1995), 1397–1404. Norgen, M.: Optimal design using stratified bianisotropic media: Application to anti-reflection coatings, J. Electro. Waves Appl. 12 (1998), 933–959. Norgen, M. and He, S.: Reconstruction of the constitutive parameters for an material in a rectangular waveguide, IEEE Trans. Microwave Theory Techniques 43(6) (1995), 1315–1321. Norgen, M. and He, S.: On the possibility of reflectionless coating of a homogeneous bianisotropic layer on a perfect conductor, Electromagnetics 17 (1997), 295–307. Rikte, S.: Reconstruction of bi-isotropic material parameters using transient electromagnetic fields, Wave Motion 28 (1998), 41–58. Saadoun, M. M. I. and Engheta, N.: A reciprocal phase shifter using noval pseudochiral or medium, Microwave Opt. Tech. Lett. 5(4) (1992), 184–187. Shabat, A. B.: Inverse scattering problem for a system of differential equations, Funktsional Anal. i Prilozhen. 9 (1975), 75–78. Siushansian, R. and LoVetri, J.: Efficient evaluation of convolution integrals arising in FDTD formulations of electromagnetic dispersive media, J. Electro. Waves Appl. 11 (1997), 101–117. Stenius, P. and York, B.: On the propagation of transients in waveguides, IEEE Antennas and Propagation Magazine 37 (1995), 39–44.

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20.

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Tretyakov, S. A.: Uniaxial omega medium as a physically realizable alternative for the perfectly matched layer (PML), J. Electro. Waves Appl. 12 (1998), 821–837. 21. Tretyakov, S. A., Sihvola, A. H., Sochava, A. A. and Simovski, C. R.: Magnetoelectric interactions in bi-anisotropic media, J. Electro. Waves Appl. 12 (1998), 481–497. 22. Tretyakov, S. A. and Sochava, A. A.: Proposed composite materials for non-reflecting shields and antenna radomes, Electron. Lett. 29 (1993), 1048–1049. 23. Viitanen, A. J. and Lindell, I. V.: Plane wave propagation in a uniaxial bianisotropic medium with an application to a polarization transformer, Int. J. Infrared Millimim. Waves 14(10) (1993), 1993–2010. 24. Zhou, X.: The Riemann–Hilbert problem and inverse scattering, SIAM J. Math. Anal. 20 (1989), 966–986. 25. Zhou, X.: Inverse scattering transform for systems with rational spectral dependence, J. Differential Equations 115 (1995), 277–303.

Mathematical Physics, Analysis and Geometry 3: 195–216, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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Second-Order Covariant Tensor Decomposition in Curved Spacetime GIANLUCA GEMELLI Dipartimento di Matematica, Università di Torino, Via Carlo Alberto n.10, I-10123 Turin, Italy. e-mail: [email protected] (Received: 8 September 1998; in final form: 22 February 2000) Abstract. Local four-dimensional tensor decomposition formulae for generic vectors and 2-tensors in spacetime, in terms of scalar and antisymmetric covariant tensor potentials, are studied within the framework of tensor distributions. Earlier first-order decompositions are extended to include the case of four-dimensional symmetric 2-tensors and new second-order decompositions are introduced. Mathematics Subject Classifications (2000): 53B50, 35Q75. Key words: Laplace operator in a curved spacetime, covariant potentials for tensor fields.

1. Introduction A classic decomposition theorem (see, e.g., [22], p. 26, [10], p. 43, [18], p. 49) states that any regular vector field in the ordinary three-dimensional space can be written as the sum of a gradient plus a divergence-free component. A similar decomposition can be considered for skew-symmetric 2-tensors [9, 13]. Several decomposition theorems are known for the Weyl tensor and, more generally, for a generic antisymmetric tensor in a curved spacetime [1, 13, 19]. Covariant decompositions of symmetric 2-tensors into their irreducible parts can be introduced on a spacelike three-dimensional manifold; such decompositions proved to be useful tools for the study of the Cauchy problem for Einstein gravitational equations (see, e.g., [8, 29, 21]). A systematic study of this problem should therefore include the missing pieces: decompositions of four-dimensional vectors and symmetric 2-tensors in curved spacetime. Moreover, since the decompositions mentioned above are first-order, i.e. they involve first derivatives of the potentials, it is also interesting to consider second-order decompositions obtained by iteration. Here we attempt to reach these goals and to revisit the whole matter within the framework of tensor distributions. In Section 2, the main definitions about vector distributions and general tensor distributions are introduced.

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In Section 3, the theory of the Laplace operator within the framework of tensor distributions in a curved spacetime is recalled and a useful result about first-order covariant decomposition of a divergence-free 4-vector (Theorem 1) is readily obtained. The same result was proved in the dual framework of C ∞ functions in [13] with the help of spinor analogues of tensor equations. In Section 4, the generalized Clebsch Theorem (Theorem 2) is proved. Such a theorem rigorously establishes a first-order covariant decomposition formula for a generic (nondivergence-free) 4-vector, a formula which is usually taken for granted as the generalization of the ordinary three-dimensional theorem. The 3+1 splitting of the four-dimensional formula, however, permits us to understand the relations between the two formulations and to determine under which additional hypothesis the ordinary three-dimensional formula is obtained exactly as a corollary from the four-dimensional one (Corollaries 2.1, 2.2 and 2.3). In Section 5, a first-order covariant decomposition formula for a skew-symmetric 2-tensor in a curved spacetime is proved (Theorem 3). Such a formula was originally introduced in flat spacetime [9]; the curved spacetime generalization was again proved within the framework of C ∞ functions and with the help of spinor techniques in [13]. The splitting of the formula is examined, which permits us to relate the electric and magnetic components of the skew-symmetric tensor to its vector potentials. The particular case when the tensor is divergence-free is considered (Corollaries 3.1 and 3.3). Moreover, a new second-order covariant decomposition formula is introduced (Corollary 3.2) as a consequence of Theorems 2 and 3. In Section 6, a four-dimensional covariant first-order decomposition formula for a symmetric 2-tensor is introduced (Theorem 4), which generalizes the usual three-dimensional transverse decomposition on a spacelike hypersurface. A partially second-order decomposition formula is also given (Corollary 4.3). Again the relations between the two formulations are studied by means of the splitting of the spacetime (Corollaries 4.1 and 4.2). In Section 7, the case of flat spacetime is examined. Here a second-order decomposition formula for a generic divergence-free symmetric 2-tensor in terms of a double 2-form potential is introduced (Theorem 5). In the particular case where the tensor under investigation is trace-free, the same property is held by the trace of the potential (Corollary 5.1). We thus obtain a complete second-order decomposition formula for a generic symmetric 2-tensor (Corollary 5.2). The splitting of such a formula permits us to establish similar three-dimensional results (Corollaries 5.3 and 5.4). Conversely, the second-order decomposition of a double 2-form in terms of a divergence-free symmetric potential, and the corresponding three-dimensional results are obtained (Theorem 6 and relative corollaries). In Section 8, some applications of the theory of potentials to electromagnetism are studied. A general theorem of the existence and uniqueness of the solutions of the Maxwell equations system is given as a consequence of Theorem 1 (Corol-

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lary 1.1). The 4-vector potentials of electromagnetism are introduced and the Lorentz gauge is defined within the framework of Theorem 3 (Corollary 3.4).

2. Tensor Distributions in a Curved Spacetime Let V4 be the spacetime of general relativity, i.e. a four-dimensional oriented pseudo-Riemannian manifold of class C 3 , whose metric g is of class C 2 and of normal hyperbolic type, with the signature − + + +. Greek indices run from 0 to 3. Units are chosen in order to have the speed of light in empty space c = 1. The Riemann curvature tensor Rαβρσ is defined by the following Ricci formula: (∇β ∇α − ∇α ∇β )V σ = Rαβρ σ V ρ

(1)

which holds for any regular vector field V . The symmetric Ricci tensor, trace of the Riemann √ tensor, is defined by Rβρ = Rαβρ α . Let η = |g| denote the unit volume 4-form (Ricci antisymmetric tensor), where is the Levi-Civita indicator; η is used to define the dual ∗F of a given antisymmetric 2-tensor F : (∗F )αβ = (1/2)ηαβ µν F µν . The dual operator is involutive. Useful properties of η are (1/2)ηαρλµ ηβ ρ σ ν = gαβ gµ[σ gν]λ + gλβ gα[σ gν]µ + gµβ gλ[σ gν]α , (1/2)ηρσ λµ ηρσ αβ = gµα gλβ − gλα gµβ .

(2)

Our investigations are of a local nature. For a point x 0 ∈ V4 , let us consider an open neighbourhood ⊂ V4 with compact closure, homeomorphic to an open sphere of R4 , such that for all x ∈ , there exists a unique arc of geodesic `(x, x 0 ) joining − x and x 0 . For x ∈ , let E + x [resp. Ex ] denote the image in of the set of future pointing timelike paths originating at x [resp. ending at x]. A set A is called pastcompact [resp. future-compact] if A ∩ Ex− [resp. A ∩ Ex+ ] is compact for any x ∈ . If a set is compact, then it is also both past-compact and future-compact. A p-tensor-distribution T on is a continuous (in an appropriate sense) linear form on the space of regular [say C ∞ (), although this assumption could be relaxed to some C k , depending on the applications] test-p-tensors U with compact support K ⊂ (for complete details see, e.g., [3, 5, 6, 14, 17]). In particular, a test-p-tensor goes smoothly to zero on the border ∂K of its support. If is the domain of a local chart, in this chart a generic tensor distribution on has components which are scalar distributions on [14, 17]. If f is a function integrable in , there is a corresponding distribution f D , defined, on the generic test function ϕ, by the following Riemann 4-volume integral: Z f D (ϕ) =

f ϕ.

(3)

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Tensor distributions corresponding to generic integrable p-tensors are defined similarly. For example, if V is an ordinary 4-vector, locally integrable, there is a corresponding 4-vector distribution V D , defined, on the generic test-4-vector U , by Z D V (U ) = Vα U α . (4)

Tensor distributions which correspond to integrable tensors are called integrable; they are also said to be equivalent to the corresponding tensor. A tensor-distribution which is not integrable is called singular. An important example of singular distribution is the Dirac mensure distribution δ6 associated to a regular hypersurface 6 ⊂ (see, e.g., [14, 15, 16, 17]). Since the space of tensor distributions includes integrable and singular tensor distributions, it is an extension of the space of ordinary integrable tensors. The support of a tensor distribution on is the smallest closed set S in outside which T is identically zero (i.e. it is zero for all test tensors with support outside S). For example, if a tensor distribution is equivalent to a tensor, its support is the support of the tensor. For the singular distribution δ6 , the support is 6. The covariant derivative of a tensor distribution is, in any case, defined by (∇T )(U ) = −T (DivU ),

(5)

where (DivU )α1 ...αp = ∇β U βα1 ...αp . With this definition, the classical properties of the covariant derivative also hold for tensor distributions, including, for example, (1). As for the covariant derivative of the singular distribution δ6 , it is possible to prove (see, e.g., [14, 17]) that there is a singular distribution δ60 , with support on 6, such that ∇α δ6 = `α δ60 , where `α is the gradient vector normal to 6. In the following, we will consider differential operators on distributions and tensor distributions with a past-compact or future-compact support in . In particular, we will often have to handle with divergence-free (4-)vector distributions with past-compact or future compact support. It is therefore useful to consider them here in some detail. Divergence-free vector distributions include integrable and singular ones. An integrable divergence-free vector distribution V D is equivalent to an ordinary regular divergence-free tensor V and has the same support. In fact, if V has a locally integrable covariant derivative, then for any test function ϕ, by definition of the derivative in the sense of distributions, we have Z D D Div(V ) (ϕ) = (DivV ) (ϕ) − Div(ϕV ) (6)

which, by the Green theorem, implies that Div(V )D is null for all test functions ϕ if V is divergence-free and regular, i.e. continuous in . In particular, V must go to zero in a regular way on the border of its support (otherwise, the integral at the right-hand side would produce a term involving the jump of V ). It is therefore

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easy to construct examples of such vector distributions: it suffices, for example, to consider the divergence V α = ∇β F αβ of a skew-symmetric potential F of class C 1 . The support of ∇F , then, is also that of the vector distribution V D be it compact, past-compact, future-compact, or other. The interesting point is that more general skew-symmetric potentials exist for any divergence-free vector distribution, be it integrable or not (see Theorem 1 in the following section). Let us now construct an example of singular divergence-free vector distribution. Consider the product (in the sense of distributions) of the Dirac mensure distribution δ6 associated with a given regular hypersurface 6 ⊂ and of a regular and integrable vector V with support S ⊂ . Such a vector distribution is singular and its support is 6 ∩ S, which thus can be supposed to be compact, past-compact or future-compact, with a suitable choice of S. By definition, for any test vector U , one has (δ6 V )(U ) = δ6 (V α Uα ).

(7)

Since the ordinary rules of derivation of a product also hold for distributions and tensor distributions, we have Div(δ6 V ) = (`α V α )δ60 + (DivV )δ6 .

(8)

Thus, the considered singular vector distribution is divergence-free if the vector V is both divergence-free and tangent to 6. It is not difficult to construct an example of such a vector field. Again, it suffices, for example, to consider V α = ∇β F αβ with F tangent to 6. Therefore, as one would expect, the space of divergence-free vector distributions with past-compact [future-compact] support is larger than that of divergencefree regular vectors with past-compact [future-compact] support. 3. The Laplace Operator in a Curved Spacetime For a tensor T of order p, the (generalized) Laplace operator is defined by [14]: (1T )α1 ...αp = ∇µ ∇ µ Tα1 ...αp +

p X

Rαk µ Tα1 ... µ ...αp +

k=0

X p

+

Rαk ραl σ Tα1 ... ρ ... σ ...αp ,

(9)

k=1, k6=l

where in the second term at the right-hand side, µ is at the kth place, while in the third term, ρ and σ ar at the kth and lth place, respectively. For example, for a scalar u, a vector V and 2-tensor T , we have, respectively, 1u = ∇ρ ∇ ρ u, (1V )α = ∇ρ ∇ ρ Vα + Rα µ Vµ , (1T )αβ = ∇ρ ∇ ρ Tαβ + Rα µ Tµβ + R µ β Tαµ + 2R α µ β ν T µν .

(10)

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The following equivalent definitions for vectors and 2-tensors, in terms of commutators, are sometimes useful: (1V )α = ∇ρ ∇ ρ Vα + (∇α ∇ρ − ∇ρ ∇α )V ρ , (1T )αβ = ∇ρ ∇ ρ Tαβ + (∇α ∇σ − ∇σ ∇α )T σ β + (∇β ∇σ − ∇σ ∇β )Tα σ .

(11)

For example, from (11) it is not difficult to see that 1 commutes with the operations of index contraction and of adjoint of a skew-symmetric 2-tensor [∗(1F ) = 1(∗F )]. We have the following lemma: LEMMA 1. For any tensor distribution v with a past-compact [ future compact] support, there is only one tensor distribution u with a past-compact [ future compact] support which is solution of 1u = v. This lemma holds since 1 is a linear hyperbolic and self-adjoint operator (see [3, 5, 14]). More generally, the same result holds for a generic hyperbolic linear self-adjoint operator L, with coefficients of class C 0 (). We have a dual version of Lemma 1 which holds within the framework of regular C ∞ ordinary functions, in a fixed local chart in the domain . One thus may either work with tensor distributions or with regular functions and tensor fields. In the following, we will state our results within the framework of distributions, letting it be understood that a dual point of view also holds true. A useful theorem is the following, which we are going to prove with the help of Lemma 1. THEOREM 1. Let V , W be two given divergence-free vector distributions (∇α V α = ∇α W α = 0) with past-compact [ future-compact] support. Then there exists a unique antisymmetric tensor distribution Fαβ with past-compact [ future-compact] support such that V β = ∇α F α β ;

Wβ = ∇α (∗F )α β ,

(12)

where (∗F )α β = (F ∗)α β = (1/2)ηα β µν Fµν is the dual of F , which is antisymmetric too. Proof. Recall that for any antisymmetric 2-tensor F αβ , we have the identity ∇α ∇β F αβ = 0 (which follows from direct √ calculation and the use of the identity on Christoffel symbols: 0α β α = ∂β log |g| or, equivalently, from (1) and the symmetry of the Ricci tensor), so the compatibility condition which assures that V and W are divergence-free is automatically satisfied as soon as we find a suitable F solution of (12). Consider ∇α (∗F )αβ ; from (2), we have the identity: ηβ ρσ γ ∇α (∗F )αβ = (1/2)ηρβγ σ ηα β µν ∇α Fµν = ∇ ρ F σ γ + ∇ σ F γρ + ∇ γ F ρσ

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and similarly for ηβ ρσ γ ∇α F αβ , replacing F with (∗F ). Consequently, we have ηβ ρσ γ ∇ρ ∇α (∗F )αβ = ∇ρ ∇ ρ F σ γ + ∇ρ ∇ σ F γρ + ∇ρ ∇ γ F ρσ . Using the Ricci formula for the inversion of iterated covariant derivatives and the antisymmetry of F , we equivalently have ηβ ρσ γ ∇ρ ∇α (∗F )αβ = (1F )σ γ + ∇ σ ∇ρ F γρ + ∇ γ ∇ρ F ρσ and similarly for ηβ ρσ γ ∇ρ ∇α F αβ , replacing F with (∗F ). Now suppose that (12) holds. In this case we have (1F )σ γ = ηνρσ γ ∇ρ Wν + ∇ σ V γ − ∇ γ V σ

(13)

or, equivalently, in terms of (∗F ): [1(∗F )]λδ = ηλδσ γ ∇σ Vγ + ∇ λ W δ − ∇ δ W γ .

(14)

In case (12) holds, the equivalent relations (13) and (14) are identities. As an equation for an unknown F , however, (13) [or, equivalently, (14)] has a unique solution, according to Lemma 1. Let us prove that this solution necessarily also satisfies (12). Let D β = ∇α (∗F )αβ − W β and we have ηβ ρσ γ Dβ = ∇ ρ F σ γ + ∇ γ F σρ + ∇ σ F γρ − ηβ ρσ γ Wβ and from (13) ηβ ρσ γ ∇ρ Dβ = 0. By saturation with ησ γ λµ , we thus have ∇ [λ D µ] = 0. Consequently, D is a solution of the equation (1D)µ − ∇ µ ∇λ D λ = 0 and since D is divergence-free, also of the equation (1D)µ = 0. This equation, according to Lemma 1, has a unique solution: the null one. We conclude that D β = 0 and thus ∇α (∗F )αβ = W β . Similarly from the equivalent equation (14), we can show that we necessarily have ∇α F αβ − V β = 0 and our lemma is proved. 2 The dual result within the framework of C ∞ functions was proved in [13], with the help of spinor analogues of tensor equations. An easy example of an application of Theorem 1 is the existence and uniqueness theorem for the solution of the Maxwell equations (see Section 8). 4. Decomposition of a Vector The classic decomposition of a vector into a gradient plus a divergence-free component has many applications in hydrodynamics and electromagnetism. This result is often called the Clebsch theorem (see, e.g., [7, 10]), and in the following we will also adopt this name. However, we have to remark that, when the result is stated within the framework of global analysis on a Riemannian manifold, the preferred denomination is the Helmholtz decomposition theorem (see, e.g., [4, 27]). Moreover, the name of Clebsh is usually used to name the transformation

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of the velocity field of a hydrodynamical system which permits us to automatically satisfy the Euler equation (see, e.g., [11]), and its extensions to general Hamiltonian systems, which permits us to represent a generic 1-form in terms of the canonical coordinates (see, e.g., [2, 12, 23, 24]). The proof of the classic Clebsch theorem in any case involves the ordinary elliptic Laplace operator. It is not difficult then to extend the classical three-dimensional decomposition formula to the case of the curved spacetime V4 , by means of the generalized Laplace hyperbolic operator. THEOREM 2. Let V and W be two vector-distribution with a past-compact [ future-compact] support and let W be divergence-free. Then there exist a unique distribution φ and a unique antisymmetric tensor distribution F , with past-compact [ future-compact] support, such that V α = ∇α φ + ∇β F β α

(15)

and that Wα = ∇β (∗F )β α .

(16)

Proof. Let φ be the solution of the equation 1φ = ∇α V α . Then the vector Vα − ∇α φ is divergence-free. From Theorem 1, then, there exists an antisymmetric Fαβ such that Wβ = ∇α (∗F )α β and that Vα − ∇α φ = ∇β F β α . Thus (16) and (15) hold and our theorem is proved. 2 As a corollary, we have the ordinary three-dimensional Clebsch theorem, as we are going to show. We first need to split the spacetime and define the three-dimensional space. Let the latin indices run from 1 to 3 and let us consider a local coordinate chart adapted to some given reference frame, i.e. such that g00 = −1, g0i = g 0i = 0. We say that a vector is ‘spatial’ (with respect to the chosen reference) if its components of index 0 are null. Let us introduce the ‘magnetic’ part spatial vector H j = (1/2)η˜ ik j Fik and the ‘electric’ part E j = (1/2)η˜ ik j (∗F )ik of a generic antisymmetric 2-tensor F , where η˜ ij k = η0 ij k is the three-dimensional spatial volume element. We have F ik = η˜ ik j H j , (∗F )ik = η˜ ik j E j , F 0i = E i , (∗F )0i = −H i .

(17)

Now let us consider the hypothesis of Theorem 2. The split version of (15) and (16) is the following: Vi = ∇i φ − η˜ kj i ∇k Hj + ∇0 Ei , Wi = −η˜ kj i ∇k Ej − ∇0 Hi , V 0 = ∇0 φ + ∇k E k , W0 = −∇k H k .

(18)

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203

In other words, we have proved the following corollary: COROLLARY 2.1. In a given local chart adapted to a reference frame, let V be a generic spatial vector distribution with a past-compact [ future-compact] support. Then there are two spatial vector distributions with past-compact [ future-compact] support H and E such that Vi = ∇i φ − (CurlH )i + ∇0 Ei ,

(19)

where (CurlH )i = η˜ i j k ∇j Hk . Here H and E are not unique, unless one introduces the supplementary conditions suggested by (18): Wi = −(CurlE)i − ∇0 Hi ,

(20)

where W is an arbitrarily fixed spatial vector. Moreover, one still can choose arbitrary values v and w to assign to ∇0 φ + ∇k E k and −∇k H k respectively, to play the role which was of V0 and W0 (which are now null) in (18). We see that (19) contains the somewhat unexpected term ∇0 Ei ; we have the ordinary form of the Clebsch Theorem if and only if such a term is null. Corollary 2.1 is in fact a four-dimensional generalization of the ordinary three-dimensional Clebsch theorem and reduces to the more familiar form in the case where the metric is independent of time. COROLLARY 2.2. In the hypothesis of Corollary 2.1, let the metric tensor be independent from x 0 . Then, if V is also independent from x 0 , Equation (19) reduces to the following form: Vi = ∇i φ − (CurlH )i .

(21)

In other words, V can be split as the sum of a gradient plus a divergence-free component. Proof. Consider (19). By hypothesis, we have ∂0 gik = 0 and in our chart we, moreover, have g00 = −1, g0i = 0. For the metric connection, this implies 00α σ = 0 and for the Riemann tensor Rα0ρ σ = 0. In this situation, from (9) one has ∇0 1 = 1∇0 , and, thus, from (13) we have that if V and W are independent from x 0 , the same happens to Ei . Therefore ∇0 Ei = ∂0 Ei = 0 and the corollary is proved. 2 Actually, Corollary 2.1 is also a curved-space generalization of the ordinary three-dimensional Clebsch theorem, which reduces to the more familiar form in the case of flat spacetime: COROLLARY 2.3. In the hypothesis of Corollary 2.1, let the spacetime be flat. Then there is a spatial vector distribution H such that Vi = ∂i φ − (curlH )i , where (curlH )i = i ∇j Hk . jk

(22)

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Proof. To prove it, we have to show that, for any given spatial V , we can find a spatial W to satisfy (19)–(20) with, moreover, E = 0. Let V be the given spatial vector and W a still generic spatial vector such that ∂k W k = 0. Let us denote by a dot the partial derivative with respect to x 0 (time derivative). From (18), we have Vi = ∂i φ − kj i ∂k Hj + E˙ i , Wi = − kj i ∂k Ej − H˙ i , v = φ˙ + ∂k E k , w = −∂k H k = 0,

(23)

where we have introduced the two arbitrary scalars v and w. These two scalars give us some more degrees of freedom to use to obtain Ei = 0. We thus can write V˙i = ∂i φ˙ − kj i ∂k H˙ j + E¨ i ˜ i + ∂i ∂k E k + E¨ i , = ∂i φ˙ + kj i ∂k Wj − 1E

(24)

where we have introduced the three-dimensional ordinary elliptic Laplace operator ˜ = ∂ k ∂k . We then have 1 ˜ i + E¨ i + ∂i v. V˙i = kj i ∂k Wj − 1E

(25)

˜ Since we have 1E = −E¨ + 1E, we obtain the following identity: 1Ei = −V˙i + kj i ∂k Wj + ∂i v.

(26)

Then, if we can choose W and v such that −V˙i + kj i ∂k Wj + ∂i v = 0,

(27)

we necessarily have Ei = 0 as reqiured. If (27) should hold, we would consequently have ∂ [k W j ] = (1/2) kj i (V˙i − ∂i v)

(28)

and, by derivation, ˜ j = kj i ∂k V˙i + ∂ j w. 1W ˙

(29)

Now let us simply choose w = 0 and consider as our auxiliary spatial vector W the solution of the equation above. If we denote by Di the right-hand side of (26), we consequently have ˜ i − ∂i ∂k D k = 0. 1D

(30)

˜ − ∂k V˙ k , it suffices to choose our v as the solution of Now, since ∂k D k = 1v k ˜ ˙ equation 1v = ∂k V to have Di as the solution of the three-dimensional homoge˜ i = 0 and, consequently, Di = 0. Thus, by (26), Ei is neous Laplace equation 1D also null as it is in turn a solution of the four-dimensional homogeneous Laplace equation 1Ei = 0. This completes the proof of our corollary. 2

SECOND-ORDER COVARIANT TENSOR DECOMPOSITION

205

5. Decomposition of a Skew-Symmetric 2-Tensor For a skew-symmetric 2-tensor Fαβ , one can introduce the following covariant decomposition formula where there appear first-order derivatives of vector potentials [9, 13]: Fαβ = ∇α Vβ − ∇β Vα + ηαβ µν ∇µ Wν .

(31)

This again is a generalization of the Clebsch theorem for vectors, since if we denote by Hαβ = 2∇[α Vβ] ,

Kαβ = ηαβ µν ∇µ Wν ,

(32)

the two components of the decomposition of F , then H is irrotational and K is divergence-free in the following sense: ηαβγ δ ∇γ Hαβ = 0,

∇α K βα = 0.

(33)

Decomposition (31) was first introduced in [9] in flat spacetime. This decomposition formula is sometimes called Clebsch representation or Clebsch transformation (see, e.g., [25, 20]). We furthermore remark that, within the framework of global analysis on a Riemannian manifold, the decomposition of differential forms into the sum of exact, co-exact and harmonic components is usuallycalled Hodge decomposition (for an extensive survey, see [26]). The Hodge decomposition generalizes the Helmholtz decomposition and again one has to solve an elliptic boundary problem. This kind of decomposition also has notable applications to the continuum theory of defects, as recently shown in [28]. Let us first show that in a curved spacetime, Equations (33) are identically satisfied as a consequence of definitions (32). For example, we have ∇α K βα = ηα β µν ∇α ∇µ Wν = (∗R)ν βγ ν W γ , where (∗R)ν βγ ν = (1/2)ην β λδ Rλδγ ν = (R∗)γ ν ν β and (∗R)ν βγ ν is null as this is equivalent to R[αβρ]σ = 0. By the same argument, we thus have ηαβγ δ ∇γ Hαβ = 0. Let us then prove (31) within the framework of tensor distributions. THEOREM 3. Let F be a given skew-symmetric 2-tensor distribution with a pastcompact [ future-compact] support and v, w two given distributions with a pastcompact [ future-compact] support. Then there exist two unique vector distributions V and W , with a past-compact [ future-compact] support, such that (31) holds and that, moreover, ∇α V α = v,

∇α W α = w.

(34)

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Proof. If (31) holds, we obtain, by differentiation, the identity ∇α F α β = ∇ρ ∇ ρ Vβ − ∇α ∇β V α = (1V )β − ∇β ∇α V α . Thus, if also (34) holds, we have (1V )β = ∇α F α β + ∇β v.

(35)

Similarly, from (31) and (34), we have the following identity for W : (1W )β = −∇α (∗F )α β + ∇β w.

(36)

In any case, (35) and (36) have, as differential equations for the unknown V and W , a unique solution, according to Lemma 1. Then, let V and W be the corresponding solutions. Let us prove that they necessarily satisfy (31) and (34). Let us introduce the following skew-symmetric tensor: Dαβ = Fαβ − 2∇[α Vβ] + ηαβ µν ∇µ Wν . By (35), we have ∇α (∗D)α β = 0. By (36), we have ∇α D α β = 0 or, in terms of ∗D, ηα β µν ∇α (∗D)µν = 0. Consequently, we have (1/2)ηβρσ γ ηα β µν ∇α (∗D)µν = ∇ ρ (∗D)σ γ + ∇ σ (∗D)γρ + ∇ γ (∗D)ρσ = 0 thus, by differentiation, we have ∇ρ ∇ ρ (∗D)σ γ + ∇ρ ∇ σ (∗D)γρ + ∇ρ ∇ γ (∗D)ρσ = 0 or, equivalently, [1(∗D)]σ γ = 0. This last equation has a unique solution (the null one), according to Lemma 1. We conclude that ∗D = 0 and D = 0 and the theorem is proved. 2 COROLLARY 3.1. In the hypothesis of Theorem 3 if, moreover, F is divergencefree, then there is a vector potential W such that we have Fαβ = ηαβ µν ∇µ Wν .

(37)

Proof. To prove this corollary we have to show that it is possible to choose v or w such that V is necessarily null. Actually, from (31) we have ∇α F αβ = (1V )β − ∇β ∇α V α = (1V )β − ∇β v = 0. It therefore suffices to choose v = 0 to have V = 0, as required.

2

An example of the application of Theorem 3 is given by the definition of 4vector potential and of the Lorentz gauge in electromagnetism (see Section 8). Note that from the splitting of (31) we have Fi0 = Ei = ∇i V0 − ∇0 Vi + (CurlW )i , (∗F )i0 = Hi = −∇i W0 + ∇0 Wi + (CurlV )i ,

(38)

SECOND-ORDER COVARIANT TENSOR DECOMPOSITION

207

which permits us identify the components of the 4-vector potentials of F with the scalar and 3-vector potentials of the electric and magnetic parts E i and H i according to the generalized Clebsch Theorem (see Corollary 2.1). Decomposition formula (31) is first-order, since it involves first covariant derivatives of the vector potentials only. However, Theorem 2 allows us to again introduce potentials for the vector potentials we have just met. We thus can prove a second-order decomposition formula, in terms of second derivatives of skewsymmetric tensor potentials. COROLLARY 3.2. Let F be a given skew-symmetric 2-tensor-distribution and let V˜ and W˜ be two given divergence-free vector distributions with a past-compact [ future-compact] support. Then there exist two unique skew-symmetric tensor distributions M and N, with a past-compact [ future-compact] support, such that Fαβ = ∇α ∇µ M µ β − ∇β ∇µ M µ α + ηαβ µν ∇µ ∇σ N σ ν

(39)

and that V˜β = ∇α (∗M)α β ,

W˜ β = ∇α (∗N)α β .

(40)

Proof. For any choice of v, w, consider the two vector potentials V and W of Theorem 2. Now let us introduce their decomposition according to Theorem 1 and associated to the given V˜ and W˜ so as to additionally satisfy (40), and let us denote their skew-symmetric potentials M and N, respectively: V α = ∇α φ + ∇µ M µ α ,

Wα = ∇α ψ + ∇µ N µ α ,

(41)

where v = ∇α V α = 1φ,

w = ∇α W α = 1ψ.

By substituting (41) into (31), the gradients disappear, independently of the choice of v and w, and we obtain (39), as required. 2 We also have the analogue of Corollary 3.1 when F is a divergence-free field. COROLLARY 3.3. In the hypothesis of Theorem 3, if F is divergence-free, then there is a skew-symmetric potential N such that Fαβ = ηαβ µν ∇µ ∇σ N σ ν .

(42)

Proof. To prove this corollary, we can make use of (39) and try to show that we necessarily have ∇µ M µ α = 0. Actually we find ∇α F α β = 1(∇µ M µ β ) = 0 and thus the result is proved. 2 Further substitution of decomposition (31) into (15) does not lead to any new interesting second-order decomposition. It simply turns out to lead to the statement

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that any vector is equal to the Laplace operator of some other vector, which is trivial as a consequence of Lemma 1. 6. Decomposition of a Symmetric 2-Tensor We are now going to introduce a covariant four-dimensional first-order decomposition of a symmetric 2-tensor distribution T . Such a decomposition generalizes the usual transverse decomposition on a spacelike hypersurface [8, 29]. Here we are going to make use of a different operator than the generalized Laplace operator. Also, for the transverse decomposition on a spacelike hypersurface one has to introduce a formally similar operator different than the ordinary three-dimensional elliptic Laplace operator. The difference is that the operator we are going to use here is hyperbolic like the generalized Laplace operator, while that of the threedimensional case is elliptic like the ordinary one. THEOREM 4. Let Tαβ be a generic symmetric 2-tensor distribution with pastcompact [ future-compact] support. Then there exist a unique vector-distribution Tα and a unique symmetric divergence-free 2-tensor distribution Kαβ , with a pastcompact [ future-compact] support, such that the following decomposition formula holds: Tαβ = ∇α Tβ + ∇β Tα + Kαβ .

(43)

Proof. For (43) to hold, Tβ must be a solution of the following differential equation: ∇ν ∇ ν Tβ + ∇ν ∇β T ν = ∇ν T ν β .

(44)

If such a solution is found, then it suffices to define Kαβ = Tαβ − 2∇(α Tβ) to have the thesis of our theorem. Actually, Equation (44) admits a unique solution within the framework of tensor distributions with a past-compact [future-compact] support (or in that of C ∞ functions), since the differential operator gαβ ∇ν ∇ ν + ∇α ∇β there involved is hyperbolic and self-adjoint. 2 This decomposition is a generalization of the transverse decomposition on a spacelike hypersurface [8, 29]. If T ik is a generic symmetric spatial 2-tensor distribution, then there is, in fact, a unique spatial vector distribution T i and a unique symmetric spatial and divergence-free 2-tensor distribution Kik such that the following decomposition holds: T ik = ∇ i T k + ∇ k T i + K ik ,

∇k K ik = 0,

(45)

and that the following equation is satisfied: ∇k T ik = ∇k ∇ k T i + ∇k ∇ i T k .

(46)

SECOND-ORDER COVARIANT TENSOR DECOMPOSITION

209

T i is well-defined as the unique solution of Equation (46), since the differential operator involved there is elliptic and self-adjoint [8, 29]. If the domain of T ik is a single ordinary three-dimensional Riemann manifold, then decomposition (46) coincides with the ordinary transverse decomposition considered in [8, 29]. Let us now look for the relationship between our four-dimensional decomposition (43) and the ordinary three-dimensional one. Consider coordinates adapted to a reference frame, like in the hypothesis of Corollary 2.1. The splitting of decomposition (43) is the following: T 00 = 2∇ 0 T 0 + K 00 , T 0i = ∇ 0 T i + ∇ i T 0 + K 0i , T ik = ∇ i T k + ∇ k T i + K ik ,

(47)

while that of system (44) is the following: ∇0 T 00 + ∇k T 0k = 2∇0 ∇ 0 T 0 + ∇k (∇ k T 0 + ∇ 0 T k ), ∇0 T 0i + ∇k T ik = ∇k ∇ k T i + ∇k ∇ i T k + ∇0 (∇ 0 T i + ∇ i T 0 ).

(48)

We have just proved the following corollary. COROLLARY 4.1. In the hypothesis of Theorem 4, in a given local chart adapted to a reference frame, the spatial 2-tensor distribution T ik admits the ordinary transverse decomposition (45), where the spatial vector T i is the solution of (46), if and only if we have ∇0 K i0 = 0. The proof of this corollary easily follows from substitution of (47) into (48). Even more trivially, if we have ∇0 K i0 = 0, then we necessarily also have ∇k K ik = 0 (since K is divergence-free), thus both (45) and (46) are automatically satisfied. We then have that the ordinary transverse decomposition on a three-dimensional manifold 6 directly follows from (43) when the metric is independent of time. COROLLARY 4.2. In a given local chart adapted to a reference frame, let the metric tensor be independent of x 0 . Then, if T is a symmetric 2-tensor distribution which is also independent of x 0 , then the spatial 2-tensor distribution T ik admits the ordinary transverse decomposition (45), where the spatial vector T i is the solution of (46). The proof follows from Corollary 4.1, by the same argument as that contained in the proof of Corollary 2.2. In this case we have ∇0 K 0i = ∂0 K 0i = 0. Let us now look for second-order potentials of a symmetric 2-tensor. First of all, the potential vector T appearing in decomposition (43) can in turn be decomposed according to Theorem 2: Tα = (1/2)∇α φ + ∇µ F µ α

(49)

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so to obtain the following second-order decomposition formula: Tαβ = ∇α ∇β φ + ∇α ∇µ F µ β + ∇β ∇µ F µ α + Kαβ .

(50)

This proves the following corollary: COROLLARY 4.3. In the hypothesis of Theorem 4, if W is a given divergencefree vector distribution with a past-compact [ future-compact] support, then there exist a unique distribution φ, a unique antisymmetric 2-tensor distribution F and a unique symmetric divergence-free 2-tensor distribution Kαβ , with a past-compact [ future-compact] support, such that Wα = ∇β (∗F )β α and that (50) holds. The splitting of (50) in turn gives the following decomposition for the spatial component of T in terms of second derivatives of the electric and magnetic parts of the second-order potential F : T ik = ∇ i ∇ k φ + ∇ i ∇0 E k + ∇ k ∇0 E i − − ∇ i (CurlH )k − ∇ k (CurlH )i + K ik ,

(51)

while, for the remaining components, we have T 00 = (∇ 0 )2 φ − 2∇ 0 ∇k E k , T 0i = ∇ 0 ∇ i φ − (∇ 0 )2 E i − ∇ 0 (CurlH )i − ∇ i ∇k E k + K 0i .

(52)

The second-order decomposition (51) can also be obtained by replacing T i in (45) with its decomposition (19) according to the generalized Clebsch theorem (see Corollary 2.1). Obviously, in the hypothesis of Corollaries 2.2 and 4.2, the term ∇ i ∇0 E k + ∇ k ∇0 E i disappears. In decomposition (43), the form of the transverse component K is generally unknown; thus the relative second-order decomposition (50) is not complete, like in the preceeding section cases. We can, however, prescribe the form of K, at least in the particular case of a flat spacetime, as we are going to show in the next section. In this case, in fact, we can introduce a double 2-form H as a second-order potential for K. 7. Decomposition of Symmetric 2-Tensors and Double 2-Forms in Flat Spacetime We recall that a 4-tensor H is called a double 2-form (or a symmetric double bivector) if it has the same algebraic properties of a curvature tensor, i.e. Hαβρσ = H[αβ]ρσ = Hαβ[ρσ ] = Hρσ αβ ,

H[αβρ]σ = 0.

(53)

THEOREM 5. Let the spacetime be flat, let K be a given divergence-free symmetric 2-tensor distribution with a past-compact [ future-compact] support. Then there

SECOND-ORDER COVARIANT TENSOR DECOMPOSITION

211

is a unique double 2-form distribution H with a past-compact [ future-compact] support such that ∂[α Hβρ]σ ν = 0 and that Kαβ = ∂µ ∂ν H µ α ν β .

(54)

Proof. First, let us note that ∂[α Hβρ]σ ν = 0 can be equivalently written as ∂µ (∗H ∗)µ α ν β = 0, where (∗H ∗)µ α ν β = (1/4) µ α λδ ν β ρσ Hλδρσ . Now let us suppose that (54) holds; we equivalently have Kαβ = (1/4) µ α λγ ν β ρσ ∂µ ∂ν (∗H ∗)λγρσ .

(55)

We thus obtain φαψ γ β λµ Kαβ = ∂[γ ∂ [φ (∗H ∗)ψ] µλ]

(56)

and, consequently, φαψ γ βλµ∂ ∂µ Kαβ = [12 (∗H ∗)]ψφλγ + +1[∂ φ ∂ (∗H ∗)φλγ + ∂ γ ∂µ (∗H ∗)ψφµλ ] + +1[∂ λ ∂µ (∗H ∗)ψφγ µ + ∂ ψ ∂ (∗H ∗)φλγ ].

(57)

Then, if we also suppose that ∂µ (∗H ∗)µ α ν β = 0, we simply have the following identity: [12 (∗H ∗)]ψφλγ = φαψ γ βλµ∂ ∂µ Kαβ .

(58)

As a differential equation for the unknown ∗H ∗, however, Equation (58) certainly admits a unique solution as a consequence of Lemma 1. The equivalent dual equation for H is (12 H )ψφλγ = ∂ γ ∂ φ K ψλ + ∂ λ ∂ ψ K φγ − ∂ λ ∂ φ K ψγ − ∂ γ ∂ ψ K φλ .

(59)

Let then H ψφλγ be the solution of (59) or, equivalently, the dual of the solution of (58). It is easy to check that H ψφλγ necessarily has all the required algebraic and differential properties. Now let Dαβ = Kαβ − ∂µ ∂ν H µ α ν β . By definition, we have ∂α D αβ = 0 and D = Dα α = K + ∂µ ∂ν H µν , where H µν = Hα µν α and K = Kαα . Moreover, from (59) we have (12 H )φλ + ∂ λ ∂ φ K = −1K φλ .

(60)

Consequently, since Kαβ is divergence-free, 12 (∂φ ∂λ H φλ + K) = 0

(61)

and we can conclude that D = 0. Furthermore, Dαβ is by definition the solution of φαψ γ βλµ∂ ∂µ Dαβ = 0 or, equivalently, ∂ γ ∂ φ D ψλ + ∂ λ ∂ ψ D φγ − ∂ λ ∂ φ D ψγ − ∂ γ ∂ ψ D φλ = 0.

(62)

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From (62) we then have 1D ψλ = ∂ λ ∂ ψ D + ∂ λ ∂γ D ψγ − ∂ ψ ∂γ D γ λ = 0. We conclude that Dαβ = 0 and our theorem is proved.

(63) 2

Moreover, we have the following corollary, whose proof easily follows from (60): COROLLARY 5.1. In the hypothesis of Theorem 5, if K αβ is trace-free, then we have (1H )αβ = −K αβ and H = 0. It is, in general, not possible to directly extend such a result to the case of curved spacetime, since in such a case one looses the possibility of rendering explicit the property of K being divergence-free by introducing the double 2-form potential H : one in fact has that ∇α ∇µ H αµβν is not necessarily null. Decompositions (54) and (50) lead to the following formula: Tαβ = ∂α ∂β φ + ∂α ∂µ F µ β + ∂β ∂µ F µ α + ∂µ ∂ν H µ α ν β .

(64)

We have just proved the following corollary. COROLLARY 5.2. In the hypothesis of Theorem 4, if the spacetime is flat and W is a given divergence-free vector distribution with a past-compact [ future-compact] support, then there exist a unique distribution φ, a unique antisymmetric 2-tensor distribution F and a unique double 2-form distribution Hαβρσ , with a past-compact [ future-compact] support, such that Wα = ∂β (∗F )β α and that (64) holds. From the splitting of (54) we have, for the spatial components of K, K ik = ∂m ∂n H mink + ∂n (H˙ 0ink + H˙ ni0k ) + H¨ 0i0k .

(65)

From (59) we have that in the particular case where K αβ is independent of x 0 , the same applies for the double 2-form potential. We thus have the following corollaries: COROLLARY 5.3. In a given local chart adapted to a reference frame in a flat spacetime, let K be a divergence-free symmetric 2-tensor distribution, independent of x 0 . Then the spatial 2-tensor distribution K ik is such that ∂i K ik = 0 and admits the following decomposition: K ik = ∂m ∂n H mink .

(66)

COROLLARY 5.4. In a given local chart adapted to a reference frame in a flat spacetime, let T be a generic symmetric 2-tensor distribution, independent of x 0 . Then the spatial 2-tensor distribution T ik admits the following decomposition: T ik = ∂ i T k + ∂ k T i + ∂m ∂n H mink .

(67)

SECOND-ORDER COVARIANT TENSOR DECOMPOSITION

213

Apparently there seem to be no obvious ways to directly prove formulae (66) and (67) in three dimensions: one cannot introduce the adjoint of a double 2-form by means of the three-dimensional Levi–Civita indicator ij k only. We can also prove the converse of Theorem 5. THEOREM 6. Let the spacetime be flat, let H be a given double 2-form distribution with past-compact [ future-compact] support such that ∂[α Hβρ]σ ν = 0. Then there is a unique divergence-free symmetric 2-tensor distribution with a pastcompact [ future-compact] support such that H ψφλγ = ∂ γ ∂ φ K ψλ + ∂ λ ∂ ψ K φγ − ∂ λ ∂ φ K ψγ − ∂ γ ∂ ψ K φλ .

(68)

Proof. Let H ψγ = Hα ψγ α and H = Hα α . Suppose (68) holds, then by contraction, one has 1K ψγ = −H ψγ − ∂ γ ∂ ψ K,

(69)

where K = Kα α . Consequently, 1K = −(1/2)H.

(70)

Thus, (69) and (70) actually are necessary conditions for (68) to hold. However, we can solve (70) with respect to the unknown scalar distribution K and subsequently (69) with respect to the unknown tensor distribution K ψγ . The results are compatible in the sense that K is the trace of the resulting K ψγ . Moreover, from (69) and (70) one has 1∂ψ K ψγ = −∂ψ H ψγ + (1/2)∂ γ H,

(71)

which is null by contraction of ∂[α Hβρ]νσ = 0. Thus K ψγ is also divergence-free. We now have to show that (69) and (70) are also sufficient. To see it, simply let D ψφλγ = H ψφλγ − ∂ γ ∂ φ K ψλ − ∂ λ ∂ ψ K φγ + ∂ λ ∂ φ K ψγ + ∂ γ ∂ ψ K φλ .

(72)

From (69) we therefore have 1D ψφλγ = 1H ψφλγ + ∂ γ ∂ φ H ψλ + ∂ λ ∂ ψ H φγ − ∂ λ ∂ φ H ψγ − ∂ γ ∂ ψ H φλ = gαβ (∂ α ∂ β H ψφλγ + ∂ γ ∂ φ H αψλβ − ∂ γ ∂ ψ H αφλβ + + ∂ λ ∂ ψ H αφγ β − ∂ λ ∂ φ H αψγ β ) (73) and, consequently, from ∂[α Hβρ]νσ = 0, we find 1D ψφλγ = gαβ ∂ α ∂ [β H λγ ]ψφ = 0, i.e. D = 0 and our theorem is proved.

(74) 2

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From (70), we also immediately have the following corollary: COROLLARY 6.1. In the hypothesis of Theorem 6, if the double-trace H of Hαβρσ is null, then the potential Kαβ is trace-free and (1K)αβ = −H αβ . Finally, if the double 2-form distribution H is independent of x 0 , the same happens to its 2-tensor-distribution potential K, which leads to the following corollary. COROLLARY 6.2. In the hypothesis of Theorem 6, if Hαβρσ is independent of x 0 , one has H ij kn = ∂ n ∂ j K ik + ∂ k ∂ i K j n − ∂ k ∂ j K in − ∂ n ∂ i K j k ,

(75)

where ∂n K in = 0. 8. An Application to Electromagnetism As an example of the application of Theorem 1, we have the existence and uniqueness theorem for weak solutions of the Maxwell equations when the electric density current vector is replaced by a generic vector-distribution. Such vector distribution could also be singular to describe, for example, regularly discontinuous currents or concentrated charges. COROLLARY 1.1. Let the electric density current 4-vector distribution J α (with ∇α J α = 0) have past-compact [ future-compact] support. Then there is a unique electromagnetic field tensor distribution with a past-compact [ future-compact] support Fαβ such that ∇α F αβ = J β ,

∇α (∗F )αβ = 0.

(76)

The proof directly follows from Theorem 1 provided we set V α = J α and W α = 0. Moreover, as an example of the application of Theorem 3, we have the following definition of the 4-vector potential: COROLLARY 3.4. Let Fαβ be a solution of (76), then, for any couple of scalardistributions v and w, there are two vector-distributions V and W such that (31) holds, V satisfies the following equation: (1V )β = Jβ + ∇β v

(77)

and W is determined by (1W )β = ∇β w.

(78)

In other words, the Maxwell equations (76) are equivalent to (77) for any choice of v, where V is the 4-vector distribution potential of the electromagnetic field F .

SECOND-ORDER COVARIANT TENSOR DECOMPOSITION

215

The expression (31) of F actually also involves the inessential vector W , which is determined by (78), i.e. by the choice of w, but which does not appear in (77). It suffices to choose w = 0 to have W = 0 and, from (31), Fαβ = ∇α Vβ − ∇β Vα .

(79)

The freedom of choice of the scalar v is instead the gauge freedom of electromagnetism. The choice v = 0 defines what we can call the Lorentz gauge. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Bampi, F. and Caviglia, G.: Third-order tensor potentials for the Riemann and Weyl tensors, Gen. Relativity Gravitation 15 (1983), 375. Baumeister, R.: Clebsh representation and variational principles in the theory of relativistic dynamical systems, Utilitas Math. 16 (1979), 43. Bruhat, Y.: The Cauchy problem, In: L. Witten (ed.), Gravitation, An Introduction to Current Research, Wiley, New York, 1962, p. 130. Cantor, M.: Boundary value problems for asymptotically homogeneous elliptic second order operators, J. Differential Equations 34 (1979), 102. Choquet-Bruhat, Y.: Hyperbolic partial differential equations on a manifold, In: C. M. DeWitt and J. A. Wheeler (eds), Battelle Rencontres, Benjamin, New York, 1968, p. 84. Choquet-Bruhat, Y., DeWitt-Morette, C. and Dillard-Bleik, M.: Analysis, Manifolds and Physics. Part I: Basics, North-Holland, Amsterdam, 1977. Crupi, G.: Considerazioni sul teorema di Clebsch e sul lemma di Finzi, Istituto Lombardo Rend. Sci. A 100 (1966), 951. Deser, S.: Covariant decomposition of symmetric tensors and the gravitational Cauchy problem, Ann. Inst. H. Poincaré 7 (1967), 149. Finzi, B.: Sul principio della minima azione e sulle equazioni elettromagnetiche che ne derivano, Rend. Sc. fis. mat. nat. Lincei 12 (1952), 378. Finzi, B. and Pastori, M.: Calcolo tensoriale e applicazioni, Zanichelli, Bologna, 1961. Gaffet, B.: On generalized vorticity-conservation laws, J. Fluid Mech. 156 (1985), 141. Goncharov, V. and Pavlov, V.: Some remarks on the physycal foundation of the Hamiltonian description of fluid motions, European J. Mech. B Fluids 16 (1997), 509. Illge, R.: On potentials for several classes of spinor and tensor fields in curved spacetimes, Gen. Relativity Gravitation 20 (1988), 551. Lichnerowicz, A.: Propagateurs et commutateurs en relativité générale, Publications Mathématiques 10, Institut des Hautes Études Scientifiques, Paris, 1961. Lichnerowicz, A.: Théorie des rayons en hydrodynamique et magnétohydrodinamique relativiste, Ann. Inst. H. Poincaré 7 (1967), 271. Lichnerowicz, A.: Ondes de choc et hypothéses de compressibilité en magnétohydrodynamique relativiste, Comm. Math. Phys. 12 (1969), 145. Lichnerowicz, A.: Magnetohydrodynamics: Waves and Shock Waves in Curved Space-Time, Math. Phys. Stud. 14, Kluwer Acad. Publ., Dordrecht, 1994. Marchioro, C. and Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids, Springer-Verlag, New York, 1994. Massa, E. and Pagani, E.: Is the Riemann tensor derivable from a tensor potential?, Gen. Relativity Gravitation 16 (1984), 805. Maugin, G.: Sur la transformation de Clebsch et la magnétohydrodynamique relativiste, C.R. Acad. Sci. Paris Sér. A-B 274 (1972), A602. Monroe, D. K.: Local transverse-traceless tensor operators in general relativity, J. Math. Phys. 9 (1981), 1994.

216 22. 23. 24. 25. 26. 27. 28. 29.

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Persico, E.: Introduzione alla fisica matematica, Zanichelli, Bologna, 1952. Rund, H.: Clebsch potentials and variational principles in the theory of dynamical systems, Arch. Ration. Mech. Anal. 65 (1977), 305. Rund, H.: Clebsch representations and relativistic dynamical systems, Arch. Ration. Mech. Anal. 71 (1979), 199. Rund, H.: Clebsch potentials in the theory of electromagnetic fields admitting electric and magnetic charge distributions, J. Math. Phys. 18 (1977), 84. Scwarz, G.: Hodge Decomposition – A Method for Solving Boundary Value Problems, Lecture Notes in Math. 1607, Springer, Berlin, 1995. Specovius-Neugebauer, M.: The Helmholtz decomposition of weighted Lr -spaces, Comm. Partial Differential Equations 15 (1990), 273. Wenzelburger, J.: A kinematical model for continuous distributions of dislocations, J. Geom. Phys. 24 (1998), 334. York, J. W., Jr.: Covariant decomposition of symmetric tensors in the theory of gravitation, Ann. Inst. H. Poincaré 21 (1974), 319.

Mathematical Physics, Analysis and Geometry 3: 217–285, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

217

Asymptotic Completeness for a Renormalized Nonrelativistic Hamiltonian in Quantum Field Theory: The Nelson Model ZIED AMMARI Centre de Mathématiques, UMR 7640 CNRS, École Polytechnique, 91128 Palaiseau Cedex, France. e-mail: [email protected] (Received: 10 January 2000; in final form: 11 July 2000) Abstract. Scattering theory for the Nelson model is studied. We show Rosen estimates and we prove the existence of a ground state for the Nelson Hamiltonian. Also we prove that it has a locally finite pure point spectrum outside its thresholds. We study the asymptotic fields and the existence of the wave operators. Finally we show asymptotic completeness for the Nelson Hamiltonian. Mathematics Subject Classifications (2000): 81U10, 81T10. Key words: quantum field theory, Mourre theory, scattering theory, asymptotic completeness.

Table of Contents 1 2

3

4

5

6

7 8

Introduction Presentation of the Model 2.1 Basic definitions and notations 2.2 Technical estimates 2.3 The Nelson model Construction of the Nelson Hamiltonian 3.1 Dressing transformation 3.2 Removal of the ultraviolet cutoff Higher Order Estimates 4.1 Rosen estimates 4.2 Number-energy estimates 4.3 Commutator estimates Spectral Theory for the Nelson Hamiltonian 5.1 HVZ theorem 5.2 Mourre estimate Construction of the Wave Operators 6.1 Asymptotic fields 6.2 Wave operators Propagation Estimates Asymptotic Completeness Appendix

218 219 219 222 227 228 229 231 234 235 245 246 250 251 252 260 260 266 268 275 281

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1. Introduction Recently there has been a renewed interest in quantum field theory models that describe a system of nonrelativistic particles interacting with a bosonic field. The main physical example is a nonrelativistic atom interacting with photons. For this model the existence of a ground state was established in [BFS], [AH]. The absence of excited states was also shown in [BFS] using a renormalization group analysis and in [BFSS], [DJ] using a positive commutator method. Another result related to the present work is [DG2] where for a model of a confined atom interacting with massive bosons the asymptotic completeness of wave operators was proved. In all these works, the model contains an ultraviolet cutoff which switches off the interaction between the nucleons and the bosonic field above a certain momentum scale. This can be justified physically by the fact that nucleons interacting with bosons of very high energy will become relativistic and in such a situation the model will anyhow loose its validity. With a cutoff these models are free of ultraviolet divergences and, hence, can be easily constructed rigorously by rather elementary methods. However the presence of a cutoff implies that the interaction term is now non local and that quantitative results depend on the choice of the cutoff scale. Therefore it would be more satisfactory to remove the ultraviolet cutoff from the model under consideration. When the interaction term is linear in the field variables, the removal of the ultraviolet cutoff was done long time ago by Nelson [Ne]. This was probably the first model which was rigorously constructed using a renormalization procedure. It consists in considering cutoff Hamiltonians Hκ , where κ is some ultraviolet cutoff parameter and applying a cutoff-dependent unitary transformation Uκ . After substracting a divergent self-energy term Eκ , the sequence of Hamiltonians Uκ (Hκ − Eκ )Uκ∗ converges in norm resolvent sense to a Hamiltonian Hˆ ∞ when κ → ∞ while Uκ converges strongly to a unitary transformation U∞ (in other words, no ∗ ˆ change of representation is necessary). The Hamiltonian H := U∞ H∞ U∞ is called the Nelson Hamiltonian. After Nelson’s paper, the Nelson model was studied by Cannon [Ca] and Fröhlich [Fr]. In this paper we consider the Nelson model for a confined atom and massive bosons and study its spectral and scattering theory. Our main result is the asymptotic completeness of the wave operators, which implies the unitarity of the S matrix. The strategy and the proofs of our paper follow closely those of [DG2], which is devoted to a similar model with an ultraviolet cutoff. Nevertheless there are new difficulties coming from the fact that the Nelson Hamiltonian is only defined as the resolvent limit of the cutoff Hamiltonians. Let us now describe the content of the paper. In Section 2 we recall classical notations related to Fock spaces, introduce some definitions and prove some extensions of Glimm–Jaffe’s Nτ estimates.

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219

In Section 3 we recall the construction of the Nelson Hamiltonian following [Ne]. In Section 4 we prove the so-called higher order estimates, following an idea of Rosen [Ro]. In Section 5 we study the spectral theory for the confined Nelson Hamiltonian. We prove an HVZ theorem and a positive commutator estimate. Section 6 is devoted to the scattering theory for the Nelson model. We show the existence of asymptotic fields and that the CCR representation they define is of Fock type, using an argument from [DG3]. In Section 7 we prove various propagation estimates for the Nelson Hamiltonian. Finally the asymptotic completeness of the wave operators is shown in Section 8. 2. Presentation of the Model In this section we define the Nelson model. We start with a review of the basic construction and the main notations related to bosonic Fock spaces. For a more detailed exposition we refer the reader to [Be], [BR], [BSZ]. In Subsection 2.2 we give some technical estimates obtained by adaptation of Nτ -estimates [GJ]. Finally in Subsection 2.3 we introduce the formal Hamiltonian of the interacting system of P confined nonrelativistic particles (nucleons) with a relativistic scalar field (mesons). In order to give sense to the formal Hamiltonian, we put a highmomentum cutoff in the interaction and we show that the cutoff Hamiltonian is a well defined selfadjoint operator. 2.1.

BASIC DEFINITIONS AND NOTATIONS

Let h be a complex Hilbert space. Let ⊗ns h denote the symmetric n-fold tensor L power of h. We introduce the bosonic Fock space by 0(h) := n>0 ⊗ns h. ⊗0s h := C, identified as subspace of 0(h), represents the space of zero-particle states. We denote by the vector (1, 0, . . .) usually called vacuum vector and by 0fin (h) the subspace of finite particle states, which is the subspace of finite sum of vectors in ⊗ns h. Among the main operators acting on 0(h), we will first recall the definitions of the most familiar as number operator N given in its spectral decomposition N|⊗ns h := n1. Creation operators, which are unbounded operators densely defined 1 on D(N 2 ), are given by p a ∗ (h)|⊗ns h := (n + 1)Sn+1 h ⊗ 1⊗ns h , where Sn denotes the orthogonal projection from ⊗n h into ⊗ns h. The annihilation operator a(h) is the adjoint of a ∗ (h). We will use the notation a ] for a or a ∗ . We define the field operator by 1 φ(h) := √ (a ∗ (h) + a(h)), h ∈ h. 2 φ(h) is essentially selfadjoint on 0fin (h). We still denote by φ(h) its closure. By functional calculus we get unitary operators called Weyl operators, defined as W (h) := eiφ(h) . We recall a useful differentiation estimate for W (h):

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Let 0 6 6 1, k(W (h1 ) − W (h2 ))uk 6 C kh1 − h2 k ((kh1 k2 + kh2 k2 ) 2 kuk + k(N + 1) 2 uk),

(2.1)

lim sup s −1 k(W (sh) − 1 − isφ(h))(N + 1)− 2 − k = 0,

(2.2)

1

s→0 khk6c

> 0.

Let h be a Hilbert space. Let f : h → h be an (unbounded) operator. We denote by d0(f) the amplification of f to the whole space 0(h) d0(f)|⊗ns h :=

n X

1⊗(j −1) ⊗ f ⊗ 1⊗(n−j ) .

j =1

Let hi , i = 1, 2 be two Hilbert spaces. Let g : h1 → h2 be a bounded operator. We define the operator 0(g) by 0(g) : h1 → h2 , 0(g)|⊗ns h1 := g⊗(n) . A less familiar operator is d0(f, g), where f, g are two operators on h1 into h2 . It is defined as in [DG2]: d0(f, g) : 0(h1 ) → 0(h2 ), n X d0(f, g)|⊗ns h1 := f⊗(j −1) ⊗ g ⊗ f⊗(n−j ) . j =1

We notice that d0(f, f) = N0(f) and if h1 = h2 , we have d0(1, g) = d0(g). If kfk 6 1, the following inequality holds kN − 2 d0(f, g)uk 6 kd0(g∗ g) 2 uk. 1

1

(2.3)

Let i1 (resp. i2 ) be the injection of h1 (resp. h2 ) into h1 ⊕ h2 . There exists a unitary transformation U identifying 0(h1 ) ⊗ 0(h2 ) with 0(h1 ⊕ h2 ), defined as follows s (p + q)! U u ⊗ v := Sp+q 0(i1 )u ⊗ 0(i2 )v, u ∈ ⊗ps h1 , v ∈ ⊗qs h2 . p!q! This transformation has the following properties: (i) U ⊗ = . (ii) Let h1 ∈ h1 , h2 ∈ h2 a ] (h1 ⊕ h2 )U = U (a ] (h1 ) ⊗ 1 + 1 ⊗ a ] (h2 )), φ(h1 ⊕ h2 )U = U (φ(h1 ) ⊗ 1 + 1 ⊗ φ(h2 )). (iii) Let fi : hi → hi , i = 1, 2 be two operators

221

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d0(f1 ⊕ f2 )U = U (d0(f1 ) ⊗ 1 + 1 ⊗ d0(f2 )), U 0(f1 ) ⊗ 0(f2 ) = 0(f1 ⊕ f2 ). We define the scattering identification operator I I : 0fin (h) ⊗ 0fin (h) → 0fin (h), s (p + q)! I u ⊗ v := Sp+q u ⊗ v, p!q!

u ∈ ⊗ps h, v ∈ ⊗qs h.

We can also define I by the following formula: I

p Y

∗

a (hi ) ⊗

i=1

q Y

∗

a (gi ) :=

i=1

q Y

∗

a (gi )

i=1

p Y

a ∗ (hi ),

hi , gi ∈ h.

i=1

Let π be the following map π : h ⊕ h → h, (h0 , h∞ ) → h0 + h∞ . Then we can √ express I as following I = 0(π )U . We notice that I is unbounded since kπ k = 2. Let i = (i0 , i∞ ) be a pair of maps from h to h. We define I (i) : 0fin (h) ⊗ 0fin (h) → 0fin (h), I (i) := I 0(i0 ) ⊗ 0(i∞ ). Let i = (i0 , i∞ ), j = (j0 , j∞ ) be two pairs of maps from h to h. We define dI (i, j ) : 0fin (h) ⊗ 0fin (h) → 0fin (h), dI (i, j ) := I (d0(i0 , j0 ) ⊗ 0(i∞ ) + 0(i0 ) ⊗ d0(i0 , j∞ )). ∗ If i0 i0∗ + i∞ i∞ 6 1 we have the estimates 1

1

∗ 2 k(N0 + N∞ )− 2 dI ∗ (i, j )uk 6 kd0(j0 j0∗ + j∞ j∞ ) uk,

(2.4)

|(u2 |dI ∗ (i, j )u1 )| 1

1

6 kd0(|j0 |) 2 ⊗ 1 u2 k kd0(|j0 |) 2 u1 k + 1 2

(2.5)

1 2

+ k1 ⊗ d0(|j∞ |) u2 k kd0(|j∞ |) u1 k. For other properties and equivalent definitions of these operators, we refer the reader to [DG3].

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Let K be an auxiliary Hilbert space. Let v ∈ B(K, K ⊗ h). We define an extended creation operator: a ∗ (v) : K ⊗ 0fin (h) → K ⊗ 0fin (h), p a ∗ (v)|K⊗⊗ns h := (n + 1) (1K ⊗ Sn+1 )v ⊗ 1⊗ns h . a ∗ (v) is closable densely defined operator since its adjoint a(v) is densely defined. We define the field operator φ(v) as in the scalar case. When h = L2 (Rd , dk), v ∈ B(K, K ⊗ h) can be represented as a function k → v(k) ∈ B(K), such that for x ∈ K, v(k)x := vx(k), k-a.e. and Z K × K 3 (x, y) → (v(k)∗ v(k)x| y)K dk = (vx|vy)K⊗h is a continuous quadratic form. A stronger condition is v ∈ L2 (Rd , B(K)), i.e.: Z kv(k)k2B(K) dk < ∞. Assume that [v1∗ (k), v2 (k 0 )] = 0, ∀k, k 0 . Then: [a(v1 ), a ∗ (v2 )] = v1∗ v2 ⊗ 10(h), [φ(v1 ), φ(v2 )] = iIm(v1∗ v2 ) ⊗ 10(h), [φ(v1 ), W (v2 )] = Im(v1∗ v2 ) ⊗ W (v2 ). 2.2.

TECHNICAL ESTIMATES

In this subsection we will collect some technical estimates which are adaptation of Glimm–Jaffe’s Nτ -estimates. We recall the symbolic annihilation and creation operators in the case of a Fock space constructed over the space of square integrable functions h := L2 (Rd , dk). Let 9 ∈ 0fin (h): 1

(a(k)9)(n) (k1 , . . . , kn ) := (n + 1) 2 9 (n+1) (k, k1 , . . . , kn ), (a ∗ (k) 9)(n) (k1 , . . . , kn ) n 1 X := n− 2 δ(k − kj ) 9 (n−1) (k1 , . . . , kˆj , . . . , kn ), j =1

where kˆj means that kj is omitted. Let S(Rd ) be the Schwartz space. We can define the monomial a(k1 ) . . . a(ks ) as an operator from 0fin (S(Rd )) into S(Rds ) ⊗ 0fin (S(Rd )). Let K be an auxiliary Hilbert space. Let w be unbounded operator from K⊗h⊗s into K ⊗h⊗r , with a domain containing K ⊗S(Rds ). A Wick monomial with symbol w is the following sesquilinear form on K ⊗ 0fin (S(Rd )) Wr,s := a ∗ (k10 ) . . . a ∗ (kr0 ) w a(k1 ) . . . a(ks ).

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223

Let ω be a positive regular function satisfying: ω ∈ C ∞ (Rd ), |∂kα ω(k)| 6 cα (1 + |k|)Nα ω(k) > m > 0.

for α ∈ Nd ,

We set Nτ := d0(ωτ ). LEMMA 2.1. Let r be a positive integer and τ := (τi )1...r be a sequence of real 1 Q numbers. For ψ ∈ D( ri=1 Nτ2i ):

r

j

2 r Z

Y 1 2 X

Y

τ Pr,j (k1 , . . . , kj ) a(ki )ψ dk1 . . . dkj , (2.6)

Nτ2i ψ =

i=1

j =1

i=1

τ where Pr,j is a sum of homogeneous functions in the variables ω(ki ) of degree Pr τ and satisfying i=1 i X τ Pr,j (k1 , . . . , kj ) = ω(ki1 )τ1 . . . ω(kir )τr , (2.7) (is )1...r ∈Sr,j

where Sr,j is the set, constructed by induction, of surjective maps i from {1, . . . , r} into {1, . . . , j }, such that is 6 s and (is )1...r−1 is in Sr−1,j or in Sr−1,j −1 and Sj,j +1 = S1,0 = ∅. Qj Proof. i=1 a(ki ) can be defined as operator on 0fin (S(Rd )) into S(Rdj ) ⊗ 0fin (S(Rd )). For ψ ∈ 0fin (S(Rd )),

j

2

Y

(k1 , . . . , kj ) → a(ki )ψ

i=1

is a function in S(Rdj ). This implies that the right-hand side of (2.6) is well defined 1

for ψ ∈ 0fin (S(Rd )). The hypothesis on ω imply that 0fin (S(Rd )) is a core for Nτ2 1 Q and for ri=1 Nτ2i . If the lemma holds for ψ ∈ 0fin (S(Rd )) then it can be extended to 1 1 Q Q ψ ∈ D( ri=1 Nτ2i ). In fact since 0fin (S(Rd )) is a core for ri=1 Nτ2i , then we can ex1 Qj Q τ 12 tend i=1 a(ki ) to bounded operator from D( ri=1 Nτ2i ) into L2 (Rdj , (Pr,j ) dk)⊗ 3 0(h). Let us prove the lemma for ψ ∈ 0fin (S(R )) by induction in r. For r = 1, 1

kNτ21 ψk2 = (ψ|Nτ1 ψ) Z = ω(k)τ1 ka(k)ψk2 dk. τ1 We see that P1,1 (k) = ω(k)τ1 . (2.6), (2.7) are satisfied for r = 1. Assume that (2.6), (2.7) hold for r. Using the fact that Nτ preserves 0fin (S(Rd )) and the

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induction hypothesis, we have

r+1

Y 1 2

Nτ2i ψ

i=1

=

r Z X

j

Y

τ Pr,j (k1 , . . . , kj )

j =1

=

r Z X

τ Pr,j

j =1

=

r Z X

1 2 τr+1

a(ki )N

i=1

2

ψ dk

2 ! 12 j j

X Y

τr+1 N + ω(k ) a(k )ψ

τr+1

dk i i

i=1

( τ Pr,j

j =1

i=1

j

2

2 ) j j

Y

1 Y

X

ω(ki )τr+1 a(ki )ψ + Nτ2r+1 a(ki )ψ dk

i=1

i=1

i=1

2 # j " j r+1 Z

Y

X X

τ τ τr+1 = Pr,j ω(ki )τr+1 + Pr,j

a(ki )ψ dk, −1 ω(kj )

j =1

i=1

i=1

where Pr,r+1 = Pr,0 = 0. Then we obtain the following iterated relation τ Pr+1,j (k1 , . . . , kj )

=

τ Pr,j (k1 , . . . , kj )

j X

τ τr+1 ω(ki )τr+1 + Pr,j . −1 (k1 , . . . , kj −1 )ω(kj )

(2.8)

i=1

Q τ We note that Pr,r (k1 , . . . , kr ) = ri=1 ω(ki )τi . It is easy to see, using induction hypothesis (2.7) for r and (2.8), that X τ Pr+1,j (k1 , . . . , kr+1 ) = ω(ki1 )τ1 . . . ω(kir+1 )τr+1 . 2 (is )1...r+1 ∈Sr+1,j

COROLLARY 2.2. Let α, ν ,τ := (τi )1...r be a sequence of real numbers. For 1 Q ψ ∈ D(Nνα ri=1 Nτ2i )

2 r

α Y 12 Nτi ψ

Nν

i=1

2 !α j j r Z

X X Y

τ ν Pr,j (k1 , . . . , kj ) Nν + = ω(ki ) a(ki )ψ ×

j =1

× dk1 . . . dkj , τ where Pr,j is the function defined by (2.7).

i=1

i=1

(2.9)

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LEMMA 2.3. Let p, q ∈ N and τ 0 := (τi0 )1...p , τ := (τi )1...q be two sequences of real numbers. Let Wr,s be a Wick monomial such that r 6 p, s 6 q. Then

p

! q

Y

Y 1 1

− − 2 2 0 Wr,s (Nτi + 1) (Nτi + 1)

i=1

6

i=1

τ 0 − 12 k(Pp,r )

τ − 12 w (Pq,s ) kB(K⊗h⊗s ,K⊗h⊗r ) .

Proof. Let ψ, φ ∈ K ⊗ 0fin (S(Rd )). (Wr,s ψ|φ) r ! s Y Y = w a(ki )ψ a(ki0 )φ i=1 i=1 K⊗h⊗r ⊗0(h) ! s r Y τ 0 21 Y τ 0 − 12 τ − 12 τ 21 0 6 (Pp,r ) w (Pq,s ) (Ps,q ) a(ki )ψ (Pp,r ) a(ki )φ i=1 i=1

s

2 ! 1 Z 2

Y

1 1 0

τ −2 τ −2 τ 6 k(Pp,r Pq,s ) w (Pq,s ) kB(K⊗h⊗s ,K⊗h⊗r ) a(ki )ψ dk ×

i=1

r

2 ! 1 Z 2

Y

τ0 0 × Pp,r a(ki )φ dk 0

i=1

q

p

Y 1 Y 1 1 1 0

τ −2 τ −2 6 k(Pp,r ) w (Pq,s ) kB(K⊗h⊗s ,K⊗h⊗r ) Nτ2i ψ × Nτ20 φ . i

i=1

i=1

This inequality shows that the quadratic form p Y

(Nτi0 + 1)− 2 Wr,s 1

i=1

q Y 1 (Nτi + 1)− 2 i=1

can be extended to a bounded operator with norm less than

τ 0 − 1

τ − 12 2

P w (Pq,s ) . p,r B(K⊗h⊗s ,K⊗h⊗r )

2

COROLLARY 2.4. Let vi ∈ B(K, K ⊗h); i = 1, . . . , n. Then there exists c > 0 such that

n n

Y Y

p ] −p− n2 (i) (N + 1) a (vi )(N + 1) kvi kB(K,K⊗h) .

6 c

i=1 i=1

n n

Y Y

p −p− n2 (ii) (N + 1) 6 c φ(vi )(N + 1) kvi kB(K,K⊗h) .

i=1

(iii)

Let r, s ∈

N, (τi0 )1...r , (τi )1...s

i=1

be sequences of real numbers. Then

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ZIED AMMARI

r

! s

Y

Y 1 1

(Nτi0 + 1)− 2 Wr,s (Nτi + 1)− 2

i=1 i=1

r

s

Y

Y τ0 τi i

6 ω(ki0 )− 2 w ω(ki )− 2

i=1

.

B(K⊗h⊗s ,K⊗h⊗r )

i=1

Proof. Clearly (i) gives (ii). For p = 0 and n = 1, (i) follows from the last lemma by taking ω = 1 and r = p = 1, s = q = 0 or r = p = 0, s = q = 1. For p, n ∈ N, commutation properties reduces the inequality to the case p = 0, n = 1. (iii) is a direct application of Lemma 2.3. 2 We recall now well known estimates, see [Ro]. LEMMA 2.5. Let b be a positive operator. Then d0(bα1 ) 6 N 1−α1 d0(b)α1 ,

where α1 6 1.

d0(bτ1 )α1 6 d0(bτ2 )α2 d0(bτ3 )α3 , where α1 = α2 + α3 , and α1 τ1 = α2 τ2 + α3 τ3 . Combining Lemma 2.5 and Lemma 2.3 we obtain a slightly more general estimate. LEMMA 2.6. Let r, s, p, q ∈ N. Let τ 0 j := (τ 0 ij )1...p , τj := (τji )1...q , j = 1 . . . 3 be sequences of real numbers such that τ 0 i1 = τ 0 i2 + τ 0 i3 , τ1i = τ2i + τ3i . Then

p

q

Y

Y

− 12 − 12 − 12 − 12 i i (N + 1) (N + 1) W (N i + 1) (N i + 1)

r,s τ2 τ3 τ 02 τ 03

i=1

6

i=1

τ 0 1 − 12 k(Pp,r )

τ1 − 12 w (Pq,s ) k.

The following estimate is an immediate application of Lemma 2.6. COROLLARY 2.7. Let r, s, α, β be positive integers such that α 6 r, β 6 s. Then the following assertion holds k(N + 1)− 6 P

r−α 2

s−β

α

β

(d0(ω) + 1)− 2 Wr,s (N + 1)− 2 (d0(ω) + 1)− 2 k

r

s

Y

Y τ0 τ i i

inf ω(ki0 )− 2 w ω(ki )− 2 .

{τi0 ,τi ∈[0,1]| ⊗s ⊗r P

τi0 =α,

τi =β}

i=1

i=1

B(K⊗h

,K⊗h

)

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A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

2.3.

THE NELSON MODEL

The Nelson model [Ne] describes a system of P nonrelativistic particles coupled to a scalar relativistic field of bosons by a local, translation invariant interaction. It exhibits a relatively mild ultraviolet divergence, and was the first QFT model, on which a renormalization procedure was rigorously carried on. We consider the atomic Hamiltonian of the system of P nonrelativistic confined particles as follows K :=

P 1 X 2 D + V (x1 , . . . , xP ). 2M j =1 j

It acts on the Hilbert space L2 (R3P , dx) which we denote in the sequel by K. We assume that V ∈ L2loc (R3P ) and V > 0. Kato’s inequality gives that K is essentially 1 self-adjoint on C0∞ (R3P ), see [RS, I–IV, Thm. X.28]. We set hxi := (|x|2 + 1) 2 . We will also assume X V >c hxi iα , α > 2. i β

β

We notice that for 0 6 β 6 1, hDiβ (K + 1)− 2 , hxiβ (K + 1)− 2 are bounded operators. The boson one particle space is the Hilbert space h := L2 (R3 , dk), where k denotes the boson momentum observable. The boson position observable −(∇k /i) will be denotes by the italic letter x. This should not be confused with the nucleon position observable denoted by the roman letter x. The free bosonic Hamiltonian is defined by the second quantization of a single boson energy. It acts on the bosonic Fock space 0(h). Hb := d0(ω), 1

ω(k) := (|k|2 + m2 ) 2 ,

m > 0.

It is essentially self-adjoint on 0fin (D(ω)). The Hilbert space of the joint system is H := K ⊗ 0(h). The Hamiltonian without interaction is given by H0 := K ⊗ 1 + 1 ⊗ Hb . This is a self-adjoint operator since K ⊗ 1 and 1 ⊗ Hb commute on H . The local translation invariant interaction between nucleons and bosons is given by the formal expression P X

ϕ(xj ),

where ϕ(x)

j =1 3 − 12

:= [2(2π ) ]

Z

e−ihk,xi (a ∗ (k) + a(−k))

dk 1

ω(k) 2

.

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The interaction term cannot be defined as an operator on H with a dense domain. 1 This comes from the fact that ω− 2 ∈ / L2 (R3 , dk), because the integral diverges for large k. This phenomenon is known as an ultraviolet problem. In order to have a well defined operator, one introduces cutoff interactions: Z χκ (k) 3 − 12 ϕκ (x) := [2(2π ) ] e−ihk,xi (a ∗ (k) + a(−k)) dk, 1 ω(k) 2 P X 1 χκ (k) Iκ := ϕκ (xj ), vκ := [(2π )3 ]− 2 e−ihk,xi . 1 2 ω(k) j =1 Here χ is a positive function in C ∞ (R3 ) such that 0 6 χ(k) 6 1, χ(k) = 1 for |k| 6 1, χ(k) = 0 for |k| > 2 and χ(−k) = χ(k). We set χκ (k) := χ(k/κ). LEMMA 2.8. One has for α > 0 (i)

d0(ω)α 6 H0α ,

(ii)

K α 6 H0α .

Proof. For ψ ∈ C0∞ (R3 ) ⊗ 0fin (S(R3 )), which is a core for H0 , one has (d0(ω)ψ, ψ) 6 (H0 ψ, ψ), (Kψ, ψ) 6 (H0 ψ, ψ). This means that d0(ω) 6 H0 and K 6 H0 . Since H0 , K and d0(ω) commute, the spectral theorem gives the inequalities announced in the lemma. 2 1

Iκ are well-defined operators on D((N + 1) 2 ) as long as κ < ∞ and they are H0 -bounded with infinitesimal bound. We set Hκ := H0 + Iκ . THEOREM 2.9. For κ < ∞, Hκ is a self-adjoint operator on D(H0 ). Proof. Using Corollary 2.4 and Lemma 2.8, we prove for c independent from λ 1

kIκ (H0 + λ)−1 k 6 c λ− 2 kvκ k. Then by the Kato–Rellich theorem, one sees that Hκ is a selfadjoint operator on D(H0 ). 2

3. Construction of the Nelson Hamiltonian In this section we recall the construction in [Ne] of the Nelson Hamiltonian. It consists in applying to the cutoff Hamiltonians Hκ a cutoff dependent unitary transformation Uκ , letting then κ to ∞.

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

3.1.

229

DRESSING TRANSFORMATION

For a fixed κ0 and κ < ∞, we define (2π )− 2 χκ (k) − χκ0 (k) 3

gκ (k) := −i

Gκ :=

P X

1 2

ω(k) +

ω (k)

k2 2M

∈ C0∞ (R3 ),

e−ihk,xj i gκ (k) ∈ B(K, K ⊗ h),

(3.1)

(3.2)

j =1

Z 1 X Cκ := |gκ (k)|2 sinhk, (xj − x` )i dk ∈ S(R3P ), 2 16j,`6P Z 1 (χκ (k) − χκ0 (k))2 P Eκ := dk − 2(2π )3 ω(k) (ω(k) + k2 ) −

P (2π )3

Z

(3.3)

2M

χκ (k) χκ (k) − χκ0 (k) dk, ω(k) (ω(k) + k2 )

(3.4)

2M

rκ (x) := −ik e−ihk,xi gκ (k),

(3.5)

Uκ := eiφ(Gκ )+iCκ ,

(3.6)

Hˆ κ := Uκ (Hκ − Eκ )Uκ∗ .

(3.7)

In order to simplify the writing of some formulas, we will replace often rκ (xj ), vκ (xj ) by rκj , vκj . LEMMA 3.1. For a fixed κ0 and for κ0 < κ < ∞, Hˆ κ is a selfadjoint operator on the domain D(H0 ) and equal to X Hˆ κ = H0 + Vκ (xi − xj ) ⊗ 1 + Iˆκ , (3.8) 16i<j 6P

where P 1 X Iˆκ := Iκ0 + Rj (rκ (xj )), 2M j =1

√ √ 1 2 1 2 a (v) + a ∗ (v) + a ∗ (v) a(v) − 2Dj a(v) − 2a ∗ (v)Dj , 2 2 Z Z Vκ (x) := Re ω(k) |gκ (k)|2 e−ihk,xi dk − 2Im g¯ κ (k) vκ (k)e−ihk,xi dk.

Rj (v) :=

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Proof. We compute Uκ Hκ Uκ∗ using commutation relations. We notice that terms coming from Hb and K have been mixed. This creates a translation invariant potential and second-order interacting terms: Uκ Hb Uκ∗ = Hb − φ(iωGκ ) + X

+

Vκ(1) (xi

16i<j 6P

where Vκ(1) (x)

Z := Re

Uκ KUκ∗ =

P − xj ) + 2

Z ω(k)|gκ (k)|2 dk,

(3.9)

ω(k)|gκ (k)|2 e−ihk,xi dk.

P 1 X (Dj + φ(ik gκ (k)e−ihk,xj i ))2 + 2M j =1

+ V (x1 , . . . , xP ). X

Uκ Iκ Uκ∗ = Iκ +

(3.10)

Vκ(2)(xi − xj ) −

16i<j 6P

P − (2π )3 where Vκ(2) (x)

Z := −2Im

Z

χκ (k) χκ (k) − χκ0 (k) dk, ω(k) ω(k) + k2 2M

g¯ κ (k) vκ (k) e−ihk,xi dk.

Using the following computation (Dj − φ(rκj ))2 √ 1 1 2 = Dj2 + a 2 (rκj ) + a ∗ (rκj ) + a ∗ (rκj ) a(rκj ) − 2Dj a(rκj ) − 2 2 Z √ ∗ j 1 2 −ihk,xj i − 2a (rκ )Dj − φ(ik e gκ ) + k 2 |gκ (k)|2 dk, 2 in the second term (3.10) and collecting similar terms together, we obtain (3.8). It is easy to see, by (3.9)–(3.10), that Uκ preserves D(H0 ) for κ < ∞, hence D(Hˆ κ ) = Uκ D(H0 ) = D(H0 ). 2

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

3.2.

231

REMOVAL OF THE ULTRAVIOLET CUTOFF

We set (2π )− 2 1 − χκ0 (k) 3

g∞ (k) := −i

G∞ :=

1

ω 2 (k) (ω(k) +

P X

k2 ) 2M

,

e−ihk,xj i g∞ .

j =1

Z

V∞ := Re

ω(k) |g∞ (k)|2 e−ihk,xi dk − Z

− 2Im C∞

g¯∞ (k)

1 1

e−ihk,xi dk,

ω(k) 2

Z 1 X |g∞ (k)|2 sinhk, (xj − x` )i dk, := 2 16j,`6P

U∞ := eiφ(G∞ )+iC∞ . LEMMA 3.2. Letting κ to ∞, we obtain the following limits: (i) (ii) (iii) (iv) (v)

Eκ → −∞, gκ → g∞ in h, Cκ → C∞ in L∞ (R3P ), Vκ → V∞ in L∞ (R3 ) + Ls (R3 ) for s ∈ ]2, +∞[, Uκ → U∞ strongly in B(H).

Proof. (i) is obvious. (ii) follows from the monotone convergence theorem. (iii) follows from (ii), since |gκ |2 converges in L1 (R3 ). Vκ(1) converges in L∞ (R3 ), which follows from the fact that its integrand converges in L1 (R3 ). Vκ(2) converges in Ls (R3 ) for s ∈ ]2, +∞[, by using Hausdorff–Young inequality. This proves (iv). Using the fact that the map h 3 v → eiφ(v) is strongly continuous and the limits (ii), (iii), we see that s-limk→∞ Uκ exists and is equal to eiφ(G∞ )+iC∞ . 2 We give now a lemma which will be useful in this section and in Section 4. LEMMA 3.3. For s ∈ [0, 1], and vi ∈ B(K, K ⊗ h), i = 1, 2 we have: (i)

s

1−s 2

− 2s

− 1−s 2

k(N + 1)− 2 a(v1 ) (H0 + 1)− a ∗ (v1 ) (N + 1)

k 6 kω

s−1 2

− 2s

v1 kB(K,K⊗h) .

(ii)

k(H0 + 1)

(iii)

k(N + 1)−s a(v1 ) a(v2 ) (H0 + 1)−1+s k 1−s 1−s 6 kω− 2 v1 kB(K,K⊗h) kω− 2 v2 kB(K,K⊗h) .

(iv)

k(H0 + 1)−s a ∗ (v1 )a ∗ (v2 )(N + 1)−1+s k s s 6 kω− 2 v1 kB(K,K⊗h) kω− 2 v2 kB(K,K⊗h) .

k 6 kω

v1 kB(K,K⊗h) .

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Proof. We see clearly that suitable choice of r, s, α, and β in Corollary 2.7 gives similar inequalities with d0(ω) in the place of H0 . Now using Lemma 2.8 we obtain (i)–(iv). 2 LEMMA 3.4. One has for vi ∈ B(K, K ⊗ h), such that [V , vi ] = 0, where V is the potential of the atomic Hamiltonian and s, β ∈ [0, 1]: (i)

1

1

s

− 12

− 12

− 2s

k(N + 1)− 2 a(v1 ) (H0 + 1)− 2 k 6 c k(V + 1)− 2 ω a ∗ (v1 ) (N + 1)

s−1 2 s−1 2

v1 kB(K,K⊗h) .

(ii)

k(H0 + 1)

(iii)

k(N + 1)−1 a(v1 )a(v2 )(H0 + 1)− 2 k βs (1−β)s s−1 s−1 6 c k(V + 1)− 2 ω 4 v1 k × k(V + 1)− 2 ω 4 v2 k.

(iv)

k(H0 + 1)− 2 a ∗ (v1 )a ∗ (v2 )(N + 1)−1 k βs (1−β)s s−1 s−1 6 c k(V + 1)− 2 ω 4 v1 k × k(V + 1)− 2 ω 4 v2 k.

(v)

k(N + 1)− 2 (H0 + 1)− 2 a ∗ (v1 )a(v2 )(H0 + 1)− 2 k 1−s s s 1−s 6 ck(V + 1)− 2 ω− 2 v1 kB(K,K⊗h) × k(V + 1)− 2 ω− 2 v2 kB(K,K⊗h) .

k 6 c k(V + 1)

ω

v1 kB(K,K⊗h) .

1

1

1

1

1

1

Proof. Using Lemma 3.3, we see that (i)–(v) are true with (H0 + 1)− 2 replaced 1−s s s s by (H0 + 1)− 2 (V +1)− 2 . We use then the fact that (V +1) 2 (H0 +1)− 2 is bounded for s ∈ [0, 1]. 2 We set for κ < ∞: X Bκ (φ) :=

ˆ Vκ (xi − xj ) + Iκ φ|φ ,

1

φ ∈ D(H02 ).

16i<j 6P

LEMMA 3.5. There exists κ0 , 0 6 a < 1, 0 6 b independent from κ, such that for κ > 2κ0 1

|Bκ (φ)| 6 akH02 φk2 + bkφk2 ,

1

φ ∈ D(H02 ).

Proof. One has using Lemma 3.3 1 1 k(H0 + λ)− 2 Iˆκ (H0 + λ)− 2 k 1

1

1

1

1

6 c(λ− 2 kω− 2 vκ0 k + kω− 4 rκ k2 + kω− 2 rκ k2 + kω− 2 rκ k). 1

(3.11)

We notice that ω− 4 rκ contains the term χκ − χκ0 , for κ > 2κ0 , which is arbitrarily small for κ0 large enough. Using this fact and (3.11) we see that there exists κ0 such that Iˆκ has a small H0 -form bound for κ > 2κ0 . The integrand of Vκ(1) , Vκ(2) contain the term χκ − χκ0 , then there exist κ0 such that Vκ(1) (resp. Vκ(2) ) has a small L∞ (R3 ) (resp. Ls (R3 ), s > 2) norm for κ > 2κ0 . After a change of variables to separate the motion of the center of mass, the term Vκ(2) becomes Vκ(3) + Vκ(4) ∈

233

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT 0

L∞ +Lq where 2 6 q < ∞. Using the Sobolev injection H 1 (R3 ) → Lq (R3 ), 2 6 q 0 6 6, we obtain, by a convenient choice of q and q 0 , that Vκ(2) is H0 -form bounded [Ne]. 2 In all the sequel a limit of a sequence of operators written without a prefix shall be understand as a norm limit. THEOREM 3.6. There is a unique selfadjoint operator Hˆ ∞ acting on H , satisfying (i)

limκ→∞ (Hˆ κ − z)−1 = (Hˆ ∞ − z)−1 ,

(ii)

s-limκ→∞ e−it Hκ = e−it H∞

ˆ

ˆ

for t ∈ R. 1

1

The domain of Hˆ ∞ satisfies D(Hˆ ∞ ) ⊂ D(Hˆ ∞2 ) = D(H02 ). Proof. The proof is based on the Theorem A.1 in the appendix. Let us apply now this theorem with X ˆ Bκ (φ) = Vκ (xi − xj ) + Iκ φ| φ . 16i<j 6P 1

Using Lemma 3.3, one has for φ ∈ D(H02 ) |Bκ (φ) − Bκ 0 (φ)| (2) −2 6 c(kVκ(1) − Vκ(1) − Vκ(2) (rκ − rκ 0 )k + 0 kL∞ + kVκ 0 kLs + kω 1

+ kω− 4 (rκ − rκ 0 )k + kω− 4 (rκ − rκ 0 )k2 ) × k(H0 + 1) 2 φk2 . 1

1

1

(3.12)

We see clearly, using Lemma 3.2(ii) and (iv) in the right-hand side of (3.12), that Bκ satisfies hypothesis of Theorem A.1. The KLMN theorem applied for B∞ , gives 1 1 that D(Hˆ ∞2 ) = D(H02 ). So this proves the theorem. 2 ∗ ˆ DEFINITION 3.7. The Hamiltonian H := U∞ H∞ U∞ is called the Nelson Hamilˆ tonian and H∞ the modified Hamiltonian.

THEOREM 3.8. One has (i) limκ→∞ (Hκ − Eκ − z)−1 = (H − z)−1 , (ii) s-limκ→∞ e−it (Hκ −Eκ ) = e−it H for t ∈ R. 1

∗ (iii) D(H ) ⊂ D(H 2 ) = U∞ D(H02 ). 1

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Proof. One has ∗ kUκ∗ (Hˆ κ − z)−1 Uκ − U∞ (Hˆ ∞ − z)−1 U∞ k ∗ 6 k(Uκ∗ − U∞ )(Hˆ κ − z)−1 k + ∗ + k(Uκ∗ − U∞ )(Hˆ ∞ − z¯ )−1 k + + k(Hˆ κ − z)−1 − (Hˆ ∞ − z)−1 k.

(3.13)

Using (3.13), Theorem 3.6 and the fact that the map h → W (h)(N + 1)− , > 0, is norm continuous, we obtain ∗ kUκ∗ (Hˆ κ − z)−1 Uκ − U∞ (Hˆ ∞ − z)−1 U∞ k 6 c(k(W (Gκ ) − W (G∞ ))(N + 1)− k + + k(Hˆ κ − z)−1 − (Hˆ ∞ − z)−1 k + + keiCκ − eiC∞ k).

The application of Lemma 3.2 completes the proof of (i). (ii) follows from the equivalence of the convergence in the strong resolvent sense and the strong convergence of unitary groups (Trotter theorem). (iii) is obvious. 2 Let χκ0 be another cutoff function and define H 0 to be the Nelson Hamiltonian constructed using the later cutoff. PROPOSITION 3.9. There exists a finite constant E, such that H 0 = H + E. Proof. We define H 00 (resp.Hˆ κ00 ) to be the Nelson Hamiltonian (resp. the cutoff modified Hamiltonian) obtained using the dressing transformation given by (2π )− 2 χκ0 (k) − χκ0 (k) 3

gκ00

:= −i

1

ω 2 (k)

ω(k) +

k2 2M

.

It is easy to see that H 0 = H 00 + E, where E is a finite constant. Using similar 00 00 calculus with (3.12), we obtain that Hˆ ∞ = Hˆ ∞ . Since U∞ = U∞ we have H = 00 0 H . Then H = H + E. 2

4. Higher Order Estimates In this section we prove some estimates which allow to bound powers of N and H0 by powers of H . They play an important role. Fröhlich has proved a higher estimates in the massless case [Fr], but they are different from what we intend to prove. Their proofs are based in the following principle of cutoff independence [Ro].

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

235

LEMMA 4.1. Let {Nj } and {Hj } be sequences of operators such that, for c independent of j, kNj ψk 6 c kHj ψk,

for ψ ∈ D(Hj ).

Suppose that Nj is self-adjoint and that Nj → N in the strong resolvent sense, where N is self-adjoint, and Hj → H in the strong graph limit. Then kNψk 6 c kH ψk,

for ψ ∈ D(H ).

Let 0 (resp. 0j ) denotes the graph of H (resp. Hj ). We recall that Hj → H in the strong graph sense if for all (ψ, ϕ) ∈ 0, there exist a sequence (ψj , ϕj ) ∈ 0j which converges to (ψ, ϕ) in H × H. The following easy first order estimate follows from the proof of Theorem 3.6. LEMMA 4.2. There exists c > 0 independent for κ such that for κ0 6 κ 6 ∞ H0 6 (Hˆ κ + c).

4.1.

ROSEN ESTIMATES

In this subsection we prove higher order estimates using a technique due to Glimm and Jaffe to prove similar estimates for the Y2 and (ϕ 4 )2 models. It has been taken up by Rosen in [Ro] for the general (ϕ 2n)2 model and Fröhlich [Fr]. This technique is based in the so called pull-through formula which is the identity that we obtain, in a formal way, when we move the resolvent through a product of annihilation operators a(ki ). But some care must be taken when we want to rigorously prove the dense subspace H0 ⊂ D(HQ 0 ) on which Q pull through formula, since we need aQ a(k ) acts as an operator and satisfies a(k )H ⊂ D(H ) and i i 0 0 i i i a(ki ) can be defined as operator on H0 H0 . We need also a resolvent control of the commutator of the modified cutoff interaction with a(ki ), which allows to define it as locally integrable function with values in bounded operators on D(H0 ). This requires H0 H0 to be dense. Let {Ji }1...j +1 be a set of disjoint subsets of {1, . . . , n} so that within each subset Ji the elements are taken in their natural order. We introduce the notation: HκJ := [a(ki1 ), . . . , [a(kij ), Iˆκ ] . . .], where J = {i1 , . . . , ij }, R(z) := (z − Hˆ κ )−1 , −1 X R` (z) := z − ω(ki ) − Hˆ κ , i

where the sum runs over i ∈ J` ∪ J`+1 ∪ · · · ∪ Jj +1 .

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LEMMA 4.3. Let α ,τ := (τi )1...r be a sequence of real numbers. For ! r Y 1 ψ ∈ D (H0 + 1)α Nτ2i i=1

2 r

Y 1

α 2 Nτi ψ

(H0 + 1)

i=1

!α j

X

τ Pr,j = (k1 , . . . , kj ) H0 + ω(ki ) + 1 ×

j =1 i=1

2 j

Y

× a(ki )ψ dk1 . . . dkj .

r Z X

(4.1)

i=1

Proof. This lemma is similar to Corollary 2.2. We prove (4.1) for ψ ∈ D(K) ⊗ 1 Q 0fin (S(R3 )) and then we extend it to ψ ∈ D((H0 + 1)α ri=1 Nτ2i ). This ex1 Qj Q tends i=1 a(ki ) to bounded operator from D((H0 + 1)α ri=1 Nτ2i ) into (H0 + Pj τ 12 −α 2 3j 2 i=1 ω(ki ) + 1) L (R , (Pr,j ) dk) ⊗ H . Pj Using the fact that (H0 + i=1 ω(ki ) + 1)−1 H0 is bounded and (4.1), we have r for ψ ∈ (H0 + 1)−1 D(N 2 )

2 Z r

p−r

Y

2

a(ki )ψ dk

N H0

i=1

2 ! r Z r

X Y p−r

2 6 c H0 + ω(ki ) + 1 a(ki )N ψ dk

i=1

i=1

2 Z r

Y

p−r

6 c a(ki )(H0 + 1)N 2 ψ dk

i=1

p

6 c kN 2 (H0 + 1)ψk2 . Q p Hence ri=1 a(ki ) can be defined as bounded operator from (H0 +1)−1 D(N 2 ) into p−r L2 (R3r , dk) ⊗ (H0 + 1)−1 D(N p 2 ). Itp is easy to see, using commutation relations and Lemma 3.1, that (N + 1)− 2 Iˆκ N 2 (H0 + 1)−1 is bounded forp κ < ∞, hence p 2 ). Iˆκ can be defined as bounded operator from (H0 + 1)−1 D(N 2 ) into D(N p So the commutator Hκ{1} acts as bounded operator from (H0 + 1)−1 D(N 2 ) into

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A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

L2 (R3 , dk1 ) ⊗ D(N that

p−1 2

). Now for J := {1, . . . , r} we can prove by induction on r p 2

Hκ{J } : (H0 + 1)−1 D(N ) → L2 R3r ,

r Y

! dki

⊗ D(N

p−r 2

).

(4.2)

i=1

For #J = 1 (4.2) is already done. Assume that (4.2) holds for #J = r. Using the Q Q p−r fact that a(kr+1 ) maps L2 (R3r , ri=1 dki ) ⊗ D(N 2 ) into L2 (R3(r+1), r+1 i=1 dki ) ⊗ p−r−1 p p−1 −1 2 3 −1 D(N 2 ) and maps (H0 +1) D(N 2 ) into L (R , dkr+1 )⊗(H0 +1) D(N 2 ), we prove (4.2). A simple computation gives: 1 X j 1 X j Hκ{1} = √ vκ0 (k1 ) + rκ (k1 )a ∗ (rκj ) + 2M 2 j =1 j =1 √ 1 j j + rκ (k1 )a(rκ ) − 2Dj rκj (k1 ), on (H0 + 1)−1 D(N 2 ), P

Hκ{1,2} =

P

P 1 X j r (k1 )rκj (k2 ), 2M j =1 κ

HκJ = 0,

on (H0 + 1)−1 D(N),

for all J such that #J > 3.

Before starting with the pull-through formula we will prove two lemma which will be useful in the sequel. LEMMA 4.4. Let r be an integer and z ∈ / σ (Hˆ κ ), then (H0 + 1)(Hˆ κ − z)−1 : D(N 2 ) → D(N 2 ), r

r

is a bijective map. r r Proof. Since (N + 1)− 2 Iˆκ N 2 (H0 + 1)−1 is bounded for κ < ∞, we see that r r (Hˆ κ − z)(H0 + 1)−1 maps D(N 2 ) into D(N 2 ). So it is enough to show that (H0 + r −r 1)N 2 (Hˆ κ − z)−1 (1 + N) 2 is bounded. We define [N, iHˆ κ ] as quadratic form on D(H0 ) [N, iHˆ κ ] =

P X

P 1 X ∗2 j ia (rκ ) − ia 2 (rκj ) + 2M j =1 j =1 √ √ + i 2Dj a(rκj ) − i 2a ∗ (rκj )Dj .

φ(ivκj0 ) +

For κ < ∞, using Corollary 2.4 and the fact that hDi(K + 1)− 2 is bounded, we see that [N, iHˆ κ ] can be defined as a bounded operator on D(H0 ). 1 ` ˆ We set ad`N . := [N, i ad`−1 N .] and adN . := [N, i.]. So adN Hκ , which is similar to ad1N Hˆ κ is defined by induction in ` as a bounded operator on D(H0 ) and equal to: 1

ad`N Hˆ κ

=

P X j =1

`

φ(i

vκj0 )

P 1 X ` `−1 ∗2 j + i 2 a (rκ ) − i` (−2)`−1 a 2 (rκj ) − 2M j =1

√ √ − (−i)` 2Dj a(rκj ) − i` 2a ∗ (rκj )Dj .

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ZIED AMMARI

Since D(N) ⊃ D(Hˆ κ ) = D(H0 ), the resolvent (z − Hˆ κ )−1 preserves the domain of N. This means that the following identity N(z − Hˆ κ )−1 = (z − Hˆ κ )−1 N + (z − Hˆ κ )−1 [N, Hˆ κ ](z − Hˆ κ )−1 , (4.3) holds in the sense of bounded operators on H . Using repeatedly (4.3) we notice that (z − Hˆ κ )−1 preserves D(N p ) and we obtain on D(N p ) N p (z − Hˆ κ )−1 = N p−1 (z − Hˆ κ )−1 N − iN p−1 (z − Hˆ κ )−1 ad1N Hˆ κ (z − Hˆ κ )−1 . (4.4) We move now all factors of N in each term to the right, we obtain the following identity between bounded operators N (z − Hˆ κ )−1 N −p = (z − Hˆ κ )−1 + p

k X

(z − Hˆ κ )−1 B` (z)N −` ,

`=1 j adN Hˆ κ

where B` (z) is a polynomial in (z − Hˆ κ )−1 , j 6 `. Using Lemma 3.3 with s = 0, we see that B` (z) is bounded for κ < ∞. Hence (z − Hˆ κ )−1 (H0 + 1) is a bijective map from D(N p ) into D(N p ). Using Hadamard’s three lines lemma [RS, I–IV] for ¯ f (ζ ) := ((z − Hˆ κ )−1 H0 N −ζ ψ, N ζ φ), in S := {ζ ∈ C, p 6 Re(ζ ) 6 p + 1}, where ψ, φ ∈ K ⊗ 0fin (h). f (ζ ) is a bounded continuous analytic function on S, satisfying |f (p + iλ)| 6 c kφk kψk,

λ ∈ R,

|f (p + 1 + iλ)| 6 c kφk kψk,

λ ∈ R.

We obtain |f (ζ )| 6 c kφk kψk,

for ζ ∈ S.

r Let 2p 6 r 6 2p + 2. Since K ⊗ 0fin (h) is a core for N 2 then (z − Hˆ κ )−1 H0 (1 + r r N)− 2 ψ ∈ D(N 2 ) and we have also r r kN 2 (z − Hˆ κ )−1 H0 (1 + N)− 2 ψk 6 c kψk, ψ ∈ K ⊗ 0fin (h). r r Then we have (H0 + 1)N 2 (Hˆ κ − z)−1 (1 + N) 2 bounded.

LEMMA 4.5. There exist > 0, b < 0 and c independent for κ (i)

1 2

1 2

kR` (b)Hκ{1} R`+1 (b)k 6 c

P X

2 !

|vκj0 (k1 )| + |rκj (k1 )ω(k1 )− | .

j =1

(ii)

1 2

1

2 kR` (b)Hκ{1,2} R`+1 (b)k

6c

P X j =1

! |rκj (k1 )ω(k1 )− rκj (k2 )ω(k2 )− |

.

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A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

Proof. Let D be a dense set of analytic vectors for K. HκJi can be defined on 1 1 2 D ⊗ 0fin (S(R3 )), then HκJi R`+1 (b) is well defined on Dκ (b) := (Hˆ κ − b) 2 D ⊗ 0fin (S(R3 )). Furthermore since D ⊗ 0fin (S(R3 )) is a core for (Hˆ κ − b), Dκ (b) is 1 dense in D((Hˆ κ − b) 2 ), which is dense in H . Hence it is enough to show (i)– (ii) in Dκ (b). Lemma 3.3 and Lemma 2.8 give the bounds uniformly in κ. Then 1

1

2 R`2 (b) HκJ R`+1 (b) extends from Dκ (b) to a bounded operator on H.

2

The following lemma is the generalized pull through formula. r

LEMMA 4.6. The following identity holds for all φ ∈ D(N 2 ): r Y

a(ki )(z − Hˆ κ )−1 φ

i=1

= R1 (z)

r Y

a(ki )φ +

i=1

+

X

R1 (z)HκJ1 R2 (z) . . . R`−1 (z)HκJ`−1 R` (z)

X

a(kj )φ +

j ∈J`

part.

+

Y

R1 (z)HκJ1 R2 (z) . . . R`−1 (z)HκJ` R(z) φ.

part.

The sum in right-hand side is taken over all the partitions of the set {1, . . . , r} into ordered subsets. Proof. We prove this lemma by induction on r. By Lemma 4.4, we know that 1 there exist ψ ∈ (H0 + 1)−1 D(N 2 ) such that φ = (z − Hˆ κ )ψ. We consider a(k1 ) as bounded operator 1

a(k1 ) : (H0 + 1)−1 D(N 2 ) → (H0 + ω(k1 ) + 1)−1 L2 (R3 , dk1 ) ⊗ H . Then we can write for κ < ∞ a(k1 )ψ = (z − ω(k1 ) − Hˆ κ )−1 (z − ω(k1 ) − Hˆ κ )a(k1 )ψ. By the justification in the beginning of this subsection, we see that Hκ{1} ψ ∈ L2 (R3 , dk1 ) ⊗ H. We have the following identity on L2 (R3 , dk1 ) ⊗ H (z − ω(k1 ) − Hˆ κ )a(k1 )ψ = a(k1 )(z − Hˆ κ )ψ + Hκ{1} ψ. This proves the pull through formula for r = 1. The formula (4.5) can be generalized by induction in r.

(4.5)

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ZIED AMMARI r

We claim that for ψ ∈ (H0 + 1)−1 D(N 2 ), Ir := {1 . . . r}, we have in ! r Y 2 3r L R , dki ⊗ H : z−

X

i=1

ω(ki ) − Hˆ κ

Y

i∈Ir

=

Y

a(ki )ψ

i∈Ir

X

a(ki )(z − Hˆ κ )ψ + HκIr ψ +

Ir =(J1 ,J2 )

i∈Ir

Y

HκJ1

a(ki )ψ,

(4.6)

i∈J2

where the sum is over all partitions (J1 , J2 ) of Ir . Let us prove (4.6), by induction r+1 on r. We have for ψ ∈ (H0 + 1)−1 D(N 2 ): Y X z− ω(ki ) − Hˆ κ a(ki )ψ i∈Ir+1

i∈Ir+1

Y Y X ˆ = a(kr+1 ) z − ω(ki ) − Hκ a(ki )ψ + Hκ{r+1} a(ki )ψ =

Y

i∈Ir

i∈Ir

a(ki )(z − Hˆ κ )ψ + Hκ{r+1}

i∈Ir+1

Y

i∈Ir

a(ki )ψ +

i∈Ir

X

+ a(kr+1 )

HκJ1

Ir =(J1 ,J2 )

Y

a(ki )ψ + a(kr+1 )HκIr ψ.

J2

Moving a(kr+1 ) through HκJ1 and HκIr and using the identity a(kr+1 )HκJ1 = HκJ1 a(kr+1 ) + HκJ1 ∪{r+1} ,

on (H0 + 1)−1 D(N

#J1 +1 2

),

we obtain (4.6). Now assume that the pull through formula holds for r, and let us prove it for r + 1. We have Y a(ki )(z − Hˆ κ )−1 φ i∈Ir+1

=

z−

X

ω(ki ) − Hˆ κ

−1

z−

i∈Ir+1

X

ω(ki ) − Hˆ κ

i∈Ir+1

Using now the iterated formula (4.6), we obtain Y a(ki )(z − Hˆ κ )−1 φ i∈Ir+1

=

z−

X i∈Ir+1

ω(ki ) − Hˆ κ

−1 Y i∈Ir+1

a(ki )φ +

Y i∈Ir+1

a(ki )ψ.

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A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

−1 X ˆ + z− ω(ki ) − Hκ HκIr+1 (z − Hˆ κ )−1 φ + i∈Ir+1

−1 X ˆ + z− ω(ki ) − Hκ

X

HκJ1

Ir+1 =(J1 ,J2 )

i∈Ir+1

Y

a(ki )(z − Hˆ κ )−1 φ.

i∈J2

Using the induction hypothesis we obtain: X Y HκJ1 a(ki )(z − Hˆ κ )−1 Ir+1 =(J1 ,J2 )

=

X

i∈J2

R1 (z)HκJ1 R2 (z) . . . R`−1 (z)HκJ`−1 R` (z)

part.

Y

a(ki ).

i∈J`

2

This completes the proof. THEOREM 4.7. Let ν 6 1, 0 6 τ and r ∈ N. Then 1

r−1

kNν2 N−τ2 ψk 6 c k(Hˆ κ + b) 2 ψk, r

r

for ψ ∈ D(Hˆ κ2 ),

(4.7)

where c, b are constants independent of κ. Proof. We prove the theorem by induction on r. The case r = 1 follows from 1 r−1 Lemma 4.2. Assume that (4.7) holds for all j 6 r. Let φ ∈ (H0 + 1)− 2 D(N 2 ) ⊂ r r D(N 2 ) and ψ := R(−b)φ. Since φ ∈ D(N 2 ), by Lemma 4.4, we see that ψ ∈ 1

r

1

j−2

2 D(Nν2 N−τ ). Clearly, we have also φ ∈ D(Nν2 N−τ2 ), j 6 r. We have 1

r

2 kNν2 N−τ ψk2

=

r Z X j =1

2 ! 12 j j

X Y

−τ Pr,j ω(ki )ν a(ki )R(−b)φ dk.

Nν +

i=1

(4.8)

i=1

(4.8) follows by Corollary 2.2. We recall that HκJ is an operator-valued function in variables ki , i ∈ J . If we denote by dJ := dki1 . . . dkip where J = {i1 , . . . , ip }, 1

1

2 then by Lemma 4.5 we have kR`2 (−b)HκJ R`+1 (−b)k ∈ L2 (R3p , dJ ). Using the pull through formula in the right-hand side of (4.8) and the fact that (Nν + 1 Pj ν 12 2 ω(k ) ) R i 1 (−b) is bounded, we obtain: i=1 1

r

2 kNν2 N−τ ψk2

2 j r Z

1

X Y

−τ Pr,j 6 c a(ki )φ dk +

R12 (−b)

j =1

i=1

242

ZIED AMMARI

+c

r XZ X

−τ Pr,j

j =1 part.

d Y

1

J

j

1

2 kR`2 (−b)Hκ ` R`+1 (−b)k2 ×

`=1

2 d+1 Y

21

Y j

× R (−b) a(k )φ dJ` + i

d+1

`=1

j

i∈Jd+1

+c

r XZ X

0

−τ Pr,j

j =1 part.

d Y

1

J

j

1

2 kR`2 (−b)Hκ ` R`+1 (−b)k2 ×

`=1 d0

1

× kR 2 (−b)φk2

Y

j

dJ`

`=1

=: I + II + III. Qj −τ We recall that Pj,j = i=1 ω(ki )−τ and by (2.7) we notice that −τ −τ Pr,j 6 c Pj,j .

(4.9) j

j

In II using (4.9) we can separate the integral in variables ki ∈ / Jd+1 and ki ∈ Jd+1 . 1

J

j

1

2 Since kR`2 (−b)Hκ ` R`+1 (−b)k ∈ L2 (R3p , dJ ), then we obtain

2 r X Z Y X Y

1

j −τ 2 II 6 c ω(ki ) Rd+1 (−b) a(ki )φ

dJd+1 .

j =1 part.

j i∈Jd+1

(4.10)

j i∈Jd+1

Now reordering the terms in the right-hand side of (4.10) and taking into account j the fact that Jd+1 6= ∅, we have

2 !− 12 j j r−1 Z

X X Y

−τ II 6 c Pj,j ω(ki )ν + b a(ki )φ dk. (4.11)

Hˆ κ +

j =1

i=1

i=1

Doing the same thing for III, we obtain 1

III 6 c kR 2 (−b)φk2 .

(4.12)

Collecting (4.11)–(4.12) we obtain r

1

2 kNν2 N−τ ψk2

6 c

r Z X j =1

+

r−1 Z X j =1

2 !− 12 j j

X Y

−τ Pr,j ω(ki )ν + b a(ki )φ dk +

Hˆ κ +

i=1

i=1

2 !− 12 j j

X Y

−τ Pj,j ω(ki )ν + b a(ki )φ dk +

Hˆ κ +

i=1

i=1

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A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

! 1 2

+ kR (−b)φk

2

2 !− 12 j j

X Y

−τ Pj,j ω(ki )ν + b a(ki )φ dk +

Hˆ κ +

r Z X

6 c

j =1

i=1

i=1

! 1

+ kR 2 (−b)φk2 .

(4.13)

(4.13) follows by (4.9). Now using the fact that Hˆ κ +

j X

!− 12 ω(ki ) + b

H0 +

ν

j X

i=1

! 12 ω(ki ) + b ν

i=1

is bounded uniformly in ki and Corollary 2.2, we obtain 1

r

2 kNν2 N−τ R(b)φk2

r X

6 c

!

k(H0 + b)

− 12

j 2

1 2

N−τ φk2 + kR (−b)φk2

j =1 r X

6 c

(4.14)

! 1 2

j−2 2

1 2

kNν N−τ φk2 + kR (−b)φk2 .

(4.15)

j =1 1

1

In fact, using the fact that N−τ (H0 + b)− 2 (1 + Nν )− 2 is bounded, we see that the right-hand side in (4.14) is less than (4.15), which holds for all φ ∈ (H0 + 1 r−1 1)− 2 D(N 2 ). Since D ⊗ 0fin (C0∞ (R3 )) ⊂ (H0 + 1)− 2 D(N 1

1

r−1 2

r−2

)

is a core for Nν2 N−τ2 , we see that (H0 + 1)− 2 D(N 1

r−2

1

r−1 2

1

r−2

) is dense in D(Nν2 N−τ2 )

and hence (4.15) holds for all φ ∈ D(Nν2 N−τ2 ). Now let r+1 φ := (Hˆ κ + b)ψ, ψ ∈ D((Hˆ κ + b) 2 )

and b > 0. Then 1

r−1 r−2 φ ∈ D((Hˆ κ + b) 2 ) ⊂ D(Nν2 N 2 ).

The induction hypothesis and (4.15) give 1

r

r+1 2 kNν2 N−τ ψk 6 c k(Hˆ κ + b) 2 ψk,

r+1 ψ ∈ D((Hˆ κ + b) 2 ).

2

244

ZIED AMMARI

COROLLARY 4.8. Let γ > 0 and < 1/r, where r ∈ N. Then Nr 6 c (Hˆ κ + b)r , 1−γ

H0

N r−1+γ 6 c (Hˆ κ + b)r ,

(4.16) (4.17)

where c and b are constants independent of κ. Proof. The inequalities follow from Lemma 2.5 and Theorem 4.7. (4.16) follows with ν = 1, τ = (1 − r)/(1 − r) in Theorem 4.7 and b = ω, τ1 = , τ2 = 1, τ3 = −(1 − r)/(1 − r), α1 = r, α2 = 1, α3 = r − 1 in Lemma 2.5. (4.17) follows with ν = 1, τ = γ /(r − 1) in Theorem 4.7 and b = ω, τ1 = 0, τ2 = 1, τ3 = −γ /(r − 1), α1 = r − 1 + γ , α2 = γ , α3 = r − 1 in Lemma 2.5 and the γ −1 fact that d0(ω)γ 6 H0 d0(ω). 2 Using the principle of the cutoff independence, formulated in Lemma 4.1, we deduce similar estimates for Hˆ ∞ . THEOREM 4.9. Let γ > 0 and < 1/r, where r ∈ N. c, b are positive constants. Then Nr 6 c (Hˆ ∞ + b)r , 1−γ

H0

N r−1+γ 6 c (Hˆ ∞ + b)r .

Theorem 4.9 for = 0 and Hadamard’s three lines lemma in [Ro] give the following corollary. COROLLARY 4.10. For r ∈ R+ there are c,b positive constants such that (N + 1)r 6 c (Hˆ ∞ + b)r . COROLLARY 4.11. For r ∈ R+ there are c,b positive constants such that (N + 1)r 6 c (H + b)r . Proof. If we prove that N r U∞ (N + 1)−r is bounded for r positive integer, the corollary follows from Corollary 4.10. We have on D(N) the identity ∗ U∞ NU∞ = N − iφ(iG∞ ) − 12 kG∞ k2 .

So U∞ preserves D(N). By iteration we have on D(N r ) r ∗ U∞ N r U∞ = N − iφ(iG∞ ) − 12 kG∞ k2 . Then we obtain the boundness of the operator N r U∞ (N + 1)−r , since r N − iφ(iG∞ ) − 12 kG∞ k2 (N + 1)−r , is bounded.

2

245

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

4.2.

NUMBER - ENERGY ESTIMATES

We say that a sequence of operators Aκ (R), R ∈ R or C is of class O(R γ ) (resp. o(R γ )) uniformly in κ, if there exists c constant independent from κ and R such that kAκ (R)k 6 cR γ (resp. kAκ (R)kR −γ → 0, when R → ∞). LEMMA 4.12. We have uniformly in κ for z in a bounded set of C \ R and m ∈ N, γ > 0: (i) (H0 + 1) 2 (N + 1) 2 +m (z − Hˆ κ )−k (N + 1)−m+k−1 ∈ O(|Im(z)|cm,k ). γ −2 1−γ (ii) (N + 1)m (z − Hˆ κ )−k (N + 1)−m+k+ 2 (H0 + 1) 2 ∈ O(|Im(z)|cm,k ). (iii) Let χ ∈ C0∞ (R) and n, q ∈ N: γ

1−γ

kN n χ(Hˆ κ )N q k < ∞. Proof. We recall the identity (4.4) on D(N k ), which had been proved in the proof of Lemma 4.4 N k (z − Hˆ κ )−1 = N k−1 (z − Hˆ κ )−1 N − iN k−1 (z − Hˆ κ )−1 ad1N Hˆ κ (z − Hˆ κ )−1 . We move now all factors of N in each term to the right, we obtain the following identity between bounded operators N k (z − Hˆ κ )−1 N −k = (z − Hˆ κ )−1 +

k X

(z − Hˆ κ )− 2 B` (z)(z − Hˆ κ )− 2 N −` , 1

1

`=1 1 1 j where B` (z) is a polynomial in (z − Hˆ κ )− 2 adN Hˆ κ (z − Hˆ κ )− 2 , j 6 `. Using Lemmas 3.3 and 4.2, we see that kB` (z)k 6 c|Im(z)|−c` , uniformly in κ. Using 1−γ γ 1 1 Corollary 4.8, (4.17), we see that kH0 2 N 2 (z − Hˆ κ )− 2 k 6 |Im(z)|− 2 , which proves (i) for k = 1. For k 6= 1, we write 1−γ

H0 2 N m+ 2 (z − Hˆ κ )−k N −m+k−1 γ

1−γ

= H0 2 N m+ 2 (z − Hˆ κ )−1 N −m γ

k Y

N m−` (z − Hˆ κ )−1 N −m+`−1 .

`=0

This proves (i) for k 6= 1. The proof of (ii) is similar to (i). (iii) follows from the higher-order estimates in Theorem 4.7 with τ = ν = 0. 2 We set: H ext := H ⊗ 0(h), Hˆ κext = Hˆ κ ⊗ 1 + 1 ⊗ d0(ω), N0 := N ⊗ 1, N∞ := 1 ⊗ N, acting in H ext . LEMMA 4.13. We have uniformly in κ and z in a bounded set of C \ R and m ∈ N

246

ZIED AMMARI

(i) (N0 + N∞ )m (z − Hˆ κext )−1 (N0 + N∞ )−m+1 ∈ O(|Im(z)|−cm ). 1 (ii) (H0ext + 1) 2 (N0 + N∞ )m (z − Hˆ κext)−1 (N0 + N∞ )−m ∈ O(|Im(z)|−cm ). Proof. The proof is analogous to the proof of Lemma 4.12. In fact we have ad`N0 +N∞ Hˆ κext = ad`N Hˆ κ ⊗ 1, (N0 + N∞ )m (z − Hˆ κext )−1 (N0 + N∞ )−m = (z − Hˆ κext )−1 +

m X

(z − Hˆ κext )− 2 B` (z) (z − Hˆ κext )− 2 (N0 + N∞ )−` , 1

1

`=1

where B` (z) ∈ O(|Imz|−c` ).

4.3.

2

COMMUTATOR ESTIMATES

Let q ∈ C0∞ (R3 ), 0 6 q 6 1, q = 1 near 0. We set q R := q(x/R). We recall that we consider h in its momentum representation L2 (R3 , dk) and x = ∇k /i. We use the following functional calculus formula, see [DG1], for χ ∈ C0∞ (R) and A a self-adjoint operator: Z i χ(A) = ∂z¯ χ(z)(z ˘ − A)−1 dz ∧ d¯z, (4.18) 2π C where χ˘ is an almost analytic extension of χ, such that χ˘|R = χ,

|∂z¯ χ(z)| ˘ 6 cn |Imz|n ,

n ∈ N.

LEMMA 4.14. Let χ ∈ C0∞ (R), then one has uniformly for κ 6 ∞ N n [χ(Hˆ κ ), 0(q R )]N m ∈ O(R −1 ). Proof. Commutation relations allow to compute [Hˆ κ , 0(q R )] as a sesquilinear form on D(H0 ), which by Nτ -estimates is a bounded operator on D(H0), when κ < ∞. We have [H0 , 0(q R )] = d0(q R , [ω, q R ]), 1 1 [φ(vκ0 ), 0(q R )] = √ a ∗ ((1 − q R )vκ0 )0(q R ) − √ 0(q R )a((1 − q R )vκ0 ), 2 2 [a 2 (rκj ), 0(q R )] = −0(q R )a((1 − q R )rκj )a((1 + q R )rκj ), [a ∗ (rκj ), 0(q R )] = a ∗ ((1 − q R )rκj )a ∗ ((1 + q R )rκj )0(q R ), 2

(4.19)

247

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

[a ∗ (rκj )a(rκj ), 0(q R )] = −a ∗ (rκj )0(q R )a((1 − q R )rκj ) + a ∗ ((1 − q R )rκj )0(q R )a(rκj ), [a ∗ (rκj )Dj , 0(q R )] = a ∗ ((1 − q R )rκj )0(q R )Dj , [Dj a(rκj ), 0(q R )] = −Dj 0(q R )a((1 − q R )rκj ). Let χ1 ∈ C0∞ (R) such that χ1 χ = χ. Using (4.18), we have N n [χ(Hˆ κ ), 0(q R )]N m = N n χ1 (Hˆ κ )[χ(Hˆ κ ), 0(q R )]N m + N n [χ1 (Hˆ κ ), 0(q R )]χ(Hˆ κ )N m Z i n = ∂z¯ χ(z)N ˘ χ1 (Hˆ κ )(z − Hˆ κ )−1 × 2π C (4.20) × [Hˆ κ , 0(q R )](z − Hˆ κ )−1 N m dz ∧ d¯z + Z i + ∂z¯ χ˘ 1 (z)N n (z − Hˆ κ )−1 [Hˆ κ , 0(q R )] × 2π C × (z − Hˆ κ )−1 χ(Hˆ κ )N m dz ∧ d¯z. Moving the power of N toward χ(Hˆ κ ), χ1 (Hˆ κ ) and then using Lemma 4.12 and Corollary 4.8, we see that it is enough to show that for b > 0, (N + 1)−n (H0 + 1 1 b)− 2 [Hˆ κ , 0(q R )](H0 + b)− 2 ∈ O(R −1 ), uniformly in κ, to have the lemma. Using now Lemma 3.4, we obtain 1 1 k(N + 1)−n (H0 + b)− 2 [Hˆ κ , 0(q R )](H0 + b)− 2 k 1

6 c(k(1 − q R ) (V + 1)− 2 vκ0 k + s

+ k(V + 1)− 2 ω

s−1 4

(1 − q R )rκ kkω

s−1 4

(1 + q R )rκ k +

+ kN −1 d0(q R , [ω, q R ])k + + k(V + 1)− 2 ω

s

s−1 2

+ k(V + 1)− 2 ω

s−1 2

s

(4.21) s

(1 − q R )rκ kk(V + 1)− 2 ω

s−1 2

rκ k +

(1 − q R )rκ kk(K + i)− 2 Dk). 1

Using the inequality (2.3) recalled in Subsection 2.1, kN −1 d0(q R , [ω, q R ])k 6 k[ω, q R ]k, and the fact that [ω, q R ] ∈ O(R −1), we see that kN −1 d0(q R , [ω, q R ])k ∈ O(R −1 ). Now for the other kind of terms we will proceed as follows. Since V > hxiα , α >

248

ZIED AMMARI s

2, we can pick µ > 0 and s < 1, such that (V + 1)− 2 hxi1+µ is bounded. Then using Lemma A.2, we obtain khxi1+µ ω

s−1 2

(1 − q R )rκ k ∈ O(R −1−µ ),

khxi1+µ ω

s−1 4

(1 − q R )rκ k ∈ O(R −1−µ ).

Hence we have: k(N + 1)−n (H0 + b)− 2 [Iˆκ , 0(q R )](H0 + b)− 2 k ∈ O(R −1−µ ). 1

1

Then the integrand in (4.20) is |Im(z)|−2 O(R −1 ). This ends the proof.

(4.22) 2

2 Let j0 ∈ C0∞ (R3 ), j∞ ∈ C ∞ (R3 ), 0 6 j0 , 0 6 j∞ , j02 + j∞ 6 1, j0 = 1 near R R R 0. We set for R > 1, j := (j0 , j∞ ), where x x R R j0 := j0 , j∞ := j∞ . R R

We set j := j 1 . LEMMA 4.15. One has uniformly for κ 6 ∞ (i) χ(Hˆ κext ) I ∗ (j R ) − I ∗ (j R ) χ(Hˆ κ ) ∈ O(R −1 ). (ii) Let χ ∈ C0∞ (R), then (N0 + N∞ )n (χ(Hˆ κext ) I ∗ (j R ) − I ∗ (j R ) χ(Hˆ κ )) N m ∈ O(R −1). Proof. The proof is similar to the previous one. Instead of (4.20), we use the identities: H0ext I ∗ (j R ) − I ∗ (j R )H0 R = dI ∗ (j R , [ω, j R ]), where [ω, j R ] = ([ω, j0R ], [ω, j∞ ]), φ(vκ0 ) ⊗ 1I ∗ (j R ) − I ∗ (j R )φ(vκ0 ) R ∗ R ˜ = φ((1 − j0R )vκ0 ) ⊗ 1I ∗ (j R ) − 1⊗φ(j ∞ vκ0 )I (j ), a ∗ (rκj )Dj ⊗ 1I ∗ (j R ) − I ∗ (j R )a ∗ (rκj )Dj R j ˜ ∗ (j∞ = a ∗ ((1 − j0R )rκj )Dj ⊗ 1I ∗ (j R ) − 1⊗a rκ )Dj I ∗ (j R ), Dj a(rκj ) ⊗ 1I ∗ (j R ) − I ∗ (j R )Dj a(rκj ) R j ˜ j a(j∞ = Dj a((1 − j0R )rκj ) ⊗ 1I ∗ (j R ) − 1⊗D rκ )I ∗ (j R ), a 2 (rκj ) ⊗ 1I ∗ (j R ) − I ∗ (j R )a 2 (rκj ) = −I ∗ (j R )a((1 − j0R )rκj )a((1 + j0R )rκj ), a ∗ (rκj ) ⊗ 1I ∗ (j R ) − I ∗ (j R )a ∗ (rκj ) 2

2

249

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT R j R j ˜ ∗ (j∞ = (−2a ∗ (j0R rκj ) ⊗ a ∗ (j∞ rκ ) − 1⊗a rκ )) × I ∗ (j R ), 2

a ∗ (rκj )a(rκj ) ⊗ 1I ∗ (j R ) − I ∗ (j R )a ∗ (rκj )a(rκj ) = a ∗ (rκj ) ⊗ 1I ∗ (j R )a((−1 + j0R )rκj ) + R j ˜ ∗ (j∞ + (a ∗ ((1 − j0R )rκj ) ⊗ 1 + 1⊗a rκ )) × I ∗ (j R )a(rκj ). We notice that for v ∈ B(K, K ⊗ h), h1 = h2 = h, we define ˜ ] (v) : K ⊗ 0(h1 ) ⊗ 0(h2 ) → K ⊗ 0(h1 ) ⊗ 0(h2 ), 1⊗a ˜ ] (v) := T −1 10(h1) ⊗ a ] (v) T , 1⊗a where T is the natural identification: T : K ⊗ 0(h1 ) ⊗ 0(h2 ) → 0(h1 ) ⊗ K ⊗ 0(h2 ). 2 Using (2.4) with j02 + j∞ 6 1, we see that (H0ext I ∗ (j R ) − I ∗ (j R )H0 )(N + 1)−1 is bounded, which shows that I ∗ (j R ) : D(H0 ) → D(H0ext ). We obtain the following identity on H

C(z) := (z − Hˆ κext)−1 I ∗ (j R ) − I ∗ (j R )(z − Hˆ κ )−1 = (z − Hˆ κext)−1 (Hˆ κext I ∗ (j R ) − I ∗ (j R )Hˆ κ )(z − Hˆ κ )−1 . Using (4.18), we obtain: χ(Hˆ κext ) I ∗ (j R ) − I ∗ (j R ) χ(Hˆ κ ) =

i 2π

Z C

∂z¯ χ˘ (z) C(z) dz ∧ d¯z.

Using Corollary 4.8 and Lemma 3.4 we obtain for b < 0 kC(b)k 6 c(k(N∞ + N0 )−1 dI ∗ (j R , [j R , ω])k + k(V + 1)− 2 ω s

+ k(V + 1)− 2 ω s

s−1 2

R j∞ vκ0 k +

s

s−1 4

(1 − j0R )rκ k kω

s

s−1 4

R j∞ rκ k kω

s

s−1 2

rκ k × k(V + 1)− 2 ω

+ k(V + 1)− 2 ω + k(V + 1)− 2 ω + k(V + 1)− 2 ω s

+ (k(V + 1)− 2 ω

s−1 4

s−1 4

(1 − j0R )vκ0 k +

(1 + j0R )rκ k + s

j0R rκ k + k(V + 1)− 4 ω s

s−1 2

s−1 2

s−1 2

s−1 4

R j∞ rκ k2 +

(1 − j0R )rκ k + s

(1 − j0R )rκ k + k(V + 1)− 2 ω

s−1 2

R j∞ rκ k) ×

1

× k(K + i)− 2 Dk + s

+ k(V + 1)− 2 ω

s−1 2

s

rκ k × k(V + 1)− 2 ω

s−1 2

R j∞ rκ k).

(4.23)

250

ZIED AMMARI

R Applying (2.4) with [j0R , ω]2 + [j∞ , ω]2 ∈ O(R −2 ), we obtain

k(b − Hˆ κext )−1 (H0extI ∗ (j R ) − I ∗ (j R )H0 )(b − Hˆ κ )−1 k ∈ O(R −1 ). For the other terms of C(b), we use (4.23), the fact that we can pick µ > 0 and s s < 1 such that (V + 1)− 2 hxi1+µ is bounded and Lemma A.2. We obtain: s

s−1 2

fR rκ k ∈ O(R −1−µ ),

s

s−1 4

fR rκ k ∈ O(R −1−µ ),

s

s−1 4

R j∞ rκ k2 ∈ O(R −1−µ),

k(V + 1)− 2 ω k(V + 1)− 2 ω k(V + 1)− 4 ω

(4.24)

R where fR denotes j∞ or 1 − j0R . Then we have

k(b − Hˆ κext )−1 × (Iˆκ ⊗ 1 I ∗ (j R ) − I ∗ (j R )Iˆκ )(b − Hˆ κ )−1 k ∈ O(R −1−µ).

(4.25)

Hence we have kC(z)k ∈ |Im(z)|−2 O(R −1 ). This proves (i). Let χ1 ∈ C0∞ (R) such that χ1 χ = χ. As in the previous lemma, we have using (4.18): (N0 + N∞ )n χ(Hˆ κext )I ∗ (j R ) − I ∗ (j R )χ(Hˆ κ )N m = (N0 + N∞ )n χ1 (Hˆ κext )(χ(Hˆ κext )I ∗ (j R ) − I ∗ (j R )χ(Hˆ κ ))N m + + (N0 + N∞ )n (χ1 (Hˆ κext)I ∗ (j R ) − I ∗ (j R )χ1 (Hˆ κ ))χ(Hˆ κ )N m Z i = ∂z¯ χ˘ (z)(N0 + N∞ )n χ1 (Hˆ κext)C(z)N m dz ∧ d¯z + 2 C Z i + ∂z¯ χ˘ 1 (z)(N0 + N∞ )n C(z)χ(Hˆ κ )N m dz ∧ d¯z. 2 C Moving (N0 + N∞ )n (resp. N m ) toward χ(Hˆ κ ) (resp. χ1 (Hˆ κext )) in the last expression and then using (i) and Lemma 4.13 we prove (ii). 2

5. Spectral Theory for the Nelson Hamiltonian We study in this section the spectral properties of both Nelson and modified Hamiltonians. In Subsection 5.1 we prove the existence of ground state for the Nelson Hamiltonian. We use essentially the fact that Hκ − Eκ are Pauli–Fierz Hamiltonian which converge in the norm resolvent sense to H . This is the subject of Theorem 5.1. In Subsection 5.2 we prove a Mourre estimate for the modified Hamiltonian, which gives that pure point spectrum is locally finite outside its thresholds.

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

5.1.

251

HVZ THEOREM

THEOREM 5.1. One has σess (H ) = [inf σ (H ) + m, +∞[, and inf σ (H ) is a discrete eigenvalue of H . Proof. Hκ is an example of a Pauli–Fierz Hamiltonian, see [DG2, Section 3]. The HVZ theorem proved in [DG2] for the Pauli–Fierz Hamiltonians gives for κ<∞ σess (Hˆ κ ) = σess (Hκ − Eκ ) = [inf σ (Hˆ κ ) + m, +∞[. Using the fact that (z − Hˆ κ )−1 → (z − Hˆ ∞ )−1 and [RS, I–IV, Thm. VIII.23], we see that lim (inf σ (Hˆ κ )) = inf σ (Hˆ ∞). (5.1) κ→∞

Let χ ∈ C0∞ (] − ∞, inf σ (Hˆ ∞) + m[), then χ ∈ C0∞ (] − ∞, inf σ (Hˆ κn ) + m[) for a sequence κn → +∞. By the HVZ theorem for Hˆ κn , χ(Hˆ κn ) is compact and by Theorem 3.6 χ(Hˆ ∞ ) is compact. So we obtain σess (Hˆ ∞ ) ⊂ [inf σ (Hˆ ∞) + m, +∞[. Let us now show that [inf σ (Hˆ ∞ ) + m, +∞[ ⊂ σess (Hˆ ∞ ). Let λ such that λ > inf σ (Hˆ ∞ ) + m. By (5.1) and the HVZ theorem for Hˆ κ , there exists a sequence κn → +∞ such that λ ∈ σess (Hˆ κn ), or equivalently µ := (λ + c)−1 ∈ σess((Hˆ κn + c)−1 ),

c 1.

Let us show that µ ∈ σess ((Hˆ ∞ + c)−1 ), i.e. :λ ∈ σess (Hˆ ∞ ): Assume the contrary and let χ ∈ C0∞ (R), such that χ(µ) = 1, χ((Hˆ ∞ + c)−1 ) compact. Let ϕκn ,j be Weyl sequences for (Hˆ κn + c)−1 at µ such that kϕκn ,j k = 1,

lim ((Hˆ κn + c)−1 − µ)ϕκn ,j = 0,

j →∞

and w- lim ϕκn ,j = 0. j →∞

One has kχ((Hˆ ∞ + c)−1 )ϕκn ,j − ϕκn ,j k 6 kχ((Hˆ κn + c)−1 ) − χ((Hˆ ∞ + c)−1 )k + kχ((Hˆ κn + c)−1 )ϕκn ,j − ϕκn ,j k. Since χ((Hˆ ∞ + c)−1 ) compact, there exists for > 0, a κ1 and j1 such that kχ((Hˆ ∞ + c)−1 )ϕκ1 ,j1 − ϕκ1 ,j1 k 6 2, kχ((Hˆ ∞ + c)−1 )ϕκ1 ,j1 k 6 . We obtain kϕκ1 ,j1 k 6 3, this gives a contradiction, if we choose < 1/3.

2

252 5.2.

ZIED AMMARI

MOURRE ESTIMATE

We denote by b the operator acting on h defined by 1 b := (∇ω . Dk + Dk . ∇ω), 2

on C0∞ (R3 ).

[ABG, Prop. 4.2.3] yields that the closure of b is the infinitesimal generator of the strongly continuous unitary group Ut associated to the vector field ∇ω in the following sense 1

Ut F := [det∇φ−t (k)] 2 F (φ−t (k)),

F ∈ S 0 (R3 ),

(5.2)

where φt is the flow of the vector field ∇ω. Moreover C0∞ (R3 ) is a core for b. We denote in the sequel by b its closure. We set B := d0(b). Clearly B is essentially selfadjoint on 0fin (C0∞ (R3 )). We denote by τ the set of thresholds, τ := σpp (Hˆ ∞ ) + mN∗ . Let S be a selfadjoint operator on H , we say that S is of class C 1 (B), see [ABG], if the map t 7→ eit B (S − z)−1 e−it B , is strongly C 1 for some z ∈ C \ σ (S). By [ABG, Lemma 6.2.9] S ∈ C 1 (B) if and only if the sesquilinear form [B, (z−S)−1 ] on D(B) is continuous for the topology of H , i.e: |((S − z)−1 ϕ, Bϕ) − (Bϕ, (S − z)−1 ϕ)| 6 c kϕk2

for ϕ ∈ D(B).

(5.3)

We recall here a well known theorem, see [ABG, Thm. 6.2.10]. THEOREM 5.2. Let S, B two selfadjoint operators acting on Hilbert space. If S ∈ C 1 (B) then (i) D(S) ∩ D(B) is dense in D(S), (ii) ([B, S]u, u) 6 c kSuk2 , u ∈ D(S) ∩ D(B), (iii) [B, (z − S)−1 ] = (z − S)−1 [B, S](z − S)−1 , where (iii) is understood as identity between bounded operators in the following sense: (z−S)−1

[B,S]

(z−S)−1

H −→ D(S) −→ D(S)∗ −→ H .

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

253

LEMMA 5.3. Hˆ κ is of class C 1 (B) if κ < ∞. Proof. We will prove (i) eit B preserves D(H0 ). (ii) |(Hˆ κ u, Bu) − (Bu, Hˆ κ u)| 6 c kH0 uk2 + kuk2 , u ∈ D(H0 ) ∩ D(B). [ABG, Thm. 6.3.4] and [ABG, Prop. 6.3.5] give that (i), (ii) implies Hˆ κ ∈ C 1 (B). Let us prove (i). It is enough to show that H0 eit B (H0 + 1)−1 e−it B is bounded to have (i). We have using (5.2) eit B d0(ω)e−it B = d0(eit b ωe−it b ) = d0(ω(φ− 2t (k))). Since ∇ω is a bounded complete P C ∞ vector field, we have |φt (k) − k| 6 c |t| uniformly in k. So this implies | N 1 ω(φt (ki )) − ω(ki )| 6 c |t| N and, hence, H0 eit B (H0 + 1)−1 e−it B is bounded. This prove (i). Let us prove (ii). We compute [Hˆ κ , iB]: (iBu, Hˆ κ u) − (iHˆ κ u, Bu) P P X 1 X j = (d0(|∇ω| )u, u) − (φ(ibvκ0 )u, u) + (ia(rκj )a(brκj )u, u) − 2M j =1 j =1 2

− (ia ∗ (rκj )a ∗ (brκj )u, u) + (ia ∗ (rκj )a(brκj ) − ia ∗ (brκj )a(rκj )u, u) − √ √ − 2(iDj a(brκj )u, u) + 2(ia ∗ (brκj )Dj u, u). A simple computation yields (i)

hxi−1 bvκ0 ∈ B(K, K ⊗ h),

(ii)

hxi−1 brκ ∈ B(K, K ⊗ h)

(iii)

−

for κ < ∞,

−1

hki hxi brκ ∈ B(K, K ⊗ h)

(5.4)

for > 0, uniformly in κ.

Now using Lemma 3.4 with β = 1 and (5.4)(i)–(ii) we obtain for u ∈ D(H0 ) ∩ D(B) |(iBu, Hˆ κ u) − (iHˆ κ u, Bu)| 1

1

6 c(krκ k k(V + 1)− 2 brκ k + k(V + 1)− 2 bvκ0 k + 1

1

+ k(K + i)− 2 Dk k(V + 1)− 2 brκ k) × (kH0 uk2 + kuk2 ). P Since V > i hxi iα , α > 2, we prove (ii). This completes the proof. LEMMA 5.4. Hˆ ∞ is of class C 1 (B).

2

254

ZIED AMMARI

Proof. Since Hˆ κ is of class C 1 (B) we know by Theorem 5.2 that (z − Hˆ κ )−1 : D(B) → D(B) and (z − Hˆ κ )−1 [Hˆ κ , iB](z − Hˆ κ )−1 = [(z − Hˆ κ )−1 , iB],

on H .

For φ ∈ D(B), one has (i(Hˆ κ + i)−1 φ, Bφ) − (iBφ, (Hˆ κ − i)−1 φ) = ((Hˆ κ + i)−1 φ, [Hˆ κ , iB](Hˆ κ − i)−1 φ). Using Lemma 3.3 and (4.17) we obtain |((Hˆ κ + i)−1 φ, Bφ) − (Bφ, (Hˆ κ − i)−1 φ)| 6 c (kd0(|∇ω|2 )(N + 1)−1 k + khki− rκ k khki− hxi−1 brκ k + + khxi−1 bvκ0 k + k(K + i)− 2 Dk khki− hxi−1 brκ k)kφk2 , 1

> 0,

where c is independent from κ. Then letting κ → ∞ and using Theorem 3.6 we obtain |((Hˆ ∞ + i)−1 φ, iBφ) − (iBφ, (Hˆ ∞ − i)−1 φ)| 6 c kφk2

for φ ∈ D(B).

This implies Hˆ ∞ ∈ C 1 (B).

2

LEMMA 5.5. We have for χ ∈ C0∞ (R) (i) (ii)

w-limκ→∞ [(Hˆ κ − i)−1 , iB] = [(Hˆ ∞ − i)−1 , iB], limκ→∞ χ(Hˆ κ )[Hˆ κ , iB]χ(Hˆ κ ) = χ(Hˆ ∞ )[Hˆ ∞ , iB]χ(Hˆ ∞ ).

Proof. (i) follows from the proof of Lemma 5.4. Let us prove (ii): χ(Hˆ κ )[Hˆ κ , iB]χ(Hˆ κ ) − χ(Hˆ κ 0 )[Hˆ κ 0 , iB]χ(Hˆ κ 0 ) = (χ(Hˆ κ ) − χ(Hˆ κ 0 ))[Hˆ κ , iB]χ(Hˆ κ ) + + χ(Hˆ κ 0 )([Hˆ κ , iB] − [Hˆ κ 0 , iB])χ(Hˆ κ ) +

(5.5)

+ χ(Hˆ κ )[Hˆ κ 0 , iB](χ(Hˆ κ ) − χ(Hˆ κ 0 )). We first claim that 1

1

lim χ(Hˆ κ )H02 = χ(Hˆ ∞ )H02 .

(5.6)

κ→∞

To have (5.6), we see using the functional calculus formula (4.18), that it is enough to show that 1

1

lim (Hˆ κ − z)−1 H02 = (Hˆ ∞ − z)−1 H02 .

κ→∞

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

255

For κ, κ 0 < ∞, we have the following operator identity on H (Hˆ κ − z)−1 − (Hˆ κ 0 − z)−1 = (Hˆ κ − z)−1 (Hˆ κ 0 − Hˆ κ )(Hˆ κ 0 − z)−1 . It follows from the proof of Theorem 3.6 that 1 1 lim (H0 + 1)− 2 (Hˆ κ 0 − Hˆ κ )(H0 + 1)− 2 = 0.

κ,κ 0 →∞

Then we obtain (5.6). We also claim that lim (H0 + 1)− 2 [Hˆ κ , iB](H0 + 1)− 2 (N + 1)−1 1

1

κ→∞

(5.7)

1 1 = (H0 + 1)− 2 [Hˆ ∞ , iB](H0 + 1)− 2 (N + 1)−1 . P In fact, using Lemma 3.4 and the fact that V > c i hxi i2 , we get

k(H0 + i)− 2 a ∗ (brκ 0 − brκ )a ∗ (rκ 0 )(H0 + i)− 2 N −1 k 1

1

6 khki− hxi−1 (brκ 0 − brκ )k khki− rκ 0 k, k(H0 + i)− 2 a(rκ 0 − rκ )a(brκ 0 )(H0 + i)− 2 N −1 k 1

1

6 khki− (rκ 0 − rκ )k khki− hxi−1 brκ 0 k, k(H0 + i)− 2 a ∗ (brκ 0 − brκ )a(rκ 0 )(H0 + i)− 2 N −1 k 1

1

6 khki− hxi−1 (brκ 0 − brκ )k khki− rκ 0 k, k(H0 + i)− 2 a ∗ (brκ 0 − brκ )Dj (H0 + i)− 2 N −1 k 1

1

1

6 khki− hxi−1 (brκ 0 − brκ )k k(K + 1)− 2 Dk, 1

1

k(H0 + i)− 2 Dj a(brκ 0 − brκ )(H0 + i)− 2 N −1 k 6 khki− hxi−1 (brκ 0 − brκ )k k(K + 1)− 2 Dk. 1

Using these estimates and (5.4), we obtain (5.7). Now using (5.5), (5.6) and (5.7) we obtain (ii). 2 We have to prove a localization estimate for [Hˆ κ , iB] similar to the one in the Lemma 4.15. LEMMA 5.6. We have uniformly in κ: χ(Hˆ κext ) [Hˆ κext , iB ext ]I ∗ (j R ) − I ∗ (j R )[Hˆ κ , iB] χ(Hˆ κ ) ∈ o(R 0 ).

256

ZIED AMMARI

Proof. We set C(z) := (z − Hˆ κext)−1 ([Hˆ κext , iB ext ]I ∗ (j R ) − I ∗ (j R )[Hˆ κ , iB])(z − Hˆ κ )−1 , where B ext := B ⊗ 1 + 1 ⊗ B. A simple computation gives [H0ext , iB ext ]I ∗ (j R ) − I ∗ (j R )[H0 , iB] = dI ∗ (j R , [|∇ω|2 , j R ]), a(rκj )a(brκj ) ⊗ 1I ∗ (j R ) − I ∗ (j R )a(rκj )a(brκj ) = I ∗ (j R )(a((j0R − 1)rκj )a(j0R brκj ) + a(rκj )a((j0 − 1)brκj )), a ∗ (rκj )a ∗ (brκj ) ⊗ 1I ∗ (j R ) − I ∗ (j R )a ∗ (rκj )a ∗ (brκj ) R R j = −(a ∗ (j0R rκj ) ⊗ a ∗ (j∞ brκj ) + a ∗ (j0R brκj ) ⊗ a ∗ (j∞ rκ ) + R j + 1 ⊗ a ∗ (j∞ rκ )a ∗ (j∞ brκj )).

We have also similar identities for a ∗ a, aDj , Dj a ∗ , replacing rκj by brκj , in the proof of Lemma 4.15. As in the proof of Lemma 4.15 we use Corollary 4.8 and Lemma 3.3. We have for β < 0: kC(β)k 6 c(k(N0 + N∞ )−1 dI ∗ (j R , [|∇ω|2 , j R ])k + + khki− hxi−1 (1 − j0R )rκ k khki− hxi−1 j0R brκ k + + khki− hxi−1 (1 − j0R )brκ k khki− hxi−1 j0R rκ k + R + khki− hxi−1 j0R rκ k khki− hxi−1 j∞ brκ k + R + khki− hxi−1 j0R brκ k khki− hxi−1 j∞ rκ k + R R + khki− hxi−1 j∞ brκ k khki− hxi−1 j∞ rκ k +

+ khki− hxi−1 brκ k khki− hxi−1 (1 − j0R )rκ k + + khki− hxi−1 rκ k khki− hxi−1 (1 − j0R )brκ k + R + khki− hxi−1 rκ k khki− hxi−1 j∞ brκ k + R + khki− hxi−1 brκ k khki− hxi−1 j∞ rκ k + R + khki− hxi−1 (1 − j0R )brκ k + khki− hxi−1 j∞ brκ k).

Using Lemma A.2, Lemma 3.3 and 5.4(iii) we obtain C(z) ∈ |Im(z)|−2 o(R 0 ),

uniformly in κ.

THEOREM 5.7. The following three assertions hold:

2

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

257

(i) Let λ ∈ R \ τ. Then there exist > 0, C0 > 0 and compact operator K0 such that 1[λ−,λ+](Hˆ ∞ )[Hˆ ∞ , iB]1[λ−,λ+] (Hˆ ∞ ) > C0 1[λ−,λ+](Hˆ ∞ ) + K0 . (ii) For all [λ1 , λ2 ] such that [λ1 , λ2 ] ∩ τ = ∅, one has pp dim 1[λ1 ,λ2 ] (Hˆ ∞ )H < ∞.

Consequently σpp(Hˆ ∞ ) can accumulate only at τ , which is a closed countable set. (iii) Let λ ∈ R \ (τ ∪ σpp (Hˆ ∞ )). Then there exists > 0, C0 > 0 such that 1[λ−,λ+](Hˆ ∞ )[Hˆ ∞ , iB]1[λ−,λ+] (Hˆ ∞ ) > C0 1[λ−,λ+](Hˆ ∞ ). Proof. We set ( n ) n X X d(λ) := inf |∇ω(ki )|2 ; τ + ω(ki ) = λ, n = 1, 2 . . . , τ ∈ σpp (Hˆ ∞ ) , ( ˜ d(λ) := inf

i=1

i=1

n X

n X

|∇ω(ki )|2 ; τ +

i=1

1µλ := [λ − µ, λ + µ],

) ω(ki ) = λ, n = 0, 1, . . . , τ ∈ σpp (Hˆ ∞ ) .

i=1

µ > 0,

d µ (λ) := infµ d(ν), ν∈1λ

˜ d˜µ (λ) := infµ d(ν), ν∈1λ

E0 := inf σ (Hˆ ∞). We will follow the logic of the proof of Mourre estimate in the case of a Pauli–Fierz Hamiltonian [DG2]. Let us recall the statements that we will prove by induction in n: H1 (n): Let > 0 and λ ∈ [E0 , E0 + nm[. There exists a compact operator K0 , an interval 1 3 λ such that 11 (Hˆ ∞ )[Hˆ ∞ , iB]11 (Hˆ ∞ ) > (d(λ) − )11 (Hˆ ∞ ) + K0 . H2 (n): Let > 0 and λ ∈ [E0 , E0 + nm[. There exists an interval 1 3 λ such that ˜ 11 (Hˆ ∞ )[Hˆ ∞ , iB]11 (Hˆ ∞ ) > (d(λ) − )11 (Hˆ ∞ ).

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H3 (n): Let µ > 0, 0 > 0. There exists δ > 0 such that for all λ ∈ [E0 , E0 + nm − 0 ], one has 11δλ (Hˆ ∞ )[Hˆ ∞ , iB]11δλ (Hˆ ∞ ) > (d˜µ (λ) − )11δλ (Hˆ ∞ ). S1 (n): τ is closed countable set in [E0 , E0 + nm]. S2 (n): For all λ1 6 λ2 6 E0 + nm with [λ1 , λ2 ] ∩ τ = ∅, one has dim 1[λ1 ,λ2 ] (Hˆ ∞ )H < ∞. pp

The sketch of the proof is given by S2 (n − 1) ⇒ S1 (n), (S1 (n), H3(n − 1)) ⇒ H1 (n), H1 (n) ⇒ H2 (n), H2 (n) ⇒ H3 (n), H1 (n) ⇒ S2 (n). H (1) and S(1) are immediate because the spectrum of Hˆ ∞ is discrete in [E0 , E0 + m[. S2 (n − 1) ⇒ S1 (n) is obvious. H1 (n) ⇒ H2 (n), H2 (n) ⇒ H3 (n) follow using arguments in [CFKS], [Mr]. The implication H1 (n) ⇒ S2 (n) is based in the Virial theorem which holds since Hˆ ∞ ∈ C 1 (B), see [ABG, Prop. 7.2.10]. So we have only to prove the implication (S1 (n), H3 (n − 1)) ⇒ H1 (n). Let χ ∈ C0∞ (R) χ(Hˆ κ ) = I (j R )1{0} (N∞ )I ∗ (j R )χ(Hˆ κ ) + I (j R )1[1,∞[ (N∞ )I ∗ (j R )χ(Hˆ κ ) = 0(q R )χ(Hˆ κ ) + I (j R )1[1,∞[ (N∞ )χ(Hˆ κext )I ∗ (j R ) + o(R 0 ).

(5.8) (5.9)

(5.8) follows from the fact that I (j R )I ∗ (j R ) = 1 and I (j R ) is bounded. (5.9) follows from the fact that 0(q R ) = I (j R )1{0} (N∞ )I ∗ (j R ),

q R = (j R )2 ; I (j R ) = 0(j R )U,

and Lemma 4.15. We notice that the term 0(q R )χ(Hˆ κ ) is compact since 0(q R )(H0 1 +1)− 2 is compact which is proved in [DG2, Lemma 4.2]. Let λ ∈ [E0 , E0 +nm[. Since S2 (n−1) ⇒ S1 (n), the set τ is closed in [E0 , E0 + nm], which gives d(λ) = supµ>0 d µ (λ). So we can choose µ such that d µ (λ) > d(λ) − 3 . H3 (n − 1) gives for λ1 < E0 + (n − 1)m µ 11δλ (Hˆ ∞ )[Hˆ ∞ , iB]11δλ (Hˆ ∞ ) > d˜ (λ1 ) − 1 δ (Hˆ ∞ ). 1 1 3 1λ

259

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

Replacing λ1 with λ − d0(ω(k)), we obtain 11δλ (Hˆ ∞ + 1 ⊗ d0(ω(k)))([Hˆ ∞, iB] + 1 ⊗ d0(|∇ω|2 )) × × 11δλ (Hˆ ∞ + 1 ⊗ d0(ω(k)))1[1,∞[ (N∞ ) > 11δλ (Hˆ ∞ + 1 ⊗ d0(ω(k))) × µ 2 × d˜ (λ − 1 ⊗ d0(ω(k))) + 1 ⊗ d0(|∇ω| ) − 1[1,∞[ (N∞ ) 3 µ > d (λ) − 1 δ (Hˆ ∞ + 1 ⊗ d0(ω(k)))(Hˆ ∞ + 1 ⊗ d0(ω))1[1,∞[ (N∞ ) 3 1λ 2 > d µ (λ) − 11δλ (Hˆ ∞ + 1 ⊗ d0(ω(k)))(Hˆ ∞ + 1 ⊗ d0(ω))1[1,∞[ (N∞ ). 3 Let χ ∈ C0∞ (R), χ1 ∈ C0∞ (R) such that χ1 χ = χ. One has uniformly in κ: χ(Hˆ κ )[Hˆ κ , iB]χ(Hˆ κ ) = 0(q R )χ(Hˆ κ )[Hˆ κ , iB]χ(Hˆ κ ) . . . + + I ∗ (j R )1[1,+∞[ (N∞ )χ(Hˆ κext )I ∗ (j R )χ1 (Hˆ κ )[Hˆ κ , iB]χ(Hˆ κ ) +

(5.10)

+ o(R 0 ) = 0(q R )χ(Hˆ κ )[Hˆ κ , iB]χ(Hˆ κ ) + + I ∗ (j R )1[1,+∞[ (N∞ )χ(Hˆ κext )I ∗ (j R )[Hˆ κ , iB]χ(Hˆ κ ) + o(R 0 )

(5.11)

= 0(q R )χ(Hˆ κ )[Hˆ κ , iB]χ(Hˆ κ ) + + I ∗ (j R )1[1,+∞[ (N∞ )χ(Hˆ κext )[Hˆ κext , iB ext ]χ(Hˆ κext )I ∗ (j R ) +

(5.12)

+ o(R 0 ). (5.10) follows by (5.9). Lemma 4.15(i) gives (5.11) and (5.12) follows by Lemma 5.6. Lemma 5.5(ii) proves that 0(q R )χ(Hˆ κ )[Hˆ κ , iB]χ(Hˆ κ ) → 0(q R )χ(Hˆ ∞ )[Hˆ ∞ , iB]χ(Hˆ ∞ ) norm limit. Now, letting κ → ∞ in the expression (5.12) which holds uniformly in κ and using the fact that 0(q R )χ(Hˆ κ )[Hˆ κ , iB]χ(Hˆ κ ) is compact, we obtain: χ(Hˆ ∞ )[Hˆ ∞ , iB]χ(Hˆ ∞ ) ext ext ext ∗ R = K1 (R) + I ∗ (j R )1[1,+∞[ (N∞ )χ(Hˆ ∞ )[Hˆ ∞ , iB ext ]χ(Hˆ ∞ )I (j ) + 0 + o(R ),

260

ZIED AMMARI

where K1 (R) is a compact operator. This gives for χ such that supp χ ⊂ [λ − δ, λ + δ[ 2 ˆ ˆ ˆ χ(H∞ )[H∞ , iB]χ(H∞ ) > d(λ) − χ 2 (Hˆ ∞ ) + K1 (R) + o(R 0 ). 3 Choosing R large enough, we obtain H1 (n). Properties (ii), (iii) are standard consequences of (i). 2

6. Construction of the Wave Operators 6.1.

ASYMPTOTIC FIELDS

In this subsection we prove the existence of asymptotic fields using the Cook method, see, e.g., [H-K]. We set ht := e−it ω(k) h, for h ∈ h and we denote by h0 the space C0∞ (R3 \ {0}). We introduce Heisenberg derivatives: IDκ := ∂t + i[Hκ , ·], ID := ∂t + i[H, ·]. Since the existence of asymptotic fields in time ±∞ is similar, we will restraint proofs of this subsection to the case +∞. THEOREM 6.1. (i) For h ∈ h the strong limits W ± (h) := s − lim eit H W (ht )e−it H t →±∞

(6.1)

exist and are called asymptotic Weyl operators. (ii) Furthermore W ± (h)(H + i)−1 = lim eit H W (ht )(H + i)−1 e−it H . t →±∞

(iii) The map h 3 h 7→ W ± (h)

is strongly continuous,

±

h 3 h 7→ W (h)(H + 1)−1

is norm continuous.

(iv) The Weyl commutation relations hold: W ± (h) W ± (g) = e 2 Im(h|g) W ± (h + g), i

W ± (h)∗ = W ± (−h). (v) The Hamiltonian preserves the asymptotic Weyl operators: eit H W ± (h)e−it H = W ± (h−t ).

(6.2)

261

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

Proof. We have the relation on H W (ht ) = e−it H0 W (h)eit H0 . Hence we can define ∂t W (ht ) as quadratic form on D(H0 ) ∂t W (ht ) = −i[H0 , W (ht )].

(6.3)

Using (6.3) and Theorem 2.9 we have, since Im(ht |vκ ) ∈ B(K), the following identity on H ∂t (eit Hκ W (ht )e−it Hκ ) = ieit Hκ Im(ht |vκ )W (ht )e−it Hκ .

(6.4)

We will first prove (6.1) and (6.2) for h ∈ h0 then we extend to h ∈ h. Let h ∈ h0 , we notice in this case that (hs |vκ ) = (hs |vκ1 ), for κ > κ1 . Since eit Hκ is a strongly continuous unitary group and using the inequality (2.1) we see that t 7→ eit Hκ Im(ht |vκ1 )W (ht )e−it Hκ is strongly measurable. Hence, by integrating (6.4) we obtain on H the identity: eit Hκ W (ht )e−it Hκ Z = W (h) + i

(6.5) t

eisHκ Im(hs |vκ1 )W (hs )e−isHκ ds,

h ∈ h0 .

0

Using Theorem 3.8 and the convergence dominated theorem, letting κ → ∞ in (6.5), we obtain eit H W (ht )e−it H

(6.6) Z

= W (h) + i

t

eisH Im(hs |vκ1 )W (hs )e−isH ds,

h ∈ h0 .

0

Moreover eit H W (ht )(H + i)−1 e−it H = W (h)(H + i)−1 + i

(6.7) Z

t

eisH Im(hs |vκ1 )W (hs )(H + i)−1 e−isH ds.

0

P Clearly there exists > 0 such that i hxi i1+ (H + i)−1 is bounded since P P 1+ ˆ (H∞ + i)−1 is bounded. Lemma A.2 gives that i hxi i−1− Im(hs |vκ1 ) ∈ i hxi i O(s −1− ). Then the existence of the limits (6.2) and consequently (6.1) for h ∈ h0 follows. Let h ∈ h and hn ∈ h0 a sequence such that limn→∞ hn = h in h. Corollary 4.11 and inequality (2.1) gives k(eisH W (hs )e−isH − eit H W (ht )e−it H ) (H + i)−1 k 6 c (k(eisH W (hn,s )e−isH − eit H W (hn,t )e−it H )(H + i)−1 k + khn − hk ).

262

ZIED AMMARI

This inequality gives the existence of (6.2) and (6.1) for h ∈ h. This shows (i), (ii). Using (2.1) and Corollary 4.11 we have keisH (W (hs ) − W (gs ))e−isH (H + i)−1 k 6 c kh − gk− .

(6.8)

Taking the limit s → ∞ in (6.8) we obtain k(W + (h) − W + (g))(H + 1)−1 k 6 c kh − gk− . 2

This proves (iii). The rest follows from simple computations. THEOREM 6.2. The five following assertions hold:

(i) There exist self-adjoint operators φ ± (h), called asymptotic fields, such that W ± (h) = eiφ

± (h)

for h ∈ h. n

(ii) For hi ∈ h, i = 1 . . . n. We have D((H + i) 2 ) ⊂ D( n Y

±

φ (hi )(H + i)

− n2

i=1

= lim e

it H

t →±∞

n Y

Qn 1

φ ± (hi )) and

φ(hi,t )e−it H (H + i)− 2 . n

i=1

(iii) The map (h1 , . . . , hn ) 7→

n Y

φ ± (hi )(H + i)− 2

n

is norm continuous.

i=1

(iv) The commutation relations hold as quadratic forms on D(φ ± (h))∩D(φ ± (g)) [φ ± (h), φ ± (g)] = iIm(h|g). (v) We have eit H φ ± (h)e−it H = φ ± (h−t ). Proof. Since s → W + (sh) is strongly continuous using Theorem 6.1(iii), (i) follows from Stone’s theorem. We intend to show the existence of the following limit for hi ∈ h, i = 1 . . . n lim eit H

t →+∞

n Y

n

φ(hi,t )e−it H (H + 1)− 2 .

(6.9)

i=1

Let hi ∈ h0 , i = 1 . . . n. As in the previous proof we have the following identity as quadratic form on D(H0 ) which extends as an operator identity on H : ! " # n n Y Y n n ∂t φ(hi,t )(N + 1)− 2 = −i H0 , φ(hi,t ) (N + 1)− 2 . (6.10) i=1

i=1

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A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

Now we compute the derivative it Hκ

∂t e

n Y

!

φ(hi,t )(Hκ − Eκ + i)

− n2 −it Hκ

e

u, v

i=1

=

∂t

n Y

! φ(hi,t )(Hκ − Eκ + i)

− n2 −it Hκ

e

−it Hκ

u, e

v +

i=1 n Y

+

! φ(hi,t )(Hκ − Eκ + i)

− n2 −it Hκ

e

−it Hκ

u, Hκ e

v −

i=1

− Hκ (Hκ − Eκ + i)

− n2 −it Hκ

e

u,

n Y

! −it Hκ

φ(hi,t )e

v .

i=1

We have (hi,t |vκ ) = (hi,t |vκ1 ) for κ > κ1 . Hence, we have on H ! n Y n ∂t eit Hκ φ(hi,t )(Hκ − Eκ + i)− 2 e−it Hκ i=1 n n Y X

= ieit Hκ

Im(hj,t |vκ1 )φ(hi,t )(Hκ − Eκ + i)− 2 e−it Hκ . n

(6.11)

j =1 i6=j

Letting κ → ∞ in (6.11) we obtain n Y

∂t eit H

φ(hi,t )(H + i)

− n2

!

e−it H

(6.12)

i=1

= ie

it H

P

n n Y X

Im(hj,t |vκ1 )φ(hi,t )(H + i)− 2 e−it H . n

j =1 i6=j

Since i hxi i−1− Im(hi,t |vκ1 ) ∈ O(t −1− ), the dominated convergence theorem gives the existence of (6.9) for h ∈ h0 . Let hi ∈ h, i = 1 . . . n and hi,` ∈ h0 sequences such that lim` hi,` = hi , in h. Using Corollary 2.4 and Corollary 4.8, we obtain the inequality

! n n

Y

isH Y −isH it H −it H − n2 e (H + 1) φ(h )e − e φ(h )e

i,s i,t

i=1 i=1

! n n

Y Y n

6 c eisH φ(hi,`,s )e−isH − eit H φ(hi,`,t )e−it H (H + 1)− 2 +

i=1 i=1 ! n X + khi − hi,` k . i=1

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Cauchy criterion for the convergence, proves the existence of the limit (6.9). To complete (ii) it suffices to show by induction in n and for u ∈ H, the existence of following limit Y n it H 1 lim lim e φ(hi,t ) − W (sh1,t ) − 1 s→0 t →+∞ s i=2 n Y n −i φ(hi,t ) (H + 1)− 2 e−it H u = 0. (6.13) i=1

We first prove (6.13) for u ∈ D((H + 1)− ), > 0, then by an argument of density we obtain (6.13) for u ∈ H . We recall (2.2): lim sup s −1 k(W (sh) − 1 − isφ(h))(N + 1)− 2 − k = 0. 1

s→0 khk6c

We see that

! n n

1

Y Y n

lim sup (W (sh1,t ) − 1) φ(hi,t ) − i φ(hi,t ) (H + 1)− 2 − = 0.

s→0 t ∈R s i=2 i=1

This completes the proof of (ii). (iii) follows from (ii) and Corollary 2.4 (ii). (iv) follows from the properties of CCR representations. 2 DEFINITION 6.3. We define the asymptotic creation and annihilation operators on D(φ ± (h)) ∩ D(φ ± (ih)) 1 a ± (h) := √ (φ ± (h) − iφ ± (ih)), 2 1 a ±∗ (h) := √ (φ ± (h) + iφ ± (ih)). 2 We denote by a ±] (h) the operator a ±∗ (h) or a ± (h). We formulate now a theorem which follows from the previous one. THEOREM 6.4. (i) a ±∗ (h) and a ± (h) are closed operators.Q n (ii) For hi ∈ h, i = 1 . . . n. We have D((H + i) 2 ) ⊂ D( ni=1 a ±] (hi )) and n Y

±]

a (hi )(H + i)

− n2

i=1

= lim e

it H

t →±∞

n Y

n

a ] (hi,t )e−it H (H + i)− 2 .

i=1

(iii) The map (h1 , . . . , hn ) 7→

n Y i=1

a ±] (hi )(H + i)− 2

n

is norm continuous.

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A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

(iv) The commutation relations hold as quadratic form on D(a ± (h))∩D(a ± (g)) [a ± (h), a ±∗ (g)] = (h|g)1, [a ± (h), a ± (g)] = [a ±∗ (h), a ±∗ (g)] = 0. (v) We have eit H a ±] (h)e−it H = a ±] (h−t ). Similar results hold for the modified Hamiltonian Hˆ ∞ . We formulate this in the following theorem. THEOREM 6.5. (i) For h ∈ h the following limit exists ∗ Wˆ ± (h) : = U∞ W ± (h)U∞ ˆ

ˆ

= s- lim eit H∞ W (ht )e−it H∞ . t →±∞

(ii) h → Wˆ ± (h) is a CCR representation. We denote by aˆ ±∗ (h), aˆ ± (h) the creation and annihilation operators associated to this representation. Q n (iii) For hi ∈ h, i = 1 . . . n. We have D((Hˆ ∞ + i) 2 ) ⊂ D( ni=1 aˆ ±] (hi )) and n Y

ˆ

aˆ ±] (hi )(Hˆ ∞ + i)− 2 = lim eit H∞ n

t →±∞

i=1

n Y

ˆ

a ] (hi,t )e−it H∞ (Hˆ ∞ + i)− 2 , n

i=1

where aˆ ±] (h) denote either aˆ ±∗ (h) or aˆ ± (h). (iv) The map (h1 , . . . , hn ) 7→

n Y

aˆ ±] (hi )(Hˆ ∞ + i)− 2

n

is norm continuous.

i=1

(v) We have ˆ

ˆ

eit H∞ aˆ ±] (h)e−it H∞ = aˆ ±] (h−t ). Proof. The existence of the strong limit follows from (6.1) and the fact that ∗ U∞ W (ht )U∞ = e−iIm(G∞ |ht ) W (ht ),

w- lim ht = 0. t →+∞

This prove (i). Theorem 6.1(iv) and (i) give i Wˆ ± (h) Wˆ ± (g) = e 2 Im(h|g) Wˆ ± (h + g),

Wˆ ± (h)∗ = Wˆ ± (−h).

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ZIED AMMARI

This proves CCR representation. (iii) is a consequence of Theorem 6.2(ii) and the fact that ∗ U∞ φ(ht )U∞ = φ(ht ) + Im(G∞ |ht ).

2

The rest follows from Theorem 6.4.

6.2.

WAVE OPERATORS

We recall the construction of the Fock subrepresentation of a CCR representation. Details can be found in [BR], [DG3]. Let g be pre-Hilbert space, and denote by g¯ its completion. We define the space of vacua associated to a CCR representation π over g¯ : Kπ := {u ∈ H | aπ (h)u = 0, h ∈ g}. PROPOSITION 6.6. (i) Kπ is a closed space. (ii) Kπ is contained in the set of analytic vectors of φπ (h), h ∈ g. Let Hπ := Kπ ⊗ 0(¯g). We define π : Kπ ⊗ 0fin (¯g) → H , π ψ ⊗ φ(h)p := φπ (h)p ψ,

h ∈ g, ψ ∈ Kπ .

PROPOSITION 6.7. The map π extends to an isometric map π : Hπ → H , satisfies π 1 ⊗ a ] (h) = aπ] (h), h ∈ g. Theorem 6.1 shows that asymptotic Weyl operators define a CCR representation. Then we define the space of vacua in our case K ± := {u ∈ H | a ± (h)u = 0, h ∈ h}. We denote by H ± the space K ± ⊗ 0(h). PROPOSITION 6.8. The following three assertions hold: (i) K ± is closed H -invariant space. Q (ii) For hi ∈ h, i = 1 . . . n. One has K ± ⊂ D( ni=1 a ±∗ (hi )). (iii) Ran 1pp (H ) ⊂ K ± .

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

267

Proof. The fact that K ± is H -invariant follows from Theorem 6.4(v). (i) and (ii) follow by Proposition 6.7. Let us prove now (iii). Let u ∈ H such that H u = Eu, one has lim eit H a(ht )e−it H u = 0,

t →±∞

since s- lim a(ht ) = 0 t →±∞

and eit H a(ht )e−it H u = (E + i) eit (H −E) a(ht )(H + i)−1 u. This means a ± (h)u = 0.

2

We define H ± := H |K ± ⊗ 1 + 1 ⊗ d0(ω), and the wave operator ± : H ± → H, ± ψ ⊗

n Y

a ∗ (hi ) :=

i=1

n Y

a ±∗ (hi )ψ,

for ψ ∈ K ± , hi ∈ h, i = 1 . . . n.

i=1

THEOREM 6.9. ± is a unitary map satisfying a ±] (h)± = ± 1 ⊗ a ] (h), H ± = ± H ± .

f or h ∈ h,

Proof. Proposition 6.8 gives that ± is isometric and satisfies properties announced in the theorem. Let prove that ± is unitary. Using [DG3, Thm. 3.3], it suffices to show that the CCR representation h → W ± (h) admits a densely defined number operator. For each finite-dimensional space f ⊂ h, we define as quadratic form the following expression n± f (u)

:=

dimf X

ka ± (hi )uk2 ,

{hi } is an orthonormal basis of f, u ∈ H .

i=1

Now we show that n± (u) := supf n± f (u) is densely defined: 2 kn± f (u)k 6

6

lim

t →±∞

dimf X

ka(hi,t )e−it H uk2

i=1

lim (e−it H u, Ne−it H u).

t →±∞

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1

We conclude using Corollary 4.11 that n± (u) 6 c k(H + b) 2 uk. Thus D(H 2 ) ⊂ D(n± ) and Ran ± = H . 2 We define an extended wave operator ˆ ext,± :

∞ M

D((Hˆ ∞ + 1) 2 ) ⊗ ⊗ns h → H , n

n=0

ˆ ext,± ψ ⊗

n Y

a ∗ (hi ) :=

i=1

n Y

n ψ ∈ D((Hˆ ∞ + 1) 2 ).

aˆ ±∗ (hi )ψ,

i=1

We set ∗ Kˆ ± := U∞ K ±,

Hˆ ± := Kˆ ± ⊗ 0(h).

Then we have a wave operator of the modified Hamiltonian: ˆ ± : Hˆ ± → H, ˆ ±ψ ⊗

n Y

a ∗ (hi ) :=

i=1

n Y

aˆ ±∗ (hi )ψ,

for ψ ∈ Kˆ ± , hi ∈ h, i = 1 . . . n.

i=1

ˆ ext,± ˆ ± . This suggests to treat sometimes ˆ ext,± as a parWe notice that = |Hˆ ± tial isometry. Another construction of the extended wave operator is given by the following theorem, see [DG2, Thm. 5.7]: ˆ ext,± ). Then one has THEOREM 6.10. (i) Let u ∈ D( ˆ

ˆ ext

ˆ ext,± u, lim eit H∞ I e−it H∞ u =

t →±∞

where I is the scattering identification operator defined in the Subsection 2.1. ext ˆ ext,± ) and the operators (ii) Let χ ∈ C0∞ (R). Then Ran χ(Hˆ ∞ ) ⊂ D( ext ext,± ext ˆ I χ(Hˆ ∞ ), χ(Hˆ ∞ ) are bounded. Moreover ˆ

ˆ ext

ext ext ˆ ext,± χ(Hˆ ∞ lim eit H∞ I e−it H∞ χ(Hˆ ∞ )= ).

t →±∞

7. Propagation Estimates We make the following notations for the Heisenberg derivatives dl0 := ∂t + i[ω(k), ·], ID0 := ∂t + i[d0(ω), ·], ˆ κ := ∂t + i[Hˆ κ , ·]. ID

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269

PROPOSITION 7.1. Let χ ∈ C0∞ (R). For R 0 > R > 1, there exists c independent from κ such that we have for κ 6 ∞ Z 1

2 12

dt |x| ˆ −it H κ

d0 1[R,R 0 ] χ(Hˆ κ ) e u 6 ckuk2 .

t t

∞

Proof. We use a standard method in scattering theory of the N-body problem [Gr], [SS]. It is based on a technical lemma, see, e.g., [DG1, Lemma B.4.1]. Let F ∈ C ∞ (R) be a cutoff function equal to 1 near ∞, to 0 near the origin, with F 0 (s) > 1[R,R 0 ] (s). We consider the observable |x| 8(t) := χ(Hˆ κ )d0 F χ(Hˆ κ ). t By Lemma A.3, it is enough to show that |x| ˆ κ 8(t) > t −1 C0 χ(Hˆ κ )d0 F 0 ID χ(Hˆ κ ) + O(t −1−µ ) t

(7.1)

uniformly in κ to have the inequality. One has |x| ˆ κ 8(t) = χ(Hˆ κ ) d0 dl0 F ID χ(Hˆ κ ) + t |x| ˆ ˆ + χ(Hκ ) Iκ , i d0 F χ(Hˆ κ ). t Using the fact that |x| c0 0 |x| dl0 F > F + O(t −2 ), t t t it is sufficient to show that the second term in the previous identity is O(t −1−µ ), µ > 0 uniformly in κ, to have (7.1). By simple commutation relations we obtain: |x| |x| φ(vκ0 ), d0 F = iφ iF vκ0 , t t |x| j |x| ∗2 j ∗ j ∗ a (rκ ), d0 F = 2 a (rκ )a F rκ , t t |x| |x| j 2 j j a (rκ ), d0 F (7.2) = −2 a(rκ )a F rκ , t t |x| ∗ j j a (rκ )a(rκ ), d0 F t

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|x| j |x| j = a ∗ (rκj )a F rκ + a ∗ F rκ a(rκj ), t t |x| |x| j j Dj a(rκ ), d0 F = Dj a F rκ , t t |x| j |x| ∗ j ∗ a (rκ ) Dj , d0 F =a F rκ Dj . t t Using the functional calculus formula (4.18) and Corollary 4.8, it is enough to estimate 1 |x| −n − 12 ˆ (N + 1) (H0 + c) Iκ , i d0 F (H0 + c)− 2 , c > 0. t Using (7.2) and Lemma 3.4, we obtain: |x| 1 −n − 12 ˆ k(N + 1) (H0 + c) Iκ , i d0 F (H0 + c)− 2 k t

|x| − 12

6 c

(V + 1) F t vκ0 +

|x| j s−1 s s−1 j −

2 4 + kω 4 rκ k

(V + 1) ω F t rκ +

|x| j s−1 s s s−1 − j −

+ k(V + 1) 2 ω 2 rκ k (V + 1) 2 ω 2 F rκ

+ t

|x| j − 12 − 2s s−1 2 + kD (K + c) k (V + 1) ω F rκ

. t

(7.3)

It remains to see that the terms

|x|

(V + 1)− 12 F |x| vκ ,

(V + 1)− 2s ω− s−1 2 F rκ 0

and t t

|x|

(V + 1)− 2s ω− s−1 4 F rκ

t are integrable for (1 − s) small enough. They are O(t −1−µ ) by (4.24). Then using Lemma A.3 we finish the proof of the estimate announced in the proposition. 2 PROPOSITION 7.2. Let χ ∈ C0∞ (R), 0 < c0 < c1 , and |x| x x 2[c0 ,c1 ] (t) := d0 − ∇ω(k), 1[c0 ,c1 ] − ∇ω(k) . t t t One has uniformly in κ 6 ∞ Z ∞ dt 1 ˆ k2[c0 ,c1 ] (t) 2 χ(Hˆ κ )e−it Hκ uk2 6 ckuk2 . t 1

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271

Proof. Let R0 (x) ∈ C ∞ be a function such that: c0 R0 (x) = 0, for |x| 6 , 2 1 R0 (x) = x 2 + c, for |x| > 2c1 , 2 2 ∇x R0 > 1[c0 ,c1 ] (|x|). We choose c1 > 2, c2 > c1 + 1 and we define the function R(x) := F (|x|)R0 (x), where F (s) = 1, if s 6 c1 , F (s) = 0, if s > c2 . We set x 1 x x b(t) := R − ∇R , − ∇ω(k) + hc . t 2 t t We consider the observable 8(t) := χ(Hˆ κ )d0(b(t))χ(Hˆ κ ). Pseudodifferential calculus gives χ(Hˆ κ )ID0 d0(b(t))χ(Hˆ κ ) |x| 1 1 ˆ > χ(Hκ ) 2[c0 ,c2 ] (t) − d0 1[2,c2 ] χ(Hˆ κ ) + O(t −2 ). t t t The first term will serves in the application of Lemma A.4 and the second is integrable along the evolution using Proposition 7.1. To complete the proof of the proposition, it suffices to show uniformly in κ that: χ(Hˆ κ )[Iˆκ , i d0(b(t))]χ(Hˆ κ ) ∈ O(t −1−µ ),

µ > 0.

(7.4)

As in the Proposition 7.1, using (7.2) and (7.3), we see that (7.4) is bounded by a sum of terms 1

k(V + 1)− 2 b(t)vκ0 k,

s

k(V + 1)− 2 ω−

1−s 2

b(t)rκ k

and k(V + 1)− 2 ω− s

1−s 4

b(t)rκ k.

By (4.24) these terms are O(t −1−µ ), µ > 0, for (1 − s) small enough. We end the proof by using Lemma A.3. 2 PROPOSITION 7.3. Let 0 < c0 < c1 , J ∈ C0∞ ({c0 < |x| < c1 }), χ ∈ C0∞ (R). For 1 6 i 6 3, one has uniformly in κ 6 ∞ 1

2 Z ∞

x 2

dt x ˆ i −it H κ

d0 J u − ∂i ω(k) + hc χ(Hˆ κ )e 6 ckuk2 .

t t t 1

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Proof. We set 2 x A := − ∇ω(k) + t −δ , t x 1 x 2 b(t) := J A J . t t Let J1 ∈ C0∞ ({c0 < |x| < c1 }), 0 6 J 6 1, J = 1 near the support of J1 . We consider the observable 8(t) := −χ(Hˆ κ ) d0(b(t))χ(Hˆ κ ). One has χ(Hˆ κ )ID0 d0(b(t))χ(Hˆ κ ) = −χ(Hˆ κ ) d0(dl0 b(t))χ(Hˆ κ ), and we have using [DG2, Lemma 6.4] χ(Hˆ κ )ID0 d0(b(t))χ(Hˆ κ ) c0 x xi ˆ > χ(Hκ ) d0 J1 − ∂i ω(k) + hc χ(Hˆ κ ) − t t t c x x x ˆ − χ(Hκ ) d0 − ∇ω, J2 − ∇ω χ(Hˆ κ ) + O(t −1−µ ). t t t t The second term is integrable along the evolution by Proposition 7.2. It’s enough to show that χ(Hˆ κ )[Iˆκ , i d0(b(t))]χ(Hˆ κ ) ∈ O(t −1−µ ),

µ > 0, uniformly in κ. 1

This follows by using (7.3), the fact that J ( xt )A 2 ∈ O(1) and Lemma A.2. Using Lemma A.3 we end the proof. 2 PROPOSITION 7.4. Let χ ∈ C0∞ (R), supported in R \ (τ ∪ σpp(Hˆ ∞ )). There exist > 0, C independent in κ and a sequence of Hˆ κ such that for κ 6 ∞, we have

2 Z ∞

0 1[0,] |x| χ(Hˆ κ ) e−it Hˆ κ u dt 6 Ckuk2 .

t t 1 Proof. We notice that Proposition 7.4 is a minimal velocity estimate for a sequence of Hˆ κ which is uniform in κ. Let χ supported near λ such that λ ∈ R \ (τ ∪ σpp (Hˆ ∞ )). Then there exists a sequence Hˆ κ such that λ ∈ R \ (τ ∪ σpp(Hˆ κ )). Lemma 5.5 in Mourre estimate section gives χ(Hˆ κ )[Hˆ κ , iB]χ(Hˆ κ ) > cκ χ 2 (Hˆ κ ).

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A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

Let > 0. Let q ∈ C0∞ (|x| 6 2) such that 0 6 q 6 1, q(x) = 1, if |x| 6 . We set B 8κ (t) := χ(Hˆ κ )0(q t ) 0(q t )χ(Hˆ κ ). t The Heisenberg derivative of 8κ (t) is B ˆ κ 8κ (t) = χ(Hˆ κ ) d0(q t , dl0 q t ) 0(q t )χ(Hˆ κ ) + hc + ID t B + χ(Hˆ κ )[Iˆκ , i0(q t )] 0(q t )χ(Hˆ κ ) + hc + t −1 t ˆ ˆ + t χ(Hκ )0(q )[Hκ , iB]0(q t )χ(Hˆ κ ) − B − t −1 χ(Hˆ κ )0(q t ) 0(q t )0(q t )χ(Hˆ κ ) t 1 2 3 =: Rκ + Rκ + Rκ + Rκ4 . We claim that Rκ2 ∈ O(t −1−µ ),

µ > 0.

(7.5)

To prove this, we use the estimates (4.22) in the proof of Lemma 4.14 to obtain χ(Hˆ κ )[Iˆκ , i0(q t )](d0(ω) + 1)− 2 ∈ O(t −1−µ ), 1

µ > 0.

(7.6)

B 1 0(q t )(d0(ω) + 1)− 2 (N + 1)−2 ∈ O(1). t

(7.7)

To prove (7.5) it suffices by Corollary 4.8 to show that 1

(d0(ω) + 1) 2

A simple explicit calculus gives 1

(d0(ω) + 1) 2

B 0(q t ) t

B 1 1 B t 2 2 = 0(q )(d0(ω) + 1) + (d0(ω) + 1) , 0(q t ) + t t + Clearly

B 1 [(d0(ω) + 1) 2 , 0(q t )]. t

B 0(q t )(N t

+ 1)−2 ∈ O(1) and

B 1 (d0(ω) + 1) , 0(q t ) = (d0(ω) + 1)− 2 d0(|∇ω|2 )0(q t ) ∈ O(1). t 1 2

(7.8)

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To estimate the last term in (7.8), we write on the n-particle sector: 1

[(d0(ω) + 1) 2 , 0(q t )]|K⊗sn h ! 12 # n " n j −1 n Y X X Y xi xi xj = q ω(ki ) + 1 , q q t t t j =1 i=1 i=1 i=j +1 =:

n X

Rj (t).

j =1

Pseudodifferential calculus gives that xk Rj (t) ∈ O(1),

uniformly in k, j.

This proves (7.7). Now (7.6) and (7.7) imply that Rκ2 ∈ O(t −1−ν ), for κ < ∞. We consider now Rκ1 . We have: 1 x 1 x t dl0 q = − − ∇ω(k), ∇q + hc + r t =: g t + r t , 2t t t t where r t ∈ O(t −2 ). We have using (2.3) and Corollary 4.8:

B t t t

χ(Hˆ κ ) d0(q , r ) 0(q )χ(Hˆ κ ) ∈ O(t −2 ).

t We set 1 B1 := χ(Hˆ κ ) d0(q t , g t )(N + 1)− 2 ,

1

B2 := (N + 1) 2

So we obtain the inequality Rκ1 > −0−1 t −1 B1 B1∗ − 0 t −1 B2 B2∗ . Using arguments in [DG2, Prop. 6.5], we obtain −B2 B2∗ > −C1 χ(Hˆ κ )0(q t )2 χ(Hˆ κ ) − Ct −1 , Z ∞ ˆ kB1 e−it Hκ uk 6 Ckuk2 . 1

Using Lemma 4.14 and Theorem 5.7, we have Rκ3 > C0 t −1 χ(Hˆ κ )0(q t )2 χ(Hˆ κ ) − Ct −2 . We have −Rκ4 6 C2 χ(Hˆ κ )0(q t )2 χ(Hˆ κ ) + Ct −2 . t

B 0(q t )χ(Hˆ κ ). t

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Collecting the four terms, we obtain ˆ κ φκ (t) > −0 t −1 B2 B2∗ + Rκ2 + Rκ3 + Rκ4 ID > (C0 − 0 C1 − C2 )t −1 χ(Hˆ κ )0(q t )2 χ(Hˆ κ ) + Ct −2 > C˜ 0 χ(Hˆ κ )t −1 0(q t )2 χ(Hˆ κ ) − R(t), where R(t) is integrable. By Lemma A.3 we obtain the inequality announced in the proposition for χ supported near a one energy level λ. Then we complete the proof for an arbitrary χ using a standard argument, see, e.g., [DG1, Proposition 4.4.7]. 2

8. Asymptotic Completeness In this section we prove the main result of this paper, which is the asymptotic completeness of the Nelson Hamiltonian. This is the subject of Theorem 8.5, where we prove Ran 1pp (H ) = K ± . THEOREM 8.1. Let q, q˜ ∈ C0∞ (R3 ) such that 0 6 q, q˜ 6 1, q, q˜ = 1 on a neighborhood of zero and q t := q( xt ). (i) The following limits exist ˆ

ˆ

0 ± (q) := s- lim eit H∞ 0(q t )e−it H∞ t →±∞

=

ˆ

ˆ

lim s- lim eit Hκ 0(q t )e−it Hκ .

κ→+∞

t →±∞

(ii) We have 0 ± (q q) ˜ = 0 ± (q)0 ± (q), ˜ 0 6 0 ± (q) 6 0 ± (q) ˜ 6 1,

if 0 6 q 6 q, ˜

[Hˆ ∞ , 0 ± (q)] = 0. (iii) We have Ran 0 ± (q) ⊂ Kˆ ± . Proof. It is sufficient using a density argument and Lemma 4.14 to show for χ ∈ C0∞ (R) the existence of the limit ˆ ˆ s- lim eit H∞ χ(Hˆ ∞ ) 0(q t )χ(Hˆ ∞ )e−it H∞ . t →±∞

(8.1)

Using Lemma A.4, we see that as for all asymptotic limits these amounts to bound Heisenberg derivatives uniformly in κ. We have on H : ˆ ˆ ∂t (eit Hκ χ(Hˆ κ ) 0(q t )χ(Hˆ κ )e−it Hκ ) ˆ ˆ = eit Hκ χ(Hˆ κ ) d0(q t , dl0 q t )χ(Hˆ κ )e−it Hκ + ˆ

ˆ

+ eit Hκ χ(Hˆ κ )[Iˆκ , i0(q t )]χ(Hˆ κ )e−it Hκ .

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By (4.22) we have uniformly in κ: χ(Hˆ κ )[Iˆκ , i0(q t )]χ(Hˆ κ ) ∈ O(t −1− ).

(8.2)

We use now an argument introduced in [DG2]: dl0 q t =

1 t g + rt , t

where gt = −

1 2

x x − ∂ω(k) ∂q + hc and t t

r t ∈ O(t −2 ).

The estimate (2.3) and Corollary 4.8 give uniformly in κ kχ(Hˆ κ )d0(q t , dl0 r t )χ(Hˆ κ )k ∈ O(t −2 ).

(8.3)

The other term will be estimated: ˆ ˆ |(e−it Hκ u, χ(Hˆ κ )d0(q t , g t )χ(Hˆ κ )e−it Hκ u)| 1 ˆ 6 kd0(|g t |) 2 χ(Hˆ κ )e−it Hκ uk2 ,

(8.4)

u ∈ H.

Using (8.2)–(8.4), Proposition 7.3 and Lemma A.4, we obtain the existence of the limit (8.1). The first statement in (ii) follows from the fact that 0(q t q˜ t ) = 0(q t )0(q˜ t ). The second statement follows by 0 6 0(q t ) 6 0(q˜ t ) 6 1

if 0 6 q 6 q. ˜

The last statement is a consequence of (i) and Lemma 4.14. Let us prove (iii). Since Hˆ ∞ , 0 ± (q) commute, 0 ± (q) preserves D(Hˆ ∞ ). Consequently D(Hˆ ∞ ) ∩ Ran 0 ± (q) is dense in Ran 0 ± (q). Since Kˆ ± is closed, it is enough to show that D(Hˆ ∞ ) ∩ Ran 0 ± (q) ⊂ Kˆ ± to prove (iii). Let v ∈ D(Hˆ ∞ ) ∩ Ran 0 ± (q), v = 0 ± (q)u. (Hˆ ∞ + b)−1 aˆ ± (h)0 ± (q)u ˆ

ˆ

= lim eit H∞ (Hˆ ∞ + b)−1 a(ht )e−it H∞ 0 ± (q)u t →±∞

ˆ ˆ = lim eit H∞ (Hˆ ∞ + b)−1 a(ht )0(q t )e−it H∞ u t →±∞

ˆ

ˆ

= lim eit H∞ (Hˆ ∞ + b)−1 0(q t )a(q t ht )e−it H∞ u. t →±∞

If h ∈ h0 , by a stationary phase argument we have q t ht ∈ o(1), t → ±∞. Using the fact that h → (Hˆ ∞ + i)−1 a(ht ) is continuous, we obtain aˆ ± (h)v = 0 for all h ∈ h. This ends the proof. 2

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277

COROLLARY 8.2. Let {qn } ∈ C0∞ (R3 ) be a decreasing sequence of functions T such that 0 6 qn 6 1, qn = 1 on a neighborhood of 0 and ∞ supp qn = {0}. n=1 Then the following limit exist and it does not depend in the choose of the sequence (i)

P0± := lim 0 ± (qn ),

(ii)

Ran P0± ⊂ Kˆ ± .

n→∞

Moreover, P0± is an orthogonal projection. Proof. The existence of the limit (i) follows from Theorem 8.1(ii) and Lemma A.5. The independence from the choose of the sequence follows from the fact that there exists an index mn such that qn > q˜mn , q˜n > qmn ; limn→∞ mn = +∞ and Theorem 8.1(ii). (ii) is a consequence of Theorem 8.1(iii) and (i). 2 2 Let j0 ∈ C0∞ (R3 ), 0 6 j0 , 0 6 j∞ , j02 + j∞ 6 1, j0 = 1 near 0. Set j := x t t t t t (j0 , j∞ ) and j := (j0 , j∞ ), where j0 := j0 ( t ), j∞ := j∞ ( xt ). We recall that I (j t ) is the operator introduced in Subsection 2.1.

THEOREM 8.3. (i) The following limits exist ˆ ext

ˆ

s- lim eit H∞ I ∗ (j t )e−it H∞ =: W˜ ± (j ), t →±∞

ˆ ˆ ext s- lim eit H∞ I (j t )e−it H∞ = W˜ ± (j )∗ . t →±∞

(ii) For a bounded Borel function F, we have ext ˜ ± W˜ ± (j ) F (Hˆ ∞ ) = F (Hˆ ∞ )W (j ).

(iii) Let q0 , q∞ ∈ C ∞ (R3 ), ∇q0 , ∇q∞ ∈ C0∞ (R3 ), 0 6 q0 , q∞ 6 1, q0 = 1 near 0. Set j˜ := (q0 j0 , q∞ j∞ ). Then 0 ± (q0 ) ⊗ 0(q∞ (∇ω(k)))W˜ ± (j ) = W˜ ± (j˜). (iv) Let q ∈ C ∞ (R3 ), ∇q ∈ C0∞ (R3 ), 0 6 q 6 1, q = 1 near 0. Then W˜ ± (j )0 ± (q) = W˜ ± (qj ),

where qj = (qj0 , qj∞ ).

(v) Let j˜ = (j˜0 , j˜∞ ) be another pair satisfying the conditions stated before the theorem. Then W˜ ± (j˜)∗ W˜ ± (j ) = 0(j˜0 j0 + j˜∞ j∞ ), 2 in particular if j02 + j∞ = 1, then W˜ ± (j ) is isometric. (vi) Let j0 + j∞ = 1. If χ ∈ C0∞ (R), then ext ˜ ± ˆ ext,± χ(Hˆ ∞ )W (j ) = χ(Hˆ ∞ ).

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Proof. To prove (i) we use the same arguments as in Theorem 8.1. Using Lemma A.4, it is enough to prove the existence of the limit ˆ ext

ˆ

ext ∗ t −it H∞ s- lim eit H∞ χ(Hˆ ∞ )I (j )e χ(Hˆ ∞ ),

(8.5)

t →±∞

for some χ ∈ C0∞ (R). We compute ˆ ext

ˆ

∂t (eit Hκ χ(Hˆ κext )I ∗ (j t )χ(Hˆ κ )e−it Hκ ) ˆ ext

= eit Hκ (χ(Hˆ κext )D0 I ∗ (j t )χ(Hˆ κ ) + ˆ + iχ(Hˆ κext )(Iˆκ ⊗ 1I ∗ (j t ) − I ∗ (j t )Iˆκ )χ(Hˆ κ ))e−it Hκ ,

where D0 is the asymmetric Heisenberg derivative ∂t + iH0ext . − .iH0 . We have D0 I ∗ (j t ) = dI ∗ (j t , d0 j t ). Pseudodifferential calculus gives 1 d0 j t = g t + r t , t g = t

t (g0t , g∞ ),

gt

1 =− 2

x x − ∂ω(k) ∂j + hc , t t

= 0, ∞

with r t ∈ O(t −2 ). Using Corollary 4.8 and (2.4) we obtain kχ(Hˆ κext )dI ∗ (j t , r t )χ(Hˆ κ )k ∈ O(t −2 ).

(8.6)

ˆ

Using now (2.5) with uti := eit Hκ ui , one obtain |(ut1 |χ(Hˆ κext )dI ∗ (j t , g t )χ(Hˆ κ )ut2 )| 6 kd0(|g0t |) 2 ⊗ 1χ(Hˆ κext )ut2 k kd0(|g0t |) 2 χ(Hˆ κ )ut1 k + 1

1

1 1 t t + k(1 ⊗ d0(|g∞ |) 2 )χ(Hˆ κext )ut2 k kd0(|g∞ |) 2 χ(Hˆ κ )ut1 k.

Then the κ-uniform integrability of the term χ(Hˆ κext )D0 I ∗ (j t )χ(Hˆ κ ) follows using Proposition 7.1. Using (4.25) we obtain uniformly in κ χ(Hˆ κext )(Iˆκ ⊗ 1I ∗ (j t ) − I ∗ (j t )Iˆκ )χ(Hˆ κ ) ∈ O(t −1−µ ). Then the existence of the limit in (i) follows. (ii) follows by Lemma 4.15. (iii) follows using the fact that lim eit d0(ω)0(q t )e−it d0(ω) = 0(q(∇ω)),

t →±∞

t 0(q0t ) ⊗ 0(q∞ )I ∗ (j t ) = I ∗ (j˜t ).

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279

(iv) is true since I ∗ (j t )0(q t ) = I ∗ ((j q)t ). (v) is a consequence of the fact t t I (j˜t )I ∗ (j t )0(j˜0t j0t + j˜∞ j∞ ).

(vi) One has ext Hˆ ∞ 1[k,∞[ (N∞ ) > mk + E0 .

Let χ ∈ C0∞ (R). There exists n0 ∈ N such that for n > n0 ext χ(Hˆ ∞ )1]n,∞[ (N∞ ) = 0.

(8.7)

We have ext ˜ ± ˆ ext,± χ(Hˆ ∞ )W (j ) ext ˜ ± ˆ ext,± 1[0,n] (N∞ )χ(Hˆ ∞ = )W (j )

(8.8)

ˆ ext ∗ t −it Hˆ ∞ = s- lim eit H∞ I 1[0,n] (N∞ )χ(Hˆ ∞ )I (j )e

(8.9)

t →±∞

ˆ

ˆ

= s- lim eit H∞ I 1[0,n] (N∞ )I ∗ (j t )e−it H∞ χ(Hˆ ∞ ). t →±∞

(8.10)

(8.8) follows from (8.7). (8.9) follows from the limit (i) and Theorem 6.10. Lemma n 4.15 and the boundeness of the operator I 1[0,n] (N∞ )(N0 + 1)− 2 gives (8.10). We use now an estimate proved in [DG3]: kI 1]n,∞[ (N∞ )I ∗ (j t )(N + 1)−1 k 6 (n + 1)−1 .

(8.11)

ext ˜ ± ˆ ext,± χ(Hˆ ∞ Since II ∗ (j t ) = 1, letting n → ∞ we obtain )W (j ) = χ(Hˆ ∞ ). This completes the proof. 2

THEOREM 8.4. Let jn = (j0,n , j∞,n ) be a sequence satisfying the hypothesis stated in the beginning of Theorem 8.3 such that j0 + j∞ = 1 and for any > 0 there exists m, ∀n > m, supp j0,n ⊂ [−, ]. Then ˆ ±∗ = w- lim W˜ ± (jn ), κ→+∞

ˆ±

K = Ran P0± . Proof. Let q ∈ C0∞ (R), 0 6 q 6 1 and q = 1 in a neighborhood of zero such that qj0,n = j0,n for n large enough. Using Theorem 8.3(iii) and Corollary 8.2 we obtain 0 ± (q) ⊗ 1 W˜ ± (jn ) = W˜ ± (jn ),

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w- lim P0± ⊗ 1 W˜ ± (jn ) − W˜ ± (jn ) = 0.

(8.12)

n→´l+∞

Let χ ∈ C0∞ (R). We have ext ˜ ± ˆ ±∗ χ(Hˆ ∞ ) = ˆ ext,± χ(Hˆ ∞ ˆ ±∗ )W (jn )

(8.13)

ext ˆ ±∗ ˆ ext,± χ(Hˆ ∞ = w- lim )P0± ⊗ 1 W˜ ± (jn )

(8.14)

ext ˜ ± = w- lim P0± ⊗ 1χ(Hˆ ∞ )W (jn )

(8.15)

ext ˜ ± = w- lim χ(Hˆ ∞ )W (jn )

(8.16)

= w- lim W˜ ± (jn )χ(Hˆ ∞ ).

(8.17)

n→∞

n→∞

n→∞

n→∞

Formula (8.13) follows from Theorem 8.3(iv). (8.14) follows by (8.12). (8.15) is ext ˆ ext,± 1Kˆ ± ⊗ 1 = ˆ± true since P0± commutes with Hˆ ∞ and that RanP0± ⊂ Kˆ ± , ± ±∗ ± ext ˆ = 1 ˆ ± ⊗ 1. (8.16) follows from the fact that P0 commutes with Hˆ ∞ ˆ and K and (8.12). (8.17) is Theorem 8.3(ii). So we conclude by a density argument that ˆ ±∗ = w- lim W˜ ± (jn ), n→+∞

P0±

ˆ ±∗ = ˆ ±∗ . ⊗1

So we obtain ˆ ±∗ = Kˆ ± ⊗ 0(h) ⊂ Ran P0± ⊗ 0(h) ⊂ Kˆ ± ⊗ 0(h). Ran Hence we prove that Kˆ ± = Ran P0± .

2

THEOREM 8.5. We have Ran 1pp (H ) = K ± . Proof. By Proposition 6.8(iii) we have Ran 1pp (Hˆ ∞ ) ⊂ Kˆ ± . Then it suffices to show that Kˆ ± ⊂ Ran 1pp (Hˆ ∞ ). Proposition 7.4 gives the existence of > 0 and a sequence Hˆ κ such that Z +∞ dt ˆ k0(q t )χ(Hˆ κ )e−it Hκ uk2 6 Ckuk2 , t 1 where χ ∈ C0∞ (R \ (τ ∪ σpp(Hˆ ∞ ))) and q ∈ C0∞ ([−, ]), q = 1 for |x| < /2. Theorem 8.1 gives that ˆ

k0(q t )χ(Hˆ κ )e−it Hκ uk → k0 ± (q)χ(Hˆ ∞ )uk = 0,

t → ±∞

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then 0 ± (q)χ(Hˆ ∞ ) = 0. So we have Ran P0± ⊂ Ran 1τ ∪σpp (Hˆ ∞ ) (Hˆ ∞ ). Theorem 5.7 gives that τ is a closed countable set and σpp (Hˆ ∞ ) can accumulate only at τ , so 1pp (Hˆ ∞ ) = 1τ ∪σpp (Hˆ ∞ ) (Hˆ ∞ ). This proves Ran 1pp (Hˆ ∞ ) = Kˆ ± . Then we prove the theorem. 2

Appendix The following theorem follows from the KLMN theorem and [RS, I–IV, Thm. VIII.25]. THEOREM A.1. Let H0 be a positive self-adjoint operator on H . Let for κ 6 ∞, 1

Bκ be quadratic forms on D(H02 ) such that 1

|Bκ (ψ, ψ)| 6 a kH02 ψk2 + b kψk2 , 1

where a < 1 uniformly in κ and Bκ → B∞ on D(H02 ). Then

1

(i) There exist for κ 6 ∞ self-adjoint operators Hκ with D(Hκ ) ⊂ D(H02 ) and 1

1

(Hκ ψ, ψ) = Bκ (ψ, ψ) + (H02 ψ, H02 ψ), (ii) (iii)

ψ ∈ D(Hκ ),

lim (z − Hκ )−1 = (z − H∞ )−1 ,

κ→∞

s- lim e−it Hκ = e−it H∞ . κ→∞

LEMMA A.2. Let F ∈ C ∞ (R), equal to 0 near the origin and bounded near ∞. we denote by F R the derivative operator F (|x|/R). We recall that x denote the nucleon position observable. One has uniformly in κ (i)

khxi−s F R vκ0 k ∈ O(R −s ),

(ii)

khxi−s ω(k)− F R rκ k ∈ O(R −s ),

> 0.

Proof. We have hxi−s vκ0 ∈ H s (R3 , B(K)). Since we have

s

R F |Dk | hxi−s vκ 6 ck|Dk |s hxi−s vκ k 0 0

R we obtain (i). Let us prove (ii). Pseudodifferential calculus gives

ω(k)− F |Dk | > 1 rκ

R

|Dk | c 1 −

6 c F > ω(k) rκ + 2 kω(k)− rκ k. R 2 R

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ZIED AMMARI

Then we obtain (ii) using (i) for ω− rκ , which is L2 (R3 , dk) uniformly in κ.

2

Let H a Hilbert space. Let {Hκ } be a sequence of self-adjoint operators on a common domain D ⊂ H. We suppose that s-limκ→∞ e−it Hκ = e−it H∞ , where H∞ is a self-adjoint operator. We have for χ ∈ C0∞ (R), using [RS, I–IV, Thm. VIII.20; I–IV, Thm. VIII.21] s- lim χ(Hκ ) = χ(H∞ ). κ→∞

LEMMA A.3. Let t → Bt ∈ B(H) and χ ∈ C0∞ (R) such that for κ 6 ∞ kBt∗ Bt χ(Hκ )k 6 ct , where ct is κ-independent locally integrable function in t. If there exist a constant c independent of κ such that for κ < ∞ Z ∞ kBt χ(Hκ )e−it Hκ uk2 dt 6 c kuk2 , u ∈ H , 1

then

Z

∞

kBt χ(H∞ )e−it H∞ uk2 dt 6 c kuk2 .

1

Proof. We have to prove, uniformly in T , that: Z T kBt χ(H∞ )e−it H∞ uk2 dt 6 c kuk2 .

(A.1)

1

We apply the dominated convergence theorem. It is enough to show that lim kBt χ(Hκ )e−it Hκ uk2 = kBt χ(H∞ )e−it H∞ uk2 ,

κ→∞

since we have kBt χ(Hκ )e−it Hκ uk2 6 c0 kBt∗ Bt χ(Hκ )kkuk2 . It is easy to see that to prove (A.1) it suffices to show w- lim χ(Hκ )Bt∗ Bt χ(Hκ ) = χ(H∞ )Bt∗ Bt χ(H∞ ). κ→∞

This follows from the hypothesis kBt∗ Bt χ(Hκ )k 6 ct for κ 6 ∞.

2

Let Hi , i = 1, 2 be two Hilbert spaces. Let Hi,κ , i = 1, 2 be two sequences of self-adjoint operators on Hi , such that Hi,κ converge in the strong resolvent sense to Hi,∞ . The Lemma A.4 follows from the proof of Lemma A.3 and [DG1, Lemma B.4.2].

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LEMMA A.4. Let t → C(t) ∈ B(H1 , H2 ) and χ ∈ C0∞ (R). We suppose that the asymmetric Heisenberg derivatives: Dκ C(t) := ∂t C(t) + i(H1,κ C(t) − C(t)H2,κ ), satisfies for κ < ∞, the following hypothesis: (i)

Dκ C(t) = B(t) + Rκ (t).

(ii)

kχ(H2,κ ) Rκ (t) χ(H1,κ )k 6 c t −1− , > 0, uniformly for κ < ∞.

(iii)

kχ(H2,κ )B(t)k 6 ct , kB(t) χ(H1,κ )k 6 ct , uniformly for κ 6 ∞, where ct is κ-independent locally integrable function in t.

(iv)

|(u2 |B(t)u1 )| 6 c

n X

kB2,j (t)u2 kkB1,j (t)u1 k,

where

j =1

Z

∞

kBi,j (t) χ(Hi,∞ )e−it Hi,∞ uk2 dt 6 c kuk2 ,

i = 1, 2 and j = 1 . . . n.

1

Then the following limit exists s- lim eit H2,∞ χ(H2,∞ ) C(t) χ(H1,∞ )e−it H1,∞ . t →+∞

Proof. Let u ∈ H1 , v ∈ H2 . We have: (u|eit2 H2,∞ χ(H2,∞ )C(t2 )χ(H1,∞ )eit2 H1,∞ ) − − (u|eit1 H2,∞ χ(H2,∞ )C(t1 )χ(H1,∞ )e−it1 H1,∞ v) = lim (u|eit2 H2,κ χ(H2,κ )C(t2 )χ(H1,κ )e−it2 H1,κ v) − κ→∞

− (u|eit1 H2,κ χ(H2,κ )C(t1 )χ(H1,κ )e−it1 H1,κ v) Z t2 = lim (e−it H2,κ u|χ(H2,κ )IDκ C(t)χ(H1,κ )e−it H1,κ v) dt κ→∞

t1

Z = lim

κ→∞

t2

(e−it H2,κ u|χ(H2,κ )(B(t) + R(t))χ(H1,κ )e−it H1,κ v) dt.

t1

Using Lebesgue dominated convergence theorem and, as in the proof of Lemma A.3, the fact that (iii) ⇒ w- lim χ(H2,κ )B(t)χ(H1,κ ) = χ(H2,∞ )B(t)χ(H1,∞ ), κ→∞

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we obtain

Z

t2

lim

κ→∞

(e−it H2,κ u|χ(H2,κ )B(t)χ(H1,κ )e−it H1,κ v) dt

(A.2)

t1

Z

t2

=

(e−it H2,∞ u|χ(H2,∞ )B(t)χ(H1,∞ )e−it H1,∞ v) dt.

t1

We have by (ii): Z t2 lim (e−it H2,κ u|χ(H2,κ )R(t)χ(H1,κ )e−it H1,κ v) dt κ→∞

t1

6 c t1− kvkkuk,

if t1 < t2 .

(A.3)

Using (iv) we obtain: Z t2 lim (e−it H2,∞ u|χ(H2,κ )B(t)χ(H1,κ )e−it H1,∞ v) dt κ→∞

(A.4)

t1

6 c

n Z X j =1

t2

−it H2,∞

kB2,j (t)e

t1

Z uk dt × 2

t2

kB1,j (t)e−it H1,∞ uk2 dt

t1

6 c kuk kvk. (A.2), (A.3) and (A.4) give the existence of the claimed limit.

2

We recall here a convergence lemma of positive operators, see, e.g., [DG2, Lemma A.3] LEMMA A.5. Let Qn be a sequence of commuting self-adjoint operators. If 0 6 Qn 6 1,

Qn+1 6 Qn ,

Qn+1 Qn = Qn+1 .

Then there exist Q a projection Q = s-lim Qn . n

Acknowledgements The author would like to thank Prof. C. Gérard for helpful discussions and for suggestions during the writing of this paper. References [ABG]

Amrein, W. O., Boutet de Monvel, A. and Georgescu, V.: Commutator Methods and Spectral Theory of N-Body Hamiltonian, Birkhäuser, Basel, 1996.

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT

[AH] [Be] [BFS] [BFSS]

[BR] [BSZ] [Ca] [CFKS] [DG1] [DG2] [DG3] [DJ] [Fr] [Gr] [GJ] [Gr] [GS] [H-K] [Mr] [Ne] [Ro] [RS] [SS]

285

Arai, A. and Hirokawa, M.: On the existence and uniqueness of ground states of a generalized spin-boson model, J. Funct. Anal. 151 (1997), 455–503. Berezin, F. A.: The Method of Second Quantization, Academic Press, New York, 1966. Bach, V., Fröhlich, J. and Sigal, I. M.: Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137 (1998), 299–395. Bach, V., Fröhlich, J., Sigal, I. M. and Soffer, A.: Positive commutators and spectrum of Pauli-Fierz Hamiltonian of atoms and molecules, Comm. Math. Phys. 207 (1999), 557–587. Brattelli, O. and Robinson, D.: Operator Algebras and Quantum Statistical Mechanics, Springer, Berlin, 1981. Baez, J. C., Segal, I. E. and Zhou, Z.: Introduction to Algebraic and Constructive Quantum Field Theory, Princeton Ser. in Phys., Princeton Univ. Press, 1992. Cannon, J. T.: Quantum field theoretic properties of a model of Nelson: Domain and eigenvector stability for perturbed linear operators, J. Funct. Anal. 8 (1971), 101–152. Cycon, H. L., Froese, R., Kirsch, W. and Simon, B.: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Springer, New York, 1987. Derezi´nski, J. and Gérard, C.: Scattering Theory of Classical and Quantum N-particle Systems, Texts and Monogr. in Phys., Springer, New York, 1997. Derezi´nski, J. and Gérard, C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians, Rev. Math. Phys. 11 (1999), 383–450. Derezi´nski, J. and Gérard, C.: Spectral and scattering theory of spatially cut-off P (φ)2 Hamiltonians, to appear in Comm. Math. Phys. Derezi´nski, J. and Jaksic, V.: Spectral theory of Pauli-Fierz Hamiltonians I, Preprint. Fröhlich, J.: Existence of dressed electron states in a class of persistent models, Fortschr. Phys. 22 (1974), 159–198. Graf, G. M.: Asymptotic completeness for N-body short-range quantum systems: A new proof, Comm. Math. Phys. 132 (1990), 73–101. Glimm, J. and Jaffe, A.: Collected Papers, Quantum Field Theory and Statistical Mechanics Expostions, Vol. I, Birkhäuser, Basel, 1985. Gross, E. P.: Ann. Phys. 19 (1962), 219–233. Greenberg, O. W. and Schweber, S. S.: Nuovo Cimento 8 (1958), 378. Høegh-Krohn, R.: Asymptotic limits in some models of quantum field theory, J. Math. Phys. 9 (1968), 2075–2079. Mourre, E.: Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys. 78 (1981), 519–567. Nelson, E.: Interaction of non-relativistic particles with a quantized scalar field, J. Math. Phys. 5 (1964), 1190–1197. Rosen, L.: The (82n )2 quantum field theory: Higher order estimates, Comm. Pure Appl. Math. 24 (1971), 417–457. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, Vols I and II, 1976, Vol. III, 1979, Vol. IV, 1978, Academic Press, London. Sigal, I. M. and Soffer, A.: The N-particle scattering problem: Asymptotic completeness for short-range systems, Ann. of Math. 126 (1987), 35–108.

Mathematical Physics, Analysis and Geometry 3: 287–303, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

287

Generating Relations of the Hypergeometric Functions by the Lie Group-Theoretic Method I. K. KHANNA, V. SRINIVASA BHAGAVAN and M. N. SINGH Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi-221005, India (Received: 31 August 2000) Abstract. In this paper, the generating relations for a set of hypergeometric functions ψα,β,γ ,m(x) are obtained by using the representation of the Lie group SL(2, C) giving a suitable interpretation to the index m in order to derive the elements of Lie algebra. The principle interest in our results lies in the fact that a number of special cases would inevitably yield too many new and known results of the theory of special functions, namely the Laguerre, even and odd generalized Hermite, Meixner, Gottlieb, and Krawtchouk polynomials. Mathematics Subject Classifications (2000): Primary 33C10, Secondary 33C45, 33C80. Key words: special functions, hypergeometric functions, Lie algebra, generating functions.

1. Introduction Recently, we have defined a set of hypergeometric functions ψα,β,γ ,m(x). It is interesting to note that the function ψα,β,γ ,m(x) is a product of binomial and hypergeometric functions. Independently, these two do not satisfy the three-term recurrence relation, whereas their product satisfies it. Thus, it is possible to derive many properties, including the ascending and descending recurrence relations, which are essential for obtaining the generating functions by the Lie group-theoretic method. In recent years, the development of advanced computers have made it necessary to study the functions with series representations from the numerical point of view. The most important functions of these type are hypergeometric in character. On account of many properties, the multiple hypergeometric functions are used in an increasing number of problems and are capable of being elegantly represented by their uses. Because of the important role which hypergeometric functions play in problems of physics and applied mathematics, the theory of generating functions has been developed in various directions and has found wide applications in various branches, e.g., Laguerre polynomials are very useful in the quantummechanical study of the hydrogen atom, Hermite polynomials have applications in the quantum-mechanical discussion of the harmonic oscillator and probabalistic distribution. Also, orthogonal polynomials are of great importance in approximation theory, physics, and the mathematical theory of mechanical quadratures, etc.

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The hypergeometric functions have also demonstrated their significance in science and technology. In this paper, we obtain generating functions for ψα,β,γ ,m(x) by using the representation of the Lie group SL(2, C) [8] giving a suitable interpretation to the index m, in order to derive the elements of Lie algebra. The principle interest in our results lies in the fact that a number of special cases would inevitably yield too many new and known results of the theory of special functions. It is worth recalling that several of the classical polynomials, namely the Laguerre, even and odd generalized Hermite, Meixner, Gottlieb and Krawtchouk polynomials [6], are derived as special cases of our results in Sections 2, 3 and 4. 2. Definition We have defined a set of hypergeometric functions h xi β m (γ )m ψα,β,γ ,m(x) = (1 − x)−m/2 2 F1 −m, α; γ ; , m! β

(2.1)

which is valid under the conditions α is a nonzero real number, γ is neither zero nor a negative integer, m is a nonnegative integer, in general, we shall insist that α, β, γ are independent of m because many properties which are valid for α, β, γ independent of m fail to be valid when α, β, γ are dependent, (v) x is any finite complex variable such that |x| < 1.

(i) (ii) (iii) (iv)

The following special cases of ψα,β,γ ,m(x) have been obtained. x ) lim ψα,1,1+γ ,m = L(γ m (x), α→∞ α

(2.2)

(γ )

where Lm (x) is the Laguerre polynomial [1]. 2 x lim ψα,1,1/2,m = (−1)m H2m (x)/22m m! α→∞ α and

lim ψα,1,3/2,m

α→∞

x2 α

(2.3)

= (−1)m H2m+1 (x)/22m m!2x,

(2.4)

where H2m(x) and H2m+1 (x) are the even and odd Hermite polynomials respectively [2]. 2 x µ lim ψα,1,(µ+1)/2,m (2.5) = (−1)m H2m (x)/22m m! α→∞ α

GENERATING RELATIONS OF HYPERGEOMETRIC FUNCTIONS

and

lim ψα,1,(µ+3)/2,m

α→∞ µ

x2 α

289

µ

= (−1)m H2m+1 (x)/22m m!,

(2.6)

µ

where H2m (x) and H2m+1 (x) are the generalized even and odd Hermite polynomials respectively [4]. (γ )m m/2 (2.7) ρ Mm (y; γ , ρ), m! provided 0 < ρ < 1, y = 0, 1, 2, . . . , where Mm (y; γ , ρ) is the Meixner polynomial [7], ψ−y,1,γ ,m(1 − ρ −1 ) =

ψ−y,1,1,m(1 − eλ ) = emλ/2 φm (y, λ),

(2.8)

where φm (y, λ) is the Gottlieb polynomial [1], (−N)m (2.9) (1 − P −1 )−m/2 Km (y; P , N), m! provided 0 < P < 1, y = 0, 1, 2, . . . , N, where Km (y; P , N) is the Krawtchouk polynomial [3]. The following recurrence relations for ψα,β,γ ,m(x) have been obtained: ψ−y,1,−N,m (P −1 ) =

d ψα,β,γ ,m(x) dx √ 1 = 2β(1 − γ − m) 1 − xψα,β,γ ,m−1 (x) + 2x(1 − x) + m(2 − x)ψα,β,γ ,m (x) ,

(2.10)

d ψα,β,γ ,m(x) dx 1 mx(β − x) = β(−γ − m) + αx + ψα,β,γ ,m(x) + x(β − x) 2(1 − x) √ + (m + 1) 1 − x ψα,β,γ ,m+1 (x).

(2.11)

These two independent differential recurrence relations determine the linear ordinary differential equation: d2 mx(β − x) d x(β − x) 2 + γβ + (m − α − 1)x − + dx (1 − x) dx m(m − 2)x(β − x) + mα + − 4(1 − x)2 m − (2.12) (γβ + (m − α − 1)x) ψα,β,γ ,m(x) = 0. 2(1 − x)

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3. Representation of SL(2, C) and Generating Functions Let sl(2, C) be the Lie algebra of a three-dimensional complex local Lie group SL(2, C), a multiplicative 2 × 2 matrix group with elements [5] a b SL(2, C) ≡ g = : a, b, c, d ∈ C , (3.1) c d which is the determinant of the matrix g, i.e. |g| = 1. A basis for sl(2, C) is provided by the matrices 1 0 0 −1 0 0 + − 3 2 j = (3.2) , j = , j = 0 0 −1 0 0 − 12 with the commutation relations [j 3 , j + ] = j + ,

[j 3 , j − ] = −j − ,

[j + , j − ] = 2j 3 .

(3.3)

Let us write the differential equation (2.12) in operator functional notation as d L x, , m dx mx(β − x) 2 = x(β − x)D + γβ + (m − α − 1)x − D+ (1 − x) m(m − 2)x(β − x) m + mα + − (γβ + (m − α − 1)x). (3.4) 2 4(1 − x) 2(1 − x) In order to use the Lie group-theoretic method, we now construct the following partial differential equation by replacing d/dx by ∂/∂x, m by y∂/∂y and ψα,β,γ ,m(x) by f (x, y): ∂2 xy 2 (β − 2 + x) ∂ 2 (β − 1)xy ∂ 2 x(β − x) 2 + − + ∂x 4(1 − x)2 ∂y 2 (1 − x) ∂x∂y ∂ + (γβ − (α + 1)x) − ∂x γβ − α(2 − x) ∂ x(β − x) − y + f (x, y) = 0. (3.5) 2(1 − x) 4(1 − x)2 ∂y Thus

∂ ∂ L = L x, , y ∂x ∂y ∂2 xy 2 (β − 2 + x) ∂ 2 (β − 1)xy ∂ 2 ≡ x(β − x) 2 + − + ∂x 4(1 − x)2 ∂y 2 (1 − x) ∂x∂y ∂ ∂ γβ − α(2 − x) x(β − x) + (γβ − (α + 1)x) y . (3.6) − + 2 ∂x 2(1 − x) 4(1 − x) ∂y

We need the following observations [3]:

GENERATING RELATIONS OF HYPERGEOMETRIC FUNCTIONS

291

OBSERVATION I. Let L(x, d/dx, n) be a linear differential operator containing a parameter n. Assuming that L is a polynomial in n, we construct a partial differential operator L(x, ∂/∂x, y∂/∂y) by substituting y∂/∂y for n. Then z = y n νn (x) is a solution of L(x, ∂/∂x, y∂/∂y) z = 0 if and only if u = νn (x) is a solution of L(x, d/dx, n)u = 0.

(3.7)

OBSERVATION II. Let G(x, y) have a convergent expansion of the form X G(x, y) = gn (x)y n , (3.8) n

where n is not necessarily a nonnegative integer. If L(x, ∂/∂x, y∂/∂y)G(x, y) = 0, then within the region of convergence of the series (3.8), u = gn (x) is a solution of (3.7). In particular, if G(x, y) is regular at x = 0, u = gn (x) is also regular at x = 0. In lieu of Observation I, we conclude that f (x, y) = y m ψα,β,γ ,m(x) is a solution of (3.5). Let us now introduce the first-order linearly independent differential operators J 3 , J − and J + each of the form A1 (x, y)

∂ ∂ + A2 (x, y) + A3 (x, y) ∂x ∂y

such that J 3 [y m ψα,β,γ ,m(x)] = am y m ψα,β,γ ,m(x), J − [y m ψα,β,γ ,m(x)] = bm y m−1 ψα,β,γ ,m−1 (x), +

J [y ψα,β,γ ,m(x)] = cm y m

m+1

(3.9)

ψα,β,γ ,m+1 (x),

where am , bm and cm are expressions in m which are independent of x and y, but not necessarily of α, β and γ . Each Ai (x, y), i = 1, 2, 3, on the other hand, is an expression in x and y which is independent of m but not necessarily of α, β and γ . By using (3.9) and recurrence relations (2.10) and (2.11), we get the following operators: ∂ γ + , ∂y 2 √ −1 1−x ∂ xy (2 − x) ∂ = − √ , β ∂x 2β 1 − x ∂y

J3 = y J−

J+ =

(3.10)

xy(β − x) ∂ ∂ y2 (γβ − αx)y √ (2β − 3βx + x 2 ) + √ + . 3/2 2(1 − x) ∂y 1 − x ∂x 1−x

Clearly, the operators J 3 , J − and J + satisfy the commutation relations [J 3 , J ± ] = ±J ± ,

[J + , J − ] = 2J 3 .

(3.11)

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It concludes that the J -operators generate a three-dimensional Lie algebra isomorphic to sl(2, C), which is the algebra of generalized Lie derivatives of a multiplier representation SL(2, C). In order to determine the multiplier representation of SL(2, C), we first compute the actions of exp(a 0 J + ), exp(b0 J − ) and exp(c0 J 3 ) on f ∈ F , where F is the complex vector space of all functions of x and y analytic in some neighbourhood of the point (x0 , y0 ) = (0, 0). Thus, if f ∈ F is analytic in a neighbourhood of (x0 , y0 ), then the values of the multiplier representations of exp(a 0 J + )f , exp(b0 J − )f and exp(c0 J 3 )f are given by [T (exp(a 0 j + ))f ](x0 , y0 ) = [exp(a 0 J + )f ](x0, y0 ) √ (1 − x0 )γ /2 [ 1 − x0 − a 0 y0 (β − x0 )]−α √ × = [ 1 − x0 − a 0 βy0 ]γ −α √ y0 [(1 − x0 )3/2 − a 0 y0 (β − x0 )]1/2 x0 1 − x0 , √ √ ×f √ , 1 − x0 − a 0 y(β − x0 ) [ 1 − x0 − a 0 βy0 ][ 1 − x0 − a 0 y0 (β − x0 )]1/2 0 0 a βy0 a y0 (β − x0 ) √ √ < 1, |x0 | < 1, 1 − x < 1; 1 − x0 0 [T (exp(b 0 j − ))f ](x0 , y0 ) √ √ (βy0 − b0 1 − x0 )(βy0 1 − x0 − b0 ) 1/2 βx0 y0 , , =f √ √ βy0 − b0 1 − x0 β 2 1 − x0 0√ b 1 − x0 b0 < 1; < 1, |x0 | < 1, √ βy0 βy0 1 − x0 0 γc 0 0 3 f (x0 , y0 ec ). [T (exp(c j ))f ](x0, y0 ) = exp 2

(3.12)

Now, in the neighbourhood of the identity, every g ∈ SL(2, C) can be expressed as g = exp(a 0 j + )) exp(b0 j − )) exp(c0 j 3 )), from which we get the operator T (g) acting on f ∈ F given by [T (g)f ](x, y) √ √ γ c0 ( 1 − x)γ [ 1 − x − a 0 y(β − x)]−α 0 = exp f (ξ, ηec ), √ 0 γ −α 2 [ 1 − x − a βy] where

√ βxy 1 − x

ξ= √ , √ [ 1 − x − a 0 y(β − x)][βy(1 + a 0 b0 ) − b0 1 − x]

(3.13)

293

GENERATING RELATIONS OF HYPERGEOMETRIC FUNCTIONS



1/2 √ [βy(1 + a 0 b0 ) − b0 1 −√x]{βy[(1√− x)3/2 − a 0 y(β − x)]+   0 0 − x) − b0 1 − x][ 1 − x − a 0 βy]} η =  + [a b y(β  √ √ β 2 [ 1 − x − a 0 βy]2 [ 1 − x − a 0 y(β − x)] and

g=

a c

b d

∈ SL(2, C).

Further, by setting b a0 = − , d we have

b0 = −cd,

exp

c0 2

= d −1

and

ad − bc = 1,

√ [d 1 − x + by(β − x)]−α [T (g)f ](x, y) = ( 1 − x) f (ξ, ηd −2 ), (3.14) √ [d 1 − x + bβy]γ −α √

where

γ

√ βxy 1 − x

ξ =

√ √ , [d 1 − x + by(β − x)][aβy + c 1 − x]  1/2 √ 3/2 [aβy + c 1 − x]{βy[d(1 − x) + by(β − x)]+ √ √  − x) + d 1 − x][d 1 − x + bβy]}  η =  + c[by(β  , √ √ β 2 d −3 [d 1 − x + bβy][d 1 − x + by(β − x)] provided

√ c 1 − x aβy < 1,

|x| < 1, by(β − x) √ d 1 − x < 1,

and

bβy √ d 1 − x < 1,

| arg(a), arg(d)| < π.

Here g lies in a sufficiently small neighbourhood of the identity element 10 01 ∈ SL2 . Equation (3.14) defines a local multiplier representation of SL(2, C) and the differential operators J + , J − , J 3 are generalized Lie derivatives of T . In terms of the J -operators, we introduce the Casimir operator [5]. C = C1,0 = J + J − + J 3 J 3 − J 3 x xy 2 (β − 2 + x) ∂ 2 (β − 1)xy ∂ 2 ∂2 = − x(β − x) 2 + + β ∂x 4(1 − x)2 ∂y 2 (1 − x) ∂x∂y ∂ ∂ γβ − α(2 − x) x(β − x) + [γβ − (α + 1)] y − + + ∂x 2(1 − x) 4(1 − x)2 ∂y γ + (γ − 2). (3.15) 4

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It can be verified that C commutes with J + , J − and J 3 . Equation (3.15) enables us to write (3.5) as Cf (x, y) =

γ (γ − 2)f (x, y). 4

(3.16)

To derive the generating functions, we search for the functions f (x, y) which satisfy (3.16). Now, we consider the following different cases: Case 1. When f (x, y) is a common elgenfunction of C and J 3 . The simultaneous equations γ (γ − 2)f (x, y) 4

(3.17)

γ J 3 f (x, y) = ν + f (x, y) 2

(3.18)

Cf (x, y) = and

admit a solution f (x, y) = y ν ψα,β,γ ,ν (x), so that [T (g)f ](x, y)

√ c 1 − x ν/2 = (1 + bc) 1+ a × aβy 1 α−γ −ν by(β − x) − 2 (ν+2α) bβy × 1+ √ 1+ √ × d 1−x d 1−x ν/2 by(β − x) by(β − x) bβy × 1+ √ × + cd 1 + √ 1+ √ d 1−x d 1−x d 1−x x √ × ψα,β,γ ,ν , (3.19) 1−x √ 1 + by(β−x) (1 + bc) 1 + c aβy d 1−x − 12 (2γ +3ν) γ +2ν

which satisfies the relation C[T (g)f ](x, y) = 14 γ (γ − 2)[T (g)f ](x, y). If ν is not an integer, Equation (3.19) has a Laurent series expansion [3]: [T (g)f ](x, y) =

∞ X

hp (g, x)y ν+p .

(3.20)

p=−∞

Considering that [T (g)f ](x, y) is regular at x = 0, then from the result of (3.8), we have hp (g, x) = jp (g)ψα,β,γ ,ν+p (x), which implies that [T (g)f ](x, y) =

∞ X p=−∞

jp (g)ψα,β,γ ,ν+p (x)y ν+p .

(3.21)

GENERATING RELATIONS OF HYPERGEOMETRIC FUNCTIONS

295

To determine jp (g), put x = 0 and equate the coefficient of y p , and we get a γ +2ν (1 + bc)−γ −ν 0(1 + ν + p) b p jp (g) = × − 0(1 + p)0(1 + ν) d " # −ν, ν + γ + p; bc × 2 F1 . (3.22) 1 + p; ad Thus, the generating function (3.20) becomes √ 1 c 1 − x ν/2 by(β − x) − 2 (ν+2α) −ν/2 (1 + bc) 1+ 1+ √ × aβy d 1−x α−γ −ν bβy by(β − x) × 1+ √ 1+ + d(1 − x)3/2 d 1−x ν/2 cd by(β − x) bβy + √ × 1+ √ 1+ √ βy 1 − x d 1−x d 1−x x √ × ψα,β,γ ,ν 1−x √ 1 + by(β−x) (1 + bc) 1 + c aβy d 1−x =

0(1 + ν + p) by p × − 0(1 + ν)0(1 + p) d p=−∞ " # −ν, ν + γ + p; bc × 2 F1 ψα,β,γ ,ν+p (x), 1 + p; ad c βy b − by(β − x) |x| < 1, 1, < < √ d √1 − x < 1, a d 1−x −π < arg(a), arg(d) < π and ad − bc = 1. (3.23) ∞ X

Deductions. (I) By putting a = d = β = y = 1 and c = 0, Equation (3.23) yields α−γ −ν/2 √ b x 1 (1 − b 1 − x)− 2 (ν+2α) 1 − √ ψα,1,γ ,ν √ 1−x 1−b 1−x ∞ X (1 + ν)p = ψα,1,γ ,ν+p (x)bp , p! p=0 √ b |b 1 − x| < 1, (3.24) √1 − x < 1. (II) Taking a = d = β = y = 1 and b = 0, we have ν/2 √ x c (1 − c 1 − x) 1 − √ ψα,1,γ ,ν √ 1−x 1−c 1−x

296

I. K. KHANNA ET AL.

=

∞ X (1 − γ − ν)p ψα,1,γ ,ν−p (x)cp , p! p=0 √ c |c 1 − x| < 1, √1 − x < 1.

(3.25)

Applications. Further, using Section 2.2, we get the following generating functions: (i)

∞ k X (ν + 1)k L(α) ν+k (x)b k=0

k!

= (1 − b)−1−α−ν exp

(ii)

∞ k X (−α − ν)k L(α) ν−k (x)c

k!

k=0

(iii)

k!22k

= (1 − b)

− 12 (2ν+2µ+1)

∞ µ X (−2ν)2k H2ν−2k (x)ck

k!

k=0

(v)

k!22k

= (1 − b)

− 12 (2ν+2µ+3)

|c| < 1;

−x 2 b x µ exp H √ , 1 − b 2ν 1−b = (1 − c)

ν

µ H2ν

√

|b| < 1;

x 1−c

,

|c| < 1;

−x 2 b x µ exp H √ , 1 − b 2ν 1−b

∞ µ X (−2ν − 1)2k H2ν+1−2k (x)ck

k!

k=0

(vii)

x , 1−c

∞ µ X (−1)k H2ν+1+2k (x)bk k=0

(vi)

= (1 − c)ν L(α) ν

|b| < 1;

∞ µ X (−1)k H2ν+2k (x)bk k=0

(iv)

x −xb L(α) , 1−b ν 1−b

= (1 − c)

ν

µ H2ν+1

|b| < 1; x

√ , 1−c

∞ X (ν + γ )k ρ k/2 Mν+k (y; γ , ρ)bk

k!

k=0

= (1 − bρ

1/2 −y−γ −ν

|bρ 1/2 | < 1,

)

(1 − bρ

−1/2 y

) Mν

ρ 1/2 − b y; γ , −1/2 , ρ −b

|bρ −1/2 | < 1,

provided that γ > 0, 0 < ρ < 1, y = 0, 1, 2, . . . ;

|c| < 1;

297

GENERATING RELATIONS OF HYPERGEOMETRIC FUNCTIONS

(viii)

∞ X (−ν)k ρ −k/2 Mν−k (y; γ , ρ)ck

k!

k=0

−1/2 ν

= (1 − cρ

) Mν

ρ 1/2 − c y; γ , −1/2 , ρ −c

|cρ −1/2 | < 1,

provided that γ > 0, 0 < ρ < 1, y = 0, 1, 2, . . . ;

(ix)

∞ X (ν + 1)k ekλ/2 φν+k (y, λ)bk

k!

k=0

= (1 − be

λ/2 y−ν

)

(1 − be

)

φν

eλ/2 − b y, log −λ/2 e −b

,

|be−λ/2 | < 1;

|beλ/2 | < 1,

(x)

−λ/2 −y−1

∞ X (−ν)k e−kλ/2 φν−k (y, λ)ck

k!

eλ/2 − c = (1 − ce ) φν y, log −λ/2 , |ce−λ/2 | < 1; e −c k ∞ X (−N + ν)k b (xi) Kν+k (y; P , N) √ k! 1 − P −1 k=0 N−y−ν √ √ y 1 − √ b −1 = 1 − b 1 − P −1 Kν (y; p − bp 1 − P −1 , N), 1−P b √ −1 |b 1 − P | < 1, √ −1 < 1; k=0

−λ/2 ν

1−P

(xii)

∞ X (−ν)k

k!

p k Kν−k (y; P , N) c 1 − P −1

√ √ = (1 − c 1 − P −1 )ν Kν (y; p − cp 1 − P −1 , N),

k=0

√ |c 1 − P −1 | < 1,

provided that 0 < P < 1, y = 0, 1, 2, . . . , N, which are the generating relations for the Laguerre, even and odd generalized Hermite, Meixner, Gottlieb, Krawtchouk and polynomials, respectively. The generating function (3.23) is valid under the condition that ν is not an integer. However, if ν is a nonnegative integer, say ν = k, the generating function can be further written as (1 + bc)

−k/2

√ 1 c 1 − x k/2 by(β − x) − 2 (k+2α) 1+ 1+ √ × aβy d 1−x

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I. K. KHANNA ET AL.

α−γ −k by(β − x) bβy 1+ + × 1+ √ d(1 − x)3/2 d 1−x k/2 cd by(β − x) bβy + √ × 1+ √ 1+ √ βy 1 − x d 1−x d 1−x x √ × ψα,β,γ ,k 1−x √ 1 + by(β−x) (1 + bc) 1 + c aβy 1−x ∞ X p! by p−k = {0(p − k + 1)}−1 × − k! d p=0 " # −k, γ + p; bc × 2 F1 ψα,β,γ ,p (x), p − k + 1; ad c βy b −1 < , ad − bc = 1. < √ a d 1 −x

(3.26)

Case 2. When f (x, y) is a common elgenfunction of the operators C and J − . Let f (x, y) be a solution of the simultaneous equations γ Cf (x, y) = (γ − 2)f (x, y) (3.27) 4 and J − f (x, y) = −f (x, y), which may be written as ∂2 xy 2 (β − 2 + x) ∂ 2 (β − 1)xy ∂ 2 x(β − x) 2 + − + ∂x 4(1 − x)2 ∂y 2 (1 − x) ∂x∂y ∂ + [γβ − (α + 1)x] − ∂x γβ − α(2 − x) ∂ x(β − x) − y + f (x, y) = 0 2(1 − x) 4(1 − x)2 ∂y and

√ xy −1 1 − x ∂ (x − 2) ∂ + √ + 1 f (x, y) = 0. β ∂x 2β 1 − x ∂y

Assuming the general solution of (3.28) and (3.29) in the form βy xy f (x, y) = exp √ K √ , 1−x 1−x and substituting this in (3.29), we get 2 d d u 2 + (γ + u) + α K(u) = 0, du du

(3.28)

(3.29)

(3.30)

(3.31)

299

GENERATING RELATIONS OF HYPERGEOMETRIC FUNCTIONS

√ where u = xy/ 1 − x. This is Kummer’s differential equation [3] and has for its solution K(u) = 1 F1 [α; γ ; −u], √ where u = xy/ 1 − x. Thus, one solution of this system is βy xy f (x, y) = exp √ . 1 F1 α; γ ; − √ 1−x 1−x

(3.32)

(3.33)

If this function is expanded in powers of y, we get exp √

βy 1−x

1 F1

∞ X ψα,β,γ ,m(x)y m α; γ ; − √ , = (γ )m 1−x m=0

xy

(3.34)

which is the generating function for ψα,β,γ ,m(x). From (3.33), we have [T (g)f ](x, y) √ √ √ = ( 1 − x)γ [d 1 − x + by(β − x)]−α [d 1 − x + bβy]α−γ × × exp(ξ )1 F1 [α; γ ; −φ],

(3.35)

where    ξ = 

√ 3/2 (aβy + c 1 − x)2 {βy[d(1 √ − x) √+ by(β − x)]+ + [bcy(β − x) + cd 1 − x][d 1 − x + bβy]} √ √ d[d 1 − x +√ bβy]2 {[d 1 − √ x + by(β − x)]× × [aβy + c 1 − x] − βxy 1 − x}

1/2    

and    φ= 

x 2 y 2 (1 − x){βy[d(1 −√ x)3/2 + by(β √ − x)]+ + [bcy(β − x) + cd 1 − x][d 1 − x + bβy]} √ √ 2 d[d 1 − x + by(β − x)] [d 1 − x +√ bβy]2 × √ √ × {[d 1 − x + by(β − x)][aβy + c 1 − x] − βxy 1 − x}

1/2    

satisfying the relation C[T (g)f ](x, y) =

1 γ (γ − 2)[T (g)f ](x, y). 4

(3.36)

Since [T (g)f ](x, y) is analytic at y = 0, it can be expanded in the form [T (g)f ](x, y) =

∞ X m=0

Pm (g)ψα,β,γ ,m(x)y m .

(3.37)

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To compute the coefficients Pm (g), we put x = 0 in (3.37) and we have ∞ aβy + c X (γ )m (d + byβ)−γ exp = Pm (g)(βy)m , d + bβy m! m=0

(3.38)

which may be written as ! −γ c βy bβy d −γ exp exp 1+ d d d 2 1 + bβy d =

∞ X (γ )m m=0

m!

Pm (g)(βy)m .

(3.39)

From the well-known generating function for the Laguerre polynomials [1], we have ∞ −xt X −1−α n (1 − t) exp L(α) (3.40) = n (x)t . 1−t n=0 On comparing (3.39) with the well-known generating function (3.40), we find b m m! −γ −1) 1 Pm (g) = d exp(c/d) − L(γ . (3.41) m (γ )m d bd Thus we have α−γ by(β − x) −α bβy 1+ √ 1+ √ exp(ξ 0 )1 F1 [α; γ ; −φ 0 ] d 1−x d 1−x ∞ 1 by m X m! = exp(c/d)Lγm−1 , ψα,β,γ ,m(x) − (γ )m bd d m=0 by < 1, d where 

ξ0 = 

and

(3.42)

1/2 by(β−x) by(β−x) cd √ √ 1 + d(1−x) 1 + √bβy 3/2 + βy 1−x 1 + d 1−x d 1−x  √ bβy 2 by(β−x) c 1−x 2 √ √ d 1 + d 1−x (1 + bc) 1 + d 1−x 1 + aβy − x

a2β 2y 2 1 +

√ c 1−x 2 aβy



1/2 by(β−x) by(β−x) cd √ √ x 2 y 2 1 + d(1−x) 1 + √bβy 3/2 + βy 1−x 1 + d 1−x d 1−x √ φ0 =  2 2  . by(β−x) bβy by(β−x) 1−x 4 d 1 + d √1−x 1 + d √1−x (1 + bc) 1 + d √1−x 1 + c aβy −x

Deductions. If √ √ a = i w, b = i/ w, √ i = −1 and β = 1,

c = 0,

√ d = −i/ w,

301

GENERATING RELATIONS OF HYPERGEOMETRIC FUNCTIONS

then we have:

wy (1 − y 1 − x) 1− √ exp − √ × 1−x 1−x−y wxy × 1 F1 α; γ ; √ √ ( 1 − x − y)(1 − y 1 − x) ∞ X m! (γ −1) = Lm (w)ψα,1,γ ,m(x)y m , (γ ) m m=0 √

−α

y

α−γ

(3.43)

or

wy (1 − y 1 − x) 1− √ exp − √ × 1−x 1−x−y wxy × 1 F1 α; γ ; √ √ ( 1 − x − y)(1 − y 1 − x) m ∞ X y −1) = L(γ (w) F [−m, α; γ ; x] √ , 2 1 m 1 − x m=0 √

−α

y

α−γ

(3.44)

which is the bilateral generating function involving the Laguerre polynomial and a certain terminating 2 F1 . 4. Applications As usual, we get the following applications from (3.34): (i)

∞ X L(λ)(x)t m m

m=0

(1 + λ)m

= exp(t) 0 F1 [−; 1 + λ; −xt];

∞ X 1 (−1)m H2m(x)t m x2t (ii) = exp(t) 0 F1 −; ; − 2m m! 1 2 2 2 2 m m=0 or ∞ X (−1)m H2m(x)z2m m=0

(iii)

(2m)!

√ = exp(z2 ) cos( 2 xz);

∞ X (−1)m H2m+1 (x)t m m=0

2x22m m!(3/2)m

3 x2t = exp(t) 0 F1 −; ; − 2 2

or ∞ X (−1)m H2m+1 (x)z2m+1 m=0

(2m + 1)!

√ = exp(z2 ) sin( 2 xz);

302 (iv)

(v)

(vi)

I. K. KHANNA ET AL. ∞ µ h µ+1 i X (−1)m H2m (x)t m 2 −; = exp(t) F t ; ; −x 0 1 2 m! µ+1 2 m m=0 ∞ µ h µ+3 i X (−1)m H2m+1 (x)t m 2 −; = x exp(−t) F t ; ; −x 0 1 2 m! µ+3 2 m m=0 ∞ X Mm (y; γ , ρ)(tρ 1/2 )m

m!

m=0

= exp(tρ 1/2 ) 1 F1 [−y; γ ; (ρ −1/2 − ρ 1/2 )t];

or, equivalently, ∞ X Mm (y; γ , ρ)zm m=0

m!

= exp(z) 1 F1 [−y; γ ; −z(1 − ρ −1 )],

provided that γ > 0, 0 < ρ < 1, y = 0, 1, 2, . . . ;

(vii)

∞ X emλ/2 φm (y, λ)t m m=0

m!

= exp(te−λ/2 ) 1 F1 [−y; 1; (eλ/2 − e−λ/2 )t],

which can be reduced to ∞ X φm (y, λ)zm m=0

(viii)

m!

= exp(z) 1 F1 [1 + y; 1; −z(1 − e−λ )t].

∞ X Km (y; P , N)

t

m

√ −1 1−P P −1 t t , = exp √ F −y; −N; − √ −1 1 1 −1 1−P 1−P

m=0

m!

which can be rewritten as ∞ X Km (y; P , N)zm m=0

m!

= exp(z) 1 F1 [−y; −N; −zP −1 ],

0 < ρ < 1, y = 0, 1, 2, . . . , N. These are all well-known generating functions in one form or another for the Laguerre, even and odd (generalized) Hermite, Meixner, Gottlieb, and Krawtchouk polynomials, respectively. Remark. The corresponding bilateral (or bilinear) generating relations for the Laguerre, even and odd Hermite, Meixner, Gottlieb, and Krawtchouk polynomials can be deduced from (3.43) by using the conditions of Section 2.

GENERATING RELATIONS OF HYPERGEOMETRIC FUNCTIONS

303

Acknowledgement The authors wish to express their sincere thanks to the referee for the kind suggestions given to improve this paper. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Rainville, E. D.: Special Functions, Macmillan, New York, 1960. Szegö, G.: Orthogonal Polynomials, 4th edn, Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, Rhode Island, 1975. Srivastava, H. M. and Manocha, H. L.: A Treatise on Generating Functions, Ellis Horwood, England, 1984. Chihara, T. S.: An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. Miller, W.: Lie Theory and Special Functions, Academic Press, New York, 1968. Khanna, I. K. and Srinivasa Bhagavan, V.: Weisner’s method to obtain generating relations for the generalized polynomial set, J. Phys. A Math. Gen. 32 (1999), 1–10. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.: Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953. Cohn, P. M.: Lie Groups, Cambridge Univ. Press, Cambridge, 1961. Vilenkin, N. Ja. and Klimyk, A. U.: Representation of Lie Groups and Special Functions, Vols. 1 and 2, Kluwer Acad. Publ., Dordrecht, 1991, 1993.

Mathematical Physics, Analysis and Geometry 3: 305–321, 2000. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

305

Gauge Fields with Generic Singularities SPYROS PNEVMATIKOS1 and DIMITRIS PLIAKIS2 1 Department of Mathematics, University of Patras, 26500 Patras, Greece.

e-mail: [email protected] 2 Department of Physics, University of Crete, 71409 Heraklion, Greece. (Received: 10 March 2000; in final form: 16 January 2001) Abstract. Let M be an n-dimensional manifold equipped with an Abelian Yang–Mills field with connection form α. We consider an external potential function V and examine the existence and regularity of the vortex lines of the form α + V dt which define the motion of a particle weakly coupled to the Yang–Mills field on M. These curves are smooth unless the curvature form dα is singular and in this paper we treat this singular case from a generic aspect. The problem reduces to the division properties for smooth functions and differential forms, the development of which constitutes the main part of the work presented here. Mathematics Subject Classifications (2000): 70S15, 58A10, 58A35. Key words: gauge theory, singularities, stratifications.

0. Introduction The motion of a light particle on a smooth manifold M in the presence of a strong Abelian gauge field with connection form α and an external potential function V , reduces to the classical variational problem on the space of paths γ on M, which asks for the stationary points of the functional I (γ ) = γ (α + V dt). The corresponding Euler–Lagrange equations could be expressed by the vector field XV which is tangent to the paths γ and is given by the equation iXV dα + dV = 0 under the assumption that the curvature form dα is nondegenerate on M. The curvature form coincides with the restriction of the canonical symplectic form of the cotangent bundle on the image of the section α: M → T ∗ M and, evidently, could appear with a degeneracy locus consisting of the points where its kernel is not trivial. Several studies have appeared in the classical context with regular fibering, originating from Dirac’s works on the canonical quantization of electromagnetic theory and, in the geometrical intrinsic aspect, by Lichnerowicz, cf. [5, 12]. Generically, this locus is stratified in smooth submanifolds and a singular fibering with vector spaces of variable dimension arises. Here, the problems caused by the successive degeneracies are tackled by essentially applying the ideas of the theory of stratified symplectic structures in a generic context, cf. [20]. We focus on the external potentials that could define dynamics everywhere on M despite the apparent degeneracies; these functions constitute an algebra, the Dirac

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algebra of the structure (M, α). More precisely, we prove that the Dirac algebra consists of those smooth functions whose differential annihilates the kernel of the curvature form on the points of the first singular stratum of the structure (M, α); also, we deduce the behavior of the motion in the neighborhood of the degeneracy locus. The problem reduces to the division properties in the context of smooth functions and smooth differential forms and the development of these properties is the main purpose of this paper. These division questions were studied in the holomorphic category in the 70’s ([14, 17, 24]). Here, we treat a special case of the divisor form, relying on a smooth version of the Hironaka division by an ideal generated by analytic functions that define the degeneracy locus. An essential role is played by the ideal of the smooth functions that identically vanish on every stratum of the degeneracy locus and its equality with the ideal generated by the functions that define this stratum; in other words, we are dealing here with a problem of the differentiable Nullstellensatz on the stratified degeneracy locus. Finally, from the generic aspect, we describe the behavior of singular Lagrangian systems on a manifold M; the singular locus of the motion is exactly the projection on M of the critical locus of the corresponding Legendre transformation between T M and T ∗ M. The Lagrangian functions L that possess a Lagrangian vector field XL defined everywhere on T M are characterized by pointwise conditions on the first stratum of the critical locus and the projection of its integral curves gives the motion on M; in particular, the trajectories with initial conditions in a stratum always remain in this stratum. In the case when the Lagrangian is defined by a generic quadratic form on M, a singular connection, generalizing the Levi-Civita connection, can be used for the determination of the geodesics in the stratified singular locus. This description leads to the study of the singular variational problem developed in [22].

1. The Motion in the Presence of an Abelian Gauge Field 1.1. We begin by illustrating our methods with the planar motion of a particle of mass m and charge q in a strong magnetic field B = curl A with vector potential A. The motion is governed by the Euler-Lagrange equations that are implied by the usual Lagrangian with the minimal coupling substitution L(x, υ) = 12 (mυ + qA(x))2 − U (x), where x, υ denote the position and velocity vectors of the particle on the plane and U (x) is an external potential. In the limit of a light particle strongly coupled with the magnetic field, the Lagrangian is of first order with respect to the velocities L(x, υ) = mqυ.A(x) − V (x)

with V (x) = U (x) − 12 (qA(x))2

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and the equations of motion are put into the appealing form of a system of linear equations B(x) × υ = −∇V (x). Evidently, the zeroes of the magnetic field raise an obstruction to the existence of motion everywhere on the plane for an arbitrary external potential. Let us suppose that the magnetic field is written as B(x, y) = φ(x, y)n where n is the unit normal vector to the (x, y)-plane in R3 and the equations of motion are written as φ(x, y)n × υ = −∇V (x, y). Clearly, the existence of dynamics everywhere on the plane is equivalent to the divisibility of the right-hand side by φ(x, y) in the ring of smooth functions on R2 . A first elementary case, which illustrates the generic situation, appears when (∂xi ∂yj φ)(0, 0) = 0,

i + j < k,

(∂xk φ)(0, 0) = 0.

Then, under a local change of coordinates (x, y) → (ξ, ζ ), we get B(ξ, ζ ) = ξ k n and, after the classical smooth division, the admissible external potentials are written as V (ξ, ζ ) = ξ k+1 V (ξ, ζ ), where V is a smooth function and, thus, the solution of the equation of motion is υ = ξ n × ∇V (ξ, ζ ) + 2V (ξ, ζ )e2 , where (e1 , e2 ) denotes the base of the (x, y)-plane. In conclusion, the above external potentials are admissible for the magnetic field, i.e. the corresponding equations of motion are everywhere defined, overpassing the obstruction raised by the zeroes of the magnetic field. Clearly, these potentials constitute an algebra, called Dirac algebra, with the bracket {V1 , V2 } = ξ k (∇V1 × ∇V2 ) · n. If we also assume that the magnetic field defined by the vector potential A(x, y) = (y p , x q ), where p, q ∈ N, g.c.d. (p, q) = 1; if p, q > 1, we observe that the equation of motion (py p−1 − qx q−1 )n × υ = −∇V (x, y)

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possesses a critical locus that is an algebraic curve with a singularity at (0, 0), unlike the simple case described above. Evidently, we have to study a more complicated division problem on this locus and this issue will be treated in the sequel in a rather general context. 1.2. Let M be a smooth n-dimensional manifold equipped with the connection form of an Abelian Yang–Mills field α: M → T ∗ M, cf. [10]. The motion of a light particle on M, strongly coupled with the Yang–Mills field, with external potential V : M → R, is defined by the Lagrangian L := iC π ∗ α + π ∗ V with π : T M → M the canonical projection and C the tautological vector field on T M. The connection form is expressed locally as ai (x) dxi α(x) = 1in

and the Lagrangian leads to Euler–Lagrange equations which are of first order: (∂xj ai (x) − ∂xi aj (x))x˙j = ∂xi V (x), i = 1, . . . , n 1j n

with the critical locus being the set of points x ∈ M where the velocities are not, a-priori, well defined: det(∂xj ai − ∂xi aj )(x) = 0. These equations are expressed intrinsically by the vector field XV where this could be defined by the equation of motion iXV !α + dV = 0 outside of the degeneracy locus of the curvature form !α = dα. The curvature form is the restriction of the canonical symplectic form on the image of the section α: M → T ∗ M and the critical locus of the equations is exactly the degeneracy locus "k (!α ) "(!α ) = k>0

with "k (!α ) = {x ∈ M/ dim kerx !α = k},

k ∈ N.

In the sequel, the manifold M is assumed to be orientable with a volume form w and a Riemannian structure with inner product , on the fibers of ∧r T ∗ M, r = 1, . . . , n, thus giving rise to the Hodge ∗-operator between the fibers of ∧r T ∗ M and ∧n−r T ∗ M. Clearly !sα (x) ∧ ∗!sα (x) = ρα,s (x)w

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with 2 ρα.s (x) = !sα (x) and, if the dimension of M is 2m or 2m + 1, then −1 (0) = "k (!α ). ρα,m k>1 −1 (0) = ∅. In the Let us note in passing the effects in the known situation when ρα,m even 2m-dimensional case, the equation of motion has a smooth solution XV on M given by

iX V w =

1 ρα,m

dV ∧ !m−1 . α

In the odd (2m + 1)-dimensional case, we introduce ∗θV = iXV w and the equation of motion is equivalent to a division identity of a 2-form by a 1-form m−1 ). θV ∧ ∗!m α = ∗(dV ∧ !α

The kernel of ∗!m α is everywhere one-dimensional and, hence, defines a regular foliation; as is known, the annihilation of dV on the kernel of !α is a necessary and sufficient condition for the existence of XV on M (cf. [12]). This in turn yields the condition which is necessary for the division question that will appear in the sequel. Furthermore, if the curvature form has constant but nonmaximal rank on M, i.e. ρα,s (x) = 0 and ρα,s+1 (x) = 0, for s < n, then since = iXV !sα , s(iXV !α ) ∧ !s−1 α the external potentials that lead to a smooth solution XV on M are given by iX V w =

ρα,s−1 ∗ (dV ∧ ∗!α ). ρα,s

The admissible external potentials here also constitute the Dirac algebra endowed with the Poisson bracket {V1 , V2 } = !α (XV1 , XV2 ). 1.3. The connection forms on an n-dimensional smooth manifold M, with the curvature form transverse to the natural stratification by the rank of the bundle ∧2 T ∗ M, constitute an open and dense set in the space of connection forms on M equipped with the Whitney C ∞ -topology (cf. [16, 20]). These are called generic connection forms. The manifold M equipped with a generic connection form α is stratified into the smooth submanifolds "k (!α ) = {x ∈ M | dim kerx !α = k},

0 k n,

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of codimension k(k − 1)/2, if nonempty, whose closure satisfies the frontier condition "k (!α ) = "k (!α ). k k

The versal unfolding lemma, established for differential forms in [20], immediately leads to a natural local classification of the generic connection forms through the following normal form on Rn given in local coordinates (x1 , . . . , xn ) with τ arbitrary injection in {1, . . . , n} by xτ (i,j ) dxi ∧ dxj + dxk+1 ∧ dxk+2 + · · · + dxn−1 ∧ dxn , k n. ωk = 1i<j k

THEOREM 1. If α is a generic connection form on an n-dimensional smooth manifold M and xo ∈ "k (!α ), there exist both a germ of diffeomorphism and a germ of smooth map, respectively, φ: (M, xo ) → (Rn , 0),

ψ: (M, xo ) → GL(n, R)

such that, for every x in a neighborhood of xo , we have (dα)(ψ(x)., ψ(x).) = φ ∗ ωk (., .). The diffeomorphism φ respects the germs of the corresponding stratification, for k k, in the following sense: "k (dα) = φ −1 ("k (ωk )). Remark 1. In the case k = 0 or 1, we find the formal expression of the classical Darboux local model which always holds for the germs of closed 2-forms of constant rank. In the case k = 2, for even dimension, we find the formal expression of the singular Martinet’s local model: dα = x1 dx1 ∧ dx2 + dx3 ∧ dx4 + · · · + dxn−1 ∧ dxn . Remark 2. After a direct calculation on the normal form, the preceding theorem suggests that the local equations of the algebraic set "k (!α ) are given through the polynomials (−1)sgn σ xσ (τ (i1 ,j1 ) . . . xσ (τ (ik ,jk ) P(i,k) = σ

where the sum is taken over all permutations of {(i1 , j1 ), . . . , (ik , jk )} for all disjoint couples (i1 , j1 ), . . . , (ik , jk ), 1 il jl k ([20]). Such polynomials are called affine polynomials and provide the required division property in the ring of smooth functions essential for handling our problem. Also, Lemma 4 suggests that every nonempty stratum "k (!α ), k > 1, is defined locally by k(k − 1)/2 homogeneous affine polynomials that generate, in the ring of germs of smooth functions on M, the ideal k (!α ) = f ∈ Cx∞o (M) | f |"k (!α ) = 0 , 1 < k n.

GAUGE FIELDS WITH GENERIC SINGULARITIES

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2. The Dirac Algebra of External Potentials 2.1. Let α be the connection form of an Abelian gauge field on a smooth manifold M. We call Dirac algebra (denoted by D(M, α)) the algebra of smooth functions V : M → R that defines dynamics everywhere on M through the equation of motion iXV !α + dV = 0 admitting a unique smooth solution XV everywhere on M, where !α is the curvature form of α. In this case, the Poisson bracket is well defined, for V1 , V2 ∈ D(M, α), by {V1 , V2 } = !α (XV1 , XV2 ). THEOREM 2. The Dirac algebra D(M, α) relative to a generic connection form α on a smooth manifold M consists of those smooth functions V on M whose differential annihilates the kernel of the curvature form !α at the points of the first singular stratum of the structure (M, α). In other words, when dim M = 2n, resp. dim M = 2n + 1, then at every point of "2 (!α ), resp. "1 (!α ), this condition is expressed as ∧ dV = 0, !n−1 α

resp. !nα ∧ dV = 0.

Proof. If dim M = 2n, the equation of motion admits a unique smooth solution XV on the open dense stratum "o (!α ). We stand at a point of "2 (!α ) and, in a neighborhood of it, we trivialize the vector bundles T M and T ∗ M. Performing the local change of coordinates and the change of frames suggested by Theorem 1, the germ of !α is expressed as ω = x1 dx1 ∧ dx2 + ω0

with ω0 = dx3 ∧ dx4 + · · · + dx2n−1 ∧ dx2n .

The equation of motion is consequently written as iX ω = η, where η denotes the corresponding smooth 1-form to −dV in the local context at the origin of R2n . Setting ∗θX = iX w, where w is a volume form, the equation of motion is written as ∗ωn θX = ∗(η ∧ ωn−1 ). Clearly ∗ωn = x1 ,

ωn−1 = iK1 iK2 w + x1 ω0n−2 ∧ dx1 ∧ dx2

with KI = ∂/∂xi , i = 1, 2. In view of the assumption, since it implies that ∗(η ∧ ωn−1 ) = x1 η , where η is a germ of a smooth 1-form, Hadamard’s lemma gives the existence of a local smooth solution on "2 (ω) and, consequently, the existence of a smooth solution of the equation of motion on "o (!α ) ∪ "2 (!α ). This further suggests the annihilation of the second member of the equation of motion on "2 (!α ) and,

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hence, by continuity, on "(!α ). Standing on a local chart centered at an arbitrary point of "(!α ), the equation of motion, in matrix form, is written as [∗!nα (x)]2 XV (x) = B(x), where B(x) is the matrix product of the second member by the comatrix of !α (x). Lemma 4, which we prove below, suggests that, in suitable coordinates, ∗!α (x) is an affine polynomial that generates the ideal of germs of smooth functions that vanish on "(!α ). This property, used in two steps, assures the smooth division of B(x) by the square of ∗!α (x), cf. [20], and this implies the local smooth solution of the equation of motion that we extend to a global smooth solution by means of a partition of unity on M. If dim M = 2n + 1, we stand at a point a ∈ "3 (!α ) and, performing the local change of coordinates and the change of frames suggested by Theorem 1, the germ of !α is expressed as ω = iR (dx1 ∧ dx2 ∧ dx3 ) + ω0 , where ω0 = dx4 ∧ dx5 + · · · + dx2n ∧ dx2n+1 and R(x) =

Rij xi ∂/∂xj

with det(Rij ) = 0.

1i,j 3

The equation of motion is consequently written as iX ω = η, where η denotes the corresponding smooth 1-form to −dV in the local context at the origin of R2n+1 . Setting θX = ∗iX w,

θR = ∗iR w,

w3 = dx1 ∧ dx2 ∧ dx3 ,

this equation is written as θX ∧ θR = ∗(η ∧ ωon−1 ) + ∗(η ∧ w3 ∧ ω0n−2 )θR . The compatibility condition is expressed as iR η = 0 and, furthermore, we have θR ∧ ∗(η ∧ ω0n−1 ) = 0. Lemma 5, which we prove below (see also de Rham’s division theorem [17]), suggests that this equation has a local smooth solution on "3 (!α ) and we conclude the same for the equation of motion. By continuity, we conclude that ∗!nα ∧ ∗(dV ∧ !n−1 α )=0 holds on M and this is exactly the condition of Lemma 5. Standing at a point of an arbitrary stratum "k (!α ), we deduce that, in suitable coordinates, ∗!α is equivalent to a 1-form whose components are affine polynomials. Therefore, we directly apply Lemma 5 and conclude that there exists a local smooth solution of the equation of motion which can be extended to a global smooth solution by means of a partition of unity on M. ✷

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313

Remark 1. The above theorem asserts that if the differential of the external potential annihilates the kernel of the curvature form !α at the points of the first singular stratum, "2 (!α ) or "1 (!α ), then it does so consequently in the remaining strata of degeneracy as well. Also, in both cases the exactness of the curvature form immediately leads to the property that if V ∈ D(M, α), then the vector field XV is tangent to every stratum "k (!α ) of M. Clearly, these vector fields constitute a subalgebra in the Lie algebra of smooth vector fields on M. Remark 2. In the case of a 2n-dimensional manifold M, under the additional on "2 (!α ) doesn’t vanish at the generic assumption that the restriction of !n−1 α point xo ∈ "2 (!α ), ([20]), through the singular local Martinet model ([9, 16]), the above theorem suggests that V ∈ D(M, α) if and only if there exist local coordinates (ξ, ζ ) around xo such that, for (ξ , ζ ) = (ξ2 , . . . , ξn , ζ2 , . . . , ζn ) and smooth functions a(ξ, ζ ), b(ξ , ζ ), V (ξ, ζ ) = ξ12 a(ξ, ζ ) + b(ξ , ζ ). 2.2. In this section we present the tools necessary for the establishment of the above theorem. In the ring R{{X}} of convergent power series of n variables, we consider ck X k G= k∈Nn

with X k = X1k1 . . . Xnkn , k = (k1 , . . . , kn ) ∈ Nn , and we say that supp G = {k ∈ Nn : ck = 0}. The lexicographic ordering of (n + 1)-uples (|k|, k1 , . . . , kn ) where |k| = k1 + · · · + kn , induces a total ordering on Nn . The initial exponent exp G is defined as the smallest element of supp G and the corresponding monomial is called the initial monomial of G. If G1 , . . . , Gs ∈ R{{X}}, we introduce 9i = (exp Gi + N ) − n

i−1

9j ,

i = 1, . . . , s,

j =1

and we note 0 = Nn −

s

9i .

i=1

THEOREM 3. Let f be the germ of a smooth function and G1 , . . . , Gs the germs of analytic functions at the origin of Rn . Then there exist germs of smooth functions Q1 , . . . , Qs , R at the origin of Rn such that Gi Qi + R f = 1is

and, in addition, the Taylor series of R satisfies that exp R ∈ 0 .

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This division theorem of a smooth function by an ideal generated by analytic functions, established in [23], constitutes the basic tool of this analysis. It generalizes the classical division theorem of an analytic function by an ideal generated by analytic functions, established by Hironaka in [8]. Now, using this result, we establish the division property of a smooth function by an ideal generated by affine polynomials in the ring of smooth functions, employed in the proof of Theorem 2. LEMMA 4. If α is a generic connection form on an n-dimensional smooth manifold M then, every nonempty stratum "k (!α ), k > 1, is defined locally by k(k − 1)/2 homogeneous affine polynomials that generate the ideal k (!α ) in the ring of germs of smooth functions on M. Proof. Theorem 1 suggests that "k (!α ) is locally diffeomorphic to the smooth stratum of the algebraic variety V (P(k,i) ), Vk := i=1,...,s

where each variety V (P (k,i) ) = {x ∈ Nn | P(k,i) (x) = 0} is defined by s = k(k − 1)/2 homogeneous affine polynomials Pk,1 , . . . , Pk,s . The definition of a homogeneous affine polynomial Pi ∈ R[x1 , . . . , xn ] implies the following expression for its initial exponent exp Pi = (ei1 , . . . , ein ),

where eij = 0 or 1.

We observe that the set determining the remainder of the division of a smooth function by affine polynomials satisfies the {exp <(i1... ir ) }, 0 = (i1 ,...,ir )

where <(i1...ij ) is the coordinate space defined by the equations xir = 0, for certain 1 ir n. We will proceed inductively on n ∈ N and obviously the assertion is true for n = 1. The division of the germ of a smooth function f by analytic functions is reduced to the formal division according to [14]. Let the formal power series fˆ of f at the origin of Rn be written as i (x) + R(x), Pi (x)Q fˆ(x) = 1is

i , R their ∈ 0 . If Qi , R are the germs of smooth functions and Q where exp R formal series, then, applying Theorem 3, we obtain Pi (x)Qi (x) + R(x). f (x) = 1is

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315

Consider the intersection of the variety V (P1 , . . . , Ps ) with a coordinate hyperplane contained in 0 , say {x1 = 0}. Let x = (x2 , . . . , xn ), l ∈ N, then R(x) = R(0, x ) + x1l Rl (x) as well as one that for an affine polynomial πi , i = 1, . . . , s, P i (x ) = Pi (0, x ) = Pi (x) − πi (x )x1 . Observe then that V (P1 , . . . , Ps ) ∩ {x1 = 0} has a dense complement in V (P1 , . . . , Ps ). The inductive hypothesis suggests that i (x) = i (x) + i (x)Q S(x)x1 . P i (x )Q P R(0, x ) = 1is

1is

In consequence, we obtain that i (x) + Pi (x)Q S(x)x1 , fˆ(x) = 1is

where exp S ∈ 0 . Proceeding in this way and following the approach of [20], we annihilate the remainder by successively removing the coordinates in 0 and applying Lojasiewicz’s theorem (cf. [13, 14]), we pass from the formal division to the smooth division. ✷ The above result leads to a proof of a de Rham–Saito type of division lemma in the smooth context like the one used in the proof of Theorem 2. In the analytic context, this property could be obtained from the classical theorems on the division of differential forms, cf. [15, 17, 24]. LEMMA 5. Let ai (x, y) dxi αy (x) = 1in

be the germ a p-parameter family of 1-forms at the origin in Rn × Rp , n > 3, whose coefficients are affine polynomials ai ∈ R[x1 , . . . , xn , y1 , . . . , yp ] and S(αy ) = {x ∈ Rn | a1 (x, y) = · · · = an (x, y) = 0} with codim S(αy ) 3. If β is the germ of a smooth p-parameter family of 2-form (resp. 1-form) at the origin in Rn × Rp satisfying the compatibility condition α ∧ β = 0, then there exists a germ at the origin of Rn × Rp of a smooth p-parameter family of 1-form θy (resp. function λy ) such that β = θ ∧ α (resp. β = λα). Proof. We start with the case of 1-forms and we consider bi (x, y) dxi β(x, y) = 1in

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expressed in local coordinates (x1 , . . . , xn , y1 , . . . , yp ) at the origin of Rn × Rp . Evidently, the compatibility condition is expressed as bi aj − bj ai = 0,

i = j.

(0)

We introduce the auxiliary parameters ξ1 , . . . , ξn and setting ai (x, y)ξi and b˜j (x, y, ξ ) = bi (x, y)ξi , a˜ j (x, y, ξ ) = i=j

i=j

and summing (0) with respect to i, we obtain a˜ j bj − b˜j aj = 0. The codimension assumption on S(αy ) and the fact that the coefficients are affine polynomials assert that, if V (aiy ) = {x ∈ Rn | ai (x, y) = 0}, we could always select the parameters in such a way that V (a˜ iy ) ∩ V (aiy ) has a dense complement in V (aiy ). This holds since the level set for an affine polynomial is a conical hypersurface through the origin with singularities that lead to a conical nowhere dense subset which is easily proved by induction on the number of variables. The preceding lemma on the smooth division by an affine ideal asserts that there exist real smooth functions λi on Rn × Rp such that bi = λi ai . Setting this back in (0), we obtain that (λj − λi )aj ai = 0 and λi = λj = λ, yielding the conclusion that β = λα. The case of 2-forms is treated by induction on the dimension n. Let bij (x, y) dxi ∧ dxj β(x, y) = 1i<j
be expressed in local coordinates (x1 , . . . , xn , y1 , . . . , yp ) at the origin of Rn × Rp . For n = 3, the affine polynomials are a combination of Pi (x) = xi ,

Pij (x) = xi xj ,

i = j,

P (x) = x1 x2 x3

and the compatibility condition is expressed as a1 b23 − a2 b13 + a3 b12 = 0.

(1)

The varieties V (aiy ) are either hyperplanes or quadratic cones and the codimension assumption implies that V (a1y ) ∩ V (a2y ) ∩ V (a3y ) always has a dense complement in V (a1y ) ∩ V (a2y ), V (a2y ) ∩ V (a3y ), V (a1y ) ∩ V (a3y ). The preceding lemma on the smooth division by an ideal generated by affine polynomials implies that b12 = h12 a1 + g12 a2 ,

b13 = h13 a1 + g13 a3 ,

b23 = h23 a2 + g23 a3 . (2)

Substituting in (1), we obtain (h12 + g23 )a1 a3 + (h13 + h23 )a1 a2 + (g13 + g12 )a2 a3 = 0.

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The same arguments lead to h12 + g23 = u123 a2 ,

h13 + h23 = u132 a3 ,

g13 + g12 = u213 a1

(3)

and this in turn leads to u123 + u132 + u213 = 0.

(4)

Now plugging (3), (4) in (1) and setting θ = −g12 dx1 + h12 dx2 + (h13 − u213 a3 ) dx3 , we obtain β = θ ∧ α. Assume now that this is true up to order n − 1. Then we split the coordinates in a manner that maintains the codimenension assumption and we embody the other coordinate in the parameters. We then set β = βn−1 + η ∧ dxn , where βn−1 =

bij dxi ∧ dxj

and

1i<j

η=

cin dxi

1i
and, accordingly, α = αn−1 + αn ∧ dxn , where αn−1 =

ai dxi .

1i
The compatibility condition is now expressed as βn−1 ∧ αn−1 + (an βn−1 − η ∧ αn−1 ) ∧ dxn = 0 and this is evidently reduced to βn−1 ∧ αn−1 = 0

(5)

an βn−1 − η ∧ αn−1 = 0.

(6)

and

The inductive hypothesis suggests that βn−1 = θn−1 ∧ αn−1 which we set in (5) and obtain (η − an θn−1 ) ∧ αn−1 = 0.

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The latter by the division of the 1-forms suggests that η = an θn−1 + f αn−1 and setting θn = θn−1 + f dxn we obtain that β = θn ∧ α.

✷

3. Generic Aspect for Singular Lagrangian Systems 3.1. Let M be the n-dimensional configuration space of a mechanical system with Lagrangian function L: T M → R and consider the Legendre transformation between T M and T ∗ M expressed as L(x, v) = (x, dv L), where dv L denote the vertical derivative. The equations of motion of the Lagrangian system (M, L) are expressed intrinsically by the Lagrangian vector field XL given by the equation iXL !L + dEL = 0, where !L is the Lagrangian form defined as the pullback of the standard symplectic form of T ∗ M through the Legendre transformation and the energy function expressed as EL (x, v) =

n

vi ∂vi L(x, v) − L(x, v).

i=1

The Lagrangian vector field XL is defined outside of the critical locus of the Legendre transformation, which coincides with the degeneracy locus "(!L ) of the Lagrangian form, i.e. points where its kernel is not trivial. The projection of this locus gives the singularities of the equations of motion on M; these singularities are called removable when the Lagrangian vector field extends smoothly everywhere on T M. A Lagrangian system (M, L) is called generic if the differential of the corresponding Legendre transformation is transverse to the Legendre stratification of the Legendre bundle E(T M), cf. [22]. In this case, a stratification in smooth submanifolds is induced on the tangent bundle T M: "k,s (!L ) = {(x, v) ∈ T M | dim ker(x,v) !L = k, dim ker(x,v) DL = s}, 0 k, s n, such that "k,s (!L ) =

i,j ∈N

"k+i,s+j (!L ).

GAUGE FIELDS WITH GENERIC SINGULARITIES

319

The versal unfolding lemma for the generic Legendre sections proved in [22], after a direct calculation, leads to the local models of the stratified degeneracy locus "k,s (!L ). "(!L ) = (k,s)=(0,0)

More precisely, the strata of this locus are defined locally, in an appropriate chart, by quadratic-affine polynomials, i.e. homogeneous polynomials, without multiple factors, each factor of which is expressed by fixing all variables but one as Ai xi2 + Bi xi + Ci with Ai , Bi , Ci ∈ R[x1 , . . . , xi−1 , xi+1 , . . . , xn ], i = 1, . . . , n. These quadratic-affine polynomials generate the ideal of smooth functions that vanish on the strata of "(!L ) in the ring of germs of smooth functions; in other words, the degeneracy locus possesses the differentiable Nullstellensatz ([22]). The following theorem is proved in an analogous way to Theorem 2; its proof is given explicitly in [22] in the general context of the singular variational problems. THEOREM 6. The singularities of an n-dimensional generic Lagrangian system (M, L) are removable if and only if, at the points of the first singular stratum "1,1 (!L ), there holds !n−1 L ∧ dEL = 0. In this case, the projection of the integrals curves of the Lagrangian vector field gives the motion on M and, in particular, the trajectories with initial conditions on a stratum "k,s (!L ), k, s ∈ N, remain always in this stratum. Remark. This situation is illustrated in the following example which could be seen as a generalization of the celebrated Weierstrass example of a variational problem having no trivial extrema. Let L(x, x , ξ, ξ ) be a Lagrangian on the tangent bundle of R2 where (x, x , ξ, ξ ) denote the coordinates in R4 and assume that dL(0) = 0, ∂ξ2 L(0) = 0,

∂ξ2 L(0) = 0,

∂ξ2ξ L(0) = 0,

∂ξ3 L(0) = 0.

Clearly, the Lagrangian satisfies the classical transversality condition and, as a consequence, the corresponding Legendre transformation satisfies this condition at the origin of R4 as well. The Malgrange preparation theorem and Morse Lemma suggest that, in suitable local coordinates, we could take L(x, x , ξ, ξ ) = ξ 3 + ξ 2 + a(x.x )ξ + b(x, x ) with a, b smooth functions and the singular locus of the corresponding Legendre transformation is given by ξ = 0. The admissibility condition given by the above theorem implies the following expression for the energy function EL (x, x , ξ, ξ ) = 2ξ 3 + ξ 2 − b(x ).

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The corresponding Lagrangian vector field is XL = ξ

∂ 1 ∂ ∂ + ξ − b (x ) ∂x ∂x 2 ∂ξ

and the projection of its integral curves gives the solution of this problem on R2 . 3.2. The case of the Lagrangian systems (M, L) with the Lagrangian defined through a quadratic form Q on the manifold M, L(x, v) = 12 Q(x)(v, v) − V (π(v)), where V is a potential function on M and π : T M → M the canonical projection, presents certain generic particularities for Q transverse to the natural stratification of the fibering of quadratic forms on M. The critical locus of the corresponding Legendre transformation is projected on the degeneracy locus "(Q) consisting of the points where the kernel of the quadratic form is not trivial. This locus is stratified in smooth submanifolds "k (Q) = {x ∈ M | dim kerx Q = k},

0 < k n,

of codimension k(k + 1)/2, if nonempty, and their closure satisfies the frontier condition. On the complement of "(Q) a unique connection of vanishing torsion associated with Q is defined, the Levi-Civita connection ∇. Let XQ (M) be the set of smooth vector fields that are tangent to the strata of "(Q) and assume, in addition, that the kernel of Q is transverse to the tangent space of "1 (Q) at every point of an open and dense set contained in "1 (Q). Then, the Levi-Civita connection extends to a smooth map : XQ (M) × XQ (M) → XQ (M) ∇ if and only if "(Q) is autoparallel. In this case, on every stratum "k (Q) the quadratic form Q induces a pseudo-Riemannian metric Qk and the above map induces the Levi-Civita connection associated with Qk , 0 k n. The classical situation concerning the geodesics on a regular pseudo-Riemannian manifold can be generalized in this singular context and thus we obtain the behavior of the geodesics in the stratified singular locus, cf. [22]. Acknowledgement This work was partially supported by the ‘C. Caratheodory’ Research Programme of the University of Patras, Greece. References 1.

Arnold, V. I. and Givental, A. B.: Symplectic geometry, In: Encyclop. Math. Sci., SpringerVerlag, New York, 1985.

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2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21.

22.

23. 24. 25. 26.

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Arnold, V. I.: Remarks concerning the Morse theory of a divergence free vector field, the averaging method and the motion of charged particle in a magnetic field, Proc. Steklov Inst. 216 (1996), 3–13. Atiyah, M. P. and Bott, R.: The Yang–Mills equations over a Riemannian surface, Proc. Trans. Roy. Soc. London A 308 (1982), 523–615. Bierstorne, M. P.: The Newton diagramm of an analytic morphism and applications to differentiable functions, Bull. Amer. Math. Soc. 9 (1983), 315–318. Dirac, P. A. M.: Generalized Hamiltonian Dynamics, Canad. J. Math. 2 (1950), 129–148. Eisenbud, D.: Commutative Algebra, Grad. Text Math. 150, Springer, New York, 1995. Flato, M., Lichnerowicz, A. and Sternheimer, D.: Deformations of Poisson brackets, Dirac brackets and applications, J. Math. Phys. 19 (1978), 1754–1762. Hironaka, H.: Resolution of singularities of an algebraic variety, Ann. of Math. 79 (1964). Hormander, L.: The Analysis of Linear Partial Differential Operators III, Grund. Math. Wiss. 274, Springer, New York, 1985. Jackiw, R.: Diverse Topics in Theoretical and Mathematical Physics, World Scientific, Singapore, 1995. Leon, M. and Rodriguez, P. R.: Generalized Classical Mechanics and Field Theory, Math. Stud. 112, North-Holland, Amsterdam, 1985. Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom. 12 (1977), 253–300. Lojasiewicz, S.: Sur le problème de la division, Studia Math. 18 (1959), 87–136. Malgrange, B.: Ideals of Differentiable Functions, Oxford Univ. Press, 1966. Malgrange, B.: Frobenius avec singularités, Publ. IHES 51 (1975). Martinet, J.: Sur les singularités de formes différentielles, Ann. Inst. Fourier 20 (1970), 95–178. Moussu, R.: Sur l’existence d’intégrales premières, Ann. Inst. Fourier 26 (1976), 171–220. Pelletier, F.: Quelques propriétés géométriques des variétés pseudo-riemanniennes singulières, Ann. Fac. Sci. Toulouse, Série Math. 4(1) (1995), 87–199. Pliakis, D.: Asymptotics for singular Euler–Lagrange and Schrödinger equations, Doctoral Dissertation, University of Crete, 2000. Pnevmatikos, S.: (i) Structures hamiltoniennes en présence de contraintes, C.R. Acad. Sci. Paris A 289 (1979), 799–802, (ii) Singularités en géométrie symplectique, Res. Notes in Math. 80 (1983), 184–215, (iii) Structures symplectiques singulières génériques, Ann. Inst. Fourier 34 (1984), 201–218. Pnevmatikos, S.: (i) Divisions et voisinages privilégiés, Publ. Université de Dijon, 1977, (ii) Lectures on the divisions in the ring of analytic functions, Erasmus Scient. Monogr. Math. Fund. Appl., 1990. Pnevmatikos, S. and Pliakis, D.: (i) Quelques propriétés génériques des formes quadratiques, C.R. Acad. Sci. Paris I 323 (1996), 447–452, (ii) On the singularities of quadratic forms, J. Geom. Phys. 34 (2000), 73–95. Pnevmatikos, S. and Pliakis, D.: Smooth divisions by analytic functions on Banach spaces, Prep. Univ. Patras, 2000. Saito, K.: Sur un lemme de de Rham, Ann. Inst. Fourier 26(2) (1976), 167–170. Taylor, M.: Partial Differential Equations, Text in Appl. Math., Springer, New York, 1996. Weinstein, A.: Symplectic Structure on the Moduli Space of Yang–Mills Fields, Floer Memorial Volume, Birkhauser, Basel, 1995.

Mathematical Physics, Analysis and Geometry 3: 323–337, 2000. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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An Electrostatics Approach to the Determination of Extremal Measures JEAN MEINGUET Université Catholique de Louvain, Institut Mathématique, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium. e-mail: [email protected] (Received: 20 February 2001) Abstract. One of the most important aspects of the minimal energy (or induced equilibrium) problem in the presence of an external field – sometimes referred to as the Gauss variation problem – is the determination of the support of its solution (the so-called extremal measure associated with the field). A simple electrostatic interpretation is presented here, which is apparently new and anyway suggests a novel, rather systematic approach to the solution. By way of illustration, the classical results for Jacobi, Laguerre and Freud weights are explicitly recovered by this alternative method. Mathematics Subject Classifications (2000): 31A15, 31A25, 78A30. Key words: logarithmic potential, external field, Gauss variation problem, electrostatic interpretation.

1. Introduction Mathematicians (and physicists!) generally ‘know’ Dirichlet’s principle. They are likely less familiar with the related W. Thomson (Lord Kelvin) principle (in electrostatics) and its special case called the Gauss variation problem (or forced equilibrium problem), which is the problem of minimizing – in the presence of a given external field – the ‘energy’ associated with any sourceless (or solenoidal) vector field in the outer region bounded by a given closed set (the so-called ‘conductor’, supposed once for all to be ‘perfect’) over which a positive (electric) charge of prescribed amount is to be distributed so as to reach equilibrium (see, e.g., [5], pp. 43–44, 55–57, or [2], pp. 46, 51). As a matter of fact, the underlying potential theory needed in the following is the theory of logarithmic potentials with external fields, whose interaction with approximation-theoretical techniques and problems in the complex plane or on the real line proved extremely fruitful in recent years. As is well known, a point charge in the plane is ‘equivalent’ to a uniformly distributed charge on a straight line – perpendicular to the plane – in R3 , such (positive or negative) point charges repelling or attracting each other according to an inverse distance law (well known

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consequence of Coulomb’s law). Gauss’s variation problem then becomes that of minimizing the (weighted) energy integral IQ (µ) :=

1 dµ(z) dµ(t) + 2 log |z − t|

Q dµ,

(1)

where the minimum is taken over all positive unit charge distributions (i.e., positive unit Borel measures) µ carried by the conductor (i.e., supp(µ) ⊆ ) while Q (defined on and real-valued) is the so-called external field (strictly speaking, such a scalar ‘field’ Q is a potential). It is known – see, e.g., [6], pp. 26–33, for the basic theorem and its detailed mathematical proof – that, under rather weak conditions of ‘admissibility’ on Q, there exists a unique solution µQ (called equilibrium or extremal measure associated with Q) of this optimization problem, which is such that the relation U (z) := µ

log

1 dµ(t) = −Q(z) + FQ , |z − t|

z ∈ SQ ,

(2)

holds quasi-everywhere (i.e., possibly up to a set of zero logarithmic capacity), where SQ := supp(µQ ) is compact of positive capacity and FQ is the so-called modified Robin constant for Q. It should be stressed that a most glaring difference with the classical equilibrium problem (for which Q = 0) is that SQ need not coincide with the outer boundary of and, in fact, can be an arbitrary subset of , possibly with positive area. Determining SQ is therefore one of the most important aspects of the energy problem (or minimization of (1)). To find the extremal measure, it then remains to solve Dirichlet problems (for the Laplace equation and the essential boundary conditions (2)) and to launch the classical recovery machinery (e.g., the Sokhotskyi– Plemelj formula for arcs and its integrated version known as the Stieltjes–Perron inversion formula of Cauchy transforms). As discovered by Mhaskar–Saff in the eighties, determining SQ amounts to minimizing over the set of possible supports the (quasi-everywhere) constant value FQ of the extremal potential. It is surprising that such an obviously hard problem can be solved explicitly under suitable convexity assumptions (satisfied by the important weights w := e−Q of Jacobi, Laguerre, and Freud), SQ being then an interval whose endpoints can be obtained by solving a (simple) integral equation. The main goal of this paper is to present (in Section 2) a novel, rather systematic approach to the determination of SQ . This mathematical method can be regarded as a modern example of ‘physical mathematics’ in the sense of Sommerfeld; it is indeed motivated by an apparently new electrostatic interpretation. By applying this alternative method, we will rediscover rather automatically the ‘classical results’ mentioned above (see Sections 3, 4 and 5).

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325

2. A Physically Oriented Approach By way of constructive illustration, we will consider here the simple, two-dimensional physical picture: a (perfect) conductor in vacuum, say the finite segment := [−1, 1] in the extended complex z-plane C, is subjected to the electrostatic field of potential 1 Q(z) := λ log , z = x + iy, (3) |z − a| due to an electric charge λ > 0 located at an exterior point, say a > 1. • Suppose first that the conductor is grounded (i.e., connected to earth), which means that it may acquire whatever charges are necessary to enable it to remain at the same potential (zero, by convention). The resulting potential thus created (that is, (3) in the presence of the grounded ) is classically – up to the proportionality factor λ – the Green function of the complement of (the so-called cut plane) with pole at a, viz., 1 − φ(a)φ(z) , φ(z) := z + z2 − 1 (4) g(z, a) = log φ(z) − φ(a) (see, √ e.g., [6], p. 110). It should be noted once for all that any expression like z2 − 1 is to be understood as the branch that behaves like z near infinity, so that w = φ(z) is simply the inverse of the well-known Joukowski conformal map z = (1/2)(w + 1/w) of the exterior of the unit disk (in the φ-plane) onto the complement of . It follows in particular that the circle (in the φ-plane): φ(z) = φ(a)eiθ ,

−π θ π,

corresponds to the ellipse (in the z-plane): z = a cos θ + i a 2 − 1 sin θ, √ with foci at z = ±1, and semiaxes a, a 2 − 1, whose polar representation can be written in the form a2 − 1 = a − cos θ, (5) ρ= a + cos where ρ denotes the distance to the pole (of polar coordinates) z = 1, is the ‘true anomaly’ and θ is the ‘eccentric anomaly’ (these terms are borrowed from celestial mechanics). The distribution µ of the charge that is induced (by electrostatic influence) on the grounded conductor by the point charge λ > 0 at a > 1 or, equivalently, −λ times the so-called balayage measure of the Dirac point mass at a onto (see, e.g., [6], pp. 81–82), is given by λ ∂ g(x, a) dx dµ(x) := − π ∂n √ a2 − 1 λ dx, x ∈ [−1, 1], √ (6a) = − π (a − x) 1 − x 2

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where dx is the arc length on , and ∂/∂n denotes differentiation in the direction of the inner normal with respect to the complement of (as a matter of fact, for obvious symmetry reasons, n may denote here either the upper or the lower normal). The concrete expression on the right in (6a) is most important for the following; it is found by taking the limit of the real part of (−iλ/π ) times the derivative of the analytic function log

1 − φ(a)φ(z) φ(z) − φ(a)

as z tends to x ∈ (−1, √ 1) from the upper half-plane, while keeping in mind that (for 1 − x 2 is positive for y = 0+ (resp. negative for y = 0−) continuation reasons) √ and that a 2 − 1 is positive for a > 1 (but negative for a < −1, see Section 3). With the change of variable x = cos θ, (6a) takes the simpler form √ λ a 2 − 1 dθ, θ ∈ [−π, π ], (6b) dµ(cos θ) = − 2π a − cos θ where dθ denotes – throughout the whole paper – arc measure on the unit circle; in view of (5), the corresponding density (or Radon–Nikodym derivative) dµ(cos θ)/dθ of the induced charge has a nice geometric interpretation. As is classically expected (see, e.g., [4], p. 230), for any grounded conductor occupying a bounded region in the presence of a point charge, the density of the induced charge will never change sign; more precisely, the total mass of the distribution µ is −λ (this can be verified by explicit integration), while λ a+1 dµ(cos θ) =− , (7) min θ dθ 2π a − 1 this minimal value being attained for θ = 0. • Suppose now that the conductor is insulated (i.e., imbedded in vacuum). If a positive unit charge is placed on it in the absence of any external field, then its equilibrium distribution µ0 (i.e., the unique positive unit Borel measure minimizing the energy integral (1) where Q = 0) is known to be the arcsine distribution dµ0 (x) =

1 √ dx, π 1 − x2

x ∈ [−1, 1],

(8a)

or, equivalently, dµ0 (cos θ) =

1 dθ, 2π

θ ∈ [−π, π ],

(8b)

that is, the normalized arc measure. The constant value F0 assumed by its (logarithmic) potential on (the so-called Robin constant) is clearly F0 := log

1 = log 2, cap()

(9)

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the logarithmic capacity cap() of a finite segment being notoriously equal to one-fourth its length. • Suppose finally that the conductor is insulated in the field of potential (3). In view of (7), it is clear that √ 2 λ a+1 a −1 a+1 dµ0 = − + dθ (10) dµQ := dµ + λ a−1 2π a − cos θ a−1 is the unique (nonnegative and absolutely continuous with respect to θ) equilibrium distribution of charges over that minimizes its potential; indeed, the definition (10) amounts simply to adding to the signed measure dµ (whose logarithmic potential plus the external field has the constant value 0 on ) the smallest multiple C dµ0 of the positive measure dµ0 (whose logarithmic potential on has the conits logarithmic stant value F0 ) that makes the resulting measure dµQ nonnegative, √ potential on , namely, the constant CF0 with C := λ (a + 1)/(a − 1), being therefore as small as possible. Provided that a+1 = 1, (11) λ −1+ a−1 which simply means that the total mass of (10) over its support is 1 (or equivalently, that the charge placed on is λ + 1), the distribution µQ is nothing but the extremal measure minimizing (1) for Q defined by (3) (after all, the potential in electricity and magnetism is identical with potential energy per unit charge, see, e.g., [4], p. 53). The (logarithmic) potential FQ of corresponding to the distribution (10) ‘normalized’ by (11) – that is, the modified Robin constant for Q (see [6], p. 27) – is thus clearly FQ = (λ + 1) log 2.

(12)

As is easily verified (by an elementary computation detailed in [6], p. 46), the potentials (9) and (12) satisfy the important relation Q dµ0 (13) FQ = F0 +

according to which −FQ is the so-called ‘F -functional’ of Mhaskar–Saff (see [6], Chap. IV) whose maximization (over the set of possible supports) is achieved by the support SQ of the extremal measure µQ ; an alternative proof of (13) follows from the successive relations implied by µ Q(x) + U (x) = 0, x ∈ [−1, 1], dµ = −λ, µ

where U denotes the logarithmic potential of the induced measure (6a), viz.: Q dµ0 = − U µ dµ0 = − U µ0 dµ = −F0 dµ

= λF0 = (1 + λ)F0 − F0 = FQ − F0

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(the only nontrivial equality sign is the second one, which is justified by the Fubini– Tonelli Theorem). • It is just a simple exercise to rewrite our results about the extremal measure in a more familiar form (see, e.g., [7], p. 773). As a matter of fact, we have only to change the interval [−1, a] into [−1, 1] via the affine transformation 2x + (1 − a) (14) 1+a without modifying the ratio of the fixed charge of amount λ > 0 at the point x = a > 1 (resp. ϑ ∈ (0, 1) at ξ = 1) to the continuous charge of amount 1 (resp. 1 − ϑ) to be distributed on [−1, 1] (resp. on its image by (14)) so as to reach equilibrium. The latter condition, viz., ξ=

ϑ , 1−ϑ combined with the normalization relation (11), yields λ=

(15)

1 + ϑ2 , 1 − ϑ2 so that the actual support of the continuous charge 1 − ϑ on the ξ -axis (i.e., the image of [−1, 1] by (14)) is a=

3−a = 1 − 2ϑ 2 . (16) 1+a As to the distribution of this charge on SQ , it readily follows from (10) – always normalized by (11) – by the affine transformation (14), owing to formulas (15), (16). It turns out that the associated Jacobian has a remarkable form, viz., 1 dξ = (ξ + 1)(ξ0 − ξ ), −1 ξ ξ0 , dθ 2 where the factor 1/2 is due to the fact that we must integrate twice along cuts if we integrate once over the unit circle. Hence, the final result √ 1 (ξ + 1)(ξ0 − ξ ) dξ, −1 ξ ξ0 , (17) dµQ (ξ ) = π(1 − ϑ) 1 − ξ2 SQ = [−1, ξ0 ],

ξ0 :=

which concludes our alternative treatment of the simplest example of explicit determination of an extremal measure that is considered in [6] (see pp. 205–206, 243), that is, the application entitled ‘Incomplete Polynomials of Lorentz’ (note, however, that our ξ is to be identified with −t in the last formula of Example 5.3 on p. 243 in [6]).

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329

3. The Extremal Measure for Jacobi Weights This quite natural generalization of the physical picture considered in Section 2 corresponds to the replacement of (3) by the electrostatic field of potential Q(z) := λ1 log

1 1 + λ2 log , |z − a1 | |z − a2 |

(18)

the electric charge λ1 > 0 (resp. λ2 > 0) being located at a point outside the conductor := [−1, 1], say a1 > 1 (resp. a2 < −1). • Since the complement of possesses an explicitly known Green function, namely (4), the potential of the total field created by the charges in (18) and the countercharges induced by influence on the conductor supposed to be grounded is classically given by the associated Green potential, viz., V (z) := λ1 g(z, a1 ) + λ2 g(z, a2 )

(19)

(see, e.g., [6], p. 124). In view of (6a), (6b) and (19), the distribution µ of the charge that is induced by (18) on the grounded conductor is given by 2 2 a − 1 a − 1 1 2 1 λ1 + λ2 dθ, dµ(cos θ) = − 2π a1 − cos θ |a2 | + cos θ θ ∈ [−π, π ]. (20) The total mass of this distribution is clearly −λ1 − λ2 , while dµ dµ dµ(cos θ) = −2π min (−1), (1) C := −2π min θ dθ dθ dθ

(21)

immediately follows from the convexity with respect to the variable cos θ of the parenthesized expression in (20) (its second derivative is indeed positive over [−1, 1]). • Suppose now that the conductor is insulated in the field of potential (18). It is clear that dµQ := dµ + C dµ0 ,

with definitions (8b), (20) and (21),

(22a)

is the unique (nonnegative and absolutely continuous with respect to θ) equilibrium distribution of charges over that minimizes its potential. However, to have a chance to solve eventually the underlying Gauss variation problem or, equivalently, to minimize the potential FQ of corresponding to the extremal measure µQ (of total mass 1!) in the presence of the external field of potential (18), the points a1 > 1 and a2 < −1 must be such that the minimal value (21) is as great as possible. This requires of the two expressions on the right in (21) to be equal (they vary indeed in opposite directions as either a1 > 1 or a2 < −1 varies), their common value being necessarily C = 1 + λ1 + λ2

(22b)

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(since the total mass of (22a) over must be 1, while the total mass of (20) is −λ1 − λ2 ). In other words, the following equations must be satisfied: a1 − 1 |a2 | + 1 + ϑ2 = 1, (23a) ϑ1 a1 + 1 |a2 | − 1 a1 + 1 |a2 | − 1 + ϑ2 = 1, (23b) ϑ1 a1 − 1 |a2 | + 1 where ϑ1 :=

λ1 , 1 + λ1 + λ2

ϑ2 :=

λ2 . 1 + λ1 + λ2

(23c)

This is equivalent to the apparently simpler nonlinear system for a1 , a2 : 2 a − 1 a12 − 1 2 , ϑ2 = , a1 > 1, a2 < −1, ϑ1 = a1 − a2 a1 − a2

(24a)

whose (unique) solution is 1 + ϑ12 − ϑ22 , √ #

a1 =

a2 = −

1 + ϑ22 − ϑ12 , √ #

(24b)

where # := [1 − (ϑ1 + ϑ2 )2 ][1 − (ϑ1 − ϑ2 )2 ],

(24c)

as it can be shown by somewhat lengthy (though elementary) computations. • To rewrite the extremal measure in a more familiar form (see, e.g., [7], pp. 772– 774), it remains only to change the interval [a2 , a1 ] into [−1, 1] via the affine transformation ξ=

2x − (a1 + a2 ) a1 − a2

(25)

without modifying the ratios of the fixed charge of amount λ1 > 0 at x = a1 > 1 (resp. ϑ1 ∈ (0, 1) at ξ = 1) and of the fixed charge of amount λ2 > 0 at x = a2 < −1 (resp. ϑ2 ∈ (0, 1) at ξ = −1) to the continuous charge of amount 1 (resp. 1 − ϑ1 − ϑ2 ) to be distributed on [−1, 1] (resp. on its image by (25)) so as to reach equilibrium; in fact, the conditions λ1 =

ϑ1 , 1 − ϑ1 − ϑ2

1 + λ1 + λ2 =

λ2 =

1 1 − ϑ1 − ϑ2

ϑ2 1 − ϑ1 − ϑ2

and (26)

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are trivially equivalent to (23c). It follows that the actual support of the continuous charge 1 − ϑ1 − ϑ2 on the ξ -axis (i.e., the image of [−1, 1] by (25)) is SQ = [ξ2 , ξ1 ],

(27a)

where ξ1 = ϑ22 − ϑ12 +

√

#,

ξ2 = ϑ22 − ϑ12 −

√

#, with definition (24c). (27b)

As to the distribution of this continuous charge on SQ , it follows from (22) by the affine transformation (25) – owing to formulas (23c), (24), (26), (27) – the final result being √ 1 (ξ − ξ2 )(ξ1 − ξ ) dξ, ξ2 ξ ξ1 , (28) dµQ (ξ ) = π(1 − ϑ1 − ϑ2 ) 1 − ξ2 in accordance with, e.g., [6] (see pp. 207 and 241). 4. The Extremal Measure for Laguerre Weights A crucial step in the approach presented in this paper is the determination of the electrostatic potential outside the grounded conductor (i.e., any given compact set of C, of positive capacity) in the presence of the given external field. In the applications considered so far, this fundamental influence problem could be solved readily owing to the explicit knowledge of the Green function (of the outer domain relative to ). On the other hand, in the remaining applications, where the external field is defined directly (at least in part) rather than via given external charges, this crucial step actually requires solving explicitly a Dirichlet boundary value problem. In the Laguerre case, the external field has for potential Q(z) := λ z + s log

1 , |z − a|

λ > 0, s 0, a < −1.

(29)

Unlike the second term, which is of the type considered before (i.e., potential of a charge s 0 located at a given point a < −1), the first term is not created by a charge but rather by a dipole at infinity (of axis 0x and of moment λ); though this ‘physical’ interpretation may prove interesting (see, e.g., [1], p. 35), we will not exploit it here, essentially because it does not hold for non-uniform fields such as the one considered in Section 5. • Now suppose that the conductor := [−1, 1] is grounded and subjected to the field of potential (29). The potential of the total field thus created outside is clearly the sum of three terms: the Green potential of the charge s√located at the point a (i.e., s times the Green function (4), where φ(a) := a − | a 2 − 1| since a < −1), the external field of potential λx, and the solution h(z) of the exterior

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Dirichlet problem: #h(z) = 0, z ∈ [−1, 1], h(z) bounded as |z| → ∞ (i.e., regularity at infinity), h(x) = −λx, x ∈ [−1, 1]. It turns out that the conformal transplant of h under the Joukowski mapping, viz., 1 1 H (w) := h w+ , w = |w|eiθ with |w| 1, θ ∈ [−π, π ] (30) 2 w can be obtained readily by separating the variables in the transplanted exterior Dirichlet problem: #H (w) = 0, |w| 1, H (w) bounded as |w| → ∞, H (eiθ ) = −λ cos θ, θ ∈ [−π, π ].

(31a) (31b) (31c)

Indeed, if we transform #H to polar coordinates |w|, θ, we get for the solution H (w) of (31a) the general form Ak cos kθ + Bk sin kθ ; A0 + B0 log |w| + |w|k k=0 now the condition (31b) of regularity at infinity (see, e.g., [4], p. 248) implies B0 = 0 and Ak = Bk = 0 for all negative integers k; the Dirichlet condition (31c) thus reduces to ∞

(Ak cos kθ + Bk sin kθ) = −λ cos θ,

θ ∈ [−π, π ],

k=0

which finally yields A1 = −λ,

B1 = 0,

Ak = Bk = 0 for k = 1.

In view of (30), the required potential of the total field created outside the grounded conductor by the external field of potential (29) is given by 1 − φ(a)φ(z) + λ z − λ 1 . (32) V (z) = s log φ(z) − φ(a) φ(z) According to the classical definition 1 ∂ V (x) dx, x ∈ [−1, 1], π ∂n we get from (32) the explicit expression √ 1 s| a 2 − 1| + λ cos θ dθ, dµ(cos θ) = − 2π |a| + cos θ dµ(x) := −

θ ∈ [−π, π ],

(33)

EXTREMAL MEASURES VIA ELECTROSTATICS

333

for the distribution µ of the charge induced by (29) on the grounded conductor . The remarkable relation (21) holds again (the parenthesized function of cos θ in (33) is indeed convex over [−1, 1]), so that we get explicitly |a| + 1 |a| − 1 − λ, s +λ . (34) C = max s |a| − 1 |a| + 1 • Suppose now that the conductor is insulated in the field of potential (29). It is clear that dµQ := dµ + C dµ0 ,

with definitions (8b), (33) and (34),

(35)

is the unique (nonnegative and absolutely continuous with respect to θ) equilibrium distribution of charges over that minimizes its potential, for any given values of the parameters λ > 0, s 0, a < −1. It turns out that a further minimization of this potential is automatically achieved if the point a < −1 is such that the two expressions on the right in (34) are equal (they vary indeed in√opposite directions as a varies); owing√ to this condition, which amounts to s = λ| a 2 − 1|, (34) reduces to C = λ|a| = λ2 + s 2 . But the total mass of (35) over must be 1, while the total mass of (33) is −s, so that necessarily C = s + 1; all these relations finally imply s+1 , a = −√ 2s + 1 √ λ = 2s + 1.

(36) (37)

• To rewrite the extremal measure in a more familiar form (see, e.g., [6], pp. 208 and 243), it remains only to change the interval [a, 1] into [0, ξ1 ] (where ξ1 is any finite positive number) via the affine transformation ξ = ξ1

x−a 1−a

(38)

without modifying the ratio of the fixed charge of amount s > 0 at the point x = a defined by (36) (resp. ϑ ∈ (0, 1) at ξ = 0) to the continuous charge of amount 1 (resp. 1 − ϑ) to be distributed on [−1, 1] (resp. its image SQ by (38)) so as to reach equilibrium. It follows that the actual support of the continuous charge 1 − ϑ :=

1 s+1

(39)

on the ξ -axis is SQ = [ξ2 , ξ1 ],

(40a)

where ξ1 1 ξ2 √ = √ =: . * s + 1 − 2s + 1 s + 1 + 2s + 1

(40b)

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JEAN MEINGUET

It should be noticed that, up to an additive (unimportant!) constant, the potential (29) on the positive ξ -axis has the simple expression *ξ − s log ξ , which depends on two independent parameters * > 0, s 0 (remember that λ was eliminated by (37), for the sake of normalization). As to the distribution of the continuous charge (39) on SQ , it readily follows from (35) – normalized by (36), (37) – by the affine transformation (38) and formulas (39), (40), the final result being √ * (ξ − ξ2 )(ξ1 − ξ ) dξ, ξ2 ξ ξ1 , (41) dµQ (ξ ) = π ξ in accordance with [6] (see p. 243). 5. The Extremal Measure for Freud Weights The external field – to which the standard conductor := [−1, 1] is subjected – has now for potential Q(z) := c|x|λ ,

c > 0, λ > 0 (and x := z).

(42)

Unlike the ‘physical’ fields considered before, it is thus directly defined by its mathematical expression rather than via given external electric charges (or dipoles). • If the conductor is grounded, the potential of the total field thus created outside is naturally obtained by adding to (42) the solution h(z) of the exterior Dirichlet problem: #h(z) = 0, z ∈ [−1, 1], h(z) bounded as |z| → ∞ (i.e., regularity at infinity), h(x) = −c|x|λ , x ∈ [−1, 1]. Here again, the conformal transplant H of h under the Joukowski mapping, which is the function defined by (30), can be obtained readily by separating the variables in the transplanted exterior Dirichlet problem: #H (w) = 0, |w| 1, H (w) bounded as |w| → ∞, H (eiθ ) = −c| cos θ |λ , θ ∈ [−π, π ].

(43a) (43b) (43c)

Indeed, if we transform #H to polar coordinates |w|, θ, we get for any solution of (43a, b) the Fourier series representation H (w) = −c

∞ k=0

A2k

cos 2kθ , |w|2k

the Dirichlet boundary condition (43c) reducing to ∞ k=0

A2k cos 2kθ = | cos θ |λ ,

θ ∈ [−π, π ]

(44a)

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EXTREMAL MEASURES VIA ELECTROSTATICS

(where the prime affecting the summation symbol means that the first term is to be taken with half weight), or equivalently, to π 2 4 (cos θ)λ cos 2kθ dθ A2k := π 0 ,(λ+1) 1 (44b) = λ−1 2 ,(λ/2+k+1),(λ/2−k+1) (see, e.g., [8], p. 263, Example 40). Owing to the reflection formula of Euler for the gamma function, this expression of A2k can be rewritten in the form A2k =

,(λ+1) sin (π λ/2) ,(k−λ/2) 1 (−1)k+1 , λ−1 2 π ,(k+λ/2) k+λ/2

(44c)

which yields (via Stirling’s formula) the asymptotic formula A2k ∼

,(λ+1) sin(π λ/2) 1 (−1)k+1 λ+1 λ−1 2 π k

as k → ∞;

(44d)

Weierstrass’s test is thus applicable, so that the Fourier series (44a) of | cos θ |λ converges uniformly and absolutely to its generating function. Since the potential V (z) of the total field created outside the grounded conductor by the external field of potential (42) has for conformal transplant (under the Joukowski mapping) V(w) = −c

∞ k=0

c cos 2kθ 1 λ A2k + λ |w|+ | cos θ |λ , |w|2k 2 |w|

|w| 1,

the distribution µ of the charge induced on is apparently given by dµ(cos θ) = −

1 2π

∞ c ∂V(w) dθ = − kA2k cos 2kθ dθ. |w|→1+ ∂|w| π k=1

lim

(45)

The total mass of µ is evidently 0 (since the lines of force of the field of potential (42) are parallel to the x-axis), while C := −2π min θ

∞ dµ(cos θ) = 2c kA2k , dθ k=1

(46)

this minimal value being attained for θ = 0 mod π – were it simply for ‘physical’ reasons (logical interpretation of the underlying problem of electrostatic influence) – whereas max θ

∞ c dµ(cos θ) =− (−1)k kA2k > 0 dθ π k=1

336

JEAN MEINGUET

is attained for θ = π/2 mod π and is finite or not according as λ > 1 or not (this follows from the properties of A2k mentioned above). It should be stressed that the trigonometric series in (45) is actually an Abel sum; however, by virtue of classical tests (substantially due to Abel) exploiting the properties (44c), (44d) of the A2k ’s, this series is convergent (except for θ = ±π/2 mod 2π , whenever λ 1), necessarily to its Abel sum. It turns out that the sum of the series in (46) can be found as an Abel sum by an explicit (but lengthy) computation, the final result being π ∞ λ 2 ,(λ/2+1/2) ; (47) kA2k = (cos θ)λ dθ = π 0 ,(λ/2),(1/2) k=1 the last expression is simply (44b) for k = 0, rewritten by means of Legendre’s duplication formula (see, e.g., [8], p. 240). Rather than give complementary details, we deem it preferable to describe briefly an alternative approach to (47), based on the modern theory of generalized functions or distributions. Consider the classical Fourier series expansion ∞ cos kθ k=1

k

= − log |2 sin(θ/2)|

(48)

whose generating function goes out of bound at θ = 0 mod 2π , while being integrable in the Lebesgue sense over the fundamental period interval (−π, π ). It is easily proved (see [3], p. 30) that the Fourier series in (48) converges in the sense of generalized functions to the function on the right-hand side, so that it may be differentiated term-by-term (in the distributional sense) any number of times, which yields in particular the distributional result: ∞

k cos θ = (log |2 sin(θ/2)|) ;

(49a)

k=1

by techniques that are standard in the theory of distributions (see, e.g., [3], p. 65, for similar results), we are led to concrete definitions of the second distributional derivative – denoted by the symbol – in (49a), viz., (log |2 sin(θ/2)|) , -(θ)

π d cos(θ/2) d -(θ) + -(−θ) dθ = − 2 sin(θ/2) dθ dθ 0π 1 [-(θ) + -(−θ) − 2-(0)] dθ, =− 2 0 4 sin (θ/2)

(49b)

where ·, · is the duality bracket between the dual topological vector spaces of periodic test functions -(θ) (i.e., infinitely differentiable functions of period 2π ) and periodic distributions (of period 2π ). The result (49b) can be extended by

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EXTREMAL MEASURES VIA ELECTROSTATICS

continuity to the function | cos(θ/2)|λ , which indeed can be regarded as the limit of a uniformly convergent sequence of test functions; it is readily verified that the values taken on this function by the two forms of the accordingly extended distribution in (49b) are simply π times the first two expressions in (47), which identity is again rigorously established. • Suppose now that the conductor is insulated in the field of potential (42). It is clear that dµQ := dµ + C dµ0 ,

with definitions (8b), (45) and (46),

(50)

is the unique (nonnegative and absolutely continuous with respect to θ) equilibrium distribution of charges over that minimizes its potential, for any given values of the parameters c > 0, λ > 0. But the total mass of (50) over must be 1, while the total mass of (45) is 0, so that necessarily C = 1,

or equivalently,

1/c = 2

∞

kA2k .

(51)

k=1

• To rewrite these results in a more familiar form (see [6], pp. 204 and 238), it remains only to change the interval [−1, 1] into SQ = [−a, a],

a > 0,

(52a)

via the linear substitution ξ = ax. SQ is the support of the extremal measure µQ relative to the external potential γ |ξ |λ ,

γ > 0,

(52b)

if and only if

√

π ,(λ/2) , 2 ,(λ/2+1/2) as it follows from (51) in view of (47). a=γ

−1/λ 1/λ

c

,

where c :=

(52c)

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

Bergman, S. and Schiffer, M.: Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York, 1953. Gårding, L.: The Dirichlet problem, Math. Intelligencer 2 (1980), 43–53. Gel’fand, I. M. and Shilov, G. E.: Generalized Functions, Vol. I, Academic Press, New York, 1964. Kellogg, O. D.: Foundations of Potential Theory, Springer, Berlin, 1929. Pólya, G. and Szeg˝o, G.: Isoparametric Inequalities in Mathematical Physics, Ann. of Math. Studies 27, Princeton Univ. Press, 1951. Saff, E. B. and Totik, V.: Logarithmic Potentials with External Fields, Grundlehren Math. Wiss. 316, Springer-Verlag, Berlin, 1997. Saff, E. B., Ullman, J. L. and Varga, R. S.: Incomplete polynomials: an electrostatics approach, In E. W. Cheney (ed.), Approximation Theory III, Academic Press, San Diego, 1980, pp. 769–782. Whittaker, E. T. and Watson, G. N.: A Course of Modern Analysis, 4th edn, Cambridge Univ. Press, London, 1927.

Mathematical Physics, Analysis and Geometry 3: 339–373, 2000. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

339

Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential ALEXANDRE STOJANOVIC Université Paris 7, Institut de Mathématiques, Physique-mathématique et géométrie, case 7012, 2, place Jussieu, 75251 Paris cedex 05, France (Received: 10 March 2000) Abstract. In this work, we develop an orthogonal-polynomials approach for random matrices with orthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, which are a generalization of Gaussian random matrices. The representation of the correlation functions in these matrix models, via the technique of quaternion determinants, makes use of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices, which essentially express them in terms of the reproducing kernel of the theory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given by Bleher and Its for the corresponding orthogonal polynomials. Mathematics Subject Classifications (2000): 15A52, 42C05, 33C45. Key words: random matrices, eigenvalues, correlation function, universality conjecture, orthogonal polynomials.

1. Introduction In this paper, we consider three types of matrix models defined in the following terms. Let Enβ be the real vector space, respectively, of real symmetric n × n matrices for β = 1, complex Hermitian n × n matrices for β = 2, and quaternion real self-dual n × n matrices for β = 4. The probability measure which defines the matrix models on each space Enβ , β = 1, 2 or 4, is given by Pnβ (dM) =

1 exp(−nTr V (M))dM, Znβ

M ∈ Enβ ,

(1.1)

where Znβ is the normalization constant, dM the Lebesgue measure on the considered space Enβ , and V the potential, which is a real polynomial of even degree, denoted by d + 1 2, with a positive leading coefficient. With this convention, d is the degree of V the derivative of V . Gaussian random matrices correspond to the case where the potential V (λ) is proportional to λ2 . The trace in the expression of the probability measure (1.1) shows that the law is invariant under the action by

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ALEXANDRE STOJANOVIC

conjugation of the orthogonal group for β = 1, the unitary group for β = 2, and the unitary symplectic group for β = 4. Thus, the eigenvectors are uniformly distributed and we are only interested in the behaviour of the eigenvalues. According to [10, 11], all the eigenvalues (λ1 , . . . , λn ) of the random matrice (1.1) are real and their probability density with respect to the Lebesgue measure on Rn is given by pβ(n) (λ1 , . . . , λn ) =

1 Qnβ

|λj − λk |β

n

exp(−nV (λj )),

(1.2)

j =1

1j
where Qnβ is the new normalization constant. The k-point correlation functions of the probability distribution (1.2) are defined by (n) (λ1 , . . . , λk ) pkβ =

(λk+1 ,...,λn )∈Rn−k

pβ(n) (λ1 , . . . , λk , λk+1 , . . . , λn ) dλk+1 . . . dλn ,

(1.3)

with k ∈ {1, . . . , n}. For the case β = 2, the expression of the correlation functions via the orthogonal polynomials technique is quite simple to obtain, whereas it causes trouble in the cases β = 1 or 4. However, in the particular case of Gaussian random matrices, F. Dyson and M. Mehta (see [10, 11] and references therein) have developed the technique of quaternion determinants to express the correlation functions (1.3) in terms of matrix kernels. The elements of a matrix kernel are expressed using Hermite polynomials which naturally appear for Gaussian random matrices. Let us recall some useful facts for this work, from the theory developed by Dyson and Mehta. A complex quaternion q is defined as a linear complex combination of the four 2 × 2 following complex matrices 0 −1 1 0 , , e1 = 1 0 0 1 0 −i i 0 , e3 = , e2 = −i 0 0 −i

1=

where i is a complex number such that i 2 = −1. Thus, a quaternion can be represented as a 2 × 2 complex matrix and we denote q = q (0) + q (1) e1 + q (2) e2 + q (3) e3 ,

with q (j ) ∈ C.

If the q (j ) are real, the quaternion is real, q (0) is the scalar part, and the purely quaternion part. The dual quaternion of (1.4) is defined by q = q (0) −

3 j =1

q (j ) ej .

3

j =1

(1.4) q (j ) ej

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

341

Let Qn = (qj k )1j,kn be an n × n quaternion matrix, the dual quaternion matrix is defined by Qn = qj k 1j,kn , with qj k = qkj , for j, k ∈ {1, . . . , n}. A quaternion matrix is self-dual if Qn = Qn . The quaternion determinant of Qn is defined by Qdet Qn = ε(σ ) σ ∈Sn

(qj1 j2 qj2 j3 ... qjr j1 )(0) ,

(1.5)

all the cycles (j1 → j2 → · · · jr → j1 ) of the decomposition in disjoint cycles of σ

where Sn is the symmetric group of order n, ε(σ ) is the signature of the permutation σ , and the (0) in the high index means that we can take the scalar part. An n × n quaternion matrix can be viewed as a 2n × 2n complex matrix, denoted C(Qn ), and from this notation we have the following result: The quaternion matrix Qn is where Z is the 2n × 2n self-dual if and only if ZC(Qn ) is antisymmetric complex, 0 1 and, in this case, we have block-diagonal matrix with n diagonal blocks −1 0 Qdet Qn = Pf(ZC(Qn ))

and

det C(Qn ) = (Qdet Qn )2 ,

where Pf is the Pfaffian of an antisymmetric complex matrix of even order. Recently, in [16], C. Tracy and H. Widom obtained general formulas for matrix kernels by using the method of generating functional (see [9] and the appendix A.17 of [11]). Their results can be quite easily reformulated with quaternion determinants (see the preprint [13] for the details). Thus, we recall their results adapted to the matrix models (1.1), we have to distinguish the cases β = 1 and β = 4. We define the function  if x > 0,  1/2 0 if x = 0, ε: x → ε(x) =  −1/2 if x < 0. Let f and g be two complex-valued functions defined on R, the convolution with the function ε is defined by ε(λ − µ)f (µ) dµ (ε f )(λ) = R

and we have the property g(ε f ) = − f (ε g). R

R

We note f for the derivative of f .

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ALEXANDRE STOJANOVIC

The case β = 1. Let (π!(n) (λ))!∈N be the orthogonal polynomials on R with respect to the weight exp(−2nV (λ)). They satisfy π!(n) (λ)πm(n) (λ)e−2nV (λ) dλ = δ!m , !, m ∈ N, R

where δ!m is the Kronecker symbol. Let us denote ψ! (λ) = π!(n) (λ)e−nV (λ) ,

! ∈ N,

(1.6)

as the associated system of orthonormal functions of L2 (R, C) and Kn (λ, µ) =

n−1

(1.7)

ψj (λ)ψj (µ),

j =0

the reproducing kernel (in order to simplify the notations, we do not indicate the n-dependence in superscripts). Let us define the coefficients ψj ψk , j, k ∈ N. (1.8) aj k = (ε ψj )ψk and cj k = R

R

PROPOSITION 1.1. For even n, consider the n × n matrix A = (aj k )0j,kn−1 which is real antisymmetric invertible. Let B = (bj k )0j,kn−1 be the inverse of A. Following [16], let us define sn (λ, µ) =

n−1

bj k ψj (λ)(ε ψk )(µ)

and

αn (λ) = 0.

(1.9)

j,k=0

For odd n, consider the (n + 1) × (n + 1) matrix A = (aj k )0j,kn , where aj k = aj k anj

if j, k ∈ {0, . . . , n − 1}, = −aj n = ψj (λ)dλ if j ∈ {0, . . . , n − 1},

(1.10)

R

= 0. ann

Then A is real antisymmetric invertible. Let B = (bj k )0j,kn be the inverse of A. Let us define sn (λ, µ) =

n−1 j,k=0

bj k ψj (λ)(ε ψk )(µ)

and

αn (λ) =

n−1

bj n ψj (λ). (1.11)

j =0

Now, we define the elements of a matrix kernel for any integer n by SnT (λ, µ) = Sn (µ, λ), Sn (λ, µ) = sn (λ, µ) + αn (λ), ∂sn (λ, µ) and Dn (λ, µ) = − ∂µ Jn (λ, µ) = ελ Sn (λ, µ) − ε αn (µ) − ε(λ − µ),

(1.12)

(1.13)

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

343

where ελ indicates that the convolution by the function ε is made with respect to the variable λ. The matrix kernel equals to Sn (λ, µ) Dn (λ, µ) σn1 (λ, µ) = . (1.14) Jn (λ, µ) SnT (λ, µ) Then, the correlation functions (1.3) are given by (n) (λ1 , . . . , λk ) pk1 (n − k)! Qdet(σn1 (λp , λq ))1p,qk , = n!

k ∈ {1, . . . , n}.

(1.15)

The case β = 4. Let (π!(n) (λ))!∈N be the orthogonal polynomials on R with respect to the weight exp(−nV (λ)). They satisfy π!(n) (λ)πm(n) (λ)e−nV (λ) dλ = δ!m , !, m ∈ N. R

Let us denote ψ! (λ) = π!(n) (λ)e−

nV (λ) 2

,

! ∈ N,

(1.16)

as the associated system of orthonormal functions of L2 (R, C) and K2n+d (λ, µ) =

2n+d−1

(1.17)

ψj (λ)ψj (µ),

j =0

the reproducing kernel. Let us define the coefficients ψj ψk and cj k = ψj (ε ψk ), j, k ∈ N. aj k = R

(1.18)

R

PROPOSITION 1.2. The 2n × 2n matrix A = (aj k )0j,k2n−1 is real antisymmetric invertible. Let B = (bj k )0j,k2n−1 be the inverse of A. Following [16], let us define Sn (λ, µ) =

2n−1

bj k ψj (λ)ψk (µ),

SnT (λ, µ) = Sn (µ, λ), (1.19)

j,k=0

∂Sn (λ, µ) Dn (λ, µ) = − ∂µ

and

In (λ, µ) = ελ Sn (λ, µ).

We define the matrix kernel Sn (λ, µ) Dn (λ, µ) . σn4 (λ, µ) = In (λ, µ) SnT (λ, µ)

(1.20)

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ALEXANDRE STOJANOVIC

Then, the correlation functions (1.3) are given by (n) (λ1 , . . . , λk ) pk4 (n − k)! Qdet(σn4 (λp , λq ))1p,qk , = n!

k ∈ {1, . . . , n}.

(1.21)

The matrix kernels are interesting when they are expressed in terms of orthogonal polynomials and notably in functions of the reproducing kernel, like in the case of the unitary invariant matrix model (the case β = 2 in (1.2)) or like in the case of Gaussain random matrices, see Chapters 6 and 7 of [11]. Because it allows to use the asymptotic formulas of orthogonal polynomials, via the Christoffel–Darboux formula, to study the asymptotic spectral behaviour of the random matrices when n → ∞. Theorems 2.1 and 2.2 give new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices. More precisely, the elements of the matrix kernels are composed of a principal part, determining the asymptotic behaviour in different asymptotic regimes, which is simply expressed in terms of the reproducing kernel and a negligible part, which is a finite sum with an nindependent number of terms which we have to estimate. In order to obtain these formulas, the polynomial aspect of the potential V is very important, because it implies that the infinite matrix of the derivation operator in the basis of orthonormal functions is multi-diagonal with d diagonals along each sides of the principal diagonal and the method of computation is entirely based on this property. Let us remark that H. Widom [17] has obtained expressions for matrix kernels in terms of the reproducing kernel for more general classes of random matrices. As an application, in Theorems 2.3 and 2.4 we prove, by using the formulas given in Theorems 2.1 and 2.2, the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with twoband quartic potential by using the asymptotics given in [2] for the corresponding orthogonal polynomials. (n) (λ) the one-point correlation function, then the Let us denote ρn (λ) = p1β density of states ρ(λ) is defined by the limit (if it exists in an appropriate topology, which is here the topology of pointwise convergence) lim ρn (λ) = ρ(λ)

(1.22)

n→+∞

and the spectrum is the support of the density of states. The study of this quantity is relative to the global regime. Now, we recall the definitions of two quantities relative to the local regime. Let us recall that the probability Rn (5) that an interval 5 does not contain any eigenvalue is given by Rn (5) =

n (−1)! !=0

!!

5!

Qdet(σnβ (λp , λq ))1p,q! dλ1 . . . dλ! .

(1.23)

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

345

The local statistic of eigenvalues in the bulk of the spectrum is defined by the limit of Rn (5), with v , 5 = λ0 , λ0 + nρn (λ0 ) where λ0 is such that ρ(λ0 ) = 0 and v is a positive fixed real. The local statistic of eigenvalues at the edge of the spectrum is defined by the limit of Rn (5), with w v 5 = λ0 − 2/3 , λ0 + 2/3 , cn cn where λ0 is a point at the edge of the spectrum and v, w and c are positive fixed reals. The local statistic of eigenvalues in the bulk of the spectrum for Gaussian random matrices is given in [9, 11] (and see the references therein). The local statistic of eigenvalues at the edge of the spectrum for Gaussian orthogonal and symplectic invariant matrix models was first derived in [15]. The universality means that these quantities do not depend on a particuliar potential V but depend on the group invariance of matrix models only. Theorems 2.3 and 2.4 establish such results for matrix models with two-band quartic potential. Another work in which the local statistics of eigenvalues both in the bulk and at the edge of the spectrum are given is [7]. Finally, we point out that since the limiting scaled correlation functions (both in the bulk and at the edge of the spectrum) are proved for matrix models with two-band quartic potential to be identical with those of Gaussian orthogonal and symplectic invariant matrix models, that such quantities as level-spacing distributions and the largest eigenvalue distributions will therefore also be identical (after proper normalization) with Gaussian random matrices (see again [15]). 2. Results THEOREM 2.1 (the case β = 1). For even n, let us define, with (1.8), sj k =

n−1

aj ! c!k ,

j, k ∈ {n − d, . . . , n − 1}.

(2.1)

!=k−d

The d × d matrix D = (sj k )n−dj,kn−1 is invertible. Let D −1 = (tj k )n−dj,kn−1 be its inverse. Let us define the coefficients

n−1 n−d−1 cj m sm! t!k , cj ! − gj k = −cj k + !=n−d

m=j −d

j, k ∈ {n − d, . . . , n − 1}, then the d × d matrix G = (gj k )n−dj,kn−1 is antisymmetric.

(2.2)

346

ALEXANDRE STOJANOVIC

For odd n, let us define, with (1.8) and (1.10), sj k =

n

aj ! c!k ,

j, k ∈ {n − d, . . . , n}.

(2.3)

!=k−d

The (d + 1) × (d + 1) matrix D = (sj k )n−dj,kn is invertible. Let D −1 = (tj k )n−dj,kn be its inverse. Let us define the coefficients

n n−d−1 gj k = −cj k + cj m sm! t!k , j, k ∈ {n − d, . . . , n}, (2.4) cj ! − m=j −d

!=n−d

then the (d + 1) × (d + 1) matrix G = (gj k )n−dj,kn is antisymmetric. Now, for any integer n, the functions (1.9) and (1.11) are given by sn (λ, µ) = Kn (λ, µ) − εµ Hn (λ, µ) + εµ Gn (λ, µ),

(2.5)

where n−1

Gn (λ, µ) =

(2.6)

gj k ψj (λ)ψk (µ),

j,k=n−d

Hn (λ, µ) =

n−1 n+d−1

(2.7)

cj k ψj (λ)ψk (µ).

k=n j =k−d

And, for even n, αn (λ) = 0, whereas for odd n, we have αn (λ) =

n−1

n−1

(cj n + gj n )ψj (λ) with cj n + gj n = −

j =n−d

cn! t!j .

(2.8)

!=n−d

THEOREM 2.2 (the case β = 4). Let us define, with (1.18), sj k =

2n−1

aj ! c!k ,

j, k ∈ {2n − d, . . . , 2n + d − 1}.

(2.9)

!=j −d

The d×d matrix D = (sj k )2n−dj,k2n−1 is invertible. Let D −1 = (tj k )2n−dj,k2n−1 be its inverse. Let us define the coefficients

2n−1 2n−1 sj m tmp (δp! − sp! ) a!k , sj ! + gj k = aj k − !=k−d

m,p=2n−d

j, k ∈ {2n, . . . , 2n + d − 1},

(2.10)

then the d ×d matrix G = (gj k )2nj,k2n+d−1 is antisymmetric. The function (1.19) is given by Sn (λ, µ) = K2n+d (λ, µ) − εµ Hn (λ, µ) + εµ Gn (λ, µ),

(2.11)

347

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

where Gn (λ, µ) =

2n+d−1

(2.12)

gj k ψj (λ)ψk (µ),

j,k=2n

Hn (λ, µ) = −

2n+2d−1 2n+d−1

(2.13)

aj k ψj (λ)ψk (µ).

k=2n+d j =k−d

Let us define the functions needed in the following theorems: s(x) =

sin π x , πx

a(x, y) =

Ai(x)Ai (y) − Ai (x)Ai(y) x−y

and b(x, y) = 12 Ai(x)(ε Ai)(y), where Ai is the Airy function (see [1]). THEOREM 2.3 (the case β = 1). Let us consider the random matrix with orthogonal invariant law (1.1), with V (λ) =

tλ2 gλ4 + , 4 8

√ such that g > 0 and t < −2 g (two-band case), then we have the following results. (i) The density of states (1.22) exists (the limit is taken in the sense of the pointwise convergence on the real axis) and equals g |λ| max(0, (Z12 − λ2 )(λ2 − Z22 )), ρ(λ) = 2π with

Z1 =

√ −t − 2 g 1/2 g

and

Z2 =

√ −t + 2 g 1/2 . g

(ii) The local statistic of eigenvalues in the bulk of the spectrum is given by v lim Rn λ0 , λ0 + n→+∞ nρn(λ0 ) v +∞ v ! (−1) ··· Qdet(τ1 (xp , xq ))1p,q! dx1 . . . dx! , = !! 0 0 !=0 with

τ1 (x, y) =

−s (x − y) x−y s(x − y) s(u)du − ε(x − y) s(x − y) 0

348

ALEXANDRE STOJANOVIC

and where v is a positive real and λ0 is such that ρ(λ0 ) = 0, i.e. λ0 ∈ (−Z2 , −Z1 )∪ (Z1 , Z2 ). More precisely, the limiting scaled correlation functions in the bulk of the spectrum is given by 1 xp xq , λ Qdet σ + + λ lim n1 0 0 n→+∞ (nρn (λ0 ))k nρn(λ0 ) nρn (λ0 ) 1p,qk = Qdet(τ1 (xp , xq ))1p,qk uniformly for (xp )1pk in any compact of Rk . (iii) For the case of the local statistic of eigenvalues at the edge of the spectrum, we take the limit with n even or n odd. Then we have w v , Zj + lim Rn Zj − n→+∞ cj n2/3 cj n2/3 v +∞ (−1)! v ··· Qdet(θ1 (xp , xq ))1p,q! dx1 . . . dx! , = !! −w −w !=0 where



θ1 (x, y) = 

x y (a

(a + b)(x, y) + b)(u, y)du − ε(x − y)

 ∂(a + b) (x, y)  , ∂y (a + b)(y, x)

−

if the limit is taken with n even or if the limit is taken with n odd and j = 1. √ Moreover, v, w are positive reals and cj = (−1)j 21/3 gZj . And if the limit is taken with n odd and j = 2, we have θ1 (x, y) 

 ∂(a + b) (x, y) . = ∂y x 1 1 ((a + b)(u, y) + Ai(u))du − ε(x − y) (a + b)(y, x) + Ai(y) y 2 2 (a + b)(x, y) + 12 Ai(x)

−

More precisely, the limiting scaled correlation functions at the edge of the spectrum is given by xp xq 1 Qdet σn1 Zj + , Zj + lim n→+∞ (cj n2/3 )k cj n2/3 cj n2/3 1p,qk = Qdet(θ1 (xp , xq ))1p,qk uniformly for (xp )1pk in any compact of Rk . THEOREM 2.4 (the case β = 4). Let us consider the random matrix with symplectic invariant law (1.1), with V (λ) =

tλ2 gλ4 + , 2 4

349 √ such that g > 0 and t < −2 2g (two-band case), then we have the following results. (i) The density of states (1.22) exists (the limit is taken in the sense of the pointwise convergence on the real axis) and equals g |λ| ρ(λ) = max(0, (Z12 − λ2 )(λ2 − Z22 )), 4π ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

with

Z1 =

√ −t − 2 2g 1/2 g

and

Z2 =

√ −t + 2 2g 1/2 . g

(ii) The local statistic of eigenvalues in the bulk of the spectrum is given by v lim Rn λ0, λ0 + n→+∞ nρn(λ0 ) v +∞ (−1)! v ··· Qdet(τ4 (xp , xq ))1p,q! dx1 . . . dx! , = !! 0 0 !=0 with

τ4 (x, y) =

s(2(x − y)) −s (2(x − y)) 2(x−y) s(u)du s(2(x − y)) 0

,

and where v is a positive real and λ0 is such that ρ(λ0 ) = 0, i.e. λ0 ∈ (−Z2 , −Z1 )∪ (Z1 , Z2 ). More precisely, the limiting scaled correlation functions in the bulk of the spectrum is given by xp xq 1 , λ0 + Qdet σn4 λ0 + lim n→+∞ (nρn (λ0 ))k nρn(λ0 ) nρn (λ0 ) 1p,qk = Qdet(τ4 (xp , xq ))1p,qk uniformly for (xp )1pk in any compact of Rk . (iii) The local statistic of eigenvalues at the edge of the spectrum is given by w v , Z + lim Rn Zj − j n→+∞ cj n2/3 cj n2/3 +∞ v (−1)! v ··· Qdet(θ4 (xp , xq ))1p,q! dx1 . . . dx! , = !! −w −w !=0 where

 ∂(a + b) (x, y)  (a + b)(x, y) − , θ4 (x, y) =  ∂y x (a + b)(u, y)du (a + b)(y, x) y 

350

ALEXANDRE STOJANOVIC

√ j

and where v, w are positive reals and cj = (−1) 2gZj . More precisely, the limiting scaled correlation functions at the edge of the spectrum is given by xp xq 1 lim Qdet σn4 Zj + , Zj + n→+∞ (cj n2/3 )k cj n2/3 cj n2/3 1p,qk = Qdet(θ4 (xp , xq ))1p,qk uniformly for (xp )1pk in any compact of Rk . Remark 2.1. Using the result of the local statistics at the edge of the spectrum, we obtain the pointwise convergence to zero of the density of states at the points of the edge of the spectrum. Remark 2.2. The formulas given by Theorems 2.1 and 2.2 allow us to recover the known formulas for Gaussian random matrices. Moreover, in this case, we can compute exactly the coefficients (1.8), (1.10) and (1.18). In order to recover the formulas as they are given in [11], we have to take another scaling convention. The case β = 1. For the weight exp(−nV (λ)), we take here exp(− 12 λ2 ) (since V is an homogeneous polynomial, a simple change of variables allows us to pass from one convention to an other). Here, the Hermite polynomials which appear are orthogonal with respect to the weight exp(−λ2 ). Then (see [14]) we have the relation p p+1 ψp−1 − ψp+1 . (2.14) ψp = 2 2 Thus

cp,p+1 = −cp+1,p = −

p+1 , 2

the other coefficients being null. Multiplying (2.14) by ε ψq and integrating, we get the relation p−1 2 ap−2,q − δp−1,q for p > 0, q 0. (2.15) apq = p p The parity of the weight implies that apq = 0, if p, q have the same parity. Further, by antisymmetry, we can suppose that p < q, then an elementary recurrence done with (2.15) gives p! q−p 1/2 ((q − 1)/2)! 2 , apq = q! (p/2)!

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

351

if p is even and q is odd, whereas apq = 0, if p is odd and q is even. Now, we compute the coefficients aj n for even j , else they are null. Recall that (see [14]) the Hermite polynomials are given by πj (λ) = √

[j/2]

(−1)! j ! (2λ)j −2! , π 2j j ! !=0 !!(j − 2!)! 1

where [x] is the integer part of the real x. Moreover, we have √ (2p)! 2 λ2p e−λ /2 dλ = 2π p . 2 p! R Thus, with even j , we get √ 1/4 √ 2π (2p)! . ψj (λ) dλ = anj = p 2 p! R Now, we can recover the known formulas. Since d = 1 and the coefficients gj k are antisymmetric, we have for the functions (2.6) and (2.7) that Gn (λ, µ) = gn−1,n−1 ψn−1 (λ)ψn−1 (µ) = 0 and

Hn (λ, µ) = −

n ψn−1 (λ)(ε ψn )(µ). 2

Further, for n odd, we compute the function (2.8), and we have αn (λ) = (cn−1,n + gn−1,n )ψn−1 (λ) = −cn,n−1 tn−1,n−1 ψn−1 (λ) and, using (2.3), we find tn−1,n−1 =

1 sn−1,n−1

=

1

, an−1,n cn,n−1

because the computation of the coefficients shows that an−1,n−2 = 0. Therefore, we get αn (λ) = −

1

ψn−1 (λ) an−1,n

1 ψn−1 (λ) R ψn−1

=

and, since n − 1 is even, we verify that λ ψn−1 . (ε ψn−1 )(λ) = 0

Thus, we recover exactly the formulas given in [11]. Now, we compute the value of the term αn (λ) in the limit of the local statistics at the edge of the spectrum. We

352

ALEXANDRE STOJANOVIC

have to rescale the weight in order to have the Hermite polynomials with respect to exp(−nλ2 ). Then, taking this new scaling into acount, the value of anj computed above gives 23/4 ψn−1 ∼ √ , n → +∞, n R and, by using the results of [14], we get √ 1 1 1 αn 2+ √ = Ai(x). lim √ 2/3 2/3 n→+∞ 2 2n 2n This is exactly the result we get in the case of the quartic potential at the point Z2 . The nullity of this term at the point Z1 is a direct consequence of the presence of n+s the sign (−1)[ 2 ] in the asymptotic formula of ψn+s , which reflects the two-band structure of the spectrum. The case β = 4. For the weight exp(−nV (λ)), we take here exp(−2λ2 ) and we have the Hermite polynomials with respect to the weight exp(−2λ2 ). Adapting, the computations of the case β = 1, we get √ Sn (λ, µ) = K2n+1 (λ, µ) + 2n + 1ψ2n (λ)(ε ψ2n+1 )(µ). Remark 2.3. In this remark, we discuss the general behaviour of the coefficients (2.2), (2.4) and (2.10) and the order of the different terms in (2.5) and (2.11), when n → +∞. The case β = 1, even n. In this case (and for more general classes of orthogonal polynomials, see e.g. [12]) we know that for the coefficients (1.8), we have cn+p,n+q = O(n), when n → +∞, for fixed p, q in Z. Further, we suppose (this needs a proof) for the coefficients (1.8) that an+p,n+q = O(1/n). We deduce that, for the coefficients (2.1) of the matrix D, we have sj k = O(1). Hence, we suppose at the very outset that det D ∼ c, with c > 0. Therefore, we have tj k = O(1). Under these hypotheses, we can conclude, for the coefficents (2.2), that gj k = O(n). Moreover, if we suppose that the orthonormal functions satisfy √ ψn+p L∞ (R) = O(n1/6) and ε ψn+p L∞ (R) = O(1/ n), we deduce that the terms (2.7) ε Hn (λ, µ) and (2.6) ε Gn (λ, µ) are O(n2/3) uniformly in λ, µ, whereas the reproducing kernel Kn (λ, µ) is of the order n, which is dominant with respect to O(n2/3). The case β = 1, odd n. In√ this case, we suppose, moreover for the coefficients = O(1/ n). Thus, we deduce for the coefficients (2.3) that (1.10), that an+p,n √ sj k = O( n). The behaviour of the coefficients tj k is more complicated. We begin by studying the behaviour of det D. Let us define the line vectors Lj = (σj k )n−dkn

if j ∈ {n − d, . . . , n},

353

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

with σj k =

n−1

aj ! c!k = O(1)

if j ∈ {n − d, . . . , n − 1}

!=k−d n−1

σnk =

and

√ an! c!k = O( n).

!=k−d

Moreover, we define C = (cnk )n−dkn . Then, the matrix D can be rewritten as D = (Lj + aj n C)n−dj n . A multilinear development gives 

.. .

  Lj −1  aj n det  C det D = det(Lj )n−dj n +  j =n−d  Lj +1 .. . n−1

     = O(n),  

√ because Lj = O(1) if j ∈ {n − d, . . . , n − 1}, Ln = O( n) and C = O(n). Thus, we suppose that det D ∼ cn, with c > 0. Under this supplementary hypothesis, the expressions given by Theorem 2.1 allow√us to show that we have gj k = O(n), if j, k < n and gj n = −gnj = −cj n + O( n), for the coefficients (2.4). Indeed, we have (−1)j +k dkj with dkj = det(spq )n−dp,qn, p=k, q=j . det D Let us define the line vectors tj k =

) L(j p = (σpq )n−dqn, q=j

if j, p ∈ {n − d, . . . , n}

C (j ) = (cnq )n−dqn, q=j

if j ∈ {n − d, . . . , n}.

and

Then, for k ∈ {n − d, . . . , n − 1}, a multilinear development gives ) dkj = det L(j p n−dpn, p=k + n−1

+

p=n−d, p=k

= O(n)

) (j ) apn det (1 − δsp )L(j s + δsp C n−dsn, s=k (2.16)

354 and

ALEXANDRE STOJANOVIC

) dnj = det L(j p n−dpn−1 + +

n−1

√ ) (j ) apn det (1 − δsp )L(j = O( n). s + δsp C n−dsn−1

p=n−d

Hence, tj k = O(1) if j ∈ {n − d, . . . , n}, k ∈ {n − d, . . . , n − 1} √ and tj n = O(1/ n). Thus, we just have to prove n−1 O(1) √ if k ∈ {n − d, . . . , n − 1}, sm! t!k = O(1/ n) if k = n, !=n−d

with m ∈ {n−2d, . . . , n−d −1}. Moreover, sm! = amn cn! +O(1), because m < n. Thus, we just have to prove √ n−1 O( n) if k ∈ {n − d, . . . , n − 1}, cn! t!k = O(1) if k = n. !=n−d

In fact, (2.8) shows that n−1 !=n−d cn! t!n = −(gnn + cnn ) = 0. For k < n, according to (2.16), we have n−1 √ (−1)!+k O( n) + apn det (1 − δsp )L(!) t!k = s + det D p=n−d, p=k

(!) + δsp C n−dsn, s=k . Thus, we just have to prove n−1

(!) cn! (−1)!+k det (1 − δsp )L(!) = 0, s + δsp C n−dsn, s=k

!=n−d

but this is the development with respect to the line indexed by k = p of det((1 − δsp )(1 − δsk )Ls + (δsp + δsk )C)n−dsn which is null, because the line C appears twofold. Hence, we have the result on the coefficients gj k . Therefore, for the function (2.8), we have αn (λ) = O(n2/3), uniformly in λ and we conclude as in the case of even n. The case β = 4. This case is similar to the first case, we just have to exchange the role of the coefficients aj k and cj k given in (1.18). Remark 2.4. We see that the main problem in the estimation of the coefficients gj k is the division by det D, because it obliges us to compute the equivalent and

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

355

not just an estimate. In order to do this in the case of the quartic potential, we compute the exact form of equivalents of the coefficients (1.8), (1.10) and (1.18), the explicit form of det D in terms of these coefficients, and then we can compute the equivalent of det D. But, despite the parity of the potential V and its low degree, the computations are long and fastidious. And, although there exist similar asymptotic formulas for orthogonal polynomials corresponding to general polynomial potential (see [5]), it seems impossible to use the same method to prove the behaviour of det D. That is why we propose the following hypothetic way to find the asymptotic behaviour of det D. We should use something like the variational approach, see [4, 8], because we have the following expression of det D in terms of normalization constants. Let us denote r β |λj − λk | w(λj ) dλj , Qr,β (w) = · · · Rr 1j
j =1

where, w is a positive function of one real variable. Then, for the case β = 1, even n, we have n Q 2 ,4 (e−2nV )Qn,1 (e−nV ) 2 , det D = 2n (n/2)!Qn,2 (e−2nV ) for the case β = 1, odd n, 2 (n + 1) Q n+1 ,4 (e−2nV )Qn,1 (e−nV ) 2 , det D = 2n 2 (((n + 1)/2)!)2 Qn,2 (e−2nV )Qn+1,2 (e−2nV ) and for the case β = 4 Qn,4 (e−nV )Q2n,1 (e−nV /2 ) 2 . det D = 22n n!Q2n,2 (e−nV ) Results from [8] show that the first term of the asymptotic developement of the 2 normalization constant is en ×constant. It is not sufficient here, thus we need for other terms. 3. Proofs of the Results We give only the proof of Theorem 2.1 for even n; the cases β = 1, odd n, and β = 4 are based on the same principles but they are technically more complicated (see [13] for details). The proof of Theorem 2.4 is similar to the proof of Theorem 2.3. Proof of Theorem 2.1. We suppose n is even. First step: By using the orthogonality property and the polynomial aspect of V , we find the relations j +d cj k ψk , j ∈ N, (3.1) ψj = k=j −d,k0

356

ALEXANDRE STOJANOVIC

and j −1

nV ψj =

j +d

cj k ψk −

cj k ψk ,

k=j +1

k=j −d,k0

Moreover, starting from the relation δj k = (3.1), we get the identity δj k =

k+d

R

j ∈ N.

ψj ψk , integrating by parts, and using

j, k ∈ N.

aj ! c!k ,

(3.2)

(3.3)

!=k−d,!0

Second step: We compute the matrix product AC in terms of coefficients (2.1) by using (3.3). In terms of block matrices, we have In−d E (3.4) AC = 0 D with D = (s!k )n−d!,kn−1

and

E = (s!k )0!n−d−1,n−dkn−1 .

(3.5)

Third step: We compute (AC)−1 . According to [16], we know that A and C are invertible, hence the matrix D (3.5) is invertible too. A block-matrix computation using (3.4) gives us 0 −ED −1 −1 (AC) = In + 0 D −1 − Id with D −1 − Id = (t!k − δ!k )n−dk,!n−1 .

(3.6)

The coefficients are equal to t!k − δ!k =

n−1

(δ!m − s!m )tmk ,

!, k ∈ {n − d, . . . , n − 1}.

m=n−d

Thus, (3.6) becomes (AC)−1 = In + (0 F ) with F =

n−1

m=n−d

(δ!m − s!m )tmk

(3.7) 0!n−1,n−dkn−1

Fourth step: We compute B = A−1 . We use B = C(AC)−1 and (3.7). Since B and C are antisymmetric, we get 0n−d 0 (3.8) with G = (gj k )n−dj,kn−1 B=C+ 0 G

357

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

and gj k =

n−1

cj m

n−1

(δm! − sm! )t!k .

(3.9)

!=n−d

m=0

Fifth step: We simplify the expression of the coefficients (3.9) to get (2.2). We have n−1

n−1 cj m (δm! − sm! )t!k gj k = m=j −d

=

n−d−1

!=n−d

cj m

m=j −d

=

n−d−1

n−1

!=n−d

cj m −

m=j −d

n−1

= −cj k +

cj ! −

!=n−d

n−1

sm! t!k +

cj m (δm! − sm! )t!k

!,m=n−d

!=n−d

n−1

n−1

(δm! − sm! )t!k +

cj m (tmk − δmk )

m=n−d n−d−1

cj m sm! t!k .

m=j −d

Sixth step: We compute Dn (λ, µ). (1.9), (1.13) and (3.8) give us n−1

Dn (λ, µ) = −

cj k ψj (λ)ψk (µ) −

j,k=0

n−1

gj k ψj (λ)ψk (µ).

(3.10)

j,k=n−d

Let us denote kn (λ, µ) =

n−1

πj(n) (λ)πj(n)(µ).

j =0

We have, with (1.7), n−1

−

cj k ψj (λ)ψk (µ)

j,k=0

∂Kn ∂Kn 1 (λ, ν)Kn (ν, µ) − Kn (λ, ν) (ν, µ) dν 2 ∂ν ∂ν ∂kn ∂kn 1 (λ, ν)kn (ν, µ) − kn (λ, ν) (ν, µ) e−n(V (λ)+V (µ)+2V (ν)) dν =− 2 R ∂ν ∂ν ∂kn 1 ∂kn (λ, µ) − (λ, µ) e−n(V (λ)+V (µ)) , = 2 ∂λ ∂µ =−

because, by using the orthognality property, we get πj(n) (ν)kn (ν, µ)e−2nV (ν) dν = πj(n) (µ), for j ∈ {0, . . . , n − 1}. R

358

ALEXANDRE STOJANOVIC

Then (3.10), with (2.6), becomes ∂kn 1 ∂kn Dn (λ, µ) = (λ, µ) − (λ, µ) e−n(V (λ)+V (µ)) − Gn (λ, µ) 2 ∂λ ∂µ and we get the result with (2.7) because ∂kn 1 ∂kn (λ, µ) − (λ, µ) e−n(V (λ)+V (µ)) 2 ∂λ ∂µ ∂Kn 1 1 ∂Kn (λ, µ) − (λ, µ) + (nV (λ) − nV (µ))Kn (λ, µ) =− 2 ∂λ ∂µ 2 1 ∂Kn ∂Kn ∂Kn (λ, µ) + (λ, µ) + (λ, µ) e−n(V (λ)+V (µ)) + =− ∂µ 2 ∂λ ∂µ 1 + (nV (λ) − nV (µ))Kn (λ, µ) 2 ∂Kn (λ, µ) + Hn (λ, µ). =− ∂µ Since, by using (3.1), we have ∂Kn ∂Kn (λ, µ) + (λ, µ) ∂λ ∂µ n−1 n−1 n+d−1 n+d−1 cj k ψj (λ)ψk (µ) + cj k ψj (λ)ψk (µ) =− j =n k=j −d

k=n j =k−d

and, by using (3.2), we have (nV (λ) − nV (µ))Kn (λ, µ) n−1 n−1 n+d−1 n+d−1 cj k ψj (λ)ψk (µ) + cj k ψj (λ)ψk (µ). = j =n k=j −d

k=n j =k−d

Proof of Theorem 2.3. Let us denote, with (2.6) and (2.7) Mn (λ, µ) = Gn (λ, µ) − Hn (λ, µ).

(3.11)

Thus, according to Lemmas 4.1 and 4.3, (3.11) has the form (here, in fact, we have d = 3) d−1

Mn (λ, µ) =

m(n) pq ψn+p (λ)ψn+q (µ),

with m(n) pq = O(n).

(3.12)

p,q=−d

And, for odd n, with (4.8), for (2.8), we have αn (λ) =

−1 p=−d

bn+p,n ψn+p (λ),

√ with bn+p,n = O( n).

(3.13)

359

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

(i) First, we show the existence of the density of states. According to (1.15) and (2.5), we have ρn (λ) =

1 1 1 1 Sn (λ, λ) = Kn (λ, λ) + (εµ Mn )(λ, λ) + αn (λ). n n n n

(3.14)

Moreover, with (4.3), (3.12) and (3.13), we have the estimates 1 |(εµ Mn )(λ, µ)| n d 1 (n) mpq ψn+p L∞ (R) ε ψn+q L∞ (R) = O(n−1/3) n p,q=−d

(3.15)

and −1 1 1 |αn (λ)| |bn+p,n |ψn+p L∞ (R) = O(n−1/3). n n p=−d

(3.16)

Thus, according to (3.14), (3.15) and (3.16), the result follows from the limit 1 Kn (λ, λ), n→+∞ n

lim ρn (λ) = lim

n→+∞

which is proved in [2]. (ii) Secondly, we compute the limits of the elements of the matrix kernel in the local limit inside the spectrum. Let λ0 be such that ρ(λ0 ) = 0, let us denote xn =

x nρn(λ0 )

and

yn =

y . nρn(λ0 )

Therefore, for each λ0 , there exists a positive real δ such that, if x and y are compact, then λ0 + xn and λ0 + yn are in the part P = (−Z2 + δ, −Z1 − δ) ∪ (Z1 + δ, Z2 − δ). The estimates are given uniformly with respect to P and by Lemma 4.1, we know that ψn+s L∞ (P ) = O(1). According to [2], we know that 1 Kn (λ0 + xn , λ0 + yn ) = s(x − y), n→+∞ nρn (λ0 ) lim

(3.17)

uniformly with respect to (x, y) belonging to a compact of R2 . Therefore, according to (3.15), (3.16) and (2.5), we get lim

n→+∞

1 Sn (λ0 + xn , λ0 + yn ) = s(x − y). nρn(λ0 )

360

ALEXANDRE STOJANOVIC

The results of [2] allow us to derive (3.17) and thus we get −1 ∂Kn (λ0 + xn , λ0 + yn ) 2 (nρn(λ0 )) ∂µ −1 ∂Kn (λ0 + xn , λ0 + yn ) = lim n→+∞ nρn (λ0 ) ∂y = −s (x − y).

lim

n→+∞

Since, with (4.3) and (3.12), we have 1 Mn (λ0 + xn , λ0 + yn ) = O(1/n), (nρn(λ0 ))2 we finally obtain 1 Dn (λ0 + xn , λ0 + yn ) = −s (x − y). n→+∞ (nρn (λ0 ))2 lim

Now, we show that

x−y

lim Jn (λ0 + xn , λ0 + yn ) =

n→+∞

s(u) du − ε(x − y).

(3.18)

0

In order to do this, we use the antisymmetry, which gives Jn (λ, λ) = 0 for all λ ∈ R. Thus, we compute the limit of Jn (λ0 + xn , λ0 + yn ) − Jn (λ0 + yn , λ0 + yn ). Let f be a function and we have the property λ 1 +∞ f (µ) dµ − f (µ) dµ. (ελ f )(λ) = 2 −∞ −∞ Thus, we have (ελ Kn )(λ0 + xn , λ0 + yn ) − (ελ Kn )(λ0 + yn , λ0 + yn ) λ0 +yn λ0 +xn Kn (ν, λ0 + yn ) dν − Kn (ν, λ0 + yn ) dν =

−∞ λ0 +xn

−∞

Kn (ν, λ0 + yn ) dν x u y 1 , λ0 + du. Kn λ0 + = nρn (λ0 ) y nρn (λ0 ) nρn(λ0 )

=

λ0 +yn

Hence, lim ((ελ Kn )(λ0 + xn , λ0 + yn ) − (ελ Kn )(λ0 + yn , λ0 + yn )) x−y x s(u − y) du = s(u) du. =

n→+∞

y

0

361

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

Moreover, we have ε(λ0 + xn − λ0 − yn ) = ε(x − y). Finally, with (4.3) and (3.12), we get |ελ (εµ Mn )(λ0 + xn , λ0 + yn ) − ελ (εµ Mn )(λ0 + yn , λ0 + yn )| d (n) m |ε ψn+p (λ0 + xn ) − pq p,q=−d

− ε ψn+p (λ0 + yn )||(ε ψn+q )(λ0 + yn )|

d

√ √ O( n)|ε ψn+p (λ0 + xn ) − ε ψn+p (λ0 + yn )| = O(1/ n)

p,q=−d

because

|ε ψn+p (λ0 + xn ) − ε ψn+p (λ0 + yn )| =

λ0 +xn

λ0 +yn

ψn+p = O(1/n),

and, similarly, we have √ |(ε αn )(λ0 + xn ) − (ε αn )(λ0 + yn )| = O(1/ n), hence we get (3.18). (iii) We see that the matrix kernel has a limit with an inhomogeneous normalization. Nevertheless, we have only to show that xp xq 1 , λ0 + Qdet σn1 λ0 + lim n→+∞ (nρn (λ0 ))k nρn(λ0 ) nρn (λ0 ) 1p,qk = Qdet(τ1 (xp , xq ))1p,qk uniformly with respect to (xp )1pk belonging to any compact of Rk . We remark that the antidiagonal terms of the matrix kernel in (1.5) are always multiplied by themselves in equal numbers. This is only a simple combinatorial result on the algebraic expression (1.5) of the quaternion determinant. (iv) Finally, we just have to prove that we can pass to the limit in the expression s Rn λ0 , λ0 + nρn(λ0 ) s n ! (−1) s xp , λ0 + ··· Qdet σ = n1 λ0 + !! nρ (λ ) n 0 0 0 !=0 xq dx1 . . . dx! , + nρn(λ0 ) 1p,q!

362

ALEXANDRE STOJANOVIC

where

σ n1 λ0 +

xp xq , λ0 + nρn(λ0 ) nρn (λ0 )

is the matrix kernel with its inhomogeneous normalization and where the derivation and integration are made with respect to the variables xp . Let us denote s (−1)! s xp xq (n) d! = , λ0 + ··· Qdet σ × n1 λ0 + !! nρn(λ0 ) nρn (λ0 ) 1p,q! 0 0 × dx1 . . . dx! , if 0 ! n and d!(n) = 0 if ! > n. Then, we have Rn λ0 , λ0 +

s nρn(λ0 )

=

+∞

d!(n) .

!=0

Further, for all ! ∈ N, we have lim d (n) n→+∞ !

= d! ,

where (−1)! d! = !!

s

s

··· 0

Qdet(τ1 (xp , xq ))1p,q! dx1 . . . dx! . 0

Moreover, since the convergence of x y , λ0 + = τ1 (x, y) lim σ n1 λ0 + n→+∞ nρn (λ0 ) nρn(λ0 ) is uniform on (0, s)2 and that the elements of τ1 (x, y) are bounded on (0, s), we deduce that the elements of x y , λ + + λ σ n1 0 0 nρn(λ0 ) nρn (λ0 ) are bounded by a constant C > 0. Thus, by using the Hadamard inequality, we get x x p q (2!C 2 )!/2 , Qdet σ , λ0 + n1 λ0 + nρn(λ0 ) nρn(λ0 ) 1p,q! hence (n) s ! d (2!C 2 )!/2. ! !! The right-hand side in the inequality is the general term of a convergent series, therefore the theorem of dominated convergence gives the result.

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

363

(v) For the case at the edge of the spectrum, we proceed essentially as inside the spectrum. By the result given in [2], we know that 1 x y Kn Zj + , Zj + = a(x, y). lim n→∞ cj n2/3 cj n2/3 cj n2/3 First, we will show that the complementary terms in the expression of the matrix kernel, which are expressed in terms of Gn (λ, µ), are negligible. But here the situation is more complicated, because the norm of uniform convergence of ψn+s on a neighbourhood of the edge of the spectrum is only O(n1/6) (see Lemma 4.1). We use the antisymmetry of the coefficients gj k . We have Gn (λ, µ) = gn−3,n−2 (ψn−3 (λ)ψn−2 (µ) − ψn−2 (λ)ψn−3 (µ)) + + gn−2,n−1 (ψn−2 (λ)ψn−1 (µ) − ψn−1 (λ)ψn−2 (µ)). In order to get Gn L∞ ((Zj −δ,Zj +δ)2 ) = O(n7/6), since gj k = O(n), it suffices to prove that ψn+p ⊗ ψn+q − ψn+q ⊗ ψn+p L∞ ((Zj −δ,Zj +δ)2 ) = O(n1/6).

(3.19)

Moreover, from the analysis of the asymptotic formulas done in [13], with δ a positive real, we have, uniformly in λ ∈ (Zj − δ, Zj + δ), 1/6 √ gλ σ0 n Ai(n2/3C(λ)) + O(1), (3.20) ψn+s (λ) = (−1) √ |C (λ)| where σ0 = (2 − j )[(n + s)/2] and 1 2/3 √ j yf (Zj + (λ − Zj )y) dy , C(λ) = (−1) (λ − Zj ) 0

with f (x) =

3 gx (−1)k (x + Z)(x 2 − Zk2 ). 4

The fact that the principal part in (3.20) does not depend on s, except for a sign which can be factorized, gives the result (3.19). In conclusion, we have 1 Gn (λ, µ)L∞ ((Zj −δ,Zj +δ)2 ) = O(n−1/6). (cj n2/3)2 The same arguments allow us to prove that 1 εµ Gn (λ, µ)L∞ ((Zj −δ,Zj +δ)2 ) = O(n−1/6) cj n2/3

364

ALEXANDRE STOJANOVIC

and ελ εµ (Gn (λ, µ) − Gn (µ, µ))L∞ ((Zj −δ,Zj +δ)2 ) = O(n−1/3 ). Now, we will study the contributions of the complementary terms in the expression of the matrix kernel, which are expressed in terms of Hn (λ, µ). From the proof of Theorem 2.1 we see that Hn (λ, µ) = 12 (αn (λ, µ) + βn (λ, µ)), where ∂Kn ∂Kn (λ, µ) + (λ, µ) ∂λ ∂µ n−1 n+d−1 = − cj k (ψj (λ)ψk (µ) + ψk (λ)ψj (µ))

αn (λ, µ) =

j =n k=j −d

and βn (λ, µ) = (nV (λ) − nV (µ))Kn (λ, µ) n+d−1 n−1 cj k (ψj (λ)ψk (µ) − ψk (λ)ψj (µ)). = j =n k=j −d

Thus, we see that βn (λ, µ) has an antisymmetric form similar to the form of Gn (λ, µ), so the same arguments prove that βn (λ, µ) is negligible. Therefore, a contribution is only given by αn (λ, µ). First, we compute this contribution to the term Dn (λ, µ) in the matrix kernel. We have x y 1 αn Zj + , Zj + lim n→+∞ (cj n2/3 )2 cj n2/3 cj n2/3 x y 1 ∂Kn Zj + , Zj + + = lim n→+∞ cj n2/3 ∂x cj n2/3 cj n2/3 x y ∂Kn Zj + , Zj + + ∂y cj n2/3 cj n2/3 ∂a ∂a (x, y) + (x, y). = ∂x ∂y By using the relation Ai (x) = xAi(x), the value of the limit becomes ∂a ∂a (x, y) + (x, y) = −Ai(x)Ai(y). ∂x ∂y In conclusion, we obtain x y 1 Dn Zj + , Zj + lim n→+∞ (cj n2/3 )2 cj n2/3 cj n2/3 1 ∂a = − (x, y) − Ai(x)Ai(y). ∂y 2

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

365

Now, we compute the contribution to the term Sn (λ, µ) in the matrix kernel. We have µ +∞ 1 αn (λ, ν) dν − αn (λ, ν) dν , εµ αn (λ, µ) = 2 −∞ µ where λ = Zj +

x cj n2/3

and

µ = Zj +

y . cj n2/3

When ν is outside a neighbourhood of the edge of the spectrum, this means that ν is in the complementary P of the part [−Z2 − δ, −Z2 + δ] ∪ [−Z1 − δ, −Z1 + δ] ∪ [Z1 − δ, Z1 + δ] ∪[Z2 − δ, Z2 + δ], then, using the fact that P ψn+s = O(n−1 ) (see the proof of Lemma 4.2 of [13]) and Lemma 4.1, the form of αn (λ, µ) shows that the restriction of the integral above to the part P is estimated by O(n) × O(n1/6) × O(n−1 ) = O(n1/6). Now, we will show that when ν is in a neighbourhood of a point B of the edge of the spectrum different of Zj , this means that B ∈ {−Z2 , −Z1 , Z1 , Z2 } − {Zj }, then the above integral restricted to such a neighbourhood is negligible. We have to estimate B+δ x ∂Kn x ∂Kn Zj + Zj + ,ν + ,ν dν ∂λ cj n2/3 ∂ν cj n2/3 B−δ B+δ x ∂Kn Zj + , ν dν + = ∂λ cj n2/3 B−δ x x , B + δ − K + , B − δ . Z + Kn Zj + n j cj n2/3 cj n2/3 By using the Christoffel–Darboux formula, we see that the absolute value of the denominator is bounded from below by |B − Zj |/2 > 0 (for δ small enough and n sufficiently large). It is this fact which allows us to estimate correctly the above quantity. Thus, we get x ,B ± δ Kn Zj + cj n2/3 ψn+1 Zj + cj nx2/3 ψn (B ± δ) − ψn Zj + cj nx2/3 ψn+1 (B ± δ) = Rn+1 Zj + c nx2/3 − (B ± δ) j

= O(1)O(n1/6)O(n1/6) = O(n1/3), √ where Rn+1 = O(1) is a coefficient of the three-terms recurrence relation of orthogonal polynomials (see [13] for details) and, in the same way, B+δ ψn+1 Zj + c nx2/3 ψn (ν) − ψn Zj + c nx2/3 ψn+1 (ν) j j Rn+1 dν = O(n1/3). 2 B−δ Zj + c nx2/3 − ν j

366

ALEXANDRE STOJANOVIC

We also have

B+δ

ψn+1 Zj + Rn+1

x cj n2/3

B−δ

because ψn+s Zj + and

B+δ B−δ

x cj n2/3

ψn (ν) − ψn Zj + Zj +

x cj n2/3

x cj n2/3

−ν

ψn+1 (ν)

dν = O(n1/3),

√ n5/6 gZj Ai (x) + O(n1/6) = O(n5/6) = (−1)σ0 |C (Zj )|

ψn+s (ν) dν = O(n−1/2), Zj + c nx2/3 − ν j

since the bounded function in front of ψn+s , which is O(1), does not change the estimate in the proof of Lemma 4.2 of [13]. Therefore, we have x y , Zj + εµ αn Zj + cj n2/3 cj n2/3

y Zj + 2/3 Zj +δ 1 cj n = − × 2 Zj −δ Zj + y2/3 cj n x ∂Kn x ∂Kn Zj + Z , ν + + , ν dν + O(n1/3) × j ∂λ cj n2/3 ∂ν cj n2/3 +δcj n2/3 y x t ∂Kn 1 Zj + − , Zj + + × = 2 −δcj n2/3 ∂x cj n2/3 cj n2/3 y x t ∂Kn Zj + , Zj + dt + O(n1/3) + ∂t cj n2/3 cj n2/3 hence, dividing by cj n2/3 and taking the limit, we get x y 1 εµ αn Zj + , Zj + lim n→+∞ cj n2/3 cj n2/3 cj n2/3 y +∞ 1 − (−Ai(x)Ai(t)) dt = 2 −∞ y = −Ai(x)(ε Ai)(y). To conclude, we obtain x y 1 Sn Zj + , Zj + lim n→+∞ cj n2/3 cj n2/3 cj n2/3 1 = a(x, y) + Ai(x)(ε Ai)(y). 2

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

367

By similar arguments, we compute x y y y − In Zj + , Zj + , Zj + lim In Zj + n→+∞ cj n2/3 cj n2/3 cj n2/3 cj n2/3 x x 1 = a(s, y) ds + Ai(s) ds(ε Ai)(y). 2 y y For the case of odd n, we have to compute the contribution of the term αn (λ). As in the case of Gaussian random matrices, this term contributes. We have αn (λ) = (cn−3,n + gn−3,n )ψn−3 (λ) + (cn−1,n + gn−1,n )ψn−1 (λ) and the computations give 1 (cn,n−3 (1 − an−1,n+2 cn+2,n−1 )), αn 1 = − (cn,n−1 + cn,n−3 an−3,n+2 cn+2,n−1 ), αn

cn−3,n + gn−3,n = − cn−1,n + gn−1,n

where αn is given in the proof of Lemma 4.3. Hence, we get (see [13] for the details) cn−3,n + gn−3,n =

√

n(−t)1/4

(1 + u)1/4 + O(n1/6) √ 2 2

n(−t)1/4

(1 + u)1/4 √ + O(n1/6). 2 2

and cn−1,n + gn−1,n =

√

Moreover, we have √ n1/6 gλ Ai(n2/3C(λ)) + O(1), ψn+s (λ) = (−1) √ |C (λ)| σ0

where

n+s , σ0 = (2 − j ) 2

Thus, we get

C(Zj ) = 0 and

C (Zj ) = cj .

x 1 ψn+s Zj + cj n2/3 cj n2/3 σ0 √gZ (−1) j 2/3 C (Zj ) −4/3 Ai n x + O(n ) + O(n−2/3) = √ cj n2/3 n cj |cj | (−1)σ0 Ai(x) + O(n−2/3). =√ 1/4 j 1/4 2n(−t) (1 + (−1) u)

368

ALEXANDRE STOJANOVIC

At the point Z1 , we conclude that 1 x αn Z1 + c1 n2/3 c1 n2/3 =

1 + u 1/4 1 n−3 n−1 ((−1)[ 2 ] + (−1)[ 2 ] ) Ai(x) + O(n−1/6) = O(n−1/6) 4 1−u

and, at the point Z2 , we have x 1 1 αn Z2 + = Ai(x) + O(n−1/6). 2/3 2/3 c2 n c2 n 2 Moreover, we have x y − (ε αn ) Zj + (ε αn ) Zj + cj n2/3 cj n2/3 x s 1 αn Zj + ds, = 2/3 cj n cj n2/3 y which allows us to conclude because the convergence of the integrand is uniform in the variable s belonging to any compact. 4. Auxiliary Results In this part, we give the results on the asymptotics behaviour of the different coefficients which allow us to prove Theorems 2.3 and 2.4 on the asymptotic regime of the matrix model with two-band quartic potential. The proofs of Lemmas 4.1 and 4.2 are based on the asymptotic formulas of [2] for orthonormal functions, then we use the standard results on asymptotics expansions of integrals of [3] and classical results on the Airy function given in [1] (see [13] for details). √ LEMMA 4.1 (the case β = 1). We define the parameter u = −2 g/t ∈ ]0, 1[. For n → +∞ and p fixed in Z, we have √ √ n√ −t( 1 + u + (−1)n+p 1 − u) + O(1), (4.1) cn+p,n+p−1 = 4 √ √ n√ −t( 1 + u − (−1)n+p 1 − u) + O(1). (4.2) cn+p,n+p−3 = 4 Moreover, we have ψn+p L∞ (J c ) = O(1) ψn+p L∞ (J ) = O(n1/6), and ε ψn+p L∞ (R) = O(n−1/2 ), where J = (Z1 − δ, Z1 + δ) ∪ (Z2 − δ, Z2 + δ),

(4.3)

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

369

δ is a positive real, and J c is the complementary of J in R. For p fixed in Z, such that n + p is even (else an,n+p = 0), we have, when n → +∞, n+p 1 2 1 (−1) 2 1 + + O 5/6 , an,n+p = n (−t)1/4 (1 − u)1/4 (1 + u)1/4 n where [x] is the integer part of the real x. And we have for n → +∞ π √ 1 1 1 + u sin(q − p)v/2 − an+p,n+q = √ sin v/2 n −t 2π 0 1 + u cos v √ 1 − u cos(q − p)v/2 n+p dv + − (−1) 1 + u cos v cos v/2 n+p n+q 1 1 (−1)n+p (−1)([ 2 ]+[ 2 ]) + + + √ 2 (1 − u)1/2 (1 + u)1/2 n −t n+s 1 1 (−1)n+p (−1)[ 2 ] + O 4/3 , + √ (1 − u2 )1/4 n n −t where s = p, if n + p is even and s = q, if n + q is even and with p, q fixed in Z, of different parity (else an+p,n+q = 0). √ LEMMA 4.2 (the case β = 4). Here the parameter u = −2 g/t belongs to the √ interval ]0, 1/ 2[. For n → +∞ and p fixed in Z, we have √ √ n√ 2n+p −2t 1 + 2u + (−1) 1 − 2u + O(1), (4.4) a2n+p,2n+p−1 = − 4 √ √ n√ 2n+p −2t 1 + 2u − (−1) 1 − 2u + O(1). (4.5) a2n+p,2n+p−3 = − 4 Moreover, we have ψ2n+p L∞ (J ) = O(n1/6),

ψ2n+p L∞ (J c ) = O(1)

and

(4.6) ε ψ2n+p L∞ (R) = O(n

−1/2

),

where J = (Z1 − δ, Z1 + δ) ∪ (Z2 − δ, Z2 + δ), δ is a positive real and J c is the complementary of J in R. And we have, for n → +∞, c2n+p,2n+q

2n+p 2n+q 1 (−1)n+p (−1)([ 2 ]+[ 2 ]) 1 √ √ =− √ + + 2 n −2t (1 − 2u)1/2 (1 + 2u)1/2

370

ALEXANDRE STOJANOVIC

√ π 1 1 1 + 2u sin(p − q)v/2 + + √ √ sin v/2 n −2t 2π 0 1 + 2u cos v √ 1 − 2u cos(q − p)v/2 2n+p dv − + (−1) √ cos v/2 1 + 2u cos v 2n+s 1 1 (−1)2n+p (−1)[ 2 ] + O , − √ (1 − 2u2 )1/4 n4/3 n −2t where s = p, if 2n + p is even and s = q, if 2n + q is even and with p, q fixed in Z, of different parity (else c2n+p,2n+q = 0). LEMMA 4.3. (the case β = 1). For the coefficients (2.2) and (2.4), when n → +∞, we have gj k = O(n),

j, k ∈ {n − 3, n − 2, n − 1}.

And, when n is odd, for the coefficients (2.4), we have √ gj n = −gnj = −cj n + O( n), j ∈ {n − 3, n − 2, n − 1}.

(4.7)

(4.8)

LEMMA 4.4. (the case β = 4). For the coefficients (2.10), when n → +∞, we have gj k = O(n),

j, k ∈ {2n, 2n + 1, 2n + 2}.

(4.9)

Proof of Lemma 4.3. We have to distinguish the cases of even and odd n. We examine the case of n even. We follow the procedure explained in Remark 2.2. According to Lemma 4.1, for the coefficients (1.8), we have, that cn+p,n+q = O(n) and

an+p,n+q = O(1/n).

Hence, for the coefficients (2.1), we get sn+p,n+q = O(1). Thus, in view of the expressions of the coefficients (2.2), it suffices to prove (4.7) that the coefficients t!k in the expressions (2.2) are O(1). Since the coefficients t!k are the elements of the inverse of the matrix D = (sj k )n−dj,kn−1 , the difficulty is to prove that det D ∼ c = 0, as n → +∞, where c is a positive constant. In order to do this, we have to compute the exact expression of det D in terms of coefficients (1.8) and use the asymptotic forms of these coefficients given by Lemma 4.1 to compute the asymptotic form of det D. We find det D = (1 − an−2,n+1 cn+1,n−2 )(1 − an−3,n cn,n−3 − an−1,n cn,n−1 − −an−1,n+2 cn+2,n−1 + an−3,n cn,n−3 an−1,n+2 cn+2,n−1 − −an−1,n cn,n−3 an−3,n+2 cn+2,n−1 ) and, using Lemma 4.1, we get 1 det D = c + O 1/3 , n

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

with

371

√ √ 2 1 1 + u + 1 − u n−2 (1 + u)1/4 − (−1)[ 2 ] (1 − u)1/4 √ > 0, c= 4 (1 + u)1/4 1 − u

where u is the parameter of Lemma 4.1. Now we examine the case of n odd. As we see in Remark 2.2, the asymptotic orders of the coefficients are different because √ of the presence of the coefficients = O(1/ n). Hence, for the coefficients (2.3), (1.10) for which we have an,n+p √ we get sn+p,n+q = O( n). Here we have to prove that det D ∼ cn = 0, when n → +∞, where c is a positive constant. In order to do this, we have to compute the exact expression of det D in terms of coefficients (1.8) and (1.10) and use the asymptotic forms of these coefficients given by Lemma 4.1 to compute the asymptotic form of det D. We find det D = αn γn , where − an−3,n )cn,n−3 + (an−1,n − an−1,n )cn,n−1 − αn = 1 + (an−3,n − an−1,n+2 cn+2,n−1 + (an−1,n − an−1,n )cn,n−3 an−3,n+2 cn+2,n−1 − − an−3,n )cn,n−3 an−1,n+2 cn+2,n−1 − (an−3,n

and cn+3,n (an−2,n+1 cn+1,n−2 − 1) − γn = an,n+3 − an,n+1 (cn+1,n−2 an−2,n+3 cn+3,n + cn+1,n ).

Thus, we see √that we just have to prove that αn and γn have the following asymptotic form c n, with c = 0. By using Lemma 4.1, we get √ αn = n(−t)1/4 c + O(n1/6), with

n−1 −1 u2 (1 + u)1/4 − (−1)[ 2 ] (1 − u)1/4 √ <0 c=√ √ √ 2 1 − u2 ( 1 + u − 1 − u)2

and γn =

√

n(−t)1/4 c + O(n1/6),

with n−1

−1 (1 + u)1/4 − (−1)[ 2 ] (1 − u)1/4 √ < 0. c =√ 2 1 − u2

Proof of Lemma 4.4. We follow the same scheme. According to Lemma 4.2, for the coefficients (1.18) we have that a2n+p,2n+q = O(n) and

c2n+p,2n+q = O(1/n).

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ALEXANDRE STOJANOVIC

Hence, for the coefficients (2.9), we get s2n+p,2n+q = O(1). Thus, in view of the expressions of the coefficients (2.10), it suffices to prove (4.9) that the coefficients t!k in the expressions (2.10) are O(1). Since the coefficients t!k are the elements of the inverse of the matrix D = (sj k )2n−dj,k2n−1 , we have to prove that det D ∼ c = 0, for n → +∞, where c is a positive constant. We compute the exact expression of det D in terms of coefficients (1.18) and use the asymptotic forms of these coefficients given by Lemma 4.2 to compute the asymptotic form of det D. We find det D = (1 − a2n−2,2n+1 c2n+1,2n−2 )(1 − a2n−3,2n c2n,2n−3 − a2n−1,2n c2n,2n−1 − − a2n−1,2n+2 c2n+2,2n−1 + a2n−3,2n c2n,2n−3 a2n−1,2n+2 c2n+2,2n−1 − − a2n−3,2n c2n,2n−1 a2n−1,2n+2 c2n+2,2n−3 ) and with Lemma 4.2, we get 1 det D = c + O 1/3 , n with

√ √ √ 1/4 √ 1/4 2 1 1 + 2u + 1 − 2u n (1 + 2u) + (−1) (1 − 2u) > 0, c= √ √ 4 (1 + 2u)1/4 1 − 2u where u is the parameter of Lemma 4.1. Acknowledgement The author is grateful to the referees and to Anne Boutet de Monvel for useful and critical remarks. References 1. 2. 3. 4. 5.

6. 7.

Abramowitz, M. and Stegun, I. A. (eds): Handbook of Mathematical Functions, Dover, New York, 1968. Bleher, P. and Its, A.: Semi-classical asymptotics of orthogonal polynomials, Riemann–Hilbert problem and universality in the matrix model, Ann. of Math. 150 (1999), 185–266. Bleistein, N. and Handelsman, R. A.: Asymptotic Expansions of Integrals, Dover, New York, 1986. Boutet de Monvel, A., Pastur, L. and Shcherbina, M.: On the statistical mechanics approach to the random matrix theory: the integrated density of states, J. Statist. Phys. 79 (1995), 585–611. Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X.: Asymptotics for polynomial orthogonal with respect to varying exponential weight, Internat. Math. Res. Notes 16 (1997), 759–782. DiFrancesco, P., Ginsparg, P. and Zinn-Justin, J.: 2D gravity and random matrices, Phys. Rep. 254 (1995), 1–133. Forrester, P. J., Nagao, T. and Honner, G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges, Nuclear Phys. B 553 [PM] (1999), 601–643.

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS

8. 9. 10. 11. 12. 13.

14. 15. 16. 17.

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Johansson, K.: On fluctuation of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), 151–204. Mehta, M. L.: Random Matrices, and the Statistical Theory of Energy Levels, Academic Press, New York, 1967. Mehta, M. L.: Matrix Theory, Selected Topics and Useful Results, Les Éditions de physique, France, 1989. Mehta, M. L.: Random Matrices, Academic Press, New York, 1991. Pastur, L. and Shcherbina, M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Statist. Phys. 86 (1997), 109–147. Stojanovic, A.: Une approche par les polynômes orthogonaux pour des classes de matrices aléatoires orthogonalement et symplectiquement invariantes : application à l’universalité de la statistique locale des valeurs propres, Preprint, www.physik.uni-bielefeld.de/bibos/preprints, 00-01-06. Szeg˝o, G.: Orthogonal Polynomials, Amer. Math. Soc., Providence, 1939. Tracy, C. A. and Widom, H.: Orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177 (1996), 727–754. Tracy, C. A. and Widom, H.: Correlation functions, cluster functions and spacing distributions for random matrices, J. Statist. Phys. 92 (1998), 809–835. Widom, H.: On the relation between orthogonal, symplectic and unitary matrix ensembles, J. Statist. Phys. 94 (1999), 347–364.

Mathematical Physics, Analysis and Geometry 3: 375–384, 2000. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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On the Solution of a Painlevé III Equation HAROLD WIDOM Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A. e-mail: [email protected] (Received: 23 November 2000) Abstract. In a 1977 paper of B. M. McCoy, C. A. Tracy and T. T. Wu there appeared for the first time the solution of a Painlevé equation in terms of Fredholm determinants of integral operators. Their proof is quite complicated. We present here one which is more straightforward and makes use of recent work of the author and C. A. Tracy. Mathematics Subject Classifications (2000): 34M65, 45B05, 35Q53. Key words: Painlevé equation, Fredholm determinant, sinh-Gordon equation.

1. Introduction In the 1977 paper of McCoy, Tracy and Wu [2] the following result was established. THEOREM. Let ∞ 2n+1 ∞ e−t uj 2 2n+1 λ ··· × ψ(t) = 2n + 1 u + uj +1 1 1 n=0 j =1 j 2n+1 1 1 uj − 1 α− 2 2n+1 uj − 1 α+ 2 + dy1 . . . dy2n+1 . × uj + 1 uj + 1 j =1 j =1 ∞

(1)

Then ψ satisfies the equation ψ (t) + t −1 ψ (t) =

1 2

sinh 2ψ + 2αt −1 sinh ψ.

(2)

This is a special case of the Painlevé III equation, and (1) gives a one-parameter family of solutions. It is clearly expressible in terms of the Fredholm determinants of the kernels 1 e−t u u − 1 α± 2 u+v u+1 acting on L2 (1, ∞). The proof in [2] is quite complicated, and the purpose of this note is to give a more straightforward one. Research supported by National Science Foundation grant DMS-9732687.

376

HAROLD WIDOM

First we give an equivalent formulation of the solution in terms of the kernel −1 e−t (x+x )/2 x − 1 2α (3) K(x, y) = x + y x + 1 acting on L2 (0, ∞). This is the representation λ λ ψ = log det I + K − log det I − K , 2 2

(4)

where K is the operator with kernel K(x, y). (This is very possibly known but seems not to have been written down in the literature before.) To derive this second representation of ψ we make the changes of variable uj = (xj + xj−1 )/2 in the multiple integral in the first representation. Then xj + 1 dxj uj + 1 = , duj = 12 (xj2 − 1) 2 , uj − 1 xj − 1 xj and the integral becomes

∞ 2n+1

−1

e−t (xj +xj )/2 × · · · 22n+1 1 (xj + xj +1 )(xj xj +1 + 1) 1 j =1 2n+1 2n+1 xj − 1 2α 2 2 (xj + 1) + (xj − 1) dx1 . . . dx2n+1 . × xj + 1 j =1 j =1 1

∞

If we denote by K± the operators on L2 (1, ∞) with kernels −1 x − 1 2α e−t (x+x )/2 (x ± 1)(y ± 1) , K± (x, y) = (x + y)(xy + 1) x+1 then the multiple integral is equal to tr K+2n+1 + tr K−2n+1 and so ψ =2

∞ (λ/2)2n+1 n=0

2n + 1

(tr K+2n+1 + tr K−2n+1 ).

Thus (4) will be a consequence of the following lemma: LEMMA. For each m 1, we have tr K+m + tr K−m = tr K m . Proof. Let f be an eigenfunction for K with eigenvalue λ, ∞ −t (x+x −1 )/2 x − 1 2α e f (y) dy = λf (x). x + y x + 1 0

ON THE SOLUTION OF A PAINLEVE´ III EQUATION

377

Then the substitutions x → x −1 , y → y −1 show that x −1 f (x −1 ) is also an eigenfunction corresponding to the same eigenvalue. Hence, any eigenfunction can be written as the (orthogonal) sum of an ‘even’ eigenfunction f+ satisfying x −1 f+ (x −1 ) = f+ (x) and an ‘odd’ eigenfunction f− satisfying x −1 f− (x −1 ) = −f− (x). (Of course, one of these is probably zero.) The change of variable y → y −1 and the relations y −1 f± (y −1 ) = ±f± (y) show that 1 −t (x+x −1 )/2 1 − x 2α e f± (y) dy x +y 1+x 0 ∞ −t (x+x −1 )/2 x − 1 2α e f± (y) dy. =± xy + 1 x+1 1 We deduce that f± are eigenfunctions corresponding to the eigenvalue λ for the operators K± on L2 (1, ∞) with kernels

1 1 x − 1 2α −t (x+x −1 )/2 ± = K± (x, y), e x +y xy + 1 x + 1 and the statement of the lemma follows.

✷

2. Proof of the Theorem Direct proofs of the fact that ψ as given by (4) satisfies the Painlevé equation when α = 0 have already been given [1, 3]. We shall make use of some of the results of [3] here and therefore follow that paper’s notation, more or less. We give the proof in stages. Part A. First, we introduce parameters r and s, define E(x)E(y) λ (rx+sx −1)/2 x − 1 α e , , K(x, y) = E(x) = 2 x +1 x +y and let K be the operator with this kernel K(x, y). (In the notation of [3], r = t1 , s = t−1 . The formulas we quote from there will be in terms of our parameters r and s. For convenience, we have changed the definition of K to conform to the notation used there.) Define ϕ(r, s) := log det(I + K) − log det(I − K). Then ψ(t) = ϕ(−t/2, −t/2). We know from [3] that ϕ satisfies the sinh-Gordon equation ∂ 2ϕ = ∂r∂s

1 2

sinh 2ϕ.

(5)

378

HAROLD WIDOM

In order to deduce (2) from this we must first find a connection between the r and s derivatives of ϕ. (When α = 0 the determinants, and so also ϕ, depend only on the product rs and (2) in this case is almost immediate.) To this end, we

y) := observe that the determinants are unchanged if K(x, y) is replaced by K(x, sK(sx, sy). This is the same as replacing E(x) by λ (rsx+x −1) sx − 1 α

E(x) = e sx + 1 . 2 Now

∂s E(x) = rx + 2α

x

s2x2 − 1

E(x),

which gives

y) = r E(x)

E(y)

+ 2α ∂s K(x,

s 2 xy − 1

E(y).

E(x) (s 2 x 2 − 1)(s 2 y 2 − 1)

y) was cancelled by its occurrence also as a factor (The denominator x + y in K(x, in both summands.) Hence ∂s log det(I + K)

= ∂s log det(I + K)

E(y)

+ 2α

−1 r E(x) = tr(I + K)

s 2 xy − 1

E(y)

E(x) . (s 2 x 2 − 1)(s 2 y 2 − 1)

(We abused the notation here by writing in the brackets the kernel of the operator that is meant.) Now we undo the variable change we made, which means we replace x by x/s and y by y/s and divide by s in the expressions for the kernels, and we obtain ∂s log det(I + K) = tr(I + K)

−1

2α xy − 1 r E(x)E(y) + E(x)E(y) . s s (x 2 − 1)(y 2 − 1)

If we had differentiated with respect to r without making a preliminary variable change, we would have obtained ∂r log det(I + K) = tr(I + K)−1 E(x)E(y). Hence, we have shown that s∂s log det(I + K) − r∂r log det(I + K)

xy − 1 E(x)E(y) . = 2α tr(I + K)−1 (x 2 − 1)(y 2 − 1)

ON THE SOLUTION OF A PAINLEVE´ III EQUATION

379

Replacing K by −K and subtracting gives the relation (we use subscript notation for derivatives)

xy − 1 E(x)E(y) . (6) rϕr − sϕs = 4α tr(I − K 2 )−1 (x 2 − 1)(y 2 − 1) This is the desired connection between the r and s derivatives of ϕ. Part B. Our goal is to show that ψ(t) = ϕ(−t/2, −t/2) satisfies (2), but because of those awkward factors −1/2 we prefer to derive the equivalent equation d d2 ϕ(t, t) + t −1 ϕ(t, t) = 2 sinh 2ϕ(t, t) − 4αt −1 sinh ϕ(t, t). 2 dt dt

(7)

We use d2 ϕ(t, t) = 2ϕrs (t, t) + ϕrr (t, t) + ϕss (t, t), dt 2 d ϕ(t, t) = ϕr (t, t) + ϕs (t, t). dt

(8)

Now we know that ϕ(r, s) satisfies the sinh-Gordon equation (5) so let us see what identity we have to derive. Set

xy − 1 E(x)E(y) . T = tr(I − K 2 )−1 (x 2 − 1)(y 2 − 1) Differentiating (6) with respect to r and s gives rϕrr + ϕr − sϕrs = 4αTr ,

rϕrs − sϕss − ϕs = 4αTs .

Therefore −(r + s)ϕrs + rϕrr + sϕss + ϕr + ϕs = 4α(Tr − Ts ), (r + s)ϕrs + rϕrr + sϕss + ϕr + ϕs = 2(r + s)ϕrs + 4α(Tr − Ts ). Setting r = s = t and using (8), we get t

d d2 ϕ(t, t) + ϕ(t, t) = 4tϕrs (t, t) + 4α(Tr − Ts )(t, t). 2 dt dt

Hence, by (5), d d2 ϕ(t, t) + t −1 ϕ(t, t) = 2 sinh 2ϕ(t, t) + 4αt −1 (Tr − Ts )(t, t). 2 dt dt It follows that (7) is equivalent to (Tr − Ts )(t, t) = − sinh ϕ(t, t).

(9)

380

HAROLD WIDOM

Part C. The functions Ei (x) , x2 − 1 Pi = (I − K 2 )−1 KEi

Ei (x) = x i E(x),

Fi (x) =

Qi = (I − K 2 )−1 Ei ,

will arise in the computations leading to the identity (9). We have

xy − 1 2 −1 Tr − Ts = tr(I − K ) (∂r − ∂s )E(x)E(y) + (x 2 − 1)(y 2 − 1)

xy − 1 2 −1 + tr(∂r − ∂s )(I − K ) E(x)E(y) . (x 2 − 1)(y 2 − 1)

(10)

Now (∂r − ∂s )E(x)E(y) = 12 (x − x −1 + y − y −1 )E(x)E(y), which gives xy − 1 (∂r − ∂s )E(x)E(y) = 2 (x − 1)(y 2 − 1)

1 2

y − x −1 x − y −1 + 2 E(x)E(y). y2 − 1 x −1

Hence, the first summand in (10) equals (Q0 , F1 ) − (Q−1 , F0 ).

(11)

Next, using the notation a ⊗ b for the operator with kernel a(x)b(y), we have ([3], p. 4) (∂r − ∂s )(I − K 2 )−1 = 12 (P0 ⊗ Q0 + Q0 ⊗ P0 − P−1 ⊗ Q−1 + Q−1 ⊗ P−1 ), and it follows that the second summand in (10) equals (Q0 , F1 )(P0 , F1 ) − (Q0 , F0 )(P0 , F0 )− − (Q−1 , F1 )(P−1 , F1 ) + (Q−1 , F0 )(P−1 , F0 ).

(12)

We introduce notations for the various inner products: Ui,j = (Qi , Fj ),

Vi,j = (Pi , Fj ).

These are analogous to the inner products ui,j = (Qi , Ej ),

vi,j = (Pi , Ej )

which play a crucial role in [3] and will here, also. We have shown that Tr − Ts = U0,1 − U−1,0 + U0,1V0,1 − U0,0 V0,0 − U−1,1 V−1,1 + + U−1,0 V−1,0 .

(13)

381

ON THE SOLUTION OF A PAINLEVE´ III EQUATION

Part D. There are relations among the various quantities appearing on the right side of (13). If we set ui = u0,i ,

vi = v0,i ,

Ui = U0,i ,

Vi = V0,i ,

then we have the recursion formulas xQi (x) − Qi+1 (x) = vi Q0 (x) − ui P0 (x), xPi (x) + Pi+1 (x) = ui Q0 (x) − vi P0 (x). (The first is formula (9) of [3], the second is obtained similarly.) Taking inner products with Ej gives the formulas ui,j +1 − ui+1,j = vi uj − ui vj ,

vi,j +1 + vi+1,j = ui uj − vi vj

(14)

of [3] and taking inner products with Fj gives the analogous formulas Ui,j +1 − Ui+1,j = vi Uj − ui Vj ,

Vi,j +1 + Vi+1,j = ui Uj − vi Vj .

(15)

Observe the special case i = j = −1 of the second part of (14): u2−1 + 1 = (1 + v−1 )2 .

(16)

(The ui,j are symmetric in i and j .) In fact ([3], p. 8), u−1 = sinh ϕ,

1 + v−1 = cosh ϕ.

We see from the above formulas that all the Ui,j and Vi,j may be expressed in terms of the Ui and Vi (with coefficients involving the ui and vi ). But notice that Fi+2 − Fi = Ei (here we use the form of Fi for the first time). This gives Ui+2 − Ui = ui , Vi+2 − Vi = vi and using this also it is clear that everything can be expressed in terms of the four unknown quantities U0 , V0 , U1 and V1 (and the ui and vi ). Using (16) also we compute that (13) equals −v−1 U1 + u−1 V1 + u−1 + + U1 V1 − U0 V0 − ((1 + v−1 )U0 − u−1 V0 )(u−1 U0 − (1 + v−1 )V0 )+ + ((1 + v−1 )U1 − u−1 V1 − u−1 )(u−1 U1 − (1 + v−1 )V1 − v−1 ). (17) Now we are going to use, as we did before, the fact that conjugation by the unitary operator f (x) → x −1 f (x −1 ) has the effect on K of interchanging r and s. Thus K is invariant under this conjugation when r = s. Since Ei is sent to E−i−1 and Fi to −F−i+1 we find that when r = s U0 = −(Q−1 , F−1 ) = −(1 + v−1 )U0 + u−1 V0 , U1 = −(Q−1 , F0 ) = −(1 + v−1 )U−1 + u−1 V−1 = −(1 + v−1 )(U1 − u−1 ) + u−1 (V1 − v−1 ).

382

HAROLD WIDOM

From these we deduce that u−1 u−1 V0 = U0 , V1 = U1 − 1. v−1 v−1

(18)

Using these we find that when r = s (17) simplifies to 2

u−1 2 (U0 − U12 ). v−1

Since u−1 = sinh ϕ, we have shown that the desired identity (9) is equivalent to U1 (t, t)2 − U0 (t, t)2 = 12 v−1 (t, t).

(19)

Part E. Let us compute d/dt of both sides of (19). Of course, d/dtU0 (t, t) = (∂r + ∂s )U0 (t, t), etc., so we begin by writing down these derivatives. We have 2(∂r + ∂s )Fi = Fi+1 + Fi−1 , 2(∂r + ∂s )Ei = Ei+1 + Ei−1 , 2 −1 2(∂r + ∂s )(I − K ) = 12 (P0 ⊗ Q0 + Q0 ⊗ P0 + P−1 ⊗ Q−1 + Q−1 ⊗ P−1 ). (For the last, see [3], p. 4.) Using these we compute 2(∂r + ∂s )Ui = 2Ui+1 + 2u0 Vi + (1 + u2−1 + (1 + v−1 )2 )Ui−1 − 2u−1 (1 + v−1 )2 Vi−1 . Taking i = 0 and 1 and using (18) and (16) we find that when r = s u−1 U0 − 2v−1 U1 , v−1 u−1 U1 − 2v−1 U0 . 2(∂r + ∂s )U1 = 2u0 v−1

2(∂r + ∂s )U0 = 2u0

Hence, (∂r + ∂s )(U12 − U02 ) = 2u0

u−1 2 (U − U02 ). v−1 1

(20)

Now we compute in a similar way (cf., [3], p. 5) 2(∂r + ∂s )vi = u−1 u0 + v−1 v0 + v0 + v−1,1 + u−1,−1 u−1 + v−1,−1 v−1 + v−2 + v−1,−1 . Again applying the operator f (x) → x −1 f (x −1 ) we find that when r = s we have u−1,−1 = u0 ,

v−1,−1 = v0 ,

so the above is 2(u−1 u0 + v−1 v0 + v0 + v−1,1 ).

v−2 = v−1,1 ,

ON THE SOLUTION OF A PAINLEVE´ III EQUATION

383

Applying the second part of (14) with i = −1, j = 0 gives v0 + v−1,1 = u−1 u0 − v−1 v0 , and so we have shown that, when r = s, u−1 v−1 . (∂r + ∂s )v−1 = 2u−1 u0 = 2u0 v−1 This relation and (20) show that U1 (t, t)2 − U0 (t, t)2 and v−1 (t, t) are equal up to a constant factor, and to deduce (19) it remains only to compute this factor. We do this by determining the asymptotics of both quantities as t → −∞. For convenience, we evaluate everything at r = s = −t and let t → +∞. We have v−1 = (I − K 2 )−1 KE, E−1 . If we were to replace (I − K 2 )−1 by I we would be left with ∞ 2 ∞ ∞ E(x)2 E(y)2 −1 2 1 y dy dx ∼ 2 E(x) dx (KE, E−1 ) = x+y 0 0 0 since the main contributions to the integrals come from neighborhoods of x = y = 1. It is an easy exercise to show that 2α ∞ 1 −t (x+x −1 ) x − 1 1 −2α −α− 2 −2t e e , (21) x + 1 dx ∼ $(α + 2 )2 t 0 and so (KE, E−1 ) ∼

λ2 $(α + 12 )2 2−4α−1 t −2α−1 e−4t . 4

The error caused by our replacement of (I − K 2 )−1 by I is of smaller order of magnitude. This follows from the fact that the square of the L2 norm of E is O(t −α−1/2 e−2t ), as shown above, and, hence, so is the operator norm of K. Thus the error, which equals ((I − K 2 )−1 K 3 E, E−1 ), is O(t −4α−2 e−8t ). Therefore we have shown v−1 ∼

λ2 $(α + 12 )2 2−4α−1 t −2α−1 e−4t . 4

Next,

U1 − U0 = (I − K 2 )−1 E, (x + 1)−1 E , U1 + U0 = (I − K 2 )−1 E, (x − 1)−1 E

and U12 − U02 is the product of these. As before, replacing (I − K 2 )−1 by I in each factor will not affect the first-order asymptotics of the product. After this replacement the first inner product becomes λ/2 times the integral in (21) but with an extra factor x + 1 in the denominator. Thus 1 λ U1 − U0 ∼ $(α + 12 )2−2α−1 t −α− 2 e−2t . 2

384

HAROLD WIDOM

After the replacement the second inner product becomes λ/2 times 2α ∞ dx −t (x+x −1 ) x − 1 e x + 1 x − 1. 0 This is a little trickier since when we make the variable change x = 1 + y to compute the asympotics, we must use the second-order approximations x +x −1 → 2 + y 2 − y 3 and (x + 1)−2α → 2−2α (1 − αy). But it is still straightforward and we find that U1 + U0 ∼

1 λ $(α + 12 )2−2α−1 t −α− 2 e−2t . 2

Thus U12 − U02 ∼

λ2 $(α + 12 )2 2−4α−2 t −2α−1 e−4t ∼ 12 v−1 . 4

We knew that (U12 − U02 )/v−1 is a constant and now we see that the constant equals 1/2. This establishes (19) and concludes the proof. References 1. 2. 3.

Bernard, D. and LeClair, A.: Differential equations for sinh-Gordon correlation functions at the free fermion point, Nuclear Phys. B 426 [FS] (1994), 534–558. McCoy, B. M., Tracy, C. A. and Wu, T. T.: Painlevé functions of the third kind, J. Math. Phys. 18 (1977), 1058–1092. Tracy, C. A. and Widom, H.: Fredholm determinants and the mKdV/sinh-Gordon hierarchies, Comm. Math. Phys. 179 (1996), 1–10.

Mathematical Physics, Analysis and Geometry 3: 385–403, 2000. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Asymptotic Value Distribution for Solutions of the Schrödinger Equation S. V. BREIMESSER and D. B. PEARSON Department of Mathematics, University of Hull, Hull HU6 7RX, England. e-mail: {s.v.breimesser;d.b.pearson}@maths.hull.ac.uk. (Received: 5 December 2000) Abstract. We consider the Dirichlet Schrödinger operator T = −(d2 /d x 2 )+V , acting in L2 (0, ∞), where V is an arbitrary locally integrable potential which gives rise to absolutely continuous spectrum. Without any other restrictive assumptions on the potential V , the description of asymptotics for solutions of the Schrödinger equation is carried out within the context of the theory of value distribution for boundary values of analytic functions. The large x asymptotic behaviour of the solution v(x, λ) of the equation Tf (x, λ) = λf (x, λ), for λ in the support of the absolutely continuous part µa.c. of the spectral measure µ, is linked to the spectral properties of this measure which are determined by the boundary value of the Weyl–Titchmarsh m-function. Our main result (Theorem 1) shows that the value distribution for v (N, λ)/v(N, λ) approaches the associated value distribution of the Herglotz function mN (z) in the limit N → ∞, where mN (z) is the Weyl–Titchmarsh mfunction for the Schrödinger operator −(d2 /d x 2 ) + V acting in L2 (N, ∞), with Dirichlet boundary condition at x = N. We will relate the analysis of spectral asymptotics for the absolutely continuous component of Schrödinger operators to geometrical properties of the upper half-plane, viewed as a hyperbolic space. Mathematics Subject Classifications (2000): 47E05, 34L05, 81Q10. Key words: Herglotz functions, hyperbolic geometry, m-function, Schrödinger operator, spectral theory, value distribution.

1. Introduction The principal aim of this paper is to provide a general analysis of the link between asymptotics for solutions of the Schrödinger equation on the half-line, at real spectral parameter λ, and spectral properties, considered in relation to the Weyl– Titchmarsh m-function and its boundary values, for the associated Schrödinger operator, in the case that this operator has absolutely continuous spectrum. We do not, here, restrict attention to the case in which there is purely absolutely continuous spectrum – rather, we shall assume that there is an absolutely continuous component to the spectrum, and we shall take λ to belong to a spectral support of the absolutely continuous component of the spectral measure. Work completed during the tenure of a University of Hull Open Scholarship. Partially supported by EPSRC.

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A secondary aim of the paper will be to exhibit the degree to which the analysis of spectral asymptotics for the absolutely continuous component of Schrödinger operators is related to, and dependent on, the geometrical properties of the complex upper half-plane, viewed as a hyperbolic space, and mappings of this space by analytic functions. In particular, we shall show how the theory of value distribution for boundary values of analytic functions provides a natural framework for the description of large x asymptotics for solutions of the Schrödinger equation. These ideas can be linked, through the use of hyperbolic metric, to an analysis of angle subtended, at points of C+ and their images under analytic maps, by Borel subsets of R. We will outline several interesting byproducts of this work, including resulting estimates of boundary behaviour and associated limiting value distribution for Herglotz functions. We begin by establishing the notation and background for the study of spectral theory and asymptotics for the Schrödinger operator on the half-line. Let a potential function V (x), defined for 0 x < ∞, be given, with V realvalued and integrable over bounded subintervals of [0, ∞). We make no special assumptions regarding the behaviour of V (x) in the limit as x → ∞. We associate with V the differential expression τ = −(d2 /d x 2 ) + V . Then τ may be used to define the self-adjoint operator T = −(d2 /d x 2 ) + V , acting in L2 (0, ∞) and subject to Dirichlet boundary condition at x = 0. Correspondingly, we may define the one-parameter family Tα = −(d2 /d x 2 ) + V of selfadjoint operators in L2 (0, ∞), subject for any α ∈ [0, π ) to the boundary condition (cos α)f (0) + (sin α)f (0) = 0; we may thus identify T0 with T . We are assuming here that the differential expression τ belongs to the limitpoint case at infinity (see [1]), in which case no boundary condition at x = +∞ is required in order to define Tα as a self-adjoint operator. The alternative assumption, that τ belongs to the limit-circle case, is known to lead to purely discrete spectrum for Tα (see [1]). Since we are concerned in this paper with the absolutely continuous part of the spectrum, we need not allow for the possibility of limit-circle at infinity. We shall normally denote by f (x, λ), in the case of real spectral parameter λ, and by f (x, z) where the spectral parameter z is complex with z ∈ C+ , respective solutions of the Schrödinger equation −

d2 f (x, λ) + V (x)f (x, λ) = λf (x, λ), d x2

(1)

−

d2 f (x, z) + V (x)f (x, z) = zf (x, z), d x2

(2)

in each case taking 0 x < ∞. In particular, define specific solutions uα (x, λ), vα (x, λ) for λ ∈ R (and correspondingly uα (x, z), vα (x, z) in the case z ∈ C+ ) subject to initial conditions uα (0, λ) = cos α, vα (0, λ) = − sin α,

uα (0, λ) = sin α, vα (0, λ) = cos α.

(3)

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In the special case α = 0, we shall usually denote u0 , v0 by u, v, respectively. Note that vα satisfies the boundary condition appropriate to Tα ; in particular, if vα (·, λ) ∈ L2 (0, ∞) then λ will be an eigenvalue of Tα . For each α ∈ [0, π ) and for all z ∈ C+ , the Weyl–Titchmarsh m-function mα (·) is defined (assuming limit-point case at infinity) by the condition that uα (·, z) + mα (z)vα (·, z) ∈ L2 (0, ∞).

(4)

Then mα is a Herglotz function, that is analytic in the upper half-plane, with strictly positive imaginary part. In the special case α = 0, we shall normally denote m0 (z) by m(z), in which case (4) becomes u(·, z) + m(z)v(·, z) ∈ L2 (0, ∞). An alternative characterisation of m is through the observation that m(z) = f (0, z) /f (0, z) for any (nontrivial) solution f (·, z) of Equation (2), such that f (·, z) ∈ L2 (0, ∞). In this paper, we shall in addition be interested in the m-function related to the differential expression τ = −(d2 /d x 2 ) + V where V (x) is defined on the truncated interval N x < ∞, for any N > 0. Taking for simplicity the case of Dirichlet boundary condition at x = N, we may define the self-adjoint operator T N = −(d2 /d x 2 ) + V acting in L2 (N, ∞), subject to boundary condition f (N) = 0. Correspondingly, solutions uN (·, z), v N (·, z) of Equation (2) with Im z > 0, may be defined subject to initial conditions uN (N, z) = 1, v N (N, z) = 0,

(uN ) (N, z) = 0, (v N ) (N, z) = 1,

and the m-function mN (·) with Dirichlet boundary condition at x = N is determined by the condition that uN (·, z) + mN (z)v N (·, z) ∈ L2 (N, ∞)

(z ∈ C+ ).

Note that mN (·) is the standard m-function for the Dirichlet Schrödinger operator −(d2 /d x 2 )+V (x+N) acting in L2 (0, ∞). An alternative characterisation of mN (·) is through the observation that mN (z) = f (N, z)/f (N, z) for any (nontrivial) square-integrable solution f (·, z) of Equation (2). Since u(·, z) + m(z)v(·, z) is just such a square-integrable solution, we can write explicitly mN (z) =

u (N, z) + m(z)v (N, z) . u(N, z) + m(z)v(N, z)

(5)

Given any Herglotz function F , a corresponding right-continuous, nondecreasing function ρ may be defined (uniquely up to an additive constant) by the so-called Herglotz representation ([2]) ∞ t 1 − 2 dρ(t), (6) F (z) = a + bz + t +1 −∞ t − z

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∞ where ρ satisfies in addition the integrability condition −∞ (t 2 + 1)−1 dρ(t) < ∞. The function ρ may then be used to define a Lebesgue–Stieltjes measure µ = dρ(t). Such a measure may be defined in particular for the m-function m(z), in which case the measure carries all of the spectral information for the Dirichlet operator T , in the sense that T is unitarily equivalent to the multiplication operator g(λ) → λg(λ) in the Hilbert space L2 (R; dρ). In that case the Lebesgue–Stieltjes measure µ = dρ is called the spectral measure of the operator T . We shall be concerned with values of the spectral parameter λ which belong to the support of the absolutely continuous component µa.c. of the measure µ. The main question to be addressed will be as follows: How can one describe the large x asymptotic behaviour of the solution v(x, λ) of Equation (1), for λ in the spectral support of µa.c. , in terms of spectral properties of this measure? Here we would expect ‘spectral properties of this measure’ to include a prescription of the boundary value m+ (λ) of the m-function, defined as m+ (λ) = limε→0+ m(λ + iε), since m+ (λ) determines both the density function 1/π Im m+ (λ) of µa.c. , and, through equations linking m(z) algebraically with other functions mα (z), the corresponding density functions for the spectral measures of the Tα . We shall also N find that the boundary value mN + (λ) of the ‘truncated’ m-function m (z) plays an important role in describing asymptotic behaviour of v(x, λ) for large x. Our key result (Theorem 1) relating asymptotics to spectral properties relates not to the solution v(x, λ) directly, but to its logarithmic derivative q(x, λ), defined by q(x, λ) = v (x, λ)/v(x, λ), where the prime denotes differentiation with respect to x. In order to achieve maximum generality, that is without imposing restrictive assumptions on the potential, the description of large x asymptotics has to be carried out within the context of the theory of value distribution. We shall explain more fully in Section 2 the background to the idea of value distribution for any real-valued (Lebesgue) measurable function F (λ). For such a function F , value distribution assigns to any pair A, S of Borel subsets of R a nonnegative (and possibly infinite) number M(A, S; F ), given by M(A, S; F ) = |A ∩ F −1 (S)|, where | · | stands for Lebesgue measure. Thus M is the Lebesgue measure of all λ ∈ A for which F (λ) ∈ S. We may now give a more precise statement to Theorem 1, in terms of asymptotic value distribution. Namely, the large x value distribution, over λ, of the logarithmic derivative q(x, λ) = v (x, λ)/v(x, λ) of the solution v of Equation (1) is given by the formula 1 N θ(m+ (λ), S) dλ = 0. lim |{λ ∈ A; q(N, λ) ∈ S}| − N→∞ π A

(7)

Here A is an arbitrary subset of the essential support of µa.c. , having finite measure, and θ(z, S), for any z ∈ C+ , denotes the angle subtended at z by the subset S of the real z-axis. We consider the asymptotic formula (7) to be unusual in several respects. For example,

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(i) Equation (7) holds for arbitrary locally integrable potentials which give rise to absolutely continuous spectrum, and hence is a result of considerable generality. For certain special classes of potential, the result may be further qualified by more detailed information regarding the dependence on λ and N of mN + (λ). The dependence of mx+ (λ) on x is subject to a first-order Riccati-type differential equation, and solutions exhibiting the so-called δ-clustering property ([3]) may be used to provide bounds for mN + (λ). Examples of classes of potentials which can be treated √ (λ) = i λ in this way include the L1 (0, ∞) class, for which one has limN→∞ mN + √ 2 so that mN + (λ) may be replaced by i λ in Equation (7), and potentials such as −kx (k 0) which give rise to absolutely continuous spectrum over the whole of R ([4]). Recent developments in the theory of slowly decreasing potentials, satisfying power bounds of the form |V (x)| const. × x −(δ+1/2) (δ > 0) as x → ∞, have shown that for these potentials the large x behaviour of v(x, λ) is governed for almost all λ > 0 by WKB-like asymptotics [5–8]. (ii) A second aspect of the asymptotic formula (7) to which we would draw attention appears to be completely new. Notice first of all that q(N, λ) = v (N, λ)/v(N, λ) may be determined from the solution of the Schrödinger equation (1) on the interval 0 x N, and subject to the appropriate initial conditions at x = 0. Hence, q(N, λ) is determined by the potential function V over this finite interval [0, N]. On the other hand, one has the following prescription for the determination of mN + (λ): let f (·, z), with Im z > 0, be any (nontrivial) solution of the Schrödinger equation (2) on the interval N x < ∞, and belonging to L2 (N, ∞); then N set mN + (λ) = limε→0+ f (N, λ + iε)/f (N, λ + iε). Thus we see that m+ (λ) is determined by the potential function V across the semi-infinite interval (N, ∞). It follows, then, that the asymptotic formula (7) expresses the convergence as N → ∞, for any choice of Borel sets A, S, of two analytic expressions to each other, of which the first (|{λ ∈ A; q(N, λ) ∈ S}|) is dependent on the potential function V only on the interval [0, N], and the second (1/π A θ(mN + (λ), S) dλ) is dependent on the potential function V only on the interval (N, ∞). The fact that these two intervals of definition of V do not overlap implies that we have, in Equation (7), the convergence of two expressions which are, in some sense, ‘independent’ of each other. This will have important consequences for our understanding of the theory of absolutely continuous spectrum for Schrödinger operators and we will return to this point in Section 4. (iii) A third aspect of the asymptotic formula (7) to which we draw attention is the role in the description of asymptotics of ideas drawn from the theory of value distribution, and the link with angle subtended. These and other ideas are related to the geometry of hyperbolic space, about which we shall have more to say later. (iv) Finally, we draw attention to the link which will become more apparent through the proof of Equation (7), between rate of convergence in (7) and regularity properties of m+ (λ).

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The paper is organised as follows: In Definition 1 of Section 2, we make more precise the notion of a value distribution function M, and introduce as an important special case (Definition 2) the value distribution function associated with any given Herglotz function F . The value distribution function for F may be obtained by integrating, with respect to the spectral parameter λ, the angle subtended by a given set S at the boundary value point F+ (λ), and is the limit of the corresponding integral over the complex line Im z = δ, as δ approaches zero. Surprisingly, the convergence of this limit is uniform over all Herglotz functions F and all subsets S of R (see Lemma 1), a result that has far-reaching consequences for value distribution and spectral theory. In Section 3, we exhibit the connection between angle subtended and the hyperbolic metric for C+ . Instead of using hyperbolic metric directly, we rely on an estimate of separation γ , defined by Equation (15), of points in C+ ; γ is related to hyperbolic metric by Equation (16), and to angle subtended by Equation (17). In Lemma 3 we arrive at one of the key estimates of value distribution theory for solutions of the Schrödinger equation, which demonstrates a precise bound for the large N convergence with respect to hyperbolic metric of the negative logarithmic derivative −f (N, z)/f (N, z) of solutions of Equation (2). Theorem 1 of Section 4 is a statement of the main result of this paper, which shows that the value distribution for v (N, λ)/v(N, λ) approaches the associated value distribution of the Herglotz function mN (·) in the limit as N → ∞. The proof of this theorem relies heavily on the results of the previous sections, and the sequence of inequalities which make up the proof allow in many cases a precise estimate of the rate of convergence. In the final Section 5, we consider some applications of the theory to specific examples of Schrödinger operators. These include examples of slowly decaying potentials, potentials unbounded below at infinity and sparse potentials. 2. Value Distribution and Herglotz Functions Given a (Lebesgue) measurable function f : R → R, the distribution of points (λ, f (λ)) of the graph of f may be described in terms of a measure M0 , given for Borel subsets S of R2 by M0 (S) = |{λ ∈ R; (λ, f (λ)) ∈ S}|. Here | · | stands for Lebesgue measure, and M0 (S) is the measure of the set of all λ such that the corresponding point (λ, f (λ)) of the graph of f belongs to the set S. In the case that S = A × S is the product of a pair of Borel subsets A, S of R, we shall write M0 (A × S) = M(A, S), where M(A, S) = |{λ ∈ A; f (λ) ∈ S}| = |A ∩ f −1 (S)|.

(8)

The mapping M: (A, S) → M(A, S), which assigns an extended real nonnegative number to pairs of Borel subsets of R, has the properties

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(i) A → M(A, S) defines a measure on Borel subsets of R, for fixed S; S → M(A, S) defines a measure on Borel subsets of R, for fixed A; (ii) M(A, R) = |A|, hence in particular the measure A → M(A, S) is absolutely continuous with respect to Lebesgue measure. We shall assume in addition that (iii) the measure S → M(A, S) is absolutely continuous with respect to Lebesgue measure. DEFINITION 1. Any mapping (A, S) → M(A, S), where A, S are Borel subsets of R, and satisfying properties (i)–(iii) above, will be called a value distribution function. It is important to note that not all distribution functions M are of the form M(A, S) = |A ∩ f −1 (S)| for some real-valued measurable function f . (If such f does exist, then property (iii) above is equivalent to the condition that |f −1 (S)| = 0 whenever |S| = 0.) There may be no function f for which M(A, S) describes the distribution of values. Nevertheless, the definition of value distribution function adopted here allows for the more general situation that M may describe a limiting value distribution for a sequence {fn } of functions, in the sense that M(A, S) = limn→∞ |A ∩ fn−1 (S)|. This wider use of terminology is especially appropriate in the description of absolutely continuous spectra. Properties (ii) and (iii) imply that any value distribution function M may be represented in terms of families of measures {µy } (y ∈ R) and S → ω(λ, S) (λ ∈ R) as ω(λ, S) dλ, M(A, S) = µy (A) dy (9) M(A, S) = A

S

with 0 ω(λ, S) 1; in the case M(A, S) = |A ∩ f −1 (S)| we can take ω(·, S) to be the characteristic function of the set f −1 (S). The theory of value distribution for Herglotz functions starts from the observation that, to any Herglotz function F , one may associate in a natural way a value distribution function M defined by the first of Equations (9), where ω is given by ω(λ, S) = lim+ ε→0

1 θ(F (λ + iε), S), π

(10)

with θ(F (λ + iε), S) again being the angle subtended at the point F (λ + iε) of the upper half-plane by the subset S of the real z-axis. DEFINITION 2. We shall refer to the function M, which is defined by M(A, S) = A ω(λ, S) dλ, with ω(λ, S) given by Equation (10), as the associated value distribution function for the Herglotz function F . Since ω, M, and related functions are dependent on F , we shall often write ω(λ, S; F ), M(A, S; F ), and so on.

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For complex argument z ∈ C+ , we define ω(·, S; F ) by ω(z, S; F ) = so that ω(λ, S; F ) = limδ→0+ ω(λ + iδ, S; F ). For almost all λ ∈ R, we have  1, if F+ (λ) is real and F+ (λ) ∈ S;   0, if F+ (λ) is real and F+ (λ) ∈ S; (11) ω(λ, S; F ) = 1   if Im F+ (λ) > 0, θ(F+ (λ), S), π

1 θ(F (z), S), π

where F+ (λ) = limε→0+ F (λ + iε) is the boundary value function for the Herglotz function F . It is known that F+ (λ) exists for almost all λ ∈ R. In the particular case that F+ is almost everywhere real, ω(·, S; F ) is the characteristic function of F+−1 (S), and we have that in that case M(A, S; F ) = |A ∩ F+−1 (S)|. If M(A, S; F ) = S µy (A) dy is the associated value distribution function for a Herglotz function F , as in the second of Equations (9), we can give an explicit construction for the family of measures {µy }. (See Pearson [9]; our construction here differs in minor details from that of Pearson [9], but is essentially the same.) Define first of all a one-parameter family of Herglotz functions {Fy } by Fy (z) = (y − F (z))−1 (y ∈ R). Then the measure µy is just the spectral measure µ = dρ defined through Equation (6) by the Herglotz representation for the function Fy . Although we have seen that both the measures ω(λ, ·) and µy in Equation (9) may be constructed explicitly in the case of a value distribution function defined by a Herglotz function, it may be difficult in practice to evaluate M(A, S; F ) through these integral formulae. This is because little is known a-priori regarding the measures µy in the absence of detailed spectral information, and on the other hand the determination of ω(λ, S; F ) through Equation (10) requires knowledge of the behaviour of the Herglotz function close to the real axis, where precise bounds are not easy to obtain. One can get round some of these difficulties by translating λ by a small increment iδ off the real axis, and making use of a remarkable estimate of the rate of convergence in the limit δ → 0+ . Define first of all a translated Herglotz function F δ by F δ (z) = F (z + iδ), with δ > 0, and set ωδ (λ, S; F ) = ω(λ, S; F δ ) = (1/π )θ(F (λ + iδ), S). An application of the Lebesgue dominated convergence theorem shows that (assuming |A| < ∞) ω(λ, S; F ) dλ. (12) M(A, S; F ) = lim+ ωδ (λ, S; F ) dλ = δ→0

A

A

The following lemma has the surprising consequence that, for any fixed A, this limit is uniform over all Borel sets S, and over all Herglotz functions F . LEMMA 1. Let A ⊂ R be a given Borel set, having finite Lebesgue measure. Then there exists a function EA : R → R (depending on A) such that the inequality δ ω (λ, S; F ) dλ − ω(λ, S; F ) dλ EA (δ) (13) A

A

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holds for any δ > 0, for any Borel set S ⊂ R, and for all Herglotz functions F . Moreover, the function EA can be chosen such that EA (δ) is a nondecreasing function of δ, with limδ→0+ EA (δ) = 0. Proof. A detailed proof of this result is given in Breimesser and Pearson, Geometrical aspects of spectral theory and value distribution for Herglotz functions, in preparation (from now on, we refer to this as BP2000). Remarks. The lemma shows that the convergence of the integral used in (12) to determine the value distribution function M is uniform over S and F . An explicit bound in (13) is obtained by setting 1 θ(λ + iδ, Ac ) dλ, (14) EA (δ) = π A where Ac is the complement of A. With this choice of function EA , (13) is optimal since equality can be attained by taking S = Ac and F (z) = z. In the case that A = (a, b) is an interval, we find δ (b − a)2 2(b − a) −1 δ tan + log 1 + , EA (δ) = π b−a π δ2 which illustrates clearly in this case the convergence to zero of EA (δ) in the limit δ → 0+ . Lemma 1, taken together with Equation (12), enables us to carry out an estimate, distribution function M(A, S; F ), through an evaluato order EA (δ), of the value δ tion of the integral A ω (λ, S; F ) dλ. Here the integrand is given, apart from the multiplicative constant 1/π , by the angle subtended by the set S at points F (λ+iδ), as λ is varied over A. In the following section, we shall see how such estimates of angle subtended may be carried out, using the geometry of the upper half-plane as an example of hyperbolic space. 3. Angle Subtended and Hyperbolic Space The angle subtended by a Borel set S ⊂ R at a point z ∈ C+ is given by 1 dt. θ(z, S) = Im t −z S For z1 , z2 ∈ C+ , θ(z1 , S) will be close to θ(z2 , S) provided z1 is close to z2 , unless z1 or z2 is close to the real axis. One can give a quantitative expression to this fact by defining an estimate of separation γ (·, ·) of points in the upper half-plane, given by γ (z1 , z2 ) = √

|z1 − z2 | √ Im z1 Im z2

(z1 , z2 ∈ C+ ).

(15)

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The estimate of separation γ is related to the hyperbolic distance d between points in C+ by the equation d(z1 , z2 ) . (16) γ (z1 , z2 ) = 2 sinh 2 (For proof of this and related results see BP2000; note that some authors use 12 d instead of d in defining hyperbolic distance. See Anderson [10] for an elementary introduction to the geometry of hyperbolic spaces, and Jost [11] for a more advanced treatment. See also Pommerenke [12].) We shall find γ as an estimate of separation of points in C+ more convenient to use than hyperbolic distance; however it should be noted that although d(·, ·) satisfies the triangle inequality and therefore defines a metric, γ (·, ·) does not. The connection between γ and angle subtended is given by the result (BP2000) that |θ(z1 , S) − θ(z2 , S)| √ , (17) γ (z1 , z2 ) = sup √ θ(z1 , S) θ(z2 , S) S where the supremum is taken over all Borel subsets of R having strictly positive Lebesgue measure. We shall need two further properties of hyperbolic metric, which we state for convenience in terms of γ rather than d, in the following lemma. LEMMA 2. Let M be a Möbius transformation, defined by M(z) =

az + b cz + d

(z ∈ C+ ),

with a, b, c, d real coefficients and ad − bc > 0. Then M leaves γ invariant, in the sense that γ (M(z1 ), M(z2 )) = γ (z1 , z2 ).

(18)

Moreover, if F is any Herglotz function, then F reduces the separation γ , in the sense that γ (F (z1 ), F (z2 )) γ (z1 , z2 ).

(19)

Proof. See BP2000. Möbius transformations with real coefficients and positive discriminant ad − bc leave C+ invariant, and the invariance of the metric d under such transformations may be considered the defining property of hyperbolic metric for C+ . To carry out an analysis of value distribution for solutions of the Schrödinger equation, we need to make use of the link between value distribution and angle subtended. As may be seen from Equation (17), this will involve estimates of the separation γ , as applied to solutions of the Schrödinger equation. The following lemma provides such an estimate, in the case that we consider the logarithmic derivative of solutions of the Schrödinger equation (2), with complex spectral parameter.

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LEMMA 3. Let u(·, z), v(·, z) be solutions of Equation (2), subject to initial conditions (3), with α = 0. Let m(1) be any constant such that Im m(1) 0. Then, for any N > 0, and for all z ∈ C+ , we have the estimate 1 v (N, z) u (N, z) + m (1) v (N, z) , (20) ,− √ γ − v(N, z) u(N, z) + m (1) v(N, z) I (I + 1) where the integral I is defined by N Im( u(x, z)v(x, z)) dx. I (N, z) = (Im z)

(21)

0

Proof. For simplicity of notation, we shall write u, v for the solutions u(·, z), v(·, z) respectively, and WN (f, g) = f (N)g (N) − g(N)f (N) for the Wronskian of two functions f, g at x = N. We make use of the standard identities ([1]) N 1 vv dx = WN (v, v ); 2iIm z 0 N (22) 1 {1 − ReWN (u, v )}. Im( uv) dx = − 2Im z 0 We shall also need the identity |W (u, v )|2 = 1 − W (u, u )W (v, v ),

(23)

where W stands for the Wronskian; Equation (23) may be verified directly by setting 1 = W (u, v)W ( u, v ) on the right-hand side, and writing out the Wronskians explicitly. A straightforward calculation, using Equation (15) and with W (u, v) = 1, shows that at x = N we have u + m (1) v 1 v 2 = W (v,v )W (u+m (1) v,u+m(1) v ) . γ − ,− N N v u + m (1) v − 4 For simplicity of notation we shall write just γ 2 for the left hand side of this equation. We consider first of all the case in which m(1) is real. The denominator on the right hand side is then of the form A + Bm(1) + C(m(1))2 , where A 0, C 0 and B is real. Hence the denominator is bounded below by A − (B 2 /4C), leading to the estimate 4 (24) γ 2 −

2 . WN (u, u )WN (v, v ) + ImWN (u, v ) We can now use Equation (23) on subtracting (ReWN (u, v ))2 from both sides and substituting for (ImWN (u, v ))2 in (24), to obtain γ2 −

4 1 , = 2 2 1 − (ReWN (u, v )) I +I

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where I = − 12 (1 − ReWN (u, v )). Lemma 3 now follows, in the case m(1) real, by using the second identity in Equation (22). In the case that m(1) = Re m(1) + iY,

with Y > 0,

there are additional terms Y 2 (WN (v, v ))2 + 2iY (ReWN (u, v ))WN (v, v ) in the square bracket in Equation (24). These terms may be taken into account by making the replacement u → u − iY v in this denominator. Equation (23) remains valid with u−iY v for u, and we may again obtain the estimate (20), with I replaced in this case by the integral N Y I = (Im z) Im(uv ) dx + WN (v, v ). 2i 0 N Writing WN (v, v ) = 2iIm z 0 vv dx, we see that I I , and we have proved Lemma 3 in the general case. COROLLARY 1. With the same notation as in the statement of Lemma 3, we have v (N, z) u (N, z) + m (1)v (N, z) ,− = 0, (25) lim γ − N→∞ v(N, z) u(N, z) + m (1)v(N, z) where convergence is uniform in m(1) for Im m(1) 0 and uniform in z for z in any fixed compact subset of C+ . Proof. Using (20) and (21), Equation (25) depends on the fact that N Im( uv) dx = +∞. lim N→∞

0

N N We show first of all that the ratio 0 Im( uv) dx/ 0 vv dx is a nondecreasing function of N for N > 0. Using the expression for these integrals in terms of Wronskians, the derivative with respect to N of this ratio is given by {(uv − vu )WN (v, v ) + vv(2 − WN (u, v ) − WN ( u, v))} ,

N 2 4(Im z) 0 vv dx where solutions u, v are evaluated at x = N. Using WN (u, v) = WN ( u, v ) = 1, the numerator in this expression may be simplified to −(v − v )2 = 4(Im v)2 . Hence

2 N Im v(N, z) d 0 Im( uv) dx = N

N , 2 dx 2 dN vv dx (Im z) |v(x, z)| 0 0

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from which it follows that the ratio of integrals cannot decrease with N. It follows N that the integral I (N, z) in Equation (21) is at least as large as const. × 0 vv dx. N Moreover, standard arguments ([1]) imply that limN→∞ 0 vv dx = +∞ in the limit-point case, with divergence uniform over compact subsets of C+ . The final conclusion of the corollary is then a straightforward consequence.

4. Asymptotic Value Distribution for Absolutely Continuous Spectra We are now ready to prove the main result regarding asymptotic value distribution in the limit N → ∞ of the logarithmic derivative q(N, λ) = v (N, λ)/v(N, λ) of the solution v(·, λ) of Equation (1). We have previously referred to this result in Equation (7). As before, mN (z) is the Weyl–Titchmarsh m-function for the Schrödinger operator −d2 /dx 2 + V acting in L2 (N, ∞), with Dirichlet boundary N condition at x = N, and mN + (λ) is the boundary value of m (z) as z approaches λ ∈ R. We let θ(z, S) stand for the angle subtended at z by a subset S of the real axis. Lebesgue measure is denoted by | · |. THEOREM 1. Let A be a Borel subset of an essential support of the absolutely continuous part µa.c. of the spectral measure µ for the Dirichlet Schrödinger operator T = −d2 /dx 2 + V , acting in L2 (0, ∞); we suppose also that A has finite Lebesgue measure. Then we have, for any Borel subset S of R, 1 v (N, λ) N ∈S − θ(m+ (λ), S) dλ = 0. (26) λ ∈ A; lim N→∞ v(N, λ) π A Proof. Let m(z) denote the Weyl–Titchmarsh m-function for the differential operator T . An essential support for µa.c. is the set of all λ ∈ R at which the boundary value m+ (λ) of m(z) exists with strictly positive imaginary part. Hence we may assume without loss of generality that Im m+ (λ) > 0 for all λ ∈ A. Let p be a positive constant. We shall show that the expression within the curly brackets is bounded in absolute value by 6p|A|, for all N sufficiently large. We first define a partition A = A0 ∪ A1 ∪ A2 ∪ · · · ∪ An of the set A into finitely many (n + 1 in number) disjoint subsets A0 , A1 , A2 , . . . , An , having the properties that |A0 | p|A|, with Aj bounded for j 1. To each j 1 we associate a constant m(j ) with Im m(j ) > 0, such that γ (m+ (λ), m(j ) ) p

(all λ ∈ Aj , j 1).

(27)

The general idea behind the construction of such a partition of A, together with associated constants m(j ) (j = 1, 2, . . . , n), is as follows. Points λ ∈ A at which |λ| or |m+ (λ)| are large, or at which Im m+ (λ) is small, are put into the set A0 . This leaves the set A\A0 . The range of m+ (λ), as λ runs over A\A0 , is contained in a compact subset C of C+ , and a partition C = C1 ∪ C2 ∪ · · · ∪ Cn of C into

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disjoint subsets can be found such that, for all j = 1, 2, . . . , n, we have z1 , z2 ∈ Cj ⇒ γ (z1, z2 ) p. (Thus each Cj must have diameter p, as measured by γ .) (j ) = m+ (λj ) for any (fixed) λj ∈ Aj , Finally, take Aj = (A\A0 ) ∩ m−1 + (Cj ) and m and (27) is satisfied. Now let δ be a positive constant such that, for arbitrary Herglotz function F , for any Borel subset S of R, and for all j = 1, 2, . . . , n, we have the bound δ p|Aj | ω (λ, S; F ) dλ − ω(λ, S; F ) dλ (28) Aj

Aj

(cf. Equation (13)). That such δ exists follows from Lemma 1. A-priori, we can define δ separately for each value of j ; thus δ is a function of j . However, by taking the minimum value of δ(j ) as j runs from 1 to n, we may assume δ is independent of j . We shall complete the proof of Equation (26) by showing that, for j 1, (i): π1 Aj θ(mN + (λ), S) dλ is close to the integral 1 u (N, λ) + m(j ) v (N, λ) , S dλ, θ π Aj u(N, λ) + m(j ) v(N, λ) and that (ii): |{λ ∈ Aj ; v (N, λ)/v(N, λ) ∈ S}| is close to the same integral for all N sufficiently large. We deal successively with (i) and (ii). Precise statements of (i) and (ii) are made in (29) and (31) below. Proof of (i). From Equation (5), we have mN + (λ) =

u (N, λ) + m+ (λ)v (N, λ) . u(N, λ) + m+ (λ)v(N, λ)

Hence, for fixed N and λ, the mapping from m+ (λ) to mN + (λ) is a Möbius trans formation with real coefficients and discriminant uv − vu = 1. From Lemma 2, which asserts the invariance of γ (·, ·) under such Möbius transformations, we see that the bound (27) implies that u (N, λ) + m(j ) v (N, λ) N p, for j 1 and λ ∈ Aj . γ m+ (λ), u(N, λ) + m(j ) v(N, λ) We can now use Equation (17) to deduce a corresponding bound for angle subtended, namely (j ) u (N, λ) + m v (N, λ) N θ(m (λ), S) − θ , S πp, + (j ) u(N, λ) + m v(N, λ) and integration with respect to λ over Aj leads to the bound 1 1 u (N, λ) + m(j ) v (N, λ) N , S dλ p|Aj |, θ(m+ (λ), S) dλ − θ π (j ) π Aj u(N, λ) + m v(N, λ) Aj (29)

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which holds for j = 1, 2, . . . , n. Proof of (ii). For j 1, define the subset Aδj of C+ , consisting of all z ∈ C+ of the form z = λ + iδ, for λ ∈ Aj . Thus Aδj is the translation of Aj by distance δ above the real z-axis. Since Aj is bounded, Aδj is contained in a compact subset of C+ . Hence the corollary to Lemma 3 implies that a positive constant N0 can be found such that, for j 1, N N0 and z ∈ Aδj , we have the bound v (N, z) u (N, z) + m (j ) v (N, z) ,− p. (30) γ − v(N, z) u(N, z) + m (j ) v(N, z) As in the case of the constant δ, we may choose N0 to be independent of j . Following a similar argument to that in the proof of (i) above, we may convert the γ estimate into an estimate of angle subtended. Setting z = λ + iδ and integrating with respect to λ over Aj , we have the bound v (N, λ + iδ) 1 θ − , −S dλ− v(N, λ + iδ) Aj π u (N, λ + iδ) + m (j ) v (N, λ + iδ) 1 p|Aj |, θ − , −S dλ − u(N, λ + iδ) + m (j ) v(N, λ + iδ) Aj π valid for j 1 and N N0 . Here −S is the set −S = {λ ∈ R; −λ ∈ S}. Each of the two integrals above is of the form Aj ωδ (λ, −S; F ) dλ for some F ; namely v (N, ·) , v(N, ·) u (N, ·) + m (j ) v (N, ·) . F = F2 = − u(N, ·) + m (j ) v(N, ·)

F = F1 = −

Using (28) in each case, we may compare the difference between these two integrals with the corresponding difference in the limit δ → 0+ . Noting that v (N, λ + iδ) 1 , −S dλ θ − lim δ→0+ π Aj v(N, λ + iδ) v (N, λ) ∈ −S = λ ∈ Aj ; − v(N, λ) v (N, λ) ∈ S , = λ ∈ Aj ; v(N, λ) we arrive at the bound (j ) 1 v (N, λ) u (N, λ) + m v (N, λ) , −S dλ ∈ S − θ − λ ∈ Aj ; (j ) v(N, λ) π Aj u(N, λ) + m v(N, λ) 3p|Aj |.

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We may use the identity θ(−ω, −S) = θ( ω, S) to rewrite this inequality in the form (j ) 1 v (N, λ) u (N, λ) + m v (N, λ) ∈ S − , S dλ θ λ ∈ Aj ; (j ) v(N, λ) π Aj u(N, λ) + m v(N, λ) 3p|Aj |,

(31)

which holds for all j 1 and N N0 , and completes the proof of (ii). Combining inequalities (29) and (31) now yields, for j 1 and N N0 , 1 v (N, λ) ∈ S − θ(mN (λ), S) dλ (32) 4p|Aj |. λ ∈ Aj ; + v(N, λ) π Aj Noting that A0 was chosen such that |A0 | p|A|, we now have, for all N N0 , N λ ∈ A; v (N, λ) ∈ S − 1 θ(m (λ), S) dλ + π v(N, λ) A n 1 v (N, λ) N ∈S − θ(m+ (λ), S) dλ λ ∈ Aj ; v(N, λ) π Aj j =0

2|A0 | + 4p

n

|Aj | 6p|A|.

(33)

j =1

Since p > 0 was arbitrary, we have verified Equation (26), and the theorem follows. Remarks. There is an interesting variant of the argument leading up to the inequality (31) in the proof of the theorem. For each j = 1, 2, . . . , n suppose that a function m(j ) (z) can be found, analytic in the lower half-plane with positive imaginary part, and such that (j )

γ (m+ (λ), m− (λ)) p (j )

(all λ ∈ Aj , j 1), (j )

where m− (λ) is the lower boundary value of m(j ) (z), defined by m− (λ) = limε→0+ m(j ) (λ − iε). Thus (27) corresponds to the special case in which each m(j ) is a constant (j ) function. Then (29) holds with m(j ) replaced by m− (λ) and (30) holds with m (j ) replaced by m (j ) ( z ). (Note that m (j ) ( z ) for Im z > 0 defines an analytic function with negative imaginary part.) We can then carry through the proof of (31) with (j ) again m(j ) replaced by m− (λ), and the conclusions (32) and (33) of the proof follow as before. This more general variant of the argument used in the proof of Theorem 1 is of particular interest in view of the possibility of taking m(j ) (z) to be (j ) the m-function for the zero potential, in which case m+ (λ) is both an upper and a lower boundary value, for λ > 0.

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On examination of the proof of Theorem 1, it will be found that all of the bounds are uniform over Borel subsets S of R. It follows that Equation (26) remains valid if we replace the single Borel set S by an arbitrary family {S N } of Borel sets, parametrised by N. If the family {S N } is chosen in such a way that N 1/π A θ(mN + (λ), S ) dλ converges to a limit as N → ∞ (this can always be done), then the measure |{λ ∈ A; v (N, λ)/v(N, λ) ∈ S N }| will converge to the same limit. We may then characterise the asymptotic value distribution of v (N, λ)/v(N, λ) by considering such families {S N } and their corresponding limiting measures. 5. Applications More detailed applications of the results of this paper will be considered elsewhere. Here, as an illustration of the main ideas, we shall refer briefly to three important classes of potential, namely slowly decaying potentials, potentials unbounded below at infinity, and sparse potentials. (i) Slowly decaying potentials. We consider in particular the class of potentials subject to a pointwise bound |V (x)| const. × x −α in the limit x → ∞, for some constant α > 1/2 (see [5–8]). Such a bound implies, for almost all λ > 0, the existence of a solution f (x, λ) of Equation (1), satisfying WKB-type asymptotics, √ and in particular having the property that limN→∞ f (N, λ)/f (N, λ) = i λ. From the results of Amrein and Pearson in [3], the solutions f (x, λ) may be used to construct families of solutions of the associated Riccati equation which exhibit the clustering property. (For definition of clustering and related ideas, √ see [3].) (λ) = i λ, with the In particular, for almost all λ > 0, we have limN→∞ mN + consequence that, in the conclusion of Theorem 1, Equation (26) may be written, for Borel subsets A of R+ having finite Lebesgue measure, √ 1 (N, λ) v ∈ S = θ(i λ, S) dλ. lim λ ∈ A; N→∞ v(N, λ) π A (ii) Potentials unbounded at infinity. We consider, as an example of this type, the class of potentials Vβ (x) = −βx 2 (β > 0). This class of potentials has been discussed in [3]. Proceeding along similar lines to (i) above, it may be shown that 1/2 N in the limit N → ∞. As a consequence, for all λ ∈ R we have mN + (λ) ∼ iβ N defining the family {S } of Borel sets by S N = NS ≡ {Nλ; λ ∈ S}, we have the asymptotic formula 1 v (N, λ) N ∈S = θ(iβ 1/2 , S) dλ. lim λ ∈ A; N→∞ v(N, λ) π A (Note also that v (N, λ)/(Nv(N, λ)) converges in value distribution as N → ∞.) (iii) Sparse potentials. Applications to sparse potentials will be further discussed in BP2000. There is an extensive literature on this class of potentials; see [13] and other references contained therein. We say that V is a sparse potential if

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there exists a sequence of intervals {(ak , bk )} ≡ {Ik }, having length lk = (bk − ak ), such that lk → ∞ as k → ∞, and such that V (x) ≡ 0 for x ∈ Ik . Given such a sequence of intervals, we have the following lemma (for proof, see BP2000): LEMMA 4. Let V be a sparse potential, and {(ak , bk )} = {Ik } any corresponding sequence of intervals on which V = 0, with length lk → ∞. Then if A and S are Borel subsets of R, with A bounded, it follows that √ 1 1 ak θ(m+ (λ), S) dλ = θ(i λ, S) dλ; lim k→∞ π A π A √ 1 (b , λ) v k lim λ ∈ A; ∈ S = θ(i λ, −S) dλ. k→∞ v(b , λ) π k

(34) (35)

A

Here mak (z) is the Dirichlet m-function for the interval [ak , ∞). If ma+k (λ) is real, then (1/π )θ(ma+k (λ), S) is to be interpreted as the characteristic function of the set {λ; ma+k (λ) ∈ S}. Equations (34) and (35) play a key role in understanding asymptotic value distribution for Schrödinger operators with sparse potentials, and its implications for spectral theory. Since we could redefine the sequence of intervals {Ik }, replacing each interval (ak , bk ) in the sequence by a pair of intervals (ak , ck ), (ck , bk ), with ck = (ak + bk )/2, Equations (34) and (35) remain valid with ak , respectively bk , replaced by ck on the left hand side. If A is a subset of R− and S is a subset of R+ for which the right hand side of (35) is nonzero, this implies that v (ck , λ)/v(ck , λ) and mc+k (λ) should have different asymptotic value distribution, for λ ∈ A. On the other hand, if in addition A is a subset of an essential support of the spectral measure µa.c. , then according to Theorem 1, the asymptotic value distributions should be the same. Hence, in fact there can be no absolutely continuous measure for λ < 0. Equations (34) and (35), together with Theorem 1, may also be used to prove, for various classes of sparse potentials, that the spectral measure for λ > 0 is purely singular. As a simple example of this argument, consider the potential V (x) = ∞ δ(x − xn ), with (xn+1 − xn ) → +∞ as n → ∞. (The theory presented in this n=1 paper can easily be extended to include such distributional potentials.) We can then , xk+1 ), and let√A ⊂ R+ be a subdefine a sequence of intervals {Ik }, with Ik = (xk√ set of an essential support of µa.c. . Noting that θ(i λ, S) = θ(i λ, −S) for λ > 0, Equations (34) and (35), together with Theorem 1, imply that v (xk , λ)/v(xk , λ) has the same asymptotic value distribution for λ ∈ A, in the limit k → ∞, whether xk is taken just to the right, or just to the left of the singularity of the potential. However, at the δ singularity x = xk , the function v (xk , λ)/v(xk , λ) has discontinuity 1. Hence, the two asymptotic distributions cannot agree, and we may deduce that in that case there is no absolutely continuous measure for λ > 0. More generally, one has the qualitative understanding that absolutely continuous spectrum is only allowed if the potential, in the regions where it is nonzero, fails to disturb the asymptotics of v (x, λ)/v(x, λ), where x is an endpoint of one of the intervals Ik .

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

7.

8. 9. 10. 11. 12. 13.

Coddington, E. A. and Levinson, N.: Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. Akhiezer, N. I. and Glazman, I. M.: Theory of Linear Operators in Hilbert Space, Pitman, London, 1981. Amrein, W. O. and Pearson, D. B.: Stability criteria for the Weyl m-function, Math. Phys. Anal. Geom. 1 (1998), 193–221. Eastham, M. S. P.: The asymptotic form of the spectral function in Sturm-Liouville problems with a large potential like −x c (c 2), Proc. Roy. Soc. Edinburgh A 128 (1998), 37–45. Kiselev, A.: Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials, Comm. Math. Phys. 179 (1996), 377–400. Christ, M., Kiselev, A. and Remling, C.: The absolutely continuous spectrum of onedimensional Schrödinger operators with decaying potentials, Math. Res. Lett. 4(5) (1997), 719–723. Christ, M. and Kiselev, A.: Absolutely continuous spectrum of one-dimensional Schrödinger operators with slowly decreasing potentials: some optimal results, J. Amer. Math. Soc. 11 (1998), 771–797. Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials, Comm. Math. Phys. 193(1) (1998), 151–170. Pearson, D. B.: Value distribution and spectral theory, Proc. London Math. Soc. 68(3) (1994), 127–144. Anderson, J. W.: Hyperbolic Geometry, Springer, London, 1999. Jost, J.: Compact Riemann Surfaces, Springer, Berlin, 1997. Pommerenke, C.: Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992. Simon, B. and Stolz, G.: Operators with singular continuous spectrum: sparse potentials, Proc. Amer. Math. Soc. 124(7) (1996), 2073–2080.

Mathematical Physics, Analysis and Geometry 3: 405–406, 2000.

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Contents of Volume 3

Volume 3

No. 1

2000

A. KHORUNZHY and G. J. RODGERS / Asymptotic Distribution of Eigenvalues of Weakly Dilute Wishart Matrices

1–31

A. PANKOV and K. PFLÜGER / On Ground-Traveling Waves for the Generalized Kadomtsev–Petviashvili Equations

33–47

BERNARD DECONINCK and HARVEY SEGUR / Pole Dynamics for Elliptic Solutions of the Korteweg–deVries Equation

49–74

YU. AMINOV and A. SYM / On Bianchi and Bäcklund Transformations of Two-Dimensional Surfaces in E 4

75–89

AMÉDÉE DEBIARD and BERNARD GAVEAU / The Ground State of Certain Coulomb Systems and Feynman–Kac Exponentials

91–100

Volume 3

No. 2

2000

M. M. KIPNIS / Periodic Ground State Configurations in a OneDimensional Hubbard Model of Statistical Mechanics 101–115 P. AGRANOVICH / Polynomial Asymptotic Representation of Subharmonic Functions in a Half-Plane 117–138 G. FELDER, Y. MARKOV, V. TARASOV and A. VARCHENKO / Differential Equations Compatible with KZ Equations 139–177 DMITRY SHEPELSKY / A Riemann–Hilbert Problem for Propagation of Electromagnetic Waves in an Inhomogeneous, Dispersive Waveguide 179–193

406 Volume 3

CONTENTS OF VOLUME 3

No. 3

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GIANLUCA GEMELLI / Second-Order Covariant Tensor Decomposition in Curved Spacetime 195–216 ZIED AMMARI / Asymptotic Completeness for a Renormalized Nonrelativistic Hamiltonian in Quantum Field Theory: The Nelson Model 217–285 I. K. KHANNA, V. SRINIVASA BHAGAVAN and M. N. SINGH / Generating Relations of the Hypergeometric Functions by the Lie Group-Theoretic Method 287–303 Volume 3

No. 4

2000

SPYROS PNEVMATIKOS and DIMITRIS PLIAKIS / Gauge Fields with Generic Singularities 305–321 JEAN MEINGUET / An Electrostatics Approach to the Determination of Extremal Measures 323–337 ALEXANDRE STOJANOVIC / Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential 339–373 HAROLD WIDOM / On the Solution of a Painlevé III Equation

375–384

S. V. BREIMESSER and D. B. PEARSON / Asymptotic Value Distribution for Solutions of the Schrödinger Equation 385–403