Math Phys Anal Geom (2008) 11:1–9 DOI 10.1007/s11040-008-9037-8
On the Flag Curvature of Invariant Randers Metrics Hamid Reza Salimi Moghaddam
Received: 10 December 2007 / Accepted: 6 February 2008 / Published online: 21 March 2008 © Springer Science + Business Media B.V. 2008
Abstract In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field. Keywords Invariant metric · Flag curvature · Randers space · Homogeneous space · Lie group Mathematics Subject Classifications (2000) 22E60 · 53C60 · 53C30
1 Introduction The geometry of invariant structures on homogeneous spaces is one of the interesting subjects in differential geometry. Invariant metrics are of these invariant structures. K. Nomizu studied many interesting properties of invariant Riemannian metrics and the existence and properties of invariant affine connections on reductive homogeneous spaces (see [14, 16]). Also some
H. R. S. Moghaddam (B) Department of Mathematics, Shahrood University of Technology, Shahrood, Iran e-mail:
[email protected]
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curvature properties of invariant Riemannian metrics on Lie groups has studied by J. Milnor [15]. So it is important to study invariant Finsler metrics which are a generalization of invariant Riemannian metrics. S. Deng and Z. Hou studied invariant Finsler metrics on reductive homogeneous spaces and gave an algebraic description of these metrics [12, 13]. Also, in [10, 11], we have studied the existence of invariant Finsler metrics on quotient groups and the flag curvature of invariant Randers metrics on naturally reductive homogeneous spaces. In this paper we study the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups. Flag curvature, which is a generalization of the concept of sectional curvature in Riemannian geometry, is one of the fundamental quantities which associate with a Finsler space. In general, the computation of the flag curvature of Finsler metrics is very difficult, therefore it is important to find an explicit and applicable formula for the flag curvature. One of important Finsler metrics which have found many applications in physics are Randers metrics (see [2, 3]). In this article, by using Püttmann’s formula [17], we give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and Lie groups. Then the Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups are studied. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field.
2 Flag Curvature of Invariant Randers Metrics on Homogeneous Spaces The aim of this section is to give an explicit formula for the flag curvature of invariant Randers metrics of Berwald type, arising from invariant Riemannian metrics, on homogeneous spaces. For this purpose we need the Püttmann’s formula for the curvature tensor of invariant Riemannian metrics on homogeneous spaces (see [17]). Let G be a compact Lie group, H a closed subgroup, and g0 a bi-invariant Riemannian metric on G. Assume that g and h are the Lie algebras of G and H respectively. The tangent space of the homogeneous space G/H is given by the orthogonal compliment m of h in g with respect to g0 . Each invariant metric g on G/H is determined by its restriction to m. The arising Ad H -invariant inner product from g on m can extend to an Ad H -invariant inner product on g by taking g0 for the components in h. In this way the invariant metric g on G/H determines a unique left invariant metric on G that we also denote by g. The values of g0 and g at the identity are inner products on g which we denote as < ., . >0 and < ., . >. The inner product < ., . > determines a positive definite endomorphism φ of g such that < X, Y >=< φ X, Y >0 for all X, Y ∈ g. Now we give the following lemma which was proved by T. Püttmann (see [17]).
On the flag curvature of invariant Randers metrics
3
Lemma 1 The curvature tensor of the invariant metric < ., . > on the compact homogeneous space G/H is given by
= 1/2 < B− (X, Y), [Z , W] >0 + < [X, Y], B− (Z , W) >0 + + 1/4 < [X, W], [Y, Z ]m > − < [X, Z ], [Y, W]m > − − 2 < [X, Y], [Z , W]m > + + < B+ (X, W), φ −1 B+ (Y, Z ) >0 − (1) − < B+ (X, Z ), φ −1 B+ (Y, W) >0 , where the symmetric resp. skew symmetric bilinear maps B+ and B− are defined by B+ (X, Y) = 1/2 [X, φY] + [Y, φ X] , B− (X, Y) = 1/2 [φ X, Y] + [X, φY] , and [., .]m is the projection of [., .] to m. ˜ be an invariant vector field on the homogeneous space G/H such that Let X ˜ = g( X, ˜ X) ˜ < 1. A case happen when G/H is reductive with g = m ⊕ h X ˜ is the corresponding left invariant vector field to a vector X ∈ m such and X that < X, X >< 1 and Ad(h)X = X for all h ∈ H (see [13] and [10]). By using ˜ we can construct an invariant Randers metric on the homogeneous space X G/H in the following way: F(xH, Y) =
˜ x, Y g(xH)(Y, Y) + g(xH) X
∀Y ∈ TxH (G/H).
(2)
Now we give an explicit formula for the flag curvature of these invariant Randers metrics. Theorem 1 Let G be a compact Lie group, H a closed subgroup, g0 a biinvariant metric on G, and g and h the Lie algebras of G and H respectively. Also let g be any invariant Riemannian metric on the homogeneous space ˜ is an G/H such that < Y, Z >=< φY, Z >0 for all Y, Z ∈ g. Assume that X ˜ X) ˜ < invariant vector field on G/H which is parallel with respect to g and g( X, ˜ ˜ H = X. Suppose that F is the Randers metric arising from g and X, 1 and X and (P, Y) is a flag in T H (G/H) such that {Y, U} is an orthonormal basis of P with respect to < ., . >. Then the flag curvature of the flag (P, Y) in T H (G/H) is given by K(P, Y) =
A , (1+ < X, Y >)2 (1− < X, Y >)
(3)
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where A = α. < X, U > +γ (1+ < X, Y), and for A we have: α = 1/4 < [φU, Y] + [U, φY], [Y, X] >0 + < [U, Y], [φY, X] + [Y, φ X] >0 + + 3/4 < [Y, U], [Y, X]m > +1/2 < [U, φ X] + [X, φU], φ −1 [Y, φY] >0 − (4) − 1/4 < [U, φY] + [Y, φU], φ −1 [Y, φ X] + [X, φY] >0 , and γ = 1/2 < [φU, Y] + [U, φY], [Y, X] >0 + + 3/4 < [Y, U], [Y, U]m > + < [U, φU], φ −1 ([Y, φY]) >0 − − 1/4 < [U, φY] + [Y, φU], φ −1 [Y, φU] + [U, φY] >0 .
(5)
˜ is parallel with respect to g, therefore F is of Berwald type and the Proof X Chern connection of F and the Riemannian connection of g coincide (see [6], page 305.), so we have R F (U, V)W = Rg (U, V)W, where R F and Rg are the curvature tensors of F and g, respectively. Let R := Rg = R F be the curvature tensor of F (or g). Also for the flag curvature we have [18]: K(P, Y) =
gY (R(U, Y)Y, U) , 2 gY (Y, Y).gY (U, U) − gY (Y, U)
(6)
∂ where gY (U, V) = 1/2 ∂s∂t (F 2 (Y + sU + tV))|s=t=0 . By a direct computation for F we get 2
gY (U, V) = g(U, V) + g(X, U).g(X, V) − ×
1 g(Y, Y)
g(X, Y).g(Y, V).g(Y, U) × g(Y, Y)3/2
g(X, U).g(Y, V) + g(X, Y).g(U, V) +
+ g(X, V).g(Y, U) .
(7)
Since {Y, U} is an orthonormal basis of P with respect to < ., . >, by using the formula (7) we have: gY (Y, Y).gY (U, U) − gY (Y, U) = (1+ < X, Y >)2 (1− < X, Y >).
(8)
Also we have: gY (R(U, Y)Y, U) = < R(U, Y)Y, U > + < X, R(U, Y)Y > . < X, U > + + < X, Y > . < R(U, Y)Y, U > + + < X, U > . < Y, R(U, Y)Y >,
(9)
now let α =< X, R(U, Y)Y >, θ =< Y, R(U, Y)Y > and γ =< R(U, Y)Y, U >.
On the flag curvature of invariant Randers metrics
5
By using Püttmann’s formula (see Lemma 1) and some computations we have: α = 1/4 < [φU, Y] + [U, φY], [Y, X] >0 + < [U, Y], [φY, X] + [Y, φ X] >0 + + 3/4 < [Y, U], [Y, X]m > +1/2 < [U, φ X] + [X, φU], φ −1 ([Y, φY]) >0 − − 1/4 < [U, φY] + [Y, φU], φ −1 ([Y, φ X] + [X, φY]) >0 ,
θ = 0,
(10)
(11)
and γ = 1/2 < [φU, Y] + [U, φY], [Y, U] >0 +3/4 < [Y, U], [Y, U]m > + + < [U, φU], φ −1 ([Y, φY]) >0 − −1/4 < [U, φY] + [Y, φU], φ −1 ([Y, φU] + [U, φY]) >0 .
(12)
Substituting (7), (8), (9), (10), (11) and (12) in the (6) completes the proof. Remark In the previous theorem, If we let H = {e} and m = g then we can obtain a formula for the flag curvature of the left invariant Randers metrics of Berwald types arising from a left invariant Riemannian metric g and a left ˜ on Lie group G. invariant vector field X If the invariant Randers metric arises from a bi-invariant Riemannian metric on a Lie group then we can obtain a simpler formula for the flag curvature, we give this formula in the following theorem. Theorem 2 Suppose that g0 is a bi-invariant Riemannian metric on a Lie group ˜ X) ˜ < 1 and X ˜ is ˜ is a left invariant vector field on G such that g0 ( X, G and X parallel with respect to g0 . Then we can define a left invariant Randers metric F as follows: ˜ x, Y . F(x, Y) = g0 (x)(Y, Y) + g0 (x) X Assume that (P, Y) is a flag in Te G such that {Y, U} is an orthonormal basis of P with respect to < ., . >0 . Then the flag curvature of the flag (P, Y) in Te G is given by K(P, Y) =
< [Y, [U,Y]], X>0 . < X,U >0 + < [Y, [U,Y]], U >0 (1+ < X, Y >0 ) . 4(1+ < X, Y >0 )2 (1− < X, Y >0 )
˜ is parallel with respect to g0 the curvature tensors of g0 and Proof Since X F coincide. On the other hand for g0 we have R(X, Y)Z = 1/4[Z , [X, Y]], therefore by substituting R in (6) and using (7) the proof is completed.
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3 Invariant Randers Metrics on Lie Groups In this section we study the left invariant Randers metrics on Lie groups and, in some special cases, find some results about the dimension of Lie groups which can admit invariant Randers metrics. These conclusions are obtained by using Yasuda–Shimada theorem. The Yasuda–Shimada theorem is one of important theorems which characterize the Randers spaces. In the year 2001, Shen’s examples of Randers manifolds with constant flag curvature motivated Bao and Robles to determine necessary and sufficient conditions for a Randers manifold to have constant flag curvature. Shen’s examples showed that the original version of Yasuda–Shimada theorem (1977) is wrong. Then Bao and Robles corrected the Yasuda–Shimada theorem (1977) and gave the correct version of this theorem, Yasuda–Shimada theorem (2001) (see [5]; for a comprehensive history of Yasuda–Shimada theorem see [4]). Suppose that M is an n-dimensional manifold endowed with a Riemannian metric g = (gij(x)) and a nowhere zero 1-form b = (b i (x)) such that b = b i (x)b j(x)gij(x) < 1. We can define a Randers metric on M as follows (13) F(x, Y) = gij(x)Y i Y j + b i (x)Y i . Next, we consider the 1-form β = b i (b j|i − b i| j)dxi , where the covariant derivative is taken with respect to Levi–Civita connection to M. Now we give the Yasuda–Shimada theorem from [4]. Theorem 3 (Yasuda–Shimada; see [4]) Let F be a strongly convex nonRiemannian Randers metric on a smooth manifold M of dimension n 2. Let gij be the underlying Riemannian metric and b i the drift 1-form. Then: (+)
F satisfies β = 0 and has constant positive flag curvature K if and only if: – –
b is a non-parallel Killing field of g with constant length; the Riemann curvature tensor of g is given by Rhijk = K 1 − b 2 ghk gij − ghj gik + + K gijb h b k − gik b h b j − − K ghjb i b k − ghk b i b j − − b i| jb h|k + b i|k b h| j + 2b h|i b j|k
(0) (–)
F satisfies β = 0 and has zero flag curvature ⇔ it is locally Minkowskian. F satisfies β = 0 and has constant negative flag curvature if and only if: – – –
b is a closed 1-form; b i|k = 1/2σ (gik − b i b k ), with σ 2 = −16K; g has constant negative sectional curvature 4K, that is, Rhijk = 4K(gij ghk − gik ghj).
On the flag curvature of invariant Randers metrics
7
Since any Randers manifold of dimension n = 1 is a Riemannian manifold from now on we consider n > 1. An immediate conclusion of Yasuda–Shimada theorem is the following corollary. Corollary 1 There is no non-Riemannian Randers metric of Berwald type with β = 0 and constant positive flag curvature. Now by using the results of [8] we obtain the following conclusions. Theorem 4 Let F n = (M, F, gij, b i ) be an n-dimensional parallelizable Randers manifold of constant positive flag curvature with β = 0 on M and complete Riemannian metric g = (gij). Then the dimension of M must be 3 or 7. Proof By using Theorem 2.2 of [8] M is diffeomorphic with a sphere of dimension n = 2k + 1. But a sphere Sm is parallelizable if and only if m = 1, 3 or 7 (see [1]). Therefore n = 3 or 7. A family of Randers metrics of constant positive flag curvature on Lie group S3 was studied by D. Bao and Z. Shen (see [7]). They produced, for each K > 1, an explicit example of a compact boundaryless (non-Riemannian) Randers spaces that has constant positive flag curvature K, and which is not projectively flat, on Lie group S3 . In the following we give some results about the dimension of Lie groups which can admit Randers metrics of constant positive flag curvature. These results show that the dimension 3 is important. Corollary 2 There is no Randers Lie group of constant positive flag curvature with β = 0, complete Riemannian metric g = (gij) and n = 3. Proof Any Lie group is parallelizable, so by attention to Theorem 4 and the condition n = 3, n must be 7. Since G is diffeomorphic to S7 and S7 can not admit any Lie group structure, hence the proof is completed. Similar to the [15] for the sectional curvature of the left invariant Riemannian metrics on Lie groups, we compute the flag curvature of the left invariant Randers metrics on Lie groups in the following theorem. Theorem 5 Let G be a compact Lie group with Lie algebra g, g0 a bi-invariant Riemannian metric on G, and g any left invariant Riemannian metric on G such that < X, Y >=< φ X, Y >0 for a positive definite endomorphism φ: g −→ g. Assume that X ∈ g is a vector such that < X, X >< 1 and F is the Randers ˜ and g as follows: metric arising from X ˜ x, Y , F(x, Y) = g(x)(Y, Y) + g(x) X ˜ is the left invariant vector field corresponding to X, and we have where X ˜ is parallel with respect to g. Let {e1 , · · · , en } ⊂ g be a g-orthonormal assumed X
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basis for g. Then the flag curvature of F for the flag P = span{ei , e j}(i = j) at the point (e, ei ), where e is the unit element of G, is given by the following formula: K(P = span{ei , e j}, ei ) =
X j. < R(e j, ei )ei , X > +(1 + Xi ). < R(e j, ei )ei , e j > , (1 + Xi )2 (1 − Xi )
where X = X k ek ,
< R(e j, ei )ei , X > = − 1/4 < [φe j, ei ], [ei , X] >0 + < [e j, φei ], [ei , X] >0
+ < [e j, ei ], [φei , X] >0 + < [e j, ei ], [ei , φ X] >0 +
+ 3/4 < [e j, ei ], [ei , X] > −
− 1/2 < [e j, φ X] + [X, φe j], φ −1 [ei , φei ] >0 + + 1/4 < [e j, φei ] + [ei , φe j], φ −1 [ei , φ X] + [X, φei ] >0
and
< R(e j, ei )ei , e j > = − 1/2 < [φe j, ei ], [ei , e j] >0 + < [e j, φei ], [ei , e j] >0 + + 3/4 < [e j, ei ], [ei , e j] > − < [e j, φe j], φ −1 [ei , φei ] >0 + + 1/4 < [e j, φei ] + [ei , φe j], φ −1 [ei , φe j] + [e j, φei ] >0 .
Proof By using Theorem 1, the proof is clear.
Now we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field. Theorem 6 There is no left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field on connected Lie groups with a perfect Lie algebra, that is, a Lie algebra g for which the equation [g, g] = g holds. Proof If a left invariant vector field X is parallel with respect to a left invariant Riemannian metric g then, by using Lemma 4.3 of [9], g(X, [g, g]) = 0. Since g is perfect therefore X must be zero. Corollary 3 There is not any left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field on semisimple connected Lie groups. Corollary 4 If a Lie group G admits a left invariant non-Riemannian Randers metric of Berwald type F arising from a left invariant Riemannian metric g and a left invariant vector field X then for sectional curvature of the Riemannian metric g we have K(X, u) 0 for all u, where equality holds if and only if u is orthogonal to the image [X, g].
On the flag curvature of invariant Randers metrics
9
Proof Since F is of Berwald type, X is parallel with respect to g. By using Lemma 4.3 of [9], ad(X) is skew-adjoint, therefore by Lemma 1.2 of [15] we have K(X, u) 0.
References 1. Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72(2), 20–104 (1960) 2. Antonelli, P.L., Ingarden, R.S., Matsumoto, M.: The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology. Kluwer, Dordrecht (1993) 3. Asanov, B.S.: Finsler Geometry, Relativity and Gauge Theories. Kluwer, Dordrecht (1985) 4. Bao, D.: Randers space forms. Period. Math. Hungar. 48(1), 3–15 (2004) 5. Bao, D., Robles, C.: On randers spaces of constant flag curvature. Rep. Math. Phys. 51(1), 9–42 (2003) 6. Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann–Finsler Geometry. SpringerVerlag, Berlin (2000) 7. Bao, D., Shen, Z.: Finsler metrics of constant positive curvature on lie group S3 . J. Lond. Math. Soc. 66(2), 453–467 (2002) 8. Bejancu, A., Farran H.R.: Randers manifolds of positive constant curvature. Internat. J. Math. Math. Sci. 2003(18), 1155–1165 (2003) 9. Brown, N., Finck, R., Spencer, M., Tapp K., Wu, Z.; Invariant metrics with nonnegative curvature on compact lie groups. Canad. Math. Bull. 50(1), 24–34 (2007) 10. Esrafilian, E., Salimi Moghaddam, H.R.: Flag curvature of invariant Randers metrics on homogeneous manifolds. J. Phys. A: Math. Gen. 39, 3319–3324 (2006) 11. Esrafilian, E., Salimi Moghaddam, H.R.: Induced invariant Finsler metrics on quotient groups. Balkan J. Geom. Appl. 11(1), 73–79 (2006) 12. Deng, S., Hou, Z.: Invariant Finsler metrics on homogeneous manifolds. J. Phys. A: Math. Gen. 37, 8245–8253 (2004) 13. Deng, S., Hou, Z.: Invariant Randers metrics on homogeneous Riemannian manifolds. J. Phys. A: Math. Gen. 37, 4353–4360 (2004) 14. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry Vol. 2. Interscience Publishers, John Wiley & Sons (1969) 15. Milnor, J.: Curvatures of left invariant metrics on lie groups. Adv. Math. 21, 293–329 (1976) 16. Nomizu, K.: Invariant affine connections on homogeneous spaces. Amer. J. Math. 76, 33–65 (1954) 17. Püttmann, T.: Optimal pinching constants of odd dimensional homogeneous spaces. Invent. Math. 138, 631–684 (1999) 18. Shen, Z.: Lectures on Finsler Geometry. World Scientific (2001)
Math Phys Anal Geom (2008) 11:11–51 DOI 10.1007/s11040-008-9038-7
Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies M. Cafasso
Received: 26 November 2007 / Accepted: 19 February 2008 / Published online: 1 April 2008 © Springer Science + Business Media B.V. 2008
Abstract We propose a method for computing any Gelfand-Dickey τ function defined on the Segal-Wilson Grassmannian manifold as the limit of block Toeplitz determinants associated to a certain class of symbols W (t; z). Also truncated block Toeplitz determinants associated to the same symbols are shown to be τ functions for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from the point of view of integrable systems and block Toeplitz operator theory. Examples of applications to algebro-geometric solutions are given. Keywords Block Toeplitz determinants · Integrable hierarchies · Grassmannians · KP · Riemann-Hilbert problems Mathematics Subject Classifications (2000) 37K10 · 47B35 1 Introduction This paper deals with the applications of block Toeplitz determinants and their asymptotics to the study of integrable hierarchies. Asymptotics of block Toeplitz determinants and their applications to physics is a developing field of research; in recent years it has been shown how to compute some physically relevant quantities (e.g. correlation functions) studying asymptotics of some block Toeplitz determinants (see [27–29]). In particular in [27] and [28] the authors, for the first time, showed effective computations for the case of block Toeplitz determinants with symbols that do not have half truncated Fourier
M. Cafasso (B) SISSA-International School for Advanced Studies, Via Beirut 2/4, 34014 Grignano, Italy e-mail: [email protected]
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series. This is of particular interest for us as, with our approach, we will be able to do the same for certain block Toeplitz determinants associated to algebro-geometric solutions of Gelfand Dickey hierarchies. Let us mention some theoretical results about (block) Toeplitz determinants we will use in this paper. Given a function γ (z) on the circle we denote T N (γ ) the Toeplitz matrix with symbol γ given by ⎛ (0) ⎞ γ . . . . . . γ (−N) ⎜ ⎟ ⎜ (1) ⎟ (−N+1) ⎟ ⎜ γ ... ... γ ⎜ ⎟ ⎟ T N (γ ) := ⎜ ⎜ ⎟ ⎜ ... ... ... ... ⎟ ⎜ ⎟ ⎝ ⎠ (N) (0) ... ... γ γ where γ (k) are the Fourier coefficients γ (z) = k γ (k) zk . We use the term block Toeplitz for the case of matrix-valued symbol γ (z). In that case the entries γ (i− j ) of the above matrix are n × n matrices themselves. We denote D N (γ ) := det T N (γ ) and we use the notation T(γ ) for the N × N matrix obtained letting N go to infinity. The main goal of the theory of Toeplitz determinants is to compute D N (γ ) as N goes to infinity and find expressions for D N (γ ) as well as for its limit in terms of Fredholm determinants. First result is due to Szegö that in 1952 gave a formula for asymptotics of D N (γ ) in the scalar case [5]. This result has been generalized by H. Widom in the 70’s ([6, 7] and [8]) for the matrix case; namely he proved that under suitable analytical assumptions it exists the limit D∞ (γ ) := lim
N→∞
D N (γ ) = det T(γ )T γ −1 N G(γ )
where G(γ ) is a normalizing constant and the operator T(γ )T(γ −1 ) is such that its determinant is well defined as a Fredholm determinant (see Section 3 for the precise statement). Once the asymptotics had been computed the next quite natural question was to find an expressions relating directly D N (γ ), and not just its asymptotics, to certain Fredholm determinants. The problem was solved many years later by Borodin and Okounkov in [9] for the scalar case and generalized, in the same year, for matrix case by E. Basor and H.Widom in [10]. For matrix valued case Borodin-Okounkov formula reads D N (γ ) = D∞ (γ ) det(I − Kγ ,N ) (here we assume G(γ ) = 1). The operator (I − Kγ ,N ) can be written explicitly in coordinates knowing certain Riemann-Hilbert factorizations of γ . Its Fredholm determinant is well defined (see Section 3 for details). Now many proofs of Borodin-Okounkov formula are known (for instance [11] contains another proof of the same formula, see also the earlier paper [12]).
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
13
In this paper we apply block-Toeplitz determinants to the computation of τ function of an (almost) arbitrary solution of Gelfand-Dickey hierarchy ∂ L j = Ln , L . + ∂t j (L differential operator of order n, j = nk). More precisely to a given point W = W (z)H+(n) in the big cell of Segal-Wilson vector-valued Grassmannian we associate a n × n matrix-valued symbol W (t; z) obtained by deforming W (z) (see formula (12)). In this way we define a sequence of N-truncated block Toeplitz determinants {τW,N (t)} N>0 which are shown to be solutions of certain rational reductions of KP; this is our First result: Every symbol W (t; z) defines through its truncated determinants a sequence {τW,N (t)} N>0 of solutions for KP such that τW,N (t) ∈ cKP1,nN ∩ cKPn,n
∀N > 0.
This result is stated in Theorem 5. Here we used the notation from [20]; given a τ function for KP with corresponding Lax pseudodifferential operator L we say that τ ∈ cKPm,n iff Lm can be written as the ratio of two differential operators of order m + n and n respectively. This sequence admits a stable limit which is shown to be equal to the Gelfand-Dickey τ function τW (t) associated to W; this quantity can be computed using Szegö-Widom’s theorem. This will give us the remarkable identity τW (t˜ ) = det PW (t˜;z) (1) where PW(t˜;z) is the Fredholm operator appearing in Szegö-Widom’s theorem (here we put t˜ instead of t to remember that, when working with W ∈ Gr(n) , times tnj multiple of n must be set to 0). Next step is the study of RiemannHilbert (also called Wiener-Hopf) factorization of symbol W (t; z) given by W (t; z) = T− (t; z)T+ (t; z)
(2)
with T− and T+ analytical in z outside and inside S1 respectively and normalized as T− (∞) = I. Here we assume that the symbol can be extended to an analytic function in a neighborhood of S1 . Using Plemelj’s results [13] we show that T− (t˜; z) must satisfy the integral equation T PW T (t˜; z) = I (t˜;z) −
(3)
and we write a solution of (2) in terms of wave function ψW (t˜; z) corresponding to W.
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M. Cafasso
In this way we arrive to our second result: Second result: Take W ∈ Gr(n) in the big cell and its corresponding τ function τW (t˜). τW (t˜) is equal to the Fredholm determinant of the homogeneous integral equation associated to (3) which is related to Riemann-Hilbert problem (2). The solution of this Riemann-Hilbert problem is unique for every value of parameters t˜ that makes τW (t˜) = 0 and can be computed by means of related wave function ψW (t˜; z). Theorem 5 explains how τW (t˜ ) can be written as a Fredholm determinant while the relation with the corresponding Riemann-Hilbert problem is described in Proposition 14. Explicit factorization of the symbols is written in Theorem 7. At the end of the paper we consider a particular class of symbols W (t; z) corresponding to algebro-geometric solutions of GelfandDickey hierarchies. We formulate an alternative Riemann-Hilbert problem equivalent to (2) and explain how to solve it using θ-functions. In this way we give concrete formulas for a wide class of symbols that do not have half truncated Fourier series. We think this is quite remarkable since concrete results for non half truncated symbols were available, till now, just for the concrete cases presented in [27] and [28]. The paper is organized as follows: –
–
–
–
–
Second section states some results about Segal-Wilson Grassmannian and related loop groups we will need in the sequel; proofs can be found in [1] and [2]. Third section states Szegö-Widom’s theorem and related results obtained by Widom in [6, 7] and [8] and the Borodin-Okounkov formula for block Toeplitz determinant [10]. In the fourth section we introduce and study the sequence of truncated determinants {τW,N (t)} N>0 and its stable limit τW (t). We want to remark that the property of stability was stated for the first time in [15] (see also [16]) and our sequence is actually a subsequence of the stabilizing chain studied in [17]; nevertheless, to our best knowledge, this is the first time that block Toeplitz determinants enter the game and also the observation that τW,N ∈ cKPn,n seems to be something new. The main results of this section are stated in Theorem 5. Fifth section is devoted to establishing the connection between integral equations formulated by Plemelj in [13] and Fredholm operator appearing in Szegö-Widom’s theorem. In the sixth section we show how to write Riemann-Hilbert factorization of W (t˜; z) in terms of wave function ψW (t˜; z); this result is stated in Theorem 7. Of course relation between Gelfand-Dickey hierarchy and factorization problem is something known; our exposition here is closely related to [14]. Moreover, knowing Riemann-Hilbert factorization of W (t˜; z), we can apply Borodin-Okounkov formula to give an expression of any τW,N (t˜ ) as Fredholm determinant and a recursion relation to go from τW,N (t˜ ) to τW,N+1 (t˜ ).
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
–
15
Last section gives explicit formulas for symbols and τ functions associated to algebro-geometric rank one solutions of Gelfand-Dickey hierarchies. Also we formulate an alternative Riemann-Hilbert problem equivalent to (2) in analogy with what has been done in [27] and [28]. We explain how to solve it using θ -functions.
2 Segal-Wilson Grassmannian and Related Loop Groups Here we recall some definitions and results from [1] and [2] that will be useful in the sequel. Definition 1 Let H (n) := L2 (S1 , Cn ) be the space of complex vector-valued square-integrable functions. We choose a orthonormal basis given by
eα,k := (0, . . . , zk , . . . , 0)T : α = 1 . . . n, k ∈ Z
and the polarization H (n) = H+(n) ⊕ H−(n) where H+(n) and H−(n) are the closed subspaces spanned by elements {eα,k } with k 0 and k < 0 respectively. In the sequel in order to avoid cumbersome notations we will write H instead of H (1) . Definition 2 ([2]) The Grassmannian Gr(H (n) ) modeled on H (n) consists of the subset of closed subspaces W ⊆ H (n) such that: – –
the orthogonal projection pr+ : W → H+(n) is a Fredholm operator. the orthogonal projection pr− : W → H−(n) is a Hilbert-Schmidt operator.
Moreover we will denote Gr(n) the subset of Gr(H (n) ) given by subspaces W such that zW ⊆ W. It’s well known [1] that through Segal-Wilson theory we can associate a solution of nth Gelfand-Dickey hierarchy to every element of Gr(n) ; this is the reason why we are interested in them. Lemma 1 ([1]) The map : H (n) −→ H ( f0 (z), . . . , fn−1 (z))T −→ f˜(z) := f0 (zn ) + . . . + zn−1 fn−1 (zn )
16
M. Cafasso
is an isometry. Its inverse is given by fk (z) =
1 −k ˜ ζ f (ζ ) n ζ n =z
where the sum runs over the nth roots of z. Proposition 1 Under the isometry we can identify Gr(n) with the subset {W ∈ Gr(H) : zn W ⊆ W} It is obvious that loop groups act on Hilbert spaces defined above by multiplications. We want to define a certain loop group L1/2 Gl(n, C) with good analytical properties acting transitively on Gr(n) ; in such a way we can obtain any W ∈ Gr(n) just acting on the reference point H+(n) with this group. Good analytical properties will be necessary as we want to construct symbols of some Toeplitz operators out of elements of this group and then apply Widom’s results (see below). Given a matrix g we denote with g its Hilbert-Schmidt norm n 2 g = gi, j 2 i, j=1
Definition 3 Given a measurable matrix-valued loop γ we define two norms γ ∞ and γ 2,1/2 as
1/2 |k| γ (k) 2 γ ∞ := ess sup γ (z) γ 2,1/2 := z =1
k
where we have Fourier expansion γ (z) =
∞
γ (k) zk .
k=−∞
Definition 4 L1/2 Gl(n, C) is defined as the loop group of invertible measurable loops γ such that γ ∞ + γ 2,1/2 < ∞. Proposition 2 ([2]) L1/2 Gl(n, C) acts transitively on Gr(n) and the isotropy group of H+(n) is the group of constant loops Gl(n, C). Proof can be found in [2], here we just mention the principal steps necessary to arrive to this result. – –
We define a subgroup Glres (H (n) ) of invertible linear maps g : H (n) → H (n) acting on Gr(H (n) ) (the restricted general linear group). We prove that every element of Glres (H (n) ) commuting with multiplication by z must belong to L1/2 Gl(n, C).
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
– – –
17
We take an element W ∈ Gr(n) and a basis {w1 , ..., wn } of the orthogonal complement of zW in W. Out of this basis, putting vectors side by side, we construct W and easily check that W = W (z)H+(n) . We verify that multiplication by W belongs to Glres (H (n) ); since it obviously commutes with multiplication by z we conclude that W (z) ∈ L1/2 Gl(n, C).
3 Szegö-Widom Theorem for Block Toeplitz Determinants In his work ([6, 7] and [8]) H. Widom expressed the limit, for the size going to infinity, of certain block Toeplitz determinants as Fredholm determinants of an operator P acting on H+(n) . Also he gave two different corollaries that allow us to compute this determinant in some particular cases. In this section we recall, without proofs, these results. Moreover we state Borodin-Okounkov formula as presented in [10] for matrix case. We begin with some notations; given a loop γ ∈ L1/2 Gl(n, C) we denote with T N (γ ) the block Toeplitz matrix given by ⎞ ⎛ (0) γ . . . . . . γ (−N) ⎟ ⎜ ⎟ ⎜ (1) (−N+1) ⎟ ⎜ γ ... ... γ ⎟ ⎜ ⎟ T N (γ ) := ⎜ ⎟ ⎜ ⎜ ... ... ... ... ⎟ ⎟ ⎜ ⎠ ⎝ γ (N) . . . . . . γ (0) where we have the Fourier expansion γ (z) = k γ (k) zk .We denote D N (γ ) its determinant. We use the notation T(γ ) for the N × N matrix obtained letting N go to infinity. Remark 1 It’s easy to see that, in the base we have chosen above for H (n) , T(γ ) is nothing but the matrix representation of pr+ ◦ γ : H+(n) −→ H+(n) Theorem 1 (Szegö-Widom theorem, [8]) Suppose γ ∈ L1/2 Gl(n, C) and
arg det γ (eiθ ) = 0 0θ 2π
Then it exists the limit D∞ (γ ) := lim
N→∞
where
D N (γ ) = det T(γ )T γ −1 N G(γ )
2π
G(γ ) = exp 1/2π 0
log det γ (e ) dθ iθ
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M. Cafasso
The proof of the theorem is contained in [8]; instead of rewriting it we simply consider the operator T(γ )T(γ −1 ) and explain the meaning of “det” in this case. Lemma 2 Consider γ1 , γ2 ∈ L1/2 Gln (n, C); we have ⎤ ⎡ (i+k) (− j−k) ⎦ T(γ1 γ2 ) − T(γ1 )T(γ2 ) = ⎣ γ1 γ2 k1
.
i, j0
Proof The (i, j )-entry of left hand side reads ∞
∞ (k− j ) (i−k) (k− j ) γ1 γ2 − γ1(i−k) γ2 k=−∞ k=0
=
−1
(k− j ) γ1(i−k) γ2
k=−∞
=
∞
(−k− j−1)
γ1(i+k+1) γ2
.
k=0
In particular choosing γ1 = γ and γ2 = γ −1 we obtain ⎤ ⎡ −1
(− j−k) ⎦ γ (i+k) γ −1 =⎣ I − T(γ )T γ k1
Definition 5
Pγ := T(γ )T γ
−1
⎡ =
⎣δij
⎛ −⎝
i, j0
γ
(i+k)
γ
−1 (− j−k)
k1
Thanks to the fact that i0 k1
γ (i+k) 2 =
⎞⎤ ⎠⎦
(4)
i, j0
k γ (k) 2 < ∞
k1
the product we have written on the right of (4) is a product of two HilbertSchmidt operators. So Pγ differs from the identity by a nuclear operator. Hence its determinant is well defined (see for instance [30]). In our notation we obtained the equality D∞ (γ ) = det(Pγ )
(5)
We will call Pγ Plemelj’s operator as it is related in a clear way with a Riemann-Hilbert factorization problem (see Section 5) already considered by Josip Plemelj in 1964 [13]. Unfortunately, in concrete cases, det(Pγ ) turns out to be really hard to compute; nevertheless we can use some shortcuts also provided by Widom in his works ([6, 7] and [8]).
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
19
Proposition 3 ([6]) Suppose that γ satisfies conditions imposed in SzegöWidom theorem and, moreover, γ (i) = 0 for i j + 1 or γ (i) = 0 for i j + 1. Then
D∞ (γ ) = Dj γ −1 G(γ ) j (6) Proposition 4 ([8]) Suppose we have a symbol γ satisfying conditions imposed in Szegö-Widom theorem. Suppose moreover that γ depends on a parameter x in such a way that the function x → γ (x) is differentiable. If γ −1 admits two Riemann-Hilbert factorizations γ −1 (z) = t+ (z)t− (z) = s− (z)s+ (z) such that t+ (z) :=
(k) k t+ z
s+ (z) :=
k0
t− (z) :=
d i log(D∞ (γ )) = dx 2π
k s(k) + z
k0
(k) k t− z
s− (z) :=
k0
Then
k s(k) − z
k0
trace
(∂z t+ )t− − (∂z s− )s+ ∂x γ dz.
(7)
Also D N (γ ) can be expressed as a Fredholm determinant as pointed out for the scalar case in [9] and generalized for matrix case in [10]. Theorem 2 (Borodin-Okounkov formula, [10]) Suppose that our symbol γ (z) satisfying conditions of Szegö-Widom’s theorem admits two Riemann-Hilbert factorizations γ (z) = γ+ (z)γ− (z) = θ− (z)θ+ (z) such that γ+ (z) :=
γ+(k) zk
θ+ (z) :=
k0
γ− (z) :=
θ+(k) zk
k0
γ−(k) zk
k0
θ− (z) :=
θ−(k) zk
k0
and G(γ ) = 1. Then for every N D N (γ ) = D∞ (γ ) det(I − Kγ ,N ) where, in coordinates, we have 0 if min{i, j} < N (Kγ ,N )ij = ∞
−1 (− j−k) −1 (i+k) θ− γ + k=1 γ− θ+
(8)
otherwise.
20
M. Cafasso
Remark 2 One can easily verify that θ−−1 γ+ is the inverse of γ− θ+−1 so that, again, we deal with operators of type T(φ)T(φ −1 ) with φ = γ− θ+−1 . Also we want to point out that the assumption G(γ ) = 1 is not necessary. The formula for G(γ ) = 1 is written in [11]; since in our case we will always have G(γ ) = 1 we wrote the formula as it was given in [10].
4 τ Functions for Constrained KP and Gelfand-Dickey Hierarchies as Block Toeplitz Determinants In order to fix notations we state some basic facts about KP hierarchy and some reductions of it. Standard references are [1] and [3]. For cKP reductions we make reference to [18–21] and [22]. Given the pseudodifferential Lax operator ∞
L := D +
uj D− j
j=1
KP hierarchy is defined as compatibility conditions of equations Lψ = zψ ∂ ψ = (L j)+ ψ j = 1 . . . ∞ ∂t j
(9)
where (L j)+ denote the differential part of j th power of L. These compatibility conditions are written in Lax form as
∂ L = Lj + , L ∂t j and should be seen as differential equations for coefficients {uj} with respect to variables {t j}. Equivalently one can introduce the dressing operator S=1+
∞
sj D− j
j=1
such that
∞ i ∞
i ψ := S e j=1 ti z = e j=1 ti z 1 + s1 z−1 + s2 z−2 + . . .
is a solution of (9). In this way KP hierarchy is rewritten in Sato form as L = SDS−1
(10) ∂ S = − Lj − S ∂t j where (L j)− = L j − (L j)+ . The first equation gives expression of {uj} in terms of {sj} and the second one gives time evolution for {sj}. Connection with Grassmannian goes this way: given W ∈ Gr one defines ∞
W(t) = e
i j=1 ti z
W.
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
21
For every values of parameters {ti } such that the orthogonal projection pr+: W → H+ is still Fredholm one defines
τW (t) := det pr+ : W → H+
and KP hierarchy can be recast as a set of differential equations for τW (Hirota bilinear form). Actually we have the remarkable formula, due to Sato, ψ(t) :=
τW (t − 1/[z]) ∞j=1 ti zi e τW (t)
(here τW (t − 1/[z]) = τW (t1 − 1/z, t2 − 1/2z2 , . . .)) that gives ψ (and then S and L) in terms of τW . Given the pseudodifferential symbol L and related tau function τW we say, using the notation of [20], that τ ∈ cKPm,n iff Lm can be written as the ratio of two differential operators of order m + n and n respectively. For n = 0 we recover the usual definition of mth Gelfand-Dickey hierarchy; already Segal and Wilson in [1] noticed that this reduction corresponds to considering points W ∈ Gr such that zm W ⊆ W, i.e. W ∈ Gr(m) . For n generic these reductions begun to be studied in 1995 by Dickey and Krichever ([18, 19]); a geometric interpretation of corresponding points in the Grassmannian has been given in [21] and [22]. Namely τW ∈ cKPm,n iff W contains a subspace W of codimension n in W such that zm W ⊆ W. Now, given a subspace W ∈ Gr(n) , we define the corresponding τW in a different way from the one used in [1]. Our approach generalizes what has been done by Itzykson and Zuber in the study of Witten-Kontsevich τ function in [15] (see also [16] and [17]). This approach allows us to define not just τW but also a sequence of {τW,N } N>0 approximating τW and such that τW,N ∈ cKP1,nN ∩ cKPn,n Suppose we have an element W we can represent this element as ⎛ w11 . . . ⎜... ... W=⎜ ⎝... ... w1n . . .
∀N.
∈ Gr(n) ; thanks to results stated in Section 2 ⎞ . . . wn1 ... ...⎟ ⎟ H (n) = W (z)H+(n) ... ...⎠ + . . . wnn
with W (z) ∈ L1/2 Gl(n, C). Also we assume that the matrix W (z) = {wij(z)}i, j=1..n satisfies ⎧ ⎪ ⎨wii = 1 + O (1/z) wij = z (O (1/z)) , i > j ⎪ ⎩ wij = O (1/z) , i < j
22
M. Cafasso
This means that we restrict to the big cell, i.e. we assume that the orthogonal projection pr+ : W −→ H+ is an isomorphism. Infact we have a base for W ∈ Gr(n) given by s z wj : s ∈ N, j = 1 . . . n where wj is the column vector (w1 j...wnj)T . Using the isomorphim : H (n) → H the corresponding base for W ∈ Gr is given by ωns+ j = zns (wj): s ∈ N, j = 1 . . . n and, as in Section 2, we have [(wj)](z) =
n
zi−1 w ji (zn )
i=1
This means that we obtain ωns+ j(z) = zns+ j−1 (1 + O (1/z)) and from this equation follows that the orthogonal projection onto H+ is an isomorphism since every ωk projects to zk−1 . For these points W ∈ Gr(n) and vectors spanning them we define the standard time evolution (KP flow) given by ωns+ j(t; z) := exp ti zi ωns+ j(z) = exp(ξ(t, z))ωns+ j(z) i>0
Now we want to define the τ function associated to W as limit for N → ∞ of some block Toeplitz determinants τW,N . Definition 6 Take M = Nn a multiple of n. τW,N (t) := det z−i ω j(t; z)dz
(11) 1i, j M=Nn
Fist of all we want to prove that τW,N is a block Toeplitz determinant and write explicitly the symbol. Lemma 3 For every j = 1 . . . n we have wj(t; z) := −1 (ω j(t, z)) = exp(ξ(t, ))wj(z)
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
23
where we denote ⎛
0 ... ... ... ⎜ . . ⎜ 1 0 .. .. ⎜ ⎜ .. .. . . := ⎜ ⎜0 1 ⎜. . . . . . . . ⎜ . . . . ⎝ . 0 .. 0 1
z
⎞
⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ .. ⎟ .⎟ ⎠ 0
Proof We simply verify that multiplication by z on Gr corresponds to multiplication by on Gr(n) through the isomorphism −1 .
Proposition 5 τW,N is the N-truncated (n × n)−block Toeplitz determinant with symbol W (t; z) := exp(ξ(t, ))W (z)
(12)
Proof Take i, j n and s, v N; the (i + sn, j + vn)-entry of the matrix in the right hand side of (11) is given by
z−i−sn ωj+vn (t; z)dz =
=
z−i−sn zvn ωj(t; z)dz
z−i+(v−s)n
wjl (t)(k) znk+l−1 dz = wji (t)(s−v)
k∈Z,l=1..n
so that the right hand side of (11) is the transposed of the N-truncated n × n block Toeplitz matrix with symbol W (t; z).
In the sequel of this paper we will call such symbols Gelfand-Dickey (GD) symbols. Now generalizing what has been done by Itzykson and Zuber in [15] we expand τW,N (t) in characters. Proposition 6 τW,N (t) =
l1 ,...,lnN 0
!
"
# ωi(−li +i−1)
χl1 ,...,lnN (X)
i
where X = diag(x1 , ..., xnN ) is related to times {ti } through Miwa’s parametrization
tk := trace X k /k
24
M. Cafasso
and
⎛
xl11 +nN−1 det ⎝ . . . 1 +nN−1 xlnN χl1 ,...,lnN (X) := ⎛ nN−1 x1 det ⎝ . . . xnN−1 nN
⎞ xl12 +nN−2 . . . xl1nN ... ... ...⎠ l2 +nN−2 nN xnN . . . xlnN ⎞ xnN−2 ... 1 1 . . . . . . . . .⎠ xnN−2 ... 1 nN
Proof We start from determinant representation (11). The (i, j )-entry of the matrix will be (−n) ω j pn+i−1 (t) n
where for every n 0 pn (t) :=
1 2πi
exp(ξ(t, z)) dz zn+1
are the classical Schur polynomials and pn (t) = 0 for every negative n. Then resumming everything we obtain ⎛ ⎞ " (−k j ) ⎝ ω j ⎠ det[ pk j +i−1 (t)]i, j=1...nN τW,N (t) = k1 ,...,knN
j
with k j 1 − j. Equivalently we write τW,N (t) =
l1 ,...,lnN 0
⎛ ⎝
"
⎞ (−l + j−1) ⎠ det[ pl j − j+i (t)]i, j=1...nN . ωj j
j
On the other hand it’s well known that under Miwa’s parametrization this last determinant can be written as χl1 ,...,lnN (X) (see for instance [15, 16]); this completes the proof.
We now assign degree 1 to every xi or, equivalently, degree m to tm for every m. For every N the function τW,N is a formal series belonging to the graded algebra C[[t1 , t2 , . . .]]. In general given A ∈ C[[t1 , t2 , . . .]] we define its degree as the minimal degree of its terms and we state the following definition of stable limit for sequences in C[[t1 , t2 , . . .]]. Definition 7 Given a sequence of formal series {A N (t) ∈ C[[t1 , t2 , . . .]], N = 0 . . . ∞} we say that the sequence admits a stable limit A(t) iff lim deg(A N (t) − A(t)) = ∞
N→∞
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
25
We want to prove that the sequence {τW,N } admits stable limit. It’s easy to see that deg(χl1 ...lnN ) =
M
li
i=1
From this easily verified property we obtain the following Lemma 4 Suppose deg(χl1 ,...,lnN ) = Q nN. Then, if the character is different from zero, we have χl1 ...lnN = χl1 ,...,l Q ,0,...,0 . Proof Suppose l j = 0, j > Q and li = 0 ∀ i > j. The j th column of the matrix [ pl j − j+i (t)] has positive subscripts l1 + j − 1, l2 + j − 2, . . . , l j. On the other hand li = Q; hence the sum of these subscripts is Q+
j−1
r
r=0
j
r
r=0
hence two subscripts must be equal, then two lines of the matrix are equal. From this corollary it follows directly the following result. Proposition 7 Up to degree Q the function τW,N (t) does not depend on N with N Q. Thanks to this proposition we deduce that it exists the stable limit τW (t) := lim τW,N (t) N→∞
(13)
On the other hand, in the sequel, we will prove that the symbol W (t) satisfies Szegö-Widom’s condition for every values of ti so that the limit in (13) exist pointwise in time parameters and can be written as a Fredholm determinant. Now, following again [15], we write a differential operator W,N (t) associated to the function τW,N (t). In the sequel we will always write D for the partial derivative with respect to t1 . We will prove that for every N the pseudodifferential operator W,N (t)D−nN satisfies Sato’s equations for the dressing and we recover the usual relation between τ and wave functions. Lemma 5 Define fs,N (t) :=
ωs(−k) pk+nN−1 (t)
k>s
Then we have
$ % τW,N (t) = Wr( f1,N (t), . . . , fnN,N (t)) := det DnN− j fi,N (t) 1i, jnN
26
M. Cafasso
Proof From Definition 6 the (i, j)-entry of matrix defining τW,N (t) is (−k) ω j pk+i−1 (t) k> j
On the other hand we have nN− j
D
fi,N (t) = D
nN− j
#
!
ωi(−k) pk+nN−1 (t)
=
k>i
ωi(−k) pk+ j−1 (t)
k>i
(using the equation D ( pm (t)) = pm−s (t)). Hence we obtained the proof. s
Definition 8 We define the differential operator W,N of order N in D as
W,N ( f ) :=
Wr( f, f1,N (t), . . . , fnN,N (t)) Wr( f1,N (t), . . . , fnN,N (t))
where f ∈ H depends in a differentiable way on {ti }i1 . Proposition 8 The following equations for time-derivatives of W,N holds:
∂
W,N = W,N Di −1
W,N − W,N (t)Di (14) W,N + ∂ti Proof It is enough to prove the equality of the two differential operators when acting on f1,N (t), ... fnN,N (t) which are nN independent solutions of the equation ( W,N )( f (t)) = 0 But this amounts to proving ∂ ∂i ( W,N ) f j,N (t) + W,N i f j,N (t) = 0 ∂ti ∂t1
∀j
which is true iff ∂ ( W,N f j,N (t)) = 0 ∀ j. ∂ti This equality is obviously satisfied.
Multiplying W,N from the right with D−nN we found a pseudodifferential operator that, in fact, gives a solution of KP equations. Definition 9 SW,N := W,N D−nN Proposition 9 SW,N is a monic pseudo-differential operator of order 0 satisfying Sato’s equation
∂ SW,N = − SW,N Di S−1 SW,N (15) W,N − ∂ti
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
Hence the monic pseudo-differential operator of order 1
LW,N := SW,N DS−1 W,N
27
(16)
satisfies the usual Lax system for KP
∂ LW,N = LkW,N + , LW,N ∂tk
(17)
Proof It is obvious that SW,N is a monic pseudo-differential operator of order 0 since W,N , which is of order nN, is normalized so that the leading term is equal to 1. Equation (15) follows directly from (14). The derivation of Lax system from Sato’s equations is well known: one has just to derive the relation LW,N SW,N = SW,N D
for tk and use the obvious relation [LW,N , LkW,N ] = 0
It remains to prove that τW,N (t) is really the τ function for these solutions LW,N (t) of KP equations. We recall the usual relations between the dressing S, the wave function ψ and τ function given by ψ(t; z) = S(t)(exp(ξ(t, z))) = exp(ξ(t, z))
τ (t − 1/[z]) τ (t)
(we recall that the notation t − 1/[z] stands for the vector with ith component equal to ti − iz1i ) All we have to prove is the following Proposition 10 ψW,N (t; z) := SW,N (exp(ξ(t, z))) = exp(ξ(t, z))
τW,N (t − 1/[z]) τW,N (t)
Proof Equivalently we prove that ( W,N ) exp(ξ(t, z)) = exp(ξ(t, z))znN
τW,N (t − 1/[z]) τW,N (t)
Since we have pn (t − 1/[z]) = pn (t) − z−1 pn−1 (t) the right hand side of the equality above can be written as ⎛ ⎞ DnN−1 f1 − z−1 DnN f1 . . . f1 − z−1 Df1 ⎠ ... ... ... det ⎝ nN−1 −1 nN −1 D f − z D f . . . f − z Df nN nN nN nN znN eξ(x,t) Wr( f1 , . . . , fnN )
(18)
28
M. Cafasso
(here derivative is with respect to t1 , we don’t write dependence on fi on t to avoid heavy notation) The left hand side can be written as ⎛ nN ξ(t,z) ⎞ . . . . . . eξ(t,z) z e ⎜ DnN f1 . . . . . . f1 ⎟ ⎟ det ⎜ ⎝ ... ... ... ... ⎠ DnN fnN . . . . . . fnN . Wr( f1 , . . . , fnN ) It is easy to check that these two expressions are equal.
We want now to study the structure of LW,N with more attention; our investigation will lead us to discover that, actually, we are dealing with rational reductions ([18, 19]) of KP. First of all we recall a useful lemma (proof can be found for instance in [31]). Lemma 6 Let {g1 , ..., gm } be a basis of linearly independent solutions of a differential operator K of order m. Then one can factorize K as K = (D + Tm )(D + Tm−1 ) · · · (D + T1 ) with Tj =
Wr(g1 , . . . , gm−1 ) . Wr(g1 , . . . , gm )
We will state now properties of symmetry for fs,N that will be useful in the sequel. Proposition 11 The following equalities hold: fs,N+1 (t) = fs−n,N (t)
(19)
fs,N (t) = Dn fs+n,N (t)
(20)
Proof We will use the equality (−k+n) ωs(−k) = ωs+n
which follows from the very definition of these coefficients. Then for (19) we have (−k−n) fs,N+1 (t) = ωs(−k) pk+n+nN−1 (t) = ωs−n pk+n+nN−1 (t) k
=
k
k (−k) ωs−n pk+nN−1 (t) = fs−n,N (t)
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
For (20) we have
Dn ( fs+n,N (t)) = Dn =
(−k) ωs+n pk+nN−1 (t) =
k
29
(−k) ωs+n pk+nN−1−n (t)
k
ωs(k−n) pk−n+nN−1 (t) = fs,N (t)
k
Theorem 3 For every N > 0 the pseudodifferential operator LW,N and its nth power LW,N = LnW,N can be factorized as • •
LW,N = L1,W,N (L2,W,N )−1 LW,N = M1,W,N (M2,W,N )−1
where all the factors are differential operators and • •
ord(L1,W,N ) = nN + 1, ord(L2,W,N ) = nN ord(M1,W,N ) = 2n, ord(M2,W,N ) = n
Hence, for every N, τW,N ∈ cKP1,nN ∩ cKPn,n . Proof The first factorization comes directly from the fact that LW,N (t) = W,N (t)D( W,N (t))−1
For the second factorization we note that we have the factorization LW,N (t) = W,N (t)Dn ( W,N (t))−1 where the first operator W,N (t)Dn has order M + n while the second (i.e.
W,N ) has order nN. Moreover as follows from (20) we have – –
W,N fi,N = 0 ∀i = 1, . . . , nN
W,N Dn fi,N = 0 ∀i = n + 1, . . . , nN
hence using Lemma 6 one can simplify factorization above as LW,N = M1,W,N (M2,W,N )−1 where M2,W,N is given explicitly by the formula M2,W,N = (D + Kn,N )(D + Kn−1,N )...(D + K1,N ) with
&
K j,N
!
#' Wr fn+1,N , . . . , fnN,N , f1,N , . . . , f j−1,N
= D log Wr fn+1,N , . . . , fnN,N , f1,N , . . . , f j,N
Theorem 4 The sequence {LW,N } N1 satisfies recursion relation LW,N+1 = T N LW,N (T N )−1
(21)
30
M. Cafasso
with T N = (D + Tn,N )(D + Tn−1,N )...(D + T1,N )
# Wr f1,N , ..., fnN,N , f1,N+1 , ... f j−1,N+1
. T j = D log Wr f1,N , ..., fnN,N , f1,N+1 , ... f j,N+1 !
Proof We observe that thanks to (19) – – –
W,N fi,N = 0 ∀i = 1, . . . , nN
W,N+1 fi,N = 0 ∀i = n + 1, . . . , nN
W,N+1 fi,N+1 = 0 ∀i = 1, . . . , n
Hence using again (6) we obtain the recursion relation
W,N+1 = T N W,N and from this last equation we recover the recursion relation for the Lax operator.
We want to point out that the first decomposition as well as the recursion formula are already known and, as pointed out in [20], come simply from the fact that we have a truncated dressing. Actually our sequence of {τW,N } N1 is a part of a sequence already studied by Dickey in [17] under the name of stabilizing chain; in that article Dickey already provided the recursion formula written above as well as some differential equations for coefficients of T N . Nevertheless, to our best knowledge, connection with block Toeplitz determinants never appeared before. Also the fact that τW,N ∈ cKPn,n is something new. It could be interesting to find recursion relations as well as differential equations for M1,W,N and M2,W,N ; we plan to do it in a subsequent work. Till now all we can do is to infer from recursion formula for Lax operator the following formula (M1,W,N+1 )−1 T N M1,W,N = (M2,W,N+1 )−1 T N M2,W,N .
(22)
Now we want to go one step further and see what happens for N → ∞. Obviously thanks to the property of stabilization stated in Proposition 7 we can define a pseudodifferential operator LW and a wave function ψW related to τW in the same way as for finite N and we will obtain a solution of KP as well. Actually a stronger statement holds. Proposition 12 Given W ∈ Gr(n) the functions τW , ψW and LW := (LW )n are respectively the τ function, the wave function and the differential operator of order n corresponding to a solution of nth Gelfand-Dickey hierarchy. Proof It is known [1] that subspaces satisfying zn W ⊆ W correspond to solutions of nth Gelfand-Dickey hierarchy. What we have to prove is that LW (t) = (LW (t))+ .
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
31
From the usual relation ∂ψW (t; z) = (LW )+ ψW (t; z) ∂tn
(23)
we obtain immediately ∂ SW + SW Dn = (LW )+ SW ∂tn so that we have to prove that ∂ SW =0 ∂tn On the other hand
! ψW (t; z) = exp(ξ(t, z)) 1 +
∞
# si (t)z
−i
1=1
where SW = 1 +
∞
si (t)D−i
1=1
Using this explicit expression for the wave function and substituting in (23) we obtain (LW )+ ψW (t; z) − z ψW (t; z) = exp (ξ(t, z)) n
∞ ∂si (t) i=1
∂tn
z−i
The left hand side of this equation lies on W(t) = exp(ξ(t, z))W for every t so that multiplying both terms for exp(−ξ(t, z)) one obtains that they belong to subspaces transverse one to the other (W and H− ), hence both of them vanish. ∂si = 0 for every i.
This means that ∂t n In virtue of this proposition, when computing τW associated to W ∈ Gr(n) , we will always omit times t jn multiple of n. Setting {t jn = 0, j ∈ N} will be important in order to be able to apply Szegö-Widom’s theorem; in this case we will write t˜ instead of t. Proposition 13 Take any W ∈ Gr(n) in the big cell of Gr(n) and a corresponding GD symbol W (t; z). Then
τW (t˜) = det PW (t˜;z) . (24) Proof All we have to prove is that conditions of Szegö-Widom’s theorem are satisfied and G(W (t˜; z)) = 1. We observe that W (t˜; z) ∈ L1/2 Gl(n, C)
∀t˜
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M. Cafasso
since we can always find W (z) ∈ L1/2 Gl(n, C) such that W = W (z)H+(n) and exp(ξ(t˜, )) is continuously differentiable (obviously when restricted to a finite number of times). Moreover det[exp(ξ(t˜, ))] = 1 since we deleted times multiple of n and det(W (z)) = 1 + O(z−1 ) by big cell assumption. This implies that we have
det W t˜; eiθ = 0 0θ 2π
and G(W (t˜; z)) = 1.
We are now in the position to state the main result of this paper. Theorem 5 Given any point (n) W (z)H+ = W ∈ Gr(n)
and corresponding GD symbol W (t; z) = exp
!∞
# ti
i
W (z)
i=1
the following facts hold true: –
{τW,N (t) := D N (W (t; z))}0 N<∞ is a sequence of τ functions for KP associated to wave function ∞ i ψW,N (t; z) = SW,N e i=1 ti z and pseudodifferential Lax operator LW,N = SW,N DS−1 W,N .
The dressing is given by the formula SW,N = W,N D−nN . – –
For every N > 0 we have τW,N ∈ cKP1,nN ∩ cKPn,n . The sequence admits stable limit τW (t˜) = lim τW,N (t˜). N→∞
τW (t˜ ) is a solution of the nth Gelfand-Dickey hierarchy and can be written as the Fredholm determinant
τW (t˜ ) = det PW (t˜;z) .
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
33
Proof The expression of the dressing as well as the expression of LW,N are given in Proposition 9. Proposition 10 gives the expression of ψW,N and proves at the same time that τW,N is the corresponding τ function. The fact that τW,N ∈ cKP1,nN ∩ cKPn,n is proven in Theorem 3 while the existence of the stable limit τW (t˜ ) is given by Propostion 7; Proposition 12 and Proposition 13 prove respectively that τW (t˜ ) is a solution of the nth Gelfand-Dickey hierarchy and that it can be written as a Fredholm determinant.
Remark 3 Also all the τW,N (t˜ ) can be expressed as Fredholm determinants; in order to give explicit expressions we need a certain Riemann-Hilbert factorization of symbol W (t˜; z). This factorization will be obtained in Section 6 and it will be exploited to express τW,N (t˜ ) as a Fredholm determinant.
5 Riemann-Hilbert Problem and Plemelj’s Integral Formula It is evident from Proposition 4 that Riemann-Hilbert decomposition of symbol γ for a block Toeplitz operator plays an important role in computing D∞ (γ ). Here we will show that actually Plemelj’s operator itself enters in a integral equation (see [13]) giving solutions of Riemann-Hilbert problem ϕ+ (z) = γ T (z)ϕ− (z).
(25)
Here ϕ+ (z) and ϕ− (z) are respectively analytical functions defined inside and outside the circle. In this section we consider a smaller class of loops; γ (z) will be a matrix-valued function that extends analytically on a neighborhood of S1 . For convenience of the reader we recall here main steps to arrive to Plemelj’s integral formula [13]. Lemma 7 Suppose that f+ (z), f− (z) are functions on S1 satisfying | f (ζ2 ) − f (ζ1 )| < |ζ2 − ζ1 |μ C for some positive constants μ, C and for every ζ1 , ζ2 ∈ S1 . Necessary and sufficient conditions for f+ (z) and f− (z) to be boundary values of analytic functions regular inside or outside S1 ⊆ C and with value c at infinity are respectively 1 f+ (ζ ) − f+ (z) dζ = 0 (26) 2πi ζ −z f− (ζ ) − f− (z) 1 dζ + f− (z) − c = 0 (27) 2πi ζ −z We have to point out that here both ζ and z lies on S1 so that one has to be careful and define (26) and (27) as appropriate limits. Namely one proves that taking ζ slightly inside or outside S1 along the normal and making it approach to the circle we obtain the same result which will be, by definition, the value of our integral. Now suppose we want to find solutions of (25); we normalize the
34
M. Cafasso
problem requiring ϕ− taking value C at infinity. Taking an appropriate linear combination of (26) and (27) and using (25) we find that ϕ− (z) must satisfy the equation T −1 γ (z)γ T (ζ ) − I 1 C = ϕ(z) − ϕ(ζ )dζ (28) 2πi ζ −z Note that here we do not have to take any limit since the integrand is well defined for every point of S1 . We also want to consider the associate homogeneous equation T −1 γ (z)γ T (ζ ) − I 1 ϕ(ζ )dζ (29) 0 = ϕ(z) − 2πi ζ −z as well as its adjoint 0 = ψ(z) +
1 2πi
γ (z)γ −1 (ζ ) − I ψ(ζ )dζ ζ −z
(30)
Obviously, as usual in Fredholm’s theory, the equations (29) and (30) either have only trivial solution or they have the same number of linearly independent solutions. Lemma 8 Consider two adjoint RH problems ϕ+ (z) = γ (z)T ϕ− (z)
(31)
ψ+ (z) = γ (z)ψ− (z)
(32)
normalized as ψ− (∞) = ϕ− (∞) = 0. Any solution ϕ− of (31) is a solution of (29) as well as any solution ψ+ of (32) is a solution of (30). Proof We just repeat computations made for non-homogeneous case.
Now we introduce a new integrable operator acting on H+(n) and prove that it is actually equal to the Plemelj’s operator. Definition 10 For every f ∈ H+(n) we define γ (z)γ −1 (ζ ) − I 1 ψ(ζ )dζ [P˜ γ ψ](z) := pr+ ψ(z) + 2πi ζ −z
(33)
where pr+ denote the projection onto H+(n) . Proposition 14 P˜ γ = Pγ .
Proof We write P˜ γ in coordinates and verify we obtain the same as in (4). To do so as in the definition of integrals (26) and (27) we compute (33) imposing
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
35
|ζ | < |z|; the formula will hold when ζ approach to S1 in the same way as in (26) and (27). For a consistency check we will prove we obtain the same result imposing |ζ | > |z|. Let’s start with |ζ | < |z|; we have γ (z)γ −1 (ζ ) − I 1 ψ(ζ )dζ = ψ(z) + 2πi ζ −z ⎛ ⎞ k
1 ζ ⎝ dζ (q) ( p) −1 p q⎠ ψ(z) + I − γ γ z ζ ψ (s) ζ s k 2πi z ζ p,q∈Z s 0 k1
Imposing k + q + s = 0 we get that this is equal to
(−k−s) (s) p−k ψ(z) + γ ( p) γ −1 ψ z p∈Z k1 s0
= ψ(z) +
(−k−s) (s) t γ (t+k) γ −1 ψ z
t∈Z k1 s0
Taking the projection on H+(n) we obtain exactly formula (4). Now for |ζ | > |z| we have 1 γ (z)γ −1 (ζ ) − I ψ(z) + ψ(ζ )dζ 2πi ζ −z ⎛ ⎞ k
1 ζ ⎝ dζ (q) p q = ψ(z) + γ ( p) γ −1 z ζ − I⎠ ψ (s) ζ s k 2πi z ζ p,q∈Z s 0 k0
Imposing q + s = k we arrive to
(k−s) (s) k+ p
(k−s) (s) t γ ( p) γ −1 ψ z = γ (t−k) γ −1 ψ z k,s0 p∈Z
k,s0 t∈Z
Taking the projection on H+(n) we obtain that this is equal to T(γ )T(γ −1 ) so that the two computations for |ζ | < |z| and for |ζ | > |z| coincide in virtue of Lemma 2
Theorem 6 Suppose we are given a symbol γ (z) analytic in a neighborhood of S1 and such that D∞ (γ ) = 0 Then the Riemann-Hilbert problem ϕ+ (z) = γ (z)T ϕ− (z) normalized as ϕ− (∞) = C admits (if existing) a unique solution. Proof Suppose we have two distinct solutions (ϕ1− , ϕ1+ ) and (ϕ2− , ϕ2+ ); taking the difference we obtain a non-trivial solution of (31). Then also (32) admits
36
M. Cafasso
non trivial solutions and the same holds for (30). But this means that we have a non zero ψ(z) ∈ H+(n) such that [Pγ ψ](z) = 0 which is impossible since det(Pγ ) = D∞ (γ ) = 0
Existence of factorization will be treated in the next section for the specific case of Gelfand-Dickey symbols. For a general treatment of the problem of existence see [13].
6 Factorization for Gelfand-Dickey Symbols Here we will prove that for Gelfand-Dickey symbols we can write the unique solution of factorization (25) in terms of data LW (t˜ ), ψW (t˜; z). We recall that LW (t˜ ) and ψW (t˜; z) are the stable limits of LW,N (t˜) and ψW,N (t˜; z). They represent the differential operator and the wave function associated to the solution τW (t˜ ). Our exposition here is closely related to [14]. At the end of the section we will use the factorization obtained to express any τW,N (t˜ ) as a Fredholm determinant. As we have written before in the proof of Proposition 12 we have the relation
(34) LW t˜ ψW t˜; z = zn ψW t˜; z
where ψW t˜; z admits asymptotic expansion
ψW t˜; z = exp ξ t˜, z 1 + O z−1 Now out of ψW we construct n time-dependent functions
ψW,i t˜; z := Di ψW t˜; z : i = 0, . . . , n − 1 belonging to the subspace W ∈ Gr. Definition 11 ⎛
1 ζ1 . . . ζ1n−1
⎞ ψW,0 t˜; ζ1 ψW,1 t˜; ζ1 . . . ψW,n−1 t˜; ζ1 ⎜ ⎟ ⎜
⎟ ⎜ψW,0 t˜; ζ2 ψW,1 t˜; ζ2 . . . ψW,n−1 t˜; ζ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ... ... ... ... ⎜ ⎟ ⎝ ⎠
ψW,0 t˜; ζn ψW,1 t˜; ζn . . . ψW,n−1 t˜; ζn
⎞−1 ⎛
⎜ ⎟ ⎜ ⎟ ⎜ 1 ζ2 . . . ζ n−1 ⎟ 2 ⎜ ⎟ ⎟ W (t˜; z) := ⎜ ⎜ ⎟ ⎜. . . . . . . . . . . . ⎟ ⎜ ⎟ ⎝ ⎠ n−1 1 ζn . . . ζn where ζi is the ith root of z.
Proposition 15 The matrix W (t˜; z) admits asympotic expansion
W t˜; = exp ξ t˜; I + O z−1
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
37
Moreover under the isomorphism −1 : H → H (n) we can write W ∈ Gr(n) as W = W (0, z)H+(n)
(35)
Proof One has to note that the ith column of matrix W (t˜; z) is nothing but −1 (ψW,i (t˜, z)) so that asymptotic expansion follows easily. Equation 35 corresponds to the fact that {zns ψW,i (0, z) : s ∈ Z} is a basis for W.
Observe that, since we also have W = W (z)H+(n) we obtain
W (0, z) = W (z) I + O z−1 .
From this equation and from Lemma 2 it follows that for every N > 0 we have
= T N W t˜; z T N I + O z−1 . T N W t˜; z I + O z−1 Now since for every N
det T N I + O z−1 =1
we will assume, without loss of generality, that W (0, z) = W (z) since this is true modulo an irrelevant term that does not affect values of determinants we want to compute. We now want to define a matrix W (t˜; z) analytic in z near 0 and with similar properties as W (t˜; z). Definition 12 Let φW (t˜; z) be the unique solution of
LW t˜ φW t˜; z = zn φW t˜; z analytic in z = 0 and such that (Di φ)(0, z) = zi : i = 0, . . . , n − 1 We define
⎛
1
ζ1 . . . ζ1n−1
⎞ φW,0 t˜; ζ1 φW,1 t˜; ζ1 . . . φW,n−1 t˜; ζ1 ⎜ ⎟ ⎜
⎟ ⎜φW,0 t˜; ζ2 φW,1 t˜; ζ2 . . . φW,n−1 t˜; ζ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ . . . . . . . . . . . . ⎜ ⎟ ⎝ ⎠
φW,0 t˜; ζn φW,1 t˜; ζn . . . φW,n−1 t˜; ζn
⎞−1 ⎛
⎜ ⎟ ⎜ ⎟ ⎜ 1 ζ2 . . . ζ n−1 ⎟ 2 ⎜ ⎟
⎟ W t˜; z := ⎜ ⎜ ⎟ ⎜. . . . . . . . . . . . ⎟ ⎜ ⎟ ⎝ ⎠ 1 ζn . . . ζnn−1
where as before ζi is the ith root of z and
φW,i t˜ := Di φW t˜; z : i = 0, . . . , n − 1.
38
M. Cafasso
Remark 4 W (t˜; z) admits regular expansion in z = 0 and Cauchy initial values we imposed on φW imply W (0; z) = I. ˜ Proposition 16 W (t˜; z)−1 W (t; z) does not depend on ti for any i. Proof It is well known that equations i ∂ n f = LW f + ∂ti satisfied by φW and ψW can be translated into matrix equations ∂ F = FM ∂ti satisfied by W (t˜; z) and W (t˜; z) one can write explicitly M in terms of co ni
efficients of LW . Hence we have +
∂ ˜ ˜ W t˜; z −1 = W t˜; z M−1 W t; z W t; z ∂ti
−1 ˜ ˜ ˜ −W t˜; z −1 W t; z W t; z MW t; z = 0 Theorem 7 Given a Gelfand-Dickey symbol
W t˜; z = exp ξ t˜, W (z) one can factorize it as $
%
˜ W t˜; z = exp ξ t˜, W −t˜, z −1 W −t ; z where the term inside the square bracket is analytic around z = ∞ and the other is analytic around z = 0. For assigned values of t˜ for which τW (t˜ ) = 0 this is the unique solution of the factorization problem (25) normalized at infinity to the identity. Proof Using the previous proposition we have
W t˜; z = exp ξ t˜, W (z) = exp ξ t˜, W (0, z)
˜ = exp ξ t˜, −t˜; z −1 W −t; z W (0; z)
˜ = exp ξ t˜, −t˜; z −1 W −t ; z Unicity of the factorization follows from Section 5. Corollary 1 For every N > 0
τW,N t˜ = τW t˜ det I − KW (t˜;z),N
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
with
KW (t˜;z),N
ij
=
⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎨
39
if min{i, j} < N
(− j−k) ⎪ ⎪ ⎪ ˜; z (i+k) W −t˜; z −1 − t W ⎪ ⎩ ∞
otherwise.
k=1
Proof It is enough to apply Borodin-Okounkov formula using factorization obtained above.
Corollary 2 For every N > 0
−1
τW,N t˜ = det T W −t˜; z T W −t˜; z N,N τW,N+1 t˜ (observe that the right hand side of this equation is an ordinary n × n determinant, not a Fredholm determinant). Proof
det I − KW (t˜;z),N τW,N t˜ =
. τW,N+1 t˜ det I − KW (t˜;z),N+1
On the other hand the operator (I − KW (t˜;z),N+1 )−1 (I − KW (t˜;z),N ) can be written as a block matrix obtained taking the identity matrix and replacing the N th block column by the N th block column of the matrix with (i, j)-entry equal to ∞
(i+k)
−1 (− j−k) W −t˜; z W −t˜; z .
k=1
Hence proof is obtained applying Lemma 2
7 Rank One Stationary Reductions and Corresponding Gelfand-Dickey Symbols We want to describe, more explicitly, GD symbols corresponding to solutions of Gelfand-Dickey hierarchies obtained by rank one stationary reductions. In order to emphasize that we are dealing with rank-one generic case instead of the standard expression Krichever locus we will speak about BurchnallChaundy locus. Definition 13 Given a point W ∈ Gr(n) we say that W stays in BurchnallChaundy locus iff the Lax operator LW of the corresponding solution satisfies [LW , MW ] = 0
40
M. Cafasso
for some differential operator MW of order m coprime with n. Without loss of generality we also assume m > n. The name we use is due to the fact that, already in 1923, Burchnall and Chaundy were the first to study algebras of commuting differential operators in [23–25] where they stated this important proposition we will use in the sequel. Proposition 17 ([23–25]) Given a pair of commuting differential operator L, M with relatively prime orders it exists an irreducible polynomial F(x, y) such that F(x, y) = xm + ... ± yn and F(L, M) = 0. This proposition in particular allows us to associate to every BurchnallChaundy solution a spectral curve defined by polynomial relation existing between the pair of commuting differential operators. From the Grassmannian point of view one can define an action A of pseudodifferential operators in variable t1 on H by A : DO × H −→ H n n n
∂ m ∂ , ϕ(z) −→ z ϕ(z) (t1 ) ∂t1n ∂zn
and, using this action, prove the following propostition Proposition 18 ([4]) Given a point W in the Burchnall-Chaundy locus one has zn W ⊆ W
(36)
b (z)W ⊆ W
(37)
where LW and MW are of order n and m respectively and b (z) is a series in z whose leading term is zm . Conversely, if W satisfies above properties, it stays in the Burchnall-Chaundy locus. Proof We just sketch the proof and make reference to Mulase’s article [4]. Suppose we are given LW and MW ; under conjugation with the dressing SW (t˜ ) we have
∂n ˜ ˜ ˜ S−1 t L t S t = W W W ∂t1n Under the action A this gives invariance of W with respect to zn while invariance with respect to b (z) is obtained acting with
˜ ˜ ˜ S−1 W t MW t S W t Viceversa given W we reconstruct the dressing SW (t˜ ); using it we define LW (t˜ ) and MW (t˜ ) conjugating pseudodifferential operators corresponding to zn and b (z). In particular observe that also zn and b (z) will satisfy the same
polynomial relation as LW (t˜ ) and MW (t˜ ).
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
Remark 5 Without loss of generality we can assume 1 b (z) dz = 0 ∀s ∈ Z. 2πi zns+1
41
(38)
Now suppose we are given an element W = W (z)H+(n) ∈ Gr(n) in the Burchnall-Chaundy locus. Using the explicit isomorphism we can construct a matrix B(z) := b () such that B(z)W ⊆ W.
(39)
Proposition 19 C(z) := W −1 (z)B(z)W (z) has the following properties: – – – –
C(z) is polynomial in z. trace(C(z)) = 0 m = maxi ( j − i + n deg Cij(z)) ∀ j = 1 . . . n The characteristic polynomial pC(z) (λ) of C(z) defines the spectral curve of the solution.
Proof Equation 39 can be equivalently written as (n) W −1 (z)B(z)W (z)H+ ⊆ H+(n)
and this means precisely that C(z) can’t have terms in z−k for any k > 0. The other properties are satisfied if and only if they are equally satisfied by B(z) so that we will prove them for B(z) instead of C(z). B(z) is traceless thanks to equation (38) and thanks to the fact that
trace k = 0 ∀k = sn The third properties is satisfied as B(z) = b () represents in H multiplication by a series whose leading term is equal to m. For the last property we observe that if F(x, y) is the polynomial defining the spectral curve, i.e. F(LW , MW ) = 0, then we will have F(diag(z, z, . . . , z), B(z)) = 0 as well; on the other hand thanks to Cayley-Hamilton theorem we have p B(z) (B(z)) = 0. Since F is irreducible and p B(z) (λ) has the same form p B(z) (λ) = λn + . . . ± zm we conclude that they are equal.
Observe that since W (z) is defined modulo multiplication on the left by invertible triangular matrices also C(z) is defined modulo conjugation by elements of the group of upper triangular invertible matrices. It was a
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remarkable observation of Schwarz [26] that actually Burchnall-Chaundy locus can be described by means of matrices with properties as in proposition 19 modulo the action of . Here we adapt the results of [26] to our situation. Namely we explain how, given C(z), one can recover W (z) and the corresponding spectral curve. Proposition 20 Given a matrix C(z) such that: – – –
C(z) is polynomial in z. trace(C(z)) = 0 m = maxi ( j − i + n deg Cij(z))
∀j = 1...n
it exists a unique W = W (z)H+(n) in Burchnall-Chaundy locus such that its spectral curve is defined by pC(z) (λ). In order to prove this proposition we need two lemmas. Lemma 9 Given a polynomial matrix C(z) such that
m = max j − i + n deg Cij(z) ∀j = 1...n i
(with m and n coprime) coefficients of characteristic polynomial pC(z) (λ) := λn + c1 (z)λn−1 + . . . + cn (z) satisfy n deg cs ms
∀s = 1, . . . , n − 1
deg cn = m Proof From n deg Ci, j m − j + i and definition of determinant follows immediately that n deg cs ms ∀s = 1, . . . , n. Strict inequality for s < n follows from the fact that m and n are coprime. For the equality
deg cn = deg det(C(z)) = m we observe that in every line there is a unique element Cij(z) such that m = j − i + n deg Cij(z); taking this unique element for every line and multiplying them we will obtain the leading term of determinant which will be of order m.
Lemma 10 The equation λn + c1 (z)λn−1 + . . . + cn (z) = 0
(40)
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
43
with ∀s = 1, . . . , n − 1
n deg cs ms
deg cn = m and n, m coprime has n distinct solutions {λi = b (ζi ),
b (ζ ) = ζ m 1 + O ζ −1
i = 1 . . . n} with
(as usual ζi is the ith root of z). Proof Imposing λi = ζim we have a solution of the equation (ζim )n + c1 (ζin )(ζim )n−1 + . . . + cn (ζin ) = 0 at the leading order mn. Then imposing λi = ζim (1 + l1 ζi−1 ) and plugging it into the equation (40) one obtains
ζim + l1 ζim−1
n
n−1
+ c1 ζin ζim + l1 ζim−1 + . . . + cn ζin = O ζimn
l1 can be found so that terms of order nm − 1 in the equation vanish; going on solving the equation term by term we obtain − j λi = ζim 1 + l jζi j<0
Clearly coefficients lj do not depend on the choice of the root ζi so that it exists b (λ) with stated properties.
Now we can prove Proposition 20. Proof We start computing the characteristic polynomial pC(z) (λ); thanks to Lemmas 9 and 10 we find n distinct roots b (ζ1 ), . . . , b (ζn ) with properties stated above. The aim is to find W (z) such that W (z)b ()W −1 (z) = C(z)
Since we have n distinct solutions {b (ζi ),
i = 1, . . . , n} of the equation
pC(z) (λ) = 0 it exists a matrix ϒ(ζ1 , . . . , ζn ) such that ⎛ b (ζ1 ) 0 ⎜ 0 b (ζ2 ) ⎜ ϒ(ζi )C(z)ϒ −1 (ζi ) = ⎜ ⎝ 0 ... 0 ...
⎞ 0 0 ⎟ ⎟ ⎟ 0 ⎠ . . . b (ζn ) ... ... .. .
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On the other hand it’s easy to observe that the matrix can be as ⎛ ⎞−1 ⎛ζ 0 . . . 0 ⎞ ⎛ 1 1 ζ1 . . . ζ1n−1 1 ζ1 . . . 0 ζ2 . . . 0 ⎟ ⎜ 1 ζ2 . . . ζ n−1 ⎟ ⎜ ⎜ 1 ζ2 . . . ⎜ ⎟ 2 ⎟ =⎜ ⎟⎜ ⎝. . . . . . . . . . . . ⎠ ⎜ ⎝ 0 . . . . . . 0 ⎠ ⎝. . . . . . . . . 1 ζn . . . ζnn−1 1 ζn . . . 0 . . . . . . ζn
diagonalized ⎞ ζ1n−1 ζ2n−1 ⎟ ⎟ ... ⎠ ζnn−1
and this means that multiplication by b (z) can be written in H+(n) as multiplication by ⎛
1 ⎜1 ⎜ ⎝. . . 1
ζ1 ζ2 ... ζn
... ... ... ...
⎞−1 ⎛b (ζ ) 0 1 ζ1n−1 ⎜ 0 b (ζ2 ) n−1 ⎟ ζ2 ⎟ ⎜ ⎜ ... ⎠ ⎝ 0 ... ζnn−1 0 ...
⎞⎛ 0 1 0 ⎟ ⎟⎜ 1 ⎟⎜ ⎝. . . ⎠ 0 1 . . . b (ζn )
... ... .. .
ζ1 ζ2 ... ζn
... ... ... ...
⎞ ζ1n−1 ζ2n−1 ⎟ ⎟ ... ⎠ ζnn−1
Hence we have ⎛
1 ⎜1 −1 ⎜ W (z) = ϒ (ζi ) ⎝ ... 1
ζ1 ζ2 ... ζn
... ... ... ...
⎞ ζ1n−1 ζ2n−1 ⎟ ⎟ ... ⎠ ζnn−1
Note that W (z) is defined modulo the action of so that, by construction, C(z) corresponds to a unique W ∈ Gr(n) such that W = W (z)H+(n) .
Remark 6 As it was pointed out by Schwarz [26], matrices C(z) with properties stated above can be used to describe points in the Grassmannian describing string solutions of Gelfand-Dickey hierarchies, i.e. solutions associated to reduction of type [L, M] = 1 This class of solutions has not been treated in this article since they do not live in Segal-Wilson Grassmannian but just on Sato’s Grassmannian constructed on the space of formal series; this means that we cannot use any more SzegöWidom theorem as the analytical requirements are not satisfied. Nevertheless some results obtained in Section 4 still hold since the property of stability for {τW,N (t)} does not depend on analytical properties of the symbol W (z). Hence one can try to apply the approach used in this article to the study of these (much less studied) string solutions; perhaps results obtained by Okounkov and Borodin in [9] for formal series and a generalization to block case can play in the setting of formal theory the same role played by Szegö-Widom theorem in this paper.
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
45
Example 1 (Symmetric n-coverings) Take a symmetric n-covering C of P1 given by equation λn = P(z) =
nk+1 "
z − aj
(41)
j=1
For this particular type of curves, choosing in a appropriate way the divisor on the curve, we can write explicitly W (z), B(z) and C(z). We start to observe that for any W corresponding to this spectral curve we have b (z)W ⊆ W with b (z) = P(zn )1/n Then it’s easy to prove that the corresponding B(z) = b () can be written as ⎛ ⎞ n−1 0 0 ... 0 z n P(z)1/n ⎜z−1/n P(z)1/n 0 . . . ⎟ 0 0 ⎜ ⎟ ⎜ ⎟ . . . . . . . . ⎜ ⎟ . . 0 . . ⎟ B(z) = ⎜ ⎜ ⎟ .. .. .. .. .. ⎜ ⎟ . . . . . ⎜ ⎟ ⎝ ⎠ 0
. . . 0 z−1/n P(z)1/n
Now we define n functions i−1 1 P(z) n wi (z) :=
, ((i−1)k z z − aj j=1
0
i = 1, . . . , n.
We take W := diag(w1 (z), ..., wn (z))
It is easy to verify that the matrix C(z) = W −1 (z)B(z)W (z) = ⎛ 0 0 ⎜z−1/n P(z)1/n w1 (z) 0 ⎜ w2 (z) ⎜ .. ⎜ . ⎜ 0 ⎜ .. ⎜ .. ⎜ . . ⎜ ⎝ 0 ...
... ... .. . .. .
0 0 .. . .. .
(z) 0 z−1/n P(z)1/n wwn−1 n (z)
z
n−1 n
⎞ n (z) P(z)1/n w w1 (z) ⎟ 0 ⎟ ⎟ .. ⎟ ⎟ . ⎟ .. ⎟ ⎟ . ⎟ ⎠ 0
is polynomial in z. It is worth noticing that this example already gives all possible double coverings; hence for any (possibly singular) hyperelliptc surface we found (assigning a particular divisor) the GD symbol of the corresponding algebro-geometric rank one solution of KdV. Example 2 (Rational solutions) As pointed out by Segal and Wilson [1], subspace of Burchnall-Chaundy locus corresponding to rational curves are given by W = W (z)H+(n) with W (z) rational in z. In particular the corresponding
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Gelfand-Dickey symbol will satisfy hypothesis given in Proposition 3 so that we recover the following (known) result. Proposition 21 Every rational solution of Gelfand-Dickey hierarchies can be written as a finite-size determinant. For instance, for n = 2, taking 1 − d2 z−1 0 W (z) = H+2 0 1 − c2 z−1 the inverse of Gelfand-Dickey symbol is equal to W −1 (t˜; z) =
⎞⎛ z ⎛ cosh z1/2 −z1/2 sinh z1/2 t2i+1 z2i t2i+1 z2i i 0 i 0 ⎜ ⎟⎜z−d2 ⎜ ⎟⎜ ⎝ ⎠⎝
2i 2i −z−1/2 sinh z1/2 cosh z1/2 0 i0 t2i+1 z i0 t2i+1 z
0 z z−c2
⎞ ⎟ ⎟ ⎠
Simply taking the residue one obtains that the corresponding τ function will be equal to ⎛
⎞ 2i+1 2i+1 cosh −d sinh i0 t2i+1 d i0 t2i+1 d ⎟ ⎜ ⎟ τW (t1 , t3 , ...) = det ⎜ ⎝
⎠ 2i+1 2i+1 cosh −c−1 sinh i0 t2i+1 c i0 t2i+1 c and recover 2-solitons solution for KdV. We want to point out that, for algebro geometric solutions treated in this section, the problem of factorization for Gelfand Dickey symbol can be easily translated into a Riemann-Hilbert problem on some cuts on the plane with constant jumps. For simplicity we reduce to the case n = 2; the procedure used here is equivalent to the one used by Its, Jin and Korepin in [27] and generalized by Its, Mezzadri and Mo in [28]. Suppose we want to solve the factorization problem
W t˜; z := exp ξ t˜, W (z) = T− t˜; z T+ t˜; z for our GD symbol with W (z) = diag(w1 (z), w2 (z)) as in Example 1; since it will appear many times we denote A the matrix √ 1 z √ A := 1− z Also we impose "
2g+1
P(z) :=
z − aj
j=1
with all aj having modulo less then 1 and a1 < a2 < . . . < a2g+1
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
47
We denote l1 , ...l g+1 the oriented intervals (a1 , a2 ), (a3 , a4 ), ...(a2g+1 , ∞). Instead of looking for T− (t˜; z) and T+ (t˜; z) we define a new matrix S(t˜; z) imposing ⎧
⎪ ⎨ S t˜; z := A exp −ξ(t˜, ) T− t˜; z ⎪
⎩ ˜
S t; z := AW (z)T+−1 t˜; z
z1
z1
Proposition 22 S(t˜; z) has the following properties: – –
It has no jumps on S1 It has jumps on intervals lj; precisely calling S L (t˜; z) and S R (t˜; z) the values of S(t˜; z) approaching from the left and approaching from the right the interval we have
01 S L t˜; z := S R t˜; z 10
–
It is invertible in any points but aj; there it has singular behaviour of type ⎛
⎝
1 1 S t˜; z ∼ (z − aj) 1 −1
–
⎞
1 0 ⎠ 0 ±1/2
S j t˜; z
with Sj(t˜; z) invertible in aj; minus is for a1 , . . . , ag , plus for the others. At infinity it behaves as ⎛ √
√ √
⎞ exp − z(t1 z + t3 z + . . .) z exp − z(t1 z + t3 z + . . .) √
√
⎠ S t˜; z ∼ ⎝ √ exp z(t1 z + t3 z + . . .) − z exp z(t1 z + t3 z + . . .)
Proof Let’s call S+ (t˜; z) and S− (t˜; z) the limiting values of S(t˜; z) approaching the unit circle from inside and outside; we have
−1 −1 ˜ ˜ ˜ ˜ S− t˜; z S−1 + t; z = A exp −ξ t, z T− t; z T+ t; z W (z)A
= A exp −ξ t˜, z exp ξ t˜, z W (z)W −1 (z)A−1 = I and this proves we haven’t any jumps on S1 .
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Writing explicitly S(t˜; z) as
√ √
⎞ ⎧ ⎛ √ ⎪ exp − z(t z + t z+. . .) z exp − z(t z+t z+. . .) ⎪ 1 3 1 3 ⎪ ⎪ ⎟ ⎜ ⎪ ⎪ ⎟ ⎪S t˜; z = ⎜ ⎪ ⎝ ⎠
⎪ ⎪ √ √ √ ⎪ ⎪ exp z(t z+t z+. . .) − z exp z(t z+t z+. . .) 1 3 1 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎪ ⎨ T− t˜; z z1 ⎪ ⎪ ⎞ ⎛ ⎪ ⎪ (P(z))1/2 ⎪ ⎪ ⎪ 1 ( g ⎪ ⎟ ⎜ ⎪ ⎪ j=0 (z−aj ) ⎟ ⎪ ⎜ ⎪ ⎟ −1 ˜
⎜ ⎪ ⎪ S t˜; z = ⎜ ⎟ T+ t ; z ⎪ ⎪ ⎜ ⎪ 1/2 ⎟ ⎪ ⎠ ⎝ (P(z)) ⎪ ⎪ ⎪ 1 −(g ⎩ (z−a ) j j=0
z1
we obtain almost immediately the other points of the proposition; the only thing we have to observe is that both T+ (t˜; z) and T− (t˜; z) are invertible inside and outside the circle respectively. This is because we have
(P(z))1/2
= det T+ t˜; z det T− t˜; z det W t˜; z = (g j=0 z − aj This condition combined with
lim det T− t˜; z = 1
z→∞
gives
det T+ t˜; z = 1
(P(z))1/2
. det T− t˜; z = (g j=0 z − aj
The Riemann-Hilbert problem given by Proposition 22 is equivalent to the one proposed in Section 6. What can be done is to write explicitly the solution S(t˜; z) using θ functions associated to the curve; this is what has been done in [27] and [28]. Actually comparing previous proposition with results obtained in Section 6 we immediately realize that √
√ ⎞ ⎛ ψW,0 t˜; z ψW,1 t˜; z
⎠ S t˜; z = ⎝ √
√
ψW,0 t˜; − z ψW,1 t˜; − z so that all we have to do in our case is to write down Baker-Akhiezer function in terms of special functions. We can carry on the same procedure for n arbitrary; the only difference will be that the jump matrices will remain
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
49
constant but more complicated; in any case the solution of this RiemannHilbert problem with constant jumps will be
⎞ ⎛ ψW,0 t˜; ζ1 ψW,1 t˜; ζ1 . . . ψW,n−1 t˜; ζ1 ⎟ ⎜ ⎜
⎟ ⎜ψW,0 t˜; ζ2 ψW,1 t˜; ζ2 . . . ψW,n−1 t˜; ζ2 ⎟ ⎟ ⎜ ⎟ S t˜; z = ⎜ ⎟ ⎜ ⎟ ⎜ . . . . . . . . . . . . ⎟ ⎜ ⎠ ⎝
˜ ˜ ˜ ψW,0 t; ζn ψW,1 t; ζn . . . ψW,n−1 t; ζn Explicit formulas involving special functions can be used here to apply Proposition 4 to our case. For instance taking the elliptic curve C given by equation w 2 = 4z3 − g2 z − g3 with uniformization given by the Weierstass ℘ function (z, w) = (℘ (u), ℘ (u)) one can write wave function as ψ(x, t, u) :=
σ (u − c − x)σ (c) exp xζ (u) − 1/2t℘ (u) σ (u − c)σ (x + c)
(here ζ and σ are Weierstrass ζ and σ function respectively, x and t correspond to the first and the third time). With some tedious computations, making the change of variables u = u(z), the right hand side of equation (7) can be obtained. It turns out that the only relevant factorization is the one given by
% $ %$ W −1 (x, t; u) = W −1 (u) (−x, −t; u) −1 (−x, −t; u) exp −x − t3 where as before (we just wrote z as a function of u) we have
1 (℘ (u))1/2 (x, t; u) := 1 −(℘ (u))1/2
−1
ψ(x, t, u) ∂x ψ(x, t, u) ψ(x, t, −u) ∂x ψ(x, t, −u)
Plugging into equation (7) we obtain d τ (x, t) = Kt + 2ζ (−c) − 2ζ (x − c) dx (here K is some constant); taking another derivative we obtain elliptic solution of KdV as expected. Acknowledgements I am grateful to A. Its for a fruitful discussion in which he pointed out that our Riemann-Hilbert problem can be reduced to a problem with constant jump as in [27, 28] and gave me some interesting hints about possible developments of this work. J. van de Leur suggested me to verify if vector-constrained reductions of KP had some relations with block Toeplitz determinant, I am grateful to him for this really useful suggestion.
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Moreover I wish to thank my advisor B.Dubrovin for his constant support and suggestions he gave me during many hours spent together discussing the preparation of this paper. This work is partially supported by the European Science Foundation Programme ‘Methods of Integrable Systems, Geometry, Applied Mathematics’ (MISGAM), the Marie Curie RTN ‘European Network in Geometry, Mathematical Physics and Applications’ (ENIGMA), and by the Italian Ministry of Universities and Researches (MUR) research grant PRIN 2006 ‘Geometric methods in the theory of nonlinear waves and their applications’.
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24. Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. Proc. Roy. Soc. London A 118, 557–583 (1928) 25. Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators II. The identity Pn = Qm . Proc. Roy. Soc. London A 134, 471–485 (1932) 26. Schwarz, A.: On solutions to the string equation. Modern Phys. Lett. 6(29), 2713–2725 (1991) 27. Its, A.R., Jin, B.Q., Korepin, V.E.: Entropy of XY Spin Chain and Block Toeplitz determinants. In: Binder, I., Kreimer, D. (eds.) Universality and Renormalization. Fields Institute Communications, vol. 50, pp. 151–183. American Mathematical Society, Providence (2007) 28. Its, A.R., Mezzadri, F., Mo, M.Y.: Entanglement entropy in quantum spin chains with finite range interaction. Math. Phys. arXiv:0708.0161v1 (2008, in press) 29. Basor, E., Ehrhardt, T.: Asymptotics of block Toeplitz determinants and the classical dimer model. Comm. Math. Phys. 274(2), 427–455 (2007) 30. Simon, B.: Trace ideals and their applications. Math. Surveys Monogr. 120 (2005) 31. Ince, E.L.: Ordinary Differential Equations. Dover, New York (1926)
Math Phys Anal Geom (2008) 11:53–71 DOI 10.1007/s11040-008-9039-6
Degree Complexity of a Family of Birational Maps Eric Bedford · Kyounghee Kim · Tuyen Trung Truong · Nina Abarenkova · Jean-Marie Maillard
Received: 13 November 2007 / Accepted: 31 March 2008 / Published online: 3 June 2008 © Springer Science + Business Media B.V. 2008
Abstract We compute the degree complexity of a family of birational mappings of the plane with high order singularities. Keywords Birational mappings · Degree complexity Mathematics Subject Classifications (2000) 37F99 · 32H50
Research of Eric Bedford was supported in part by the NSF. Research of Nina Abarenkova supported in part by a Russian Academy of Sciences/CNRS program. E. Bedford (B) · T. T. Truong Department of Mathematics, Indiana University, Bloomington, IN 47405, USA e-mail: [email protected] T. T. Truong e-mail: [email protected] K. Kim Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA e-mail: [email protected] N. Abarenkova Laboratory of Mathematical Problems of Physics, Petersburg Department, Steklov Institute of Mathematics, 27, Fontanka, 191023, St. Petersburg, Russia e-mail: [email protected] J.-M. Maillard Lab. de Physique Théorique et de la Matière Condensée, Université de Paris 6, Tour 24, 4ème étage, case 121, 4, Place Jussieu, 75252 Paris Cedex 05, France e-mail: [email protected]
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1 Introduction Birational maps have been found to arise in lattice statistical mechanics, for instance in vertex models or in spin-edge models [17, 28, 36]. These are fundamental non-linear symmetries of the parameter space that arise from natural (geometrical) symmetries of the lattice, combined with the so-called inversion relation [29–31]. In the case of the Yang-Baxter integrability, the analysis of these symmetries can lead efficiently to a parameterization of the Yang-Baxter equations in terms of selected algebraic varieties [27, 32]. More generally, beyond Yang-Baxter integrability, these birational maps have to be compatible with the phase diagram of the model [33], and, for instance, with the renormalization group. It is important to note that for non-YangBaxter-integrable lattice models, these birational maps can still be integrable [14]. The connection between birational mappings and lattice statistical mechanics is discussed in greater length, for instance, in [9, 15] (Maillard, unpublished manuscript). Furthermore, and far beyond lattice statistical mechanics, one even can consider these birational transformations, per se, since they naturally correspond to a very important class of discrete dynamical systems, namely the reversible [18, 34, 35] discrete dynamical systems. One such map gives rise to a family ka,b of birational maps of the plane (see [2, 20–22]). Dynamical properties of this family have been studied in a number of works [1–9, 11, 17, 23]. Recall the quantity 1 δ(k) := lim deg kn n , n→∞
which is the exponential rate of growth of the iterates of k. This is variously known as the degree complexity, the dynamical degree, or the algebraic entropy of k. When b = 0 and a is generic, δ(ka,b ) is the largest root of the polynomial x3 − x2 − 2x − 1. When b = 0 and a is generic, δ(ka,0 ) is the largest root of x2 − x − 1. The form of a map can change radically under birational equivalence: a simpler form for ka,0 which was obtained in [19] made it more accessible to detailed analysis (see [10, 12]). A basic property is that k is reversible in the sense that k = j ◦ ι is a composition of two involutions. In this case, j corresponds to lattice symmetry, and ι corresponds to matrix inversion. In this paper we give (a birationally equivalent version of) k as a composition of involutions in a new way. This shows how ka,b fits naturally into a larger family of maps. Namely, for any rational function F(y) = P(y)/Q(y), we define the involutions
x−1 y j F (x, y) = (−x + F(y), y), ι(x, y) = 1 − x − , −y − 1 − , y x−1 and the family of birational maps is given by k F = j F ◦ ι. When F is constant, the family k F is birationally equivalent to ka,0 , and when F is linear, k F
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is equivalent to ka,b . In this paper we determine the structure and degree complexity for the maps k F : Theorem 1 Let p (resp. q) denote the degree of P (resp. Q). If p < q, then for generic parameters δ(k F ) = q + 1. Otherwise, if p − q 0 is even, then for generic parameters δ(k F ) is the largest root of the polynomial x2 − ( p + 1)x − (q + 1). If p − q 0 is odd, then for generic parameters δ(k F ) is the largest root of x3 − px2 − ( p + q + 1)x − (q + 1). When k F is not generic, the growth rate δ(k F ) decreases (i.e. F → δ(k F ) is lower semicontinuous in the Zariski topology). One of the interesting things about the family is to know which parameters are not generic as well as the corresponding values of δ(k F ) is decreased. The exceptional values of a for the family ka,0 , as well as the corresponding values of δ(ka,0 ), were found by Diller and Favre [24]. Similarly, the exceptional values of (a, b ) are given in [11]. Here we look at the maximally exceptional parameters for the case where F is cubic. These are the cubic maps with the slowest degree growth and give a 2 complex parameter family of maps which are (equivalent to) automorphisms: Theorem 2 If F(y) = ay3 + ay2 + b y + 2, a = 0, then k F is an automorphism of a compact, complex surface Z . Further, the degrees of knF grow quadratically, and k F is integrable. We will analyze the family k F by inspecting the blowing-up and blowingdown behavior. That is, there are exceptional curves, which are mapped to points; and there are points of indeterminacy, which are blown up to curves. As was noted by Fornæss and Sibony [25], if there is an exceptional curve whose orbit lands on a point of indeterminacy, then the degree is not multiplicative: (deg(k F ))n = deg(knF ). The approach we use here is to replace the original domain P2 by a new manifold X . That is, we find a birational map ϕ : X → P2 , and we consider the new birational map k˜ = ϕ ◦ k F ◦ ϕ −1 . There is a well defined map k˜ ∗ : Pic(X ) → Pic(X ), and the point is to choose X so that the induced map k˜ satisfies (k˜ ∗ )n = (k˜ n )∗ . By the birational invariance of δ (see [16] and [24]) we conclude that δ(k F ) is the spectral radius of k˜ ∗ . This method has also been used by Takenawa [37–39]. The general existence of such a map k˜ when δ(k) > 1 was shown in [24]. We comment that the construction of X and k˜ can yield further information about the dynamics of k (see, for instance, [13] and [11]). The bulk of this paper is devoted to proving Theorem 1. After a division, we may rewrite F(y) = an yn + · · · + a1 y + a0 + P(y)/Q(y), where deg(P) < deg(Q). In our treatment below, we first do the cases where F(y) is a polynomial, which is in some sense the most singular and most difficult case because it involves iterated blowups to depth n. We give general properties of the map k F in Section 2. In Section 3, we describe the iterated blowup process
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in some detail. In Section 4 we carry out the blowup process to regularize k F in the case where F is a polynomial of even degree n. What we do here is to determine the action of the induced map k∗F on Pic(X ); and δ(k F ) is the spectral radius of k∗F . The case n odd is distinct, and we carry it out in Section 5. In Section 6 we handle the case where F(y) = P(y)/Q(y) with deg(P) deg(Q). That is, we present the blowup procedure, and we determine k∗F . We will see in Section 6 that the blowup process for the cases q = 0 and q p are essentially independent, since the blowup operations are performed in different places. After Sections 2–6, it is not hard to put these separate analyses together to cover the general case. The Picard group in the general case is generated by the elements produced in the independent cases, and the induced linear transformation k∗F maps them the same way it does when they are independent. Thus it is just a matter of bookkeeping to combine the two cases. Since Sections 2–5 and Section 6 are the two parts that need to be put together, and since it would be repetitive to do them simultaneously, we omit the details. The exceptional cases are also of considerable interest, but many cases arise, and it is not easy to handle them efficiently, so we do not address this issue here. As an example, however, we treat in Section 7 the case where the coefficients are as non-generic as possible when p = 3, q = 0. This leads to a proof of Theorem 2, which gives a family of automorphisms for which the degrees grow quadratically.
2 The Maps Let us set F(z) = nj=0 a j z j with n 2 and an = 0. The map k = j F ◦ ι is the composition of the two involutions defined above. The map k = [k0 : k1 : k2 ] is given in homogeneous coordinates as n k0 = x0 x1 − x20 x2 n+1 k1 = xn−1 (x2 + x0 ) + x2 0 (x1 − x0 )
2 n−1
k2 = x 2 x 0 x 1 − x 0
n
n− j 2 j a j x0 x1 − x20 x2 − x0 x1 − x1 x2
j=0
x22 − x0 x1 − x1 x2 .
(1)
Each coordinate function has degree 2n + 1, which means that deg(k) = (x0 − x1 )3n−1 x22 x20 − 2 deg(F) + 1. Since the jacobian of this map is x3n−3 0 x0 x1 − x1 x2 we have four exceptional curves : C1 := {x0 = 0}, C2 := {x0 = x1 }, C3 := {x2 = 0}, C4 := −x20 + x0 x1 + x1 x2 = 0 . When n 2 and a0 = 2, the exceptional hypersurfaces are mapped as: k : C4 → [1 : −1 + a0 : 0] ∈ C3
and C1 ∪ C2 ∪ C3 → e1 .
The points of indeterminacy for k are e1 := [0 : 1 : 0], e2 := [0 : 0 : 1], and e01 := [1 : 1 : 0].
(2)
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Fig. 1 n 2. Exceptional curves and points of indeterminacy
Figure 1 shows the relative position of the points of indeterminacy (dots with circles around them), exceptional curves, and the critical images (big dots). The information that C1 , C2 , C3 → e1 is not drawn for lack of space. The sort of singularity that will be the most difficult to deal with arises from the exceptional curve C1 → e1 ∈ C1 . In local coordinates near e1 , this looks like
tn + · · · tn−1 + · · · k[t : 1 : y] = :1: . (3) an (−y)n + · · · an (−y)n−1 + · · · For this, we will perform the blowups described in Section 3.
iterated −1 −1 is given as The inverse map k−1 = k−1 : k : k 0 1 2 n−1 n ˇ k−1 0 = x0 x2 F − x0 (x0 + x1 ) ⎛ k−1 1
= (x0 + x2 ) ⎝
n
⎞2 n− j j a j x0 x2
−
xn−1 0 (x0
+ x1 )⎠
j=0
2 n−1 n−1 ˇ x k−1 = x x + x x + x x + x ) F x − (x 2 0 1 1 2 0 2 0 2 0 0 where Fˇ = xn0 F(x2 /x0 ) =
n j=0
n− j j
a j x0 x2 . The jacobian for the inverse map is
2 ˇ ˇ xn+1 x3n−3 x22 xn0 + xn−1 − (x0 + x2 ) xn0 + xn−1 0 0 x1 − F 0 0 x1 − F The exceptional curves for k−1 are C j, 1 j 4, where ˇ C1 = C1 , C2 := xn0 + xn−1 x − F = 0 , C3 = C3 , 1 0 C4 := xn+1 − (x0 + x2 ) xn0 + xn−1 x1 − Fˇ = 0 . 0
0
k−1 : C1 ∪ C3 → e1 , C2 → e2 , and C4 → e01 ∈ C3 ,
(4)
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3 Blowups and Local Coordinate Systems In this section we discuss iterated blowups, and we explain the choices of local coordinates which will be useful in the sequel. Let π : X → C2 denote the complex manifold obtained by blowing up the origin e = (0, 0); the space is given by X = ((t, y), [ξ : η]) ∈ C2 × P ; tη = yξ , and π is projection to C2 . Let E := π −1 (e) denote the exceptional fiber over the origin, and note that π −1 is well defined over C2 − e. The closure in X of the y-axis (π −1 ({t = 0} − e)) corresponds to the hypersurface {ξ = 0} ⊂ X. On the complement {ξ = 0} set u = t and η = y/t. Then (u, η) defines a coordinate system on X \ {t = 0}, with a point being given by ((t, y), [1 : y/t]) = ((u, uη), [1 : η]). We will use the notation (u, η) L . On the set t = 0, the coordinate projection π is given in these coordinates as π L (u, η) L = (u, uη) = (t, y) ∈ C2 .
(5)
Figure 2a illustrates this blowup with emphasis on the relation between the point e and the lines t = 0 and y = 0 which contain it. The space X is drawn twice to show two choices of coordinate system; the dashed lines show where each coordinate system fails to be defined. The left hand copy of X shows the u, η-coordinate system in the complement of t = 0. The right hand side shows a different choice of coordinate; we would choose this coordinate system to work in a neighborhood of the point p1 := E ∩ {t = 0}. In the u, η coordinate system (on the upper left side of Fig. 2a), the η-axis (u = 0) represents the exceptional fiber E ∼ = P1 . The line γη = {(s, η) L : 2 s ∈ C} projects to the line {y = ηt} ⊂ C , and (0, η) L = E ∩ γη . It follows that E ∩ {y = 0} = (0, 0) L in this coordinate system. On the upper right side of Fig. 2a, we define a (ξ, v)-coordinate system on the complement of t-axis (y = 0): π R : (ξ, v) R = (t/y, y) → (vξ, v) ∈ C2 .
(6)
The exceptional fiber E is given by ξ -axis (v = 0). Next we blow up p1 = E ∩ {t = 0} = {ξ = v = 0} = (0, 0) R . Let P1 denote the exceptional fiber over p1 . The choice of a local coordinate system depends on the center of next blowup. Suppose the third blowup center is an intersection of two exceptional fibers p2 := E ∩ P1 . For this we are led to the (u, η)- coordinate system, as on the left side of Fig. 2a. Thus we have a local coordinate system on the complement of {t = 0} ∪ {y = 0}; (u1 , η1 )1 = t/y, y2 /t → (u1 , u1 η1 ) R → u21 η1 , u1 η1 ∈ C2 . (7) This (u1 , η1 )-coordinate system is defined only off the axes (t = 0) ∪ (y = 0); the new exceptional fiber P1 is given by the η1 -axis. Now we define a sequence of iterated blowups which will let us deal with the singularity (3). We start with the blowup space X as in Fig. 2b, and we continue inductively for 2 j n by setting pj := E ∩ Pj−1 and letting Pj be
Degree complexity of a family of birational maps Fig. 2 a Two choices of local coordinate systems. b Blowup of p1 in (u1 , η1 )-coordinates. c n-th iterated blowup
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η
ξ
π
π
a
η
b
c
the exceptional fiber. For each 2 j n, we use the left-hand coordinate system of Fig. 2a, which corresponds to (5). Thus we have the coordinate projection πj : Pj → C2 : j+1 j π j : (u, η)j → u j+1 η, u jη = (t, y) ∈ C2 , π −1 /t . j (t, y) = (u, η) = t/y, y (8)
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This coordinate system is defined off of {y = 0} ∪ {t = 0} ∪ P1 ∪ · · · ∪ Pj−1 . A point (0, η = c)j ∈ Pj is the landing point of the curve u → (u, c) j as u → 0, which projects to the curve u → (t(u) = u j+1 c, y(u) = u jc) ∈ C2 . In Fig. 2c, the exceptional fibers Pj, 1 j n are drawn with their fiber coordinates y j+1 /t j. 4 Mappings with q = 0 and n = Even We define a complex manifold πX : X → P2 by blowing up points e1 , q, p1 , . . . , pn−1 in the following order: (a) (b) (c) (d)
blow up e1 = [0 : 1 : 0] and let E1 denote the exceptional fiber over e1 , blow up q := E1 ∩ C4 and let Q denote the exceptional fiber over q, blow up p1 := E1 ∩ C1 and let P1 denote the exceptional fiber over p1 , blow up pj := E1 ∩ Pj−1 with exceptional fiber Pj for 2 j n − 1.
The iterated blow-up of p1 , . . . , pn−1 is exactly the process described in Section 3, so we will use the local coordinate systems defined there. That is, in a neighborhood of Q we use a (ξ1 , v1 ) = (t 2 /y, y/t) coordinate system. For E1 and Pj, 1 j n − 1 we use local coordinate systems defined in (6–8). We use homogeneous coordinates by identifying a point (t, y) ∈ C2 with [t : 1 : y] ∈ P2 . Let kX : X → X denote the induced map on the complex manifold X . In the next few lemmas, we will show that kX maps the exceptional fibers as shown in Fig. 3. Lemma 1 Under the induced map kX , the blowup fibers E1 and Pn−1 are mapped to themselves: kX : E1 ξ → −ξ/(ξ + 1) ∈ E1 Pn−1 ηn−1 → ηn−1 /(1 + an ηn−1 ) ∈ Pn−1 .
(9)
Proof First let us work on E1 . We use the local coordinate system defined in (6), so a point in the exceptional fiber E1 is (ξ, 0) R . To see the forward image of E1 we consider a nearby point (ξ, v) R → (vξ, v) with small v and we have kX (ξ, 0) R = limv→0 kX (ξ, v) R . By (1) we see that k [vξ : 1 : v] = [vξ + · · · : 1 + · · · : −v(ξ + 1) + · · · ] Fig. 3 The space X and the action of kX
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where we use · · · to indicate the higher order terms in v. As in Fig. 2a, the coordinate of the landing point in E1 is given by the ratio of t- and ycoordinates. Thus we have kX |E1 : ξ → lim k0 /k2 = lim (vξ + · · · )/(−v(ξ + 1) + · · · ) = −ξ/(ξ + 1). v→0
v→0
Now we determine the behavior of kX on Pn−1 . A fiber point (0, ηn−1 ) ∈ Pn−1 is the landing point of the arc u → (u, ηn−1 ) as u → 0. To show that kX maps Pn−1 to Pn−1 , we need to evaluate: −1 lim kX (u, ηn−1 ) = lim πn−1 ◦ k ◦ πn−1 (u, ηn−1 ).
u→0
u→0
−1 Using the formulas for πn−1 and πn−1 in (8), we obtained the desired limit.
Now we may use similar calculations to show that kX : Pj → Pn−1 ; we fix a point (0, η j) ∈ Pj and show the existence of the limit −1 lim kX u, η j = lim πn−1 ◦ k ◦ π j u, η j . u→0
u→0
Doing this, we find that the line C1 and all blowup fibers Pj, j = 1, . . . , n − 2 are all exceptional for both kX and k−1 X . And C2 is exceptional for kX : kX : C1 , C2 , P1 , · · · , Pn−2 → 1/an ∈ Pn−1 n−1 k−1 /an ∈ Pn−1 X : C1 , P1 , · · · , Pn−2 → (−1)
(10)
Combining (9–10) it is clear that the indeterminacy locus of kX consists of three points e2 , e01 , and (−1)n−1 /an ∈ Pn−1 . Lemma 2 If n is even, then the orbits of the exceptional curves C1 , C2 , P1 , . . . , Pn−2 are disjoint from the indeterminacy locus. Proof By Lemma 1, the orbit of 1/an in Pn−1 is {1/an , 1/(2an ), 1/(3an ), . . . } ⊂ Pn−1 . This is disjoint from the indeterminacy locus since it does not contain point −1/an in Pn−1 . A computation as in the proof of Lemma 1 shows that kX maps Q ↔ C3 according to: kX : Q ξ1 → [1 : a0 − ξ1 : 0] ∈ C3 , C3 [x0 : x1 : 0] → −x1 /x0 ∈ Q.
(11)
Lemma 3 If a0 = 2/m for all m > 0 then the indeterminacy locus of kX and the forward orbit of C4 under the induced map kX are disjoint. If a0 = 2/m for some m > 0, we have k2m−1 C4 = e01 . X Proof Since the forward image of C4 is [1 : −1 + a0 : 0] ∈ C3 , using (11) we have that k2m−1 C4 = [1 : ma0 − 1 : 0] ∈ C3 . Since the unique point of X
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indeterminacy in C3 is e01 , for C4 to be mapped to a point of indeterminacy, a0 must satisfy ma0 − 1 = 1 for some m 0. The following theorem comes directly from previous lemmas. Theorem 3 Suppose that n is even and a0 = 2/m for all integers m 0. Then no orbit of an exceptional curve contains a point of indeterminacy. Let us recall the Picard group Pic(X ), which is the set of all divisors in X , modulo linear equivalence, which means that D1 ∼ D2 if D1 − D2 is the divisor of a rational function. Pic(P2 ) is 1-dimensional and generated by the class of any line (hyperplane) H, and a basis of Pic(X ) is given by the class of a general hyperplane HX := π ∗ H, together with all of the blowup fibers E1 , Q, P1 , . . . , Pn−1 . If r is a rational function on X , then the pullback k∗X r := r ◦ kX is just the composition. To pull back a divisor, we just pull back its defining functions. This gives the pullback map k∗X : Pic(X ) → Pic(X ). Thus from (9–10) we see that the pullback of E1 is E1 and the pulling back of most of basis elements are trivial, that is k∗X Pj = 0 for all j = 1, . . . , n − 2. Next we pull back HX . Since k has degree 2n + 1 we have k∗ H = (2n + 1)H in Pic(P2 ). Now we pull back by πX∗ to obtain: (2n + 1)HX = πX∗ (2n + 1)H = πX∗ (k∗ H).
(12)
∗
A line is given by {h := α0 x0 + α1 x1 + α2 x2 = 0}, so k H is the divisor defined by h ◦ k = j α jk j. To write this divisor as a linear combination of basis elements HX , E1 , Q, P1 , . . . , Pn−1 , we need to check the order of vanishing of h ◦ k at all of these sets. Let us start with the coordinate system πX (ξ, v) = [vξ : 1 : v] near E1 , defined in Section 3. Using the expression for k given in Section 2 we see that α0 k0 + α1 k1 + α2 k2 vanishes to order n in v. It follows that πX∗ k∗ H vanishes at E1 with multiplicity n. Similar computations for all other basis elements gives us πX∗ k∗ H = k∗X HX + nE1 + (n + 1)Q + (n + 1) j jPj. Combining with (12) we have k∗X HX = (2n + 1)HX − nE1 − (n + 1)Q − (n + 1)
n−1
jPj.
(13)
j=1
Similarly, we obtain: k∗X : Q → HX − E1 − Q − P1 − 2P2 − · · · − (n − 1)Pn−1 Pn−1 → 2HX − E1 − Q − P1 − 2P2 − · · · − (n − 1)Pn−1 .
(14)
Theorem 4 q = 0 and n = even. Suppose F(z) = nj=1 a j z j is an even degree polynomial associated with j F . If a0 = 2/m for any positive integer m, then the degree complexity is the largest root of the quadratic polynomial x2 − (n + 1)x − 1. Proof Since P1 , . . . , Pn−2 are mapped to 0 under the action on cohomology, it suffices to consider the action restricted to HX , E1 , Q, and Pn−1 .
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By (13, 14) the matrix representation of k∗X , restricted to the ordered basis {HX , E1 , Q, Pn−1 }, is ⎞ ⎛ 2n + 1 0 1 2 ⎜ −n 1 −1 −1 ⎟ ⎟. ⎜ ⎝ −n − 1 0 −1 −1 ⎠ −n2 + 1 0 −n + 1 −n + 1 The characteristic polynomial is x(x − 1)(x2 − (n + 1)x − 1).
5 Mappings with q = 0 and n = Odd Let us start with the space X from Section 4. When n is odd, we see from (10) that the image of all exceptional lines of kX coincide with a point of indeterminacy in pn ∈ Pn−1 . Let πY : Y → P2 be the complex manifold obtained by blowing up X at the point pn , and let Pn denote the exceptional fiber over pn . In the un−1 , ηn−1 coordinate system, pn has coordinate (0, 1/an )n−1 . Thus, at Pn , we use the coordinate projection: πn : Y (u, η)n → (un (uη + 1/an ), un−1 (uη + 1/an )) ∈ C2 . Most computations in the previous section remain valid for n odd. Thus Lemma 3, (9) and (11) are still valid for the induced map kY : Y → Y . Under kY curves C1 , C2 , P1 , . . . , Pn−3 are still exceptional: kY : C1 , C2 , P1 , . . . , Pn−3 → −an−1 /a2n ∈ Pn 2 k−1 Y : C1 , P1 , . . . , Pn−3 → (an−1 − (n − 1)an )/an ∈ Pn .
(15)
The blowup fibers Pn and Pn−2 form a two cycle, kY : Pn ↔ Pn−2 and Pn−1 is mapped to itself as before. It follows that the points of indeterminacy for kY are e2 , e01 and (an−1 − (n − 1)an )/a2n ∈ Pn . For all m 0, we have 2 2 k2m Y : Pn −an−1 /an → (2man − (2m + 1)an−1 )/an ∈ Pn
(16)
The induced action of k on Y is pictured in Fig. 4. As a consequence of (15) and (16) we have: Fig. 4 The space Y and the action of kY
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Lemma 4 If n is odd, and if (2m + 2)an−1 = (2m + n − 1)an
(17)
for all m 0, then the forward orbits of C1 , C2 , P1 , . . . , Pn−3 under kY do not contain any point of indeterminacy. Combining Lemmas 3 and 4 we have Theorem 5 Suppose that n is odd, a0 = 2/m for all m > 0, and an−1 = (n − 1)an /2. Then the forward orbits of exceptional curves do not contain any points of indeterminacy. To determine kY , we use the basis {HY , E1 , Q, P1 , . . . , Pn } for Pic(Y ). Now the exceptional lines C1 , C2 , P1 , . . . , Pn−2 are mapped to Pn . Let {C1 } ∈ Pic(Y ) denote the class of the strict transform of C1 , i.e., the closure in Y of πY−1 (C1 − centers of blowup). (The curve C2 does not pass through any center of blowup, so with the same notation we have {C2 } = HY ∈ Pic(Y ).) In order to write {C1 = (x0 = 0)} in terms of our basis, we note first that πY−1 C1 = C4 ∪ E1 ∪ Q ∪ P1 ∪ · · · ∪ Pn−1 , i.e., the pullback function x0 ◦ πY vanishes on all of these curves. Thus we have to compute the multiplicities of vanishing. At Pn−1 , for instance, we consider the (un−1 , ηn−1 ) coordinate system defined in (8), and we see that k∗Y x0 vanishes to order n at Pn−1 = (un−1 = 0). Similarly we can compute the multiplicities for E1 , Q, P1 , . . . , Pn−2 and Pn , so HY = πY∗ C1 = {C1 } + E1 + Q + 2P1 + 3P2 + · · · + nPn−1 + nPn . It follows that k∗Y Pn = {C1 } + {C2 } +
n−2
Pj = 2HY − E1 − Q −
j=1
n−2
jPj − nPn−1 − nPn .
j=1
For the rest of basis entries we have k∗Y : HY → (2n + 1)HY − nE1 − (n + 1)Q − (n + 1)
n−1
jPj − n2 Pn
j=1
Q → HY − E1 − Q − P1 − 2P2 − · · · − (n − 1)Pn−1 − (n − 1)Pn , E1 → E1 ,
Pn−2 → Pn ,
and Pn−1 → Pn−1 .
Theorem 1: q = 0 and n = odd. If a0 = 2/m for all m > 0, then the degree complexity is the largest root of the cubic polynomial x3 − nx2 − (n + 1)x − 1. Proof The classes of the exceptional fibers P1 , · · · , Pn−3 are all mapped to 0, and exceptional fibers E1 and Pn−1 are simply interchanged. It follows that to get the spectral radius of k∗Y we only need to consider 4 × 4 matrix with ordered basis {HX , Q, Pn−2 , Pn } and the spectral radius is given by the largest root of x3 − nx2 − (n + 1)x − 1.
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6 Mappings with p q q Now we consider the case F(w) = P(w)/ =1 (w − β ), where the degree of P is no greater than q. In this case we have a limit λ0 = limw→∞ F(w), and λ0 = 0 q if p = q, and λ0 = 0 if p < q. We see that E ( jF ) = =1 {y = β }. In case p q, we have E (k F ) = C2 ∪ C3 ∪ C4 ∪
q
D
=1
where D = ι{y = β }. This is different from the previous case (in Section 2) in several ways: (1) C1 is no longer exceptional; (2) C2 is mapped to e2 instead of e1 , and (3) D , 1 q, are exceptional. As before, we start by blowing up e1 to create an exceptional fiber E1 , and as before, C3 maps to a point q ∈ E1 , which is indeterminate. So we also blow up q to obtain an exceptional fiber Q. Finally, we blow up the indeterminate point e2 to create an exceptional fiber E2 . Figure 5b shows the mapping of exceptional curves under the induced map k at this stage. We note that the intersection
λ
b
a
λ
c
d
Fig. 5 a p q. Exceptional curves and points of indeterminacy. b p = q. Mapping of the exceptional fibers after first blowups. c p = q. Mapping of the exceptional fibers at the second stage. d The case p < q
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E2 ∩ C4 ∩ D is indeterminate, and it corresponds to fiber coordinate equal to zero. We have drawn the case λ0 = 0, in which case the exceptional image of C2 never encounters a point of indeterminacy because λ0 ↔ 1 is a 2-cycle. As before, we see that the orbit of C4 never encounters the point of indeterminacy C2 ∩ C3 ∩ C4 if the parameters are generic. Specifically, as in Section 4, we need F(0) = 2/m for any positive integer m. It remains to look at the orbits of the curves D . If we use the coordinate system (s, ξ ) → [s : 1 : sξ ] for E1 , then the induced map on E1 maps to E1 and is the involution ξ → −(ξ + 1). The image of D is then ξ = β ∈ E1 , which is mapped to −(1 + β ). On the other hand, k−1 F : {y = β } → {ξ = −(β + 1)} ∈ E1 , which means that −(β + 1) is indeterminate for k F . So now at the second stage, we blow up the points β and −(β + 1) in E1 , 1 q. The exceptional curves for the induced map are now C4 , C2 , and D , 1 q. Figure 5c shows how the exceptional fibers map at this stage. We see that for generic parameters, these orbits do not meet the indeterminacy locus. Now we will describe the behavior of k∗F on the Picard group in terms of the ordered basis L, E1 , Q, E2 , A , B , 1 q. We see that k∗F acts as: E1 ↔ E1 , E2 ↔ E2 , Q → C3 , B → A → D + B .
(18)
To make use of (18) we must write C3 and D in terms of our basis. In P2 , we have L = C3 . Now if we move “up,” taking the pullback π ∗ as we make the various blowups, we find that at the second stage we have L = C3 + E 1 + Q +
q (A + B ), =1
and this gives us C3 in terms of our basis. Similarly, we start with D = 2L in P2 since D has degree 2. After the first stage of blowups, we have D + E1 + E2 + Q = 2L. For the second stage of blowups, we see from Fig. 5b that one of the centers of blowup (the one that produces B ) belongs to both E1 and D . Thus we have an “extra” B : q D + B + E1 + E2 + (Bs + As ) + Q = 2L. s=1
This gives D in terms of our basis. Finally, we need to express k∗F L in terms of our basis. We have {k−1 L} = (2q + 3)L in P2 because k F has degree 2q + 3. Now when we blow up e1 , for instance, we will obtain a fiber E1 with multiplicity. To determine the multiplicity we work in local coordinates (s, η) → [sη E1 = {s = 0}. : 1 : s] near We write a generic line as L = a j x j , so k−1 a jx j = a jk j = 0 . In
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the (s, η) coordinates, we have a jk j[sη : 1 : s] = sq+1 ϕ(s, η), where ϕ(0, η) is not identically zero. Thus the multiplicity of E1 is q + 1, or −1 k F L + (q + 1)E1 = (2q + 3)L at the next level. Repeating this argument for the various blowup fibers, we find −1 k L + (q + 1)E1 + (q + 1)E2 + (q + 2)Q + ((q + 1)A + (q + 2)B ) = (2q + 3)L, (19) +
{k−1 F L}
which gives us in terms of our basis. Using (18–19), we may write k∗F on the Picard group, and we find that its characteristic polynomial is x2 − (q + 1)x − (q + 1). This proves Theorem 1 in the case p = q. Next we consider the case where p < q, which means that λ0 = 0. Thus after the second stage of blowups, we see in Fig. 5c that C2 now maps to the point of indeterminacy 0 ∈ E2 . Now we blow up 0 ∈ E2 , creating a new fiber G. We find that on our new manifold, the curve C2 is no longer exceptional and maps onto G. Thus we add G to our ordered basis in the Picard group. The action of k∗F is changed in the following ways. First, we now have E2 → E2 , G → C2 = L − E2 − G. Next, there is a change in D . Since G was obtained by blowing up the (transversal) intersection point of D and E2 , the expression D = 2L− E2 − · · · is changed to D = 2L − E2 − 2G − · · · . Last, we subtract an extra (2q + 2)G from k∗F (L). With this new expression for k∗F , we obtain the characteristic polynomial −(x − q − 1)(x − 1)3 (x + 1)x2q , and this proves Theorem 1 in the case p < q. If n = 1, we have F(w) = aw + P(w)/Q(w), where deg(P) deg(Q). The situation is like what we have just done in this section, with the added fact that C2 → [0 : a : 1] → e2 ∈ I(k F ). We blow up [0 : a : 1] and e2 , creating new fibers M and E2 . The induced map behaves like C2 → ∗ ∈ M ↔ E 2 . For generic parameters, the orbit of C2 does not encounter the indeterminacy locus. To finish the proof, now, we go back and repeat the earlier parts of Section 6. Proof of Theorem 1 In order to prove Theorem 1 in general, we first do Section 6, which covers the cases n = 0 and n − 1. If n 2, we go back to Section 4 or Section 5, according to whether n is even or odd. The associated Picard group will be larger because of the iterated blowups over the point
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0 ∈ E1 . However, the fibers arising from the iterated blowup are disjoint from the blowups in Section 6 and so they still map the same as in Sections 4, 5, and the multiplicities of the pullback of a general line are the same. This gives us k∗F in this case. Finding the characteristic polynomial in this case gives us Theorem 1.
7 Degree 3: A Family of Automorphisms Let us consider the 2 parameter family of maps k = j F ◦ ι where F(z) = az3 + az2 + bz + 2 with a = 0. We consider the complex manifold πZ : Z → P2 obtained by blowing up 6 points e2 , e01 , p4 , p5 , p6 , r in the complex manifold Y constructed in Section 5. As we construct the blowups, we will let E2 , E01 , P4 , P5 , P6 and R denote the exceptional fibers over e2 , e01 , p4 , p5 , p6 , and r respectively. Specifically, we blow up e2 and e01 and then: p4 := −1/a ∈ P3 , p5 := (2 − b )/a ∈ P4 , p6 := (2b − 2 − a)/a2 ∈ P5 , and r := 0 ∈ E2 ∩ {x1 = 0}. We define the local coordinate system in a similar way we define local coordinates in Section 3. Using these local coordinates we can easily verify that under the induced map kZ we have C1 → P4 → C1 , E2 → P5 → E2 , C4 → E01 → C4 , and C2 → P6 → R → C2
and all mappings are dominant and holomorphic. For example, let us consider E2 . We may use coordinates w, ζ which are mapped by π E2 : (x, ζ ) → [w : wζ : 1] ∈ P2 . Thus E2 = (w = 0) is given by ζ -axis in this coordinate system and by considering limw→0 π P−15 ◦ k ◦ π E2 (w, ζ ) we find: kZ : E2 ζ → (2b − a − ζ − 1)/a2 ∈ P5 . The mapping among the exceptional fibers is shown in Fig. 6. What is not shown is that R → C2 and E01 → C4
Theorem 6 Suppose F(z) = az3 + az2 + bz + 2 with a = 0. Then the induced map kZ is biholomorphic. Proof Since kZ and k−1 Z have no exceptional hypersurface, indeterminacy locus for kZ is empty. It follows that kZ is an automorphism of Z .
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Fig. 6 The space Z and action of fZ
Repeating the argument in previous two sections, we have that k∗Z acts on each basis element as follows: HZ → 7HZ − 3E1 − 4P1 − 8P2 − 9P3 − 10P4 − 10P5 − −10P6 − 3E2 − 6R − 4Q − 4E01 , E1 → E1 , P1 → P3 → P1 , and P2 → P2 , P4 → HZ − E1 − 2P1 − 3P2 − 3P3 − 3P4 − 3P5 − 3P6 − E2 − R − Q, P5 → E2 , P6 → HY − E2 − R − E01 , E2 → P5 , and E01 → P6 , Q → HZ − E1 − P1 − 2P2 − 2P3 − 2P4 − 2P5 − 2P6 − Q − E01 , E01 → 2HZ − E1 − P1 − 2P2 − 2P3 − 2P4 − 2P5 − 2P6 − −E2 − 2R − 2Q − E01 . Theorem 7 Suppose F(z) = az3 + az2 + bz + 2 with a = 0. Then the degree of kn = k ◦ · · · ◦ k grows quadratically, and k is integrable. Proof All the eigenvalues of the characteristic polynomial of k∗Z have modulus one. The largest Jordan block in the matrix representation of k∗Z is a 3 × 3 block corresponding to the eigenvalue 1. Thus the growth rate of the powers of the matrix is quadratic. Integrability follows from more general results: Gizatullin [26] showed that if the growth rate is quadratic, then there is an invariant fibration by elliptic curves. In this case, we can give an explicit invariant. If we define φ = φ1 /φ2 to be the quotient of the following two polynomials; φ1 [x0 : x1 : x2 ] = x20 x22 , φ2 [x0 : x1 : x2 ] = −2x40 + 4x30 x1 − (2 + a)x20 x21 + 2ax1 x22 (x0 + x2 ) − −2b (x30 x2 − x20 x1 x2 ), then φ ◦ k = φ.
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References 1. Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: From YangBaxter equations to dynamical zeta functions for birational transformations. Statistical physics on the eve of the 21st century. Ser. Adv. Statist. Mech. 14, 436–490 (1999) 2. Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: Rational dynamical zeta functions for birational transformations. Phys. A 264, 264–293, chao-dyn/9807014 (1999) 3. Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: Topological entropy and complexity for discrete dynamical systems. Phys. Lett. A 262, 44–49, chao-dyn/9806026 (1999) 4. Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Maillard, J.-M.: Growth complexity spectrum of some discrete dynamical systems. Phys. D 130(1–2), 27–42 (1999) 5. Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Maillard, J.-M.: Real topological entropy versus metric entropy for birational measure-preserving transformations. Phys. D 144(3–4), 387–433 (2000) 6. Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: Real Arnold complexity versus real topological entropy for birational transformations. J. Phys. A 33(8), 1465–1501 (2000) 7. Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: Topological entropy and Arnold complexity for two-dimensional mappings. Phys. Lett. A 262(1), 44–49 (1999) 8. Abarenkova, N., Anglès d’Auriac, J.-C., Boukraa, S., Maillard, J.-M.: Elliptic curves from finite order recursions or non-involutive permutations for discrete dynamical systems and lattice statistical mechanics. Eur. Phys. J. B, 647–661 (1998) 9. Anglès d’Auriac, J.-C., Boukraa, S., Maillard, J.-M.: Functional relations in lattice statistical mechanics, enumerative combinatorics and discrete dynamical systems. Ann. Comb. 3, 131–158 (1999) 10. Bedford, E., Diller, J.: Real and complex dynamics of a family of birational maps of the plane: the golden mean subshift. Amer. Math. J. 127(3), 595–646 (2005) 11. Bedford, E., Diller, J.: Dynamics of a two parameter family of plane birational maps: maximal entropy. J. Geom. Anal. 16(3), 409–430 (2006) 12. Bedford, E., Diller, J.: Real dynamics of a family of plane birational maps: trapping regions and entropy zero. arXiv.math/0609113 (2006) 13. Bedford, E., Kim, K.H.: Dynamics of rational surface automorphisms: linear fractional recurrences. arXiv:math/0611297 (2007) 14. Bellon, M.P., Maillard, J.-M., Viallet, C.-M.: Quasi-integrability of the sixteen vertex model. Phys. Lett. B 281, 315–319 (1992) 15. Bellon, M.P., Maillard, J.-M., Viallet, C.-M.: Dynamical systems from quantum integrability. In: Maillard, J.-M. (ed.) Proceedings of the Conference “Yang-Baxter Equations in Paris”, pp. 95–124 World Scientific, Singapore (1993) (also published as a supplement of Int. J. Mod. Phys.) 16. Bellon, M., Viallet, C.: Algebraic entropy. Comm. Math. Phys. 204, 425–437 (1999) 17. Bose, R.C., Mesner, D.M.: On linear associative algebras corresponding to association schemes of partially balanced designs. Ann. Math. Statist. 10, 21–38 (1959) 18. Bouamra, M., Boukraa, S., Hassani, S., Maillard, J.-M.: Post-critical set and preserved meromorphic two-forms. J. Phys. A 38, 7957–7988, nlin.CD/0505024 v1 (2005) 19. Boukraa, S., Hassani, S., Maillard, J.-M.: Product of involutions and fixed points. Alg. Rev. Nucl. Sci. 2, 1–16 (1998) 20. Boukraa, S., Maillard, J.-M.: Factorization properties of birational mappings. Phys. A 220, 403–470 (1995) 21. Boukraa, S., Maillard, J.-M., Rollet, G.: Almost integrable mappings. Int. J. Mod. Phys. B8, 137–174 (1994) 22. Boukraa, S., Maillard, J.-M., Rollet, G.: Integrable mappings and polynomial growth. Phys. A 209, 162–222 (1994) 23. Boukraa, S., Maillard, J.-M., Rollet, G.: Determinental identities on integrable mappings. Int. J. Mod. Phys. B8, 2157–2201 (1994)
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Math Phys Anal Geom (2008) 11:73–86 DOI 10.1007/s11040-008-9040-0
Phase Vortex: A Dynamical System Approach Luis Fernando Mello · Denis de Carvalho Braga
Received: 22 February 2008 / Accepted: 8 April 2008 / Published online: 20 May 2008 © Springer Science + Business Media B.V. 2008
Abstract In this paper we study the flow lines defined by integral curves of the current field Im ψ ∗ ∇ψ associated to a complex scalar field ψ near a general phase vortex. This study naturally leads to the center–focus problem. Sufficient conditions for spiral behavior of the flow lines near a vortex in terms of the Lyapunov numbers are given. Keywords Phase vortex · Flow lines · Center–focus problem · Lyapunov numbers · Degenerate center Mathematics Subject Classifications (2000) 34D99 · 34C07 · 34C60
1 Introduction Consider a complex scalar function (wave function) ψ(x, y) = ρ(x, y) exp iχ(x, y) = u(x, y) + iv(x, y),
(1)
with ψ(0, 0) = 0 and assume that ψ does not depend on time. For a given ψ, it is well defined the vector field (current field) J (x, y) = Im ψ ∗ ∇ψ = ρ 2 ∇χ = u(x, y)∇v(x, y) − v(x, y)∇u(x, y).
L. F. Mello (B) Instituto de Ciências Exatas, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, CEP 37.500-903, Itajubá, Minas Gerais, Brazil e-mail: [email protected] D. de Carvalho Braga Instituto de Sistemas Elétricos e Energia, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, CEP 37.500-903, Itajubá, Minas Gerais, Brazil e-mail: [email protected]
(2)
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The origin is a singularity (equilibrium point) of J and it is called a vortex. In the context of quantum mechanics J is the probability current. For more details of interpretation of ψ as well as of J see the recent paper [5] and references therein. There is a large amount of work devoted to the vortex models within the two main areas where such objects are studied: the Bose– condensate physics with vortices and the field theory of superfluid media. Concerning vortices in condensates, the reader is referred to [10, 18] while concerning vortices in superfluid media, the reader is referred to [3, 7, 17]. The aim of this paper is the study of the generic behavior of the flow lines, which are the integral curves of J , near the vortex at the origin from the dynamical system point of view. The vortex is called a center if there exists a neighborhood V of the origin filled by closed flow lines and it is called a focus if the flow lines spiral in or out of the vortex in V. This paper is based on the recent article of M. Berry [5] and answers the following two questions that are not clear in the Berry’s paper: 1. What are the hypotheses under which the flow lines of the current field circulate the vortex? 2. In the case where the Berry’s analysis fails (K = 0 in equation (9) of [5]), what can be said about the behavior of the flow lines near the vortex? The results of the present paper extend in a different direction the analysis in [5]. Our main goal is the study of isolated degenerate vortices. In Section 2 we show that the vortex at the origin is either a center or a focus under a generic condition. The geometrical behavior of the flow lines near the vortex is studied in Section 3 via Lyapunov numbers. Some concluding remarks are presented in Section 4. An overview on the Lyapunov numbers is presented in the Appendix.
2 Flow Lines near the Vortex In this section we will study the trajectories of the planar differential equation X = J (X ),
(3)
near the vortex at the origin. Here X = (x, y). As ψ(0, 0) = 0 one has u(0, 0) = v(0, 0) = 0. This implies that the sets A = u−1 (0) and B = v −1 (0) have intersection at the origin. The description of the vortex in terms of u(x, y) = 0 and v(x, y) = 0 is essentially the Clebsch–variable representation of vortices [14]. The following general hypothesis is fundamental for what follows, which we assume henceforth: T: The sets A and B are regular curves that meet transversally at the origin. Lemma 1 Under the hypothesis T the current field (2) has a vortex at the origin which is either a center or a focus.
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Proof Write the Taylor expansion of the functions u and v near the vortex at the origin v(x, y) = cx + dy + O ||(x, y)||2 . u(x, y) = ax + by + O ||(x, y)||2 , From (2) the current field is given by J (x, y) = (bc − ad)y + O ||(x, y)||2 ,
(ad − bc)x + O ||(x, y)||2 .
Therefore the Jacobian matrix of J (x, y) at the origin has the form ⎛ ⎞ 0 bc − ad ⎠ DJ (0, 0) = ⎝ ad − bc 0 and its eigenvalues are λ1,2 = ±iω0 , where ω0 = ad − bc. But ∇u(0, 0) = (a, b ) and ∇v(0, 0) = (c, d) are linearly independent (condition (T)). Thus ω0 = 0 and the origin is either a center or a focus. If u(x, y) = ax + by and v(x, y) = cx + dy are linear with ad − bc = 0 then J (x, y) = ((bc − ad)y, (ad − bc)x). This current field has a vortex at the origin which is a center under the hypothesis T. Therefore the possibility to find a vortex of focus-type is due to the presence of the higher-order terms, or equivalently due to the nonlinearity of the current field. Without the validity of the general hypothesis T the flow lines of the current field not necessarily circulate the vortex. As an example, take the wave function ψ(x, y) = y + i(x2 + y2 ). The current field given by J (x, y) = (2xy, y2 − x2 ) has an isolated vortex at the origin called dipole. The flow lines of this current field are illustrated in Fig. 1. Another example is given by the Fig. 1 Flow lines of the current field J (x, y) = 2xy, y2 − x2
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wave function ψ(x, y) = (y − x2 ) + iy whose current field J (x, y) = (2xy, −x2 ) has whole the y-axis as singular points. The problem of to distinguish whether an equilibrium point of a planar vector field with pure imaginary eigenvalues is a center or a focus is called the center-focus problem. For analytic vector field it was solved by Lyapunov, who gave a set of polynomial conditions on the coefficients of the vector field in order to have an equilibrium point of center type. These expressions are called Lyapunov numbers (constants). There is a large literature on algorithms to compute these constants and the reader is referred to the paper of Gasull and Torregrosa [9] and references therein. Without loss of generality take ω0 = 1. Consider (3) in the form x = −y + P(x, y), y = x + Q(x, y),
(4)
where P and Q have Taylor expansions at the origin beginning with quadratic terms at least. Differential equation (4) has the form ∞
dr Rn (φ)rn , = dφ n=2
(5)
in polar coordinates (r, φ), where Rn (φ) are homogeneous trigonometric polynomials [9]. Denote by r(φ, r0 ) the solution of (5) by r = r0 at φ = 0. Thus r(φ, r0 ) = r0 +
∞
αn (φ)r0n ,
n=2
for r small, where αn (0) = 0 for all n 2. The Poincaré map (first return map) can be defined as P (r0 ) = r(2π, r0 ) = r0 +
∞
αn (2π )r0n .
(6)
n=2
The first n such that ln = αn (2π ) = 0 is always odd (see [9], p. 164), thus n = 2k + 1 and l2k+1 is called the k-th Lyapunov number of (4). In this case the vortex is called a weak focus of order k. The sign of the Lyapunov number l2k+1 quantifies the spiralling at the vortex: outwards when l2k+1 > 0 and inwards when l2k+1 < 0. If αk (2π ) = 0 for all k 2 then the origin is a center. The next well–known lemma gives a simple sufficient condition for the vortex to be a center. Lemma 2 If the wave function ψ is either holomorphic or antiholomorphic in the complex sense then the vortex at the origin is a center.
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Proof The divergence of J is given by ∇ · J (x, y) = ∇ · u(x, y)∇v(x, y) − v(x, y)∇u(x, y) = u(x, y) v(x, y) − v(x, y) u(x, y) ≡ 0, since u and v are harmonic functions. The lemma follows from Liouville’s Theorem [2] with says that the area of the phase plane is preserved by the flow of a vector field whose divergence vanishes. In [5] Berry gives a method for the calculation of the first Lyapunov number l3 (denoted by K) in terms of the Taylor expansion of the wave function at the vortex (see [5], equation (11) p. L747). In the next section we extend the analysis of Berry for other Lyapunov numbers.
3 Phase Vortex and Lyapunov Numbers In the Appendix we give some explicit expressions for the calculation of the Lyapunov numbers used in this section. Consider the class of wave functions of the form ψ(r, φ) = exp(imφ) rm f r2 , (7) which represent waves with a well–defined orbital angular momentum quantum number m [5]. Here f r2 is a smooth function of r2 . The case where m = 1 and f r2 = 1 + i br2 , b = 0, was analyzed in [5], resulting that if b > 0 (resp. b < 0) then the vortex is a repelling (resp. attractor) focus. The following proposition gives an example of wave function with a vortex at the origin which is a weak focus of arbitrary order. Proposition 1 Fix a positive integer k 1. If m = 1 and k f r2 = 1 + ia r2 , where a = 0, then the current field J associated to the wave function (7) has a vortex at the origin which is a weak focus of order k. Proof It is immediate that the wave function (7) has the form k
k
ψ(x, y) = x − ay x2 + y2 + i y + ax x2 + y2 . The current field J defines the following differential equations k 2k x = −y + 2akx x2 + y2 − a2 y x2 + y2 , k 2k y = x + 2aky x2 + y2 + a2 x x2 + y2 .
(8)
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In polar coordinates the above differential equations have the form r = 2ak r2k+1 , φ = 1 + a2r4k . Thus differential equation (5) can be written as dr r2k+1 = 2ak . dφ 1 + a2r4k Therefore we have l3 = 0, . . . , l2k−1 = 0,
l2k+1 = 4πak = 0.
The proposition is proved. In terms of the notations in the Appendix the proposition can be proved as follows. Let w = x + iy. Define the functions k s(x, y) = x2 + y2 , f (w, w) ¯ = u (w + w)/2, ¯ (w − w)/2i ¯ , g(w, w) ¯ = v (w + w)/2, ¯ (w − w)/2i ¯ , ¯ (w − w)/2i ¯ . h(w, w) ¯ = s (w + w)/2, ¯ (w − w)/2i ¯ , G(w, w) ¯ = J (w + w)/2, It follows that G(w, w) ¯ = 2 f (w, w) ¯
∂g ∂f (w, w) ¯ − 2g(w, w) ¯ (w, w). ¯ ∂ w¯ ∂ w¯
By a long but simple calculation one has G(w, w) ¯ = i 1 + a2 |w|2k w + 2ka w|w|2k .
(9)
From (19) and (9) the proposition is proved since the differential equation (3) ¯ In this case we have can be written as w = G(w, w). l3 = 0, . . . , l2k−1 = 0,
l2k+1 = 2ak = 0.
By induction on m Proposition 1 can be extended to the following one. Proposition 2 Fix a positive integer k 1. If m 1 and k f r2 = 1 + ia r2 , where a = 0, then the current field J associated to the wave function (7) has a vortex at the origin which is a weak focus of order k. Propositions 1 and 2 give examples of wave functions whose current fields have vortices at the origin which are weak foci of order k. As a consequence the flow lines spiral more and more slowly in (a < 0) or out (a > 0) the vortex
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Fig. 2 Flow lines of the current field (8) for a = −1 and k = 2
as k increases and the distance between successive windings decreases as r2k+1 near the vortex. The vortex is a center if and only if a = 0. The case studied by Berry [5] can be obtained from Proposition 1 taking k = 1. The flow lines of the current field (8) near the vortex at the origin for a = −1 and k = 2 are illustrated in Fig. 2. Since a < 0 the vortex is a weak attractor focus of order 2. In Fig. 3 are depicted the flow lines of the current field defined by the 2 wave function (7) with m = 2 and f r2 = 1 + i r2 , that is, a = 1 and k = 2. According to Proposition 2 the vortex at the origin is a weak repelling focus of order 2.
Fig. 3 Flow lines of the current field induced by (7) with m = 2 and 2 f r2 = 1 + i r2
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Consider a long solenoid of small transverse cross section endowed with a magnetic flux . The limit configuration when the cross section becomes vanishingly small while the magnetic flux enclosed is kept constant is called a magnetic string. The magnetic field vanishes everywhere except inside the magnetic string and, as there is no Lorentz force, charged particles around the string are not affected by it, according to the classical mechanics point of view. Nevertheless, there is a nontrivial quantum scattering due to the Aharonov– Bohm effect. See references [13] and [16]. The nontrivial scattering mentioned above can be studied from the following family of current fields k δ δ y x , (10) J (x, y) = ,− −1 + M k x2 + y2 k x2 + y2 where 0 δ = e /(2π c) 1/2 is the flux parameter, M and e are the mass and the charge of the particle, respectively, and 0 k < ∞ is associated to the energy 2 k2 /2M for a stationary state. See [12, 13, 16] for more details. Consider the modified current field −M 2 k 2 J˜ (x, y) = (11) (x + y2 )J (x, y) = −y + x + y2 , x , δ δ obtained from the current field (10) as δ = 0. It follows that the current field J in (10) has a vortex of center–type at the origin if and only if the modified current field J˜ in (11) has a vortex of center–type at the origin. Direct calculation leads to l3 = l5 = l7 = 0 (see the Appendix), and the vortex at the origin is a center for all 0 < δ 1/2. See Fig. 4.
Fig. 4 Flow lines of the current field (10) for 0 < δ 1/2
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Fig. 5 Flow lines of the current field J (x, y) = (−y, x + k(x2 + y2 )) for k = 0
For δ = 0 there is no vortex at the origin and the flow lines of the current field (10) are parallel. These vortices were predicted 30 years ago and observed in an analogue experiment with water waves [4]. Also, the conclusion that the vortex at the origin is of center-type for 0 < δ 1/2 can be obtained from the vanishing of the divergence of the current field. Another example of physical interest is the wave function ψ(x, y) = (x + iy) exp(iky), k ∈ R, of a simple dislocated wave which is an approximation to a solution of a Helmholtz equation [15]. The current field is given by J (x, y) = (−y, x + k(x2 + y2 )). Direct calculation leads to l3 = l5 = l7 = 0 (see the Appendix), and the vortex at the origin is a center for all k ∈ R. If k = 0 whole R2 − {0, 0} is filled by closed flow lines. But if k = 0 the neighborhood of the vortex filled by closed flow lines is bounded. See Fig. 5. Theorem 1 Consider the wave function ak 2 x x + y2 +aG(x, y) + 2 ak 2 2 +i x− y x + y + aH(x, y) , 2
ψ(x, y) = −y −
(12)
where G and H are analytic functions having Taylor expansions at the origin beginning with terms of degree 4 at least, k 1 is an integer number and a ∈ R. Then the first Lyapunov number is l3 = 4ak and the vortex at the origin is a center if and only if a = 0.
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Proof The differential equations defined by the current field obtained from the wave function (12) are given by x = −y + ak x x2 + y2 + aP(x, y, a), y = x + ak y x2 + y2 + aQ(x, y, a),
(13)
where 2 1 P(x, y, a) = − a2k−1 y x2 + y2 + 1 − ak xy G(x, y) + 4 ∂G 1 k 2 1 k 2 2 2 (x, y) − + a (3x + y )H(x, y) − x − a y x + y 2 2 ∂x ∂H ∂H 1 k 2 2 − y+ a x x +y (x, y) + aG(x, y) (x, y) − 2 ∂x ∂x − aH(x, y) Q(x, y, a) =
∂G (x, y), ∂x
2 1 2k−1 2 a x x + y2 + (1 + ak xy)H(x, y) − 4 ∂G 1 1 (x, y) − − ak (x2 + 3y2 )G(x, y) − x − ak y x2 + y2 2 2 ∂y ∂H 1 ∂H − y + a k x x2 + y2 (x, y) + aG(x, y) (x, y) − 2 ∂y ∂y − aH(x, y)
∂G (x, y). ∂y
With the notations of the Appendix one has A=
0 −1 , 1 0
q=
C(x, x, x) =
i , 1
p=
i/2 , 1/2
B(x, x) =
0 , 0
2ak 3x1 y1 z1 + x2 y2 z1 + x2 y1 z2 + x1 y2 z2 . 2ak x2 y1 z1 + x1 y2 z1 + x1 y1 z2 + 3x2 y2 z2
¯ which gives G21 = 8ak . From (20) the first Therefore G21 = p, C(q, q, q) k Lyapunov number is l3 = 4a .
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In the above example l3 = 4ak = 0 if a = 0 but the number of the limit cycles bifurcating from the vortex can be high. In fact the number of the limit cycles bifurcating from the vortex is at most k − 1. See the recent paper Gasull and Giné (unpublished manuscript).
4 Concluding Remarks A vortex is a monodromic vortex if there is no flow line tending to the vortex with definite tangent at this point. When the current field is analytic and the vortex is monodromic then it is either a center or a focus. For a general vortex the following two open problems generalize the two initial questions in Section 1: 1. Monodromic Problem: determine hypotheses (on the wave function) under which the isolated vortex of the current field is monodromic; 2. Stability Problem: once is known that the vortex is monodromic, decide whether it is a center or a focus. In this paper we give partial answers for the above two problems. We have shown that the study of the flow lines of the current field near a phase vortex under the general hypothesis T naturally leads to the center–focus problem. Here we have analyzed the case where the linear part of the current field at the vortex is nondegenerate. One possible direction of research is the study of the flow lines of the current field near a vortex when the hypothesis T is not satisfied, but the flow lines circulate the vortex. For example, the case where the linear part of the current field at the vortex is degenerate can be of interest. In this paper we have studied only planar wave functions and flow lines that circulate the vortex (the center–focus problem). Of course wave functions in 3–dimensional space as well as other types of vortex are of interest, particularly the knotted vortices [6].
Appendix: An Overview on Lyapunov Numbers The beginning of this section is a review of the projection method given in [11] for the calculation of the Lyapunov numbers l3 and l5 . The method is easily adapted to the calculation of any Lyapunov number. The expressions of the Lyapunov numbers l7 and l9 can be found in [19] and [20]. Other equivalent definitions and algorithmic procedures to write the expressions for the Lyapunov numbers for two dimensional systems can be found in Andronov et al. [1], Gasull and Torregrosa [9] and Farr et al. [8], among others. Consider the differential equations x = F(x),
(14)
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where x ∈ R2 and F is of class C∞ . Suppose (14) has an equilibrium point x = x0 where the Jacobian matrix A has a pair of purely imaginary eigenvalues λ1,2 = ±iω0 , ω0 > 0. Denoting the variable x − x0 also by x, one has 1 1 1 B(x, x) + C(x, x, x) + D(x, x, x, x) 2 6 24 1 1 + E(x, x, x, x, x) + K(x, x, x, x, x, x) 120 720 1 L(x, x, x, x, x, x, x) + O ||x||8 , + 5040
F(x) = Ax +
where, for i = 1, 2,
2 ∂ 2 Fi (ξ ) Bi (x, y) = ∂ξ j ∂ξk j,k=1
x j yk , Ci (x, y, z) =
2 j,k,l=1
ξ =0
∂ 3 Fi (ξ ) ∂ξ j ∂ξk ∂ξl
(15)
x j yk zl ,
ξ =0
and so on for Di , Ei , Ki and Li . Let p, q ∈ C be vectors such that 2
Aq = iω0 q, A p = −iω0 p, p, q =
2
p¯ i qi = 1,
(16)
i=1
where A is the transposed matrix. Any vector y ∈ R2 can be represented as ¯ where w = p, y ∈ C. The phase plane can be parameterized by y = wq + w¯ q, w, w, ¯ by means of x = H(w, w), ¯ where H : C2 → R2 has a Taylor expansion of the form 1 h jk w jw¯ k + O(|w|8 ), (17) H(w, w) ¯ = wq + w¯ q¯ + j!k! 2 j+k7
with h jk ∈ C2 and h jk = h¯ kj. Substituting this expression into (14) we obtain the following differential equation ¯ . (18) Hw w + Hw¯ w¯ = F H(w, w) The complex vectors hij are obtained solving the system of linear equations defined by the coefficients of (18), taking into account the coefficients of F, so that system (18) writes as follows w = iω0 w +
1 1 1 G21 w|w|2 + G32 w|w|4 + G43 w|w|6 + O |w|8 , 2 12 144 (19)
with G jk ∈ C. The first Lyapunov number l3 is defined by l3 = where
1 Re G21 , 2
(20)
¯ (2iω0 I2 − A)−1 B(q, q) − 2B q, A−1 B(q, q) ¯ , G21 = p, C q, q, q¯ + B q,
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and I2 is the unit 2 × 2 matrix. Defining H32 as ¯ h31 + H32 = 6B h11 , h21 + B h¯ 20 , h30 + 3B h¯ 21 , h20 + 3B q, h22 + 2B q, ¯ h21 + + 6C q, h11 , h11 + 3C q, h¯ 20 , h20 + 3C q, q, h¯ 21 + 6C q, q, ¯ h20 , h11 + C q, ¯ q, ¯ h30 + D q, q, q, h¯ 20 + 6D q, q, q, ¯ h11 + + 6C q, ¯ 21 h21 , ¯ q¯ − 6G21 h21 − 3G ¯ q, ¯ h20 + E q, q, q, q, + 3D q, q, the second Lyapunov number l5 is given by l5 =
1 Re G32 , 12
(21)
where G32 = p, H32 . The third Lyapunov number l7 is defined by l7 =
1 Re G43 , 144
(22)
where G43 = p, H43 and H43 is given by H43 = 12B h11 , h32 +6B h20 , h¯ 32 +3B h¯ 20 , h41 +18B h21 , h22 + ¯ h42 + + 12B h¯ 21 , h31 + 4B h30 , h¯ 31 + B h¯ 30 , h40 +4B q, h33 +3B q, + 36C h11 , h11 , h21 + 36C h11 , h20 , h¯ 21 + 12C h11 , h¯ 20 , h30 + + 3C h20 , h20 , h¯ 30 + 18C(h20 , h¯ 20 , h21 + 36C(q, h11 , h22 + + 12C q, h20 , h¯ 31 +12C q, h¯ 20 , h31 +36C q, h21 , h¯ 21 +4C(q, h30 , h¯ 30 + ¯ h32 + 24C q, ¯ h11 , h31 + 18C q, ¯ h20 , h22 + + 6C q, q, h¯ 32 + 12C q, q, ¯ h21 , h21 + 12C q, ¯ h¯ 21 , h30 + 3C q, ¯ q, ¯ h41 + ¯ h¯ 20 , h40 + 18C q, + 3C q, + 24D q, h11 , h11 , h11 + 36D q, h11 , h20 , h¯ 20 + 36D q, q, h11 , h¯ 21 + + 6D q, q, h20 , h¯ 30 + 18D q, q, h¯ 20 , h21 + 4D q, q, q, h¯ 31 + ¯ h11 , h21 + 36D q, q, ¯ h20 , h¯ 21 + ¯ h22 + 72D q, q, + 18D q, q, q, ¯ h¯ 20 , h30 + 12D q, q, ¯ q, ¯ h31 + 36D q, ¯ h11 , h11 , h20 + + 12D q, q, ¯ q, ¯ h11 , h30 + 18D q, ¯ q, ¯ h20 , h21 + ¯ h20 , h20 , h¯ 20 + 12D q, + 9D(q, ¯ q, ¯ q, ¯ h40 + 12E q, q, q, h11 , h¯ 20 + E q, q, q, q, h¯ 30 + + D q, ¯ h¯ 21 + 36E q, q, q, ¯ h11 , h11 + 18E q, q, q, ¯ h20 , h¯ 20 + + 12E q, q, q, q, ¯ q, ¯ h21 + 36E q, q, ¯ q, ¯ h11 , h20 + 4E q, q, ¯ q, ¯ q, ¯ h30 + + 18E q, q, q, ¯ q, ¯ q, ¯ h20 , h20 + 3K q, q, q, q, q, ¯ h¯ 20 + 12K q, q, q, q, ¯ q, ¯ h11 + + 3E q, ¯ q, ¯ q¯ − ¯ q, ¯ q, ¯ h20 + L q, q, q, q, q, + 6K q, q, q, ¯ 32 h21 + 3G21 h32 + 2G ¯ 21 h32 . − 6 2G32 h21 + G
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Acknowledgements The first author developed this work under the project CNPq 473747/2006-5. The second author is supported by CAPES. This work was finished while the first author visited Universitat Autònoma de Barcelona, supported by CNPq grant 210056/2006-1. The authors thank the referee for the comments and suggestions which allowed them to improve the presentation of this paper.
References 1. Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A.G.: Theory of Bifurcations of Dynamic Systems on a Plane. Halsted, Wiley, New York (1973) 2. Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (1989) 3. Arovas, D.P., Freire, J.A.: Dynamical vortices in superfluid films. Phys. Rev. B 55, 1068–1080 (1997) 4. Berry, M.V., Chambers, R.G., Large, M.D., Upstill, C., Walmsley, J.C.: Wavefront dislocations in the Aharonov–Bohm effect and its water wave analogue. Eur. Phys. J. 1, 154–162 (1980) 5. Berry, M.V.: Phase vortex spirals. J. Phys. A 38, L745–L751 (2005) 6. Berry, M., Dennis, M.: Knotted and linked phase singularities in monochromatic waves. Proc. Roy. Soc. A 457, 2251–2263 (2001) 7. Damski, B., Sacha, K.: Changes of the topological charge of vortices. J. Phys. A 36, 2339–2345 (2003) 8. Farr, W.W., Li, C., Labouriau, I.S., Langford, W.F.: Degenerate Hopf bifurcation formulas and Hilbert 16th problem. SIAM J. Math. Anal. 20, 13–30 (1989) 9. Gasull, A., Torregrosa, J.: A new approach to the computation of the Lyapunov constants. Comput. Appl. Math. 20, 149–177 (2001) 10. Guilleumas, M., Graham, R.: Off–axis vortices in trapped Bose-condensed gases: Angular momentum and frequency splitting. Phys. Rev. A 64, 033607 (2001) 11. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York (1998) 12. Mello, L.F., Ribeiro, Y.C.: The Aharonov–Bohm effect: mathematical aspects of the quantum flow. Appl. Math. Sci. 1, 383–394 (2007) 13. Moreira, Jr. E.S.: Aspects of classical and quantum motion on a flux cone. Phys. Rev. A 58, 1678–1686 (1998) 14. Morrison, P.J.: Hamiltonian description of the ideal fluid. Rev. Modern Phys. 70, 467–521 (1998) 15. Nye, N.F., Berry, M.V.: Dislocations in waves trains. Proc. Roy. Soc. London Ser. A 336, 165–190 (1974) 16. Olariu, S., Popescu, I.I.: The quantum effects of eletromagnetic fluxes. Rev. Modern Phys. 57, 339–436 (1985) 17. Penna, V., Rasetti, M., Spera, M.: Quantum dynamics of 3D–vortices. Contemp. Math. 219, 173–193 (1998) 18. Pitaevskii, L., Stringari, S.: Bose–Einstein Condensation. Clarendon, Oxford (2003) 19. Sotomayor, J., Mello, L.F., Braga, D.C.: Bifurcation analysis of the Watt governor system. Comput. Appl. Math. 26, 19–44 (2007) 20. Sotomayor, J., Mello, L.F., Braga, D.C.: Lyapunov coefficients for degenerate Hopf bifurcations. arXiv:0709.3949v1 [math.DS] (2007)
Math Phys Anal Geom (2008) 11:87–116 DOI 10.1007/s11040-008-9041-z
Algebraic Theory of Linear Viscoelastic Nematodynamics Arkady I. Leonov
Received: 8 April 2008 / Accepted: 8 April 2008 / Published online: 3 June 2008 # Springer Science + Business Media B.V. 2008
Abstract This paper consists of two parts. The first one develops algebraic theory of linear anisotropic nematic “N-operators” build up on the additive group of traceless second rank 3D tensors. These operators have been implicitly used in continual theories of nematic liquid crystals and weakly elastic nematic elastomers. It is shown that there exists a non-commutative, multiplicative group N6 of Noperators build up on a manifold in 6D space of parameters. Positive N-operators, which in physical applications hold thermodynamic stability constraints, do not generally form a subgroup of group N6. A three-parametric, commutative transversal-isotropic subgroup S3 N6 of positive symmetric nematic operators is also briefly discussed. The special case of singular, non-negative symmetric Noperators reveals the algebraic structure of nematic soft deformation modes. The second part of the paper develops a theory of linear viscoelastic nematodynamics applicable to liquid crystalline polymer. The viscous and elastic nematic components in theory are described by using the Leslie–Ericksen–Parodi (LEP) approach for viscous nematics and de Gennes free energy for weakly elastic nematic elastomers. The case of applied external magnetic field exemplifies the occurrence of nonsymmetric stresses. In spite of multi-(10) parametric character of the theory, the use of nematic operators presents it in a transparent form. When the magnetic field is absent, the theory is simplified for symmetric case with six parameters, and takes an extremely simple, two-parametric form for viscoelastic nematodynamics with possible soft deformation modes. It is shown that the linear nematodynamics is always reducible to the LEP-like equations where the coefficients are changed for linear memory functionals whose parameters are calculated from original viscosities and moduli. Keywords Liquid crystals . Nematodynamics . Nematic operators . Transversal isotropy . Polymers . Viscoelasticity
A. I. Leonov (*) Department of Polymer Engineering, The University of Akron, Akron, OH 44325-0301, USA e-mail: [email protected]
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Mathematics Subject Classification (2000) SC: 08A62 . 20E32 . 47B99 . 74A20
1 Introduction The properties of low molecular weight liquid crystals (LMW LCs) have been well investigated experimentally and theoretically [3]. Nematic orientational order of their molecules, with the lowest, axial symmetry of molecular groups is the most common for LMW LC. On the macroscopic level, it is convenient to describe this order by the unit vector n called director, with new degree of freedom, internal rotations. The equilibrium elastic properties of LMW nematics, with the Frank stress proportional to the space gradient of director, r n, as well as their flow properties are well described on macroscopic level by the Leslie–Ericksen–Parodi (LEP) theory called nematodynamics [3]. In spite of 25 years of study, the properties of liquid crystalline polymers (LCP’s) have not been well understood as their LMW counterparts. Common LCP’s consist of very rigid macromolecules. As in case of low molecular weight liquid crystals (LMW LCs), there are two classes of LCP: (1) lyotropic, LCP solutions in LMW solvents, and (2) thermotropic, LCP melts. Commercial thermotropic LCP’s, such as Titan (Eastman) with melting temperature Tm =330°C and Zenith 6000 (Dupont), Tm =345°C, are the random polyesters with rigid main-chain mesogenic groups. These LCP’s have very narrow time and temperature intervals between beginning the crystal melting and onset of polymer degradation, where the liquid crystalline phase exists. Even in the case of rigid LCP macromolecules, their contour length is higher (and commonly much higher) than the macromolecule end-to-end distance. Therefore the low molecular flexibility, always existing in LCP’s gives rise to their “molecular elasticity” [10] because of possible variation in the end-to-end distance of macromolecules. At low polymer concentrations in lyotropic LCP’s the effect of molecular elasticity can be neglected and these LCP’s could be treated as molecular suspensions of rigid rods [4]. The majority of experimental data for LCP’s were obtained from rheological experiments in simple shearing with the absence of the external magnetic and electrical fields [10]. The degradation of commercial thermotropic LCP’s is the main problem for their rheological studies. Therefore to make easy rheological experiments some model thermotropic LCP’s were synthesized with introducing either flexible spacers in the main chain or mesogenic side groups. Such LCP’s have been widely used in rheological experiments (e.g. see [8, 19]). In these more flexible model LCP’s the effects of Frank elasticity occur at the end of relaxation with forming a specific “texture” during very long postrelaxation time. Some new reliable results have also been recently obtained in rheological studies of Titan and Zenith. It should also be mentioned that there also exists a class of highly deformable nematic cross-linked elastomers where rigid chemical fragments were inserted into very flexible silicon rubber chains [21]. The concept of space–time evolution of director is also commonly utilized to describe the nematic order and internal rotations of LCP rigid macromolecules or their rigid chemical fragments. Nevertheless, this theory is not much developed. One of the main questions is whether the molecular elasticity is important for all the types of LCP. If it is, complicated viscoelastic phenomena should be involved in the
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theory, describing along with director evolution also the contribution of anisotropic (molecular) elastic and viscous forces. Papers [17, 18] developed a thermodynamic theory for linear anisotropic viscoelasticity for LC polymers, using along with state variables also their space gradients, which created an awkward, almost no testable description. Several molecular or semi-phenomenological theories were also developed to model the lyotropic LCP’s, when employing the same state variables, n, r n and T as in case of LMW LC’s. The papers [6, 10, 15] typically used and elaborated the Doi’s long rigid rod statistical approach. To describe LCP’s an extended Poisson–Bracket approach [1] was also elaborated and reduced to the Doi theory in the homogeneous (mono-domain) limit. To take into account a flexibility of LCP chains in model thermotropic LCP’s nematic Rouse-like molecular theories [13, 20] were developed. Currently these theories could be used for description of linear viscoelastic LCP properties, however, they have not been experimentally tested. In spite of many simplifying physical assumptions, the theories [13, 20] are still very complicated, so their extension even to a weakly nonlinear case seems improbable. The Doi rigid rod theory has been extensively tested [10]. For lyotropic LCP’s of small and moderate concentrations these tests demonstrated a success of Doi theory. However, both the modeling and industrial thermotropic LCP’s are not described by the Doi theory. Surprisingly, the Doi rigid rod theory is incapable to describe the rheological behavior of real “rigid rod” polymers whose contour length is closed to the end-to-end macromolecular distance (e.g. see [5, 16]). An unusual effect of soft deformation modes with ideally no resistance in certain directions was predicted in paper [7] for anisotropic elastic solids and observed for cross-linked nematic elastomers as predicted by a particular equilibrium theory [21]. A possible fundamental reason for occurrence of these soft modes is that large fluctuations typical for nematics in equilibrium, move these systems almost to the boundary of their thermodynamic stability where the free energy is effectively minimized not only with respect to the state variables but also with respect to material parameters [14]. Following this idea the “marginal stability” concept has been introduced and formally used in papers [11, 12] for describing the soft modes for both viscous LMW LC’s and nematic elastomers. It is now well recognized that the continual theories of viscous nematodynamics and nematic elasticity describe well the respective behavior of LMW LC’s and nematic elastomers. It seems that the theoretical description of LCP dynamics is also needed a similar general continual (or “field”) theory, which could consistently describe their specific viscoelastic nematic properties at least in weakly nonlinear limit. Developing such a theory with many sets of nematic relaxation modes seems currently unrealistic. Even in the simplest Maxwell approximation, similar to nematic dumbbell theory in statistical treatment, with one set of nematic relaxations, such a theory has not been developed. It seems that the awkward common tensor/matrix formulation of operations in existing viscous and elastic nematic theories is a primary reason for the lack of such a theory. This common formulation does not allow displaying a simple algebraic structure of these theories and makes difficult (if possible) developing a general viscoelastic theory for describing linear or nonlinear dynamic behavior of LCP’s.
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The paper consists of two parts. The first part reveals the algebraic structure of the nematic theories and presents it in a simple form. The second part develops a continuum theory of linear viscoelastic nematodynamics of Maxwell type with a single set of anisotropic relaxations in mono-domain case. Unlike the dumbbell limit in theories [13, 20], using the algebraic approach makes possible to treat this problem in all generality. Such a general approach also allows us to employ the concept of marginal stability and implement the algebraic soft mode analysis that highly simplifies the results. This approach also makes possible to develop a weakly nonlinear theory, which could model dynamic behavior of LCP’s.
2 Algebra of Nematic Operators 2.1 Definitions and General Properties ^
Consider the additive group X of traceless 3D second rank Cartesian tensors x ¼ ^ xij 2 X : trx ¼ 0 defined on the field ofT real numbers. This group can be ^ ^ ^ ^ ^ decomposed in the sum, X ¼ X s þ X a , ð X s X a ¼ 0Þ of two additive subgroups ^ ^ ^ ^ of symmetric X s and asymmetric X α tensors, so that 8x 2 X : x ¼ xs þ xa , xs 2 X s ^ and xa 2 X a . ^ ^ ^ We introduce on X linear, axially symmetric operations X ! X characterized by a given unit vector n (director). A linear operation invariant relative to transformation n ! n, is called nematic operation (or simply N-operator). The implicit definition of N-operation in the common tensor presentation is: h i . y ¼ r0 xs þ r1 nn xs þ xs nn 2nn xs : nn þ r2 nn δ 3 xs : nn þ r3 nn xa xa nn s y ¼ r4 nn xs xs nn r5 nn xa þ xa nn : a
ð1:1Þ Hereafter nn ij ¼ nj nj, the symbol · means the tensor (matrix) multiplication, the symbol: denotes the trace operation, d is the unit tensor, and rk are six independent ordered basis parameters, characterizing the operation in (1.1), denoted as: r = (r0, ^ ^ r1,..., r5). Because of (1.1) y 2 X s , y 2 X a , which justifies that operation in (1.1) s a ^ ^ transforms X ! X . Relations of (1.1) type were first introduced in the vector form in paper [2] for weak elastic gels and have been used in the tensor form in papers [11, 12] as constitutive equations for viscous and weakly elastic nematic cases. The possible non-nematic term ∼xa was excluded from (1.1) because of physical arguments (e.g. see [11, 12]). We now denote the N-operator as Nr n , and symbolically present (1.1) as y ¼ Nr n x. A particular but physically significant Onsager N-operator (or ONoperator) is defined as: Nor n Nr n jr4 ¼r3 :
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^ For any N-operator Nr n one can introduce on X the quadratic form, a scalar P, defined as: 2 2 P x Nr n x r0 xs þ 2r1 nn : x2s þ ðr2 2r1 Þ nn : xs 4r3 nn : xs xa 2r5 nn : x2a : r3 ðr3 r4 Þ=2:
ð1:2Þ o form Po has potential In case of ON operator Nor n , when . P→P , the quadratic . o o properties: 2y ¼ @P @xs , 2y ¼ @P @xa . Operator Nr n is said to be positive if ^ s a o 8x 2 X : P x Nr ðnÞ x > 0. The same holds for ON operator N n r . We now show that (1) N operator Nr n is positive (P>0) iif: eþ : r2R 6
r0 > 0;
r0 þ r1 > 0;
3=2r0 þ r2 > 0;
. ðr0 þ r1 Þr5 > ðr3 r4 Þ2 4 r3*2 ;
(1.3,2; ð1:31; 3; 4Þ) 1,2,3,4 n . and that (2) any positive N-operator Nr n has inverse, N1 r Using an orthogonal transformation, one can choose a coordinate system whose axis 1 is directed along the director. In this coordinate system (1.1) and (1.2), written in component form, are reduced to: s s s s y11 ¼ ðr0 þ 2=3r2 Þx11 ; y22 ¼r0 x22 x11 r2 =3; y33 ¼r0 x33 x11 r2 =3; y23 ¼ r0 x23 ð1:11aÞ (1.11a)
ys1k ¼ ðr0 þ r1 Þxs1k þ r3 xa1k ðk ¼ 2; 3Þ ya1k ¼ r4 xs1k þ r5 xa1k
h i P ¼ ð3=2r0 þ r2 Þx211 þ 2r0 ðx22 þ x11 =2Þ2 þ x2s23 i Xh þ2 ðr0 þ r1 Þx2s1k þ 2r3* xa1k xs1k þ r5 x2a1k
(1.11b) ð1:11bÞ
ð1:21Þ (1.21)
k¼2;3
Here the traceless condition, x11 þ x22 þ x33 ¼ 0, has been used to exclude x33 from (1.2). Demanding P>0 and using independence of terms in (1.2) yields inequalities (1.31,2,3,4). Note that the inequality (1.34) results in the inequality: ðr0 þ r1 Þr5 > r3 r4 :
ð1:35Þ (1.3. 5) of xij When (1.31,2,3,5) holds for equations (1.1), there is a unique linear dependence 1 on yij, which means the existence of a unique inverse operation N n . Note that r inequalities (1.31,2,3,4,5) hold for ON-operators Nor n when r3* ¼ r3 ðP ! Po Þ. Resolving equations (1.1) does not, however, necessitate that the parameters r 2 eþ defined by inequalities (1.31,2,3,4). The necessary and R6 belong to the manifold R 6 sufficient conditions for this resolution are: r0 ≠0, 3=2r0 þ r2 6¼ 0, ðr0 þ r1 Þr5 þ r3 r4 6¼ 0. Nevertheless, it is more convenient to use the positive conditions of the resolution: r 2 Rþ 6 :r0 > 0;
3=2r0 þ r2 > 0;
ðr0 þ r1 Þr5 þ r3 r4 > 0:
ð1:31; 3; 5Þ) (1.31,3,5
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N-operator is called N+-operator if its basis parameters satisfy the inequalities eþ Rþ , i.e. N+-operators are not necessarily positive, whereas (1.31,3,5). Evidently, R 6 6 any positive N-operator is N+- operator. 2.2 Basis N-operators and their Multiplicative Properties The tensor/matrix presentations of basis N-operators ak n ðk ¼ 0; 1; . . . ; 5Þ in (1) are explicitly defined via fourth rank numerical tensors (or simply 4-tensors) fak ðnÞgijab as:
ð0Þ ð1:41Þ fa0 gijαβ ¼ aijαβ ¼ 1 2 δiα δjβ þ δiβ δjα 2 3δij δαβ (1.41) . ð1 Þ ? ? ? n n þδ n n þ δ n n þδ n n δ? fa1 ð n Þgijαβ¼ aijαβ ¼ 1 2 δ? j β i β j α i α iα jα iβ jβ ij ¼ δij ni nj
ð1:42Þ (1.42)
ð2Þ fa2 ðnÞgijαβ ¼ aijαβ ¼ ni nj 1 3δij nα nβ 1 3δαβ
ð1:43Þ (1.43)
ð3Þ fa3 ðnÞgijαβ ¼ aijαβ ¼ 1 2 δiα nj nβ þ δjα ni nβ δiβ nj nα δjβ ni nα
(1.44) ð1:44Þ
ð4Þ fa4 ð n Þgijαβ ¼ aijαβ ð n Þ ¼ 1 2 δiα nj nβ þ δiβ nj nα δjα ni nβ δjβ ni nα ¼ fa3 ð n Þgαβij
(1.45) ð1:45Þ
ð 5Þ fa5 ðnÞgijαβ ¼ aijαβ ðnÞ ¼ 1 2 δiβ nj nα δiα nj nβ þ δjα ni nβ δjβ ni nα
ð1:46Þ (1.46)
The basis four-tensors in (1.4 1–1.4 6) are traceless with respect to the first and the second pairs of indices, i.e. ak n iiαβ ¼ ak n ijαα ¼ 0ðk ¼ 0; ::; 5Þ. The following symmetry properties hold for the basis four-tensors: n (1.51) ak n ijαβ ¼ ak n jiαβ ¼ ak n ijβα ¼ ak nÞ αβij ðk ¼ 0; 1; 2Þ ð1:51Þ a3 n ijαβ ¼ a3 n jiαβ ¼ a3 n ijβα
(1.52) ð1:52Þ
a4 n ijαβ ¼ a3 n jiαβ ¼ a3 n ijβα
ð1:53Þ (1.53)
a5 n ijαβ ¼ a5 n jiαβ ¼ a5 n ijβα ¼ a5 n αβij :
ð1:54Þ (1.54)
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The symmetry properties (1.51–1.54) of the basis four-tensors ak n ðk ¼ 0; ::; 5Þ show that they represent irreducible set of traceless 4-th rank tensors. The products of the basis tensors of different ranks are defined as: ðsÞ
ðrÞ
ðr Þ
as ð nÞ&ar ð nÞ ) aijαβ aβανγ ;
ar ðnÞ&x ) aijαβ xβα ; etc:
ð1:6Þ
This defines the sense of operation • symbolically used in “Section 2.2”. The products ai ð nÞ& aj ðnÞ, established directly are presented in Table 1. It is seen that except i,j=0,1,2, the multiplication of basis tensors is non-commutative, e.g. a0 & a3 ¼ a3 6¼ a3 & a0 ¼ 0 2.3 Multiplicative Group of N-operators Using basis operators ak n , (1.1) can be rewritten in the operator form: y ¼ Nr ðnÞ& x; Nr ðnÞ
5 X
rk ak ð nÞ;
(1.71) ð1:71Þ
k¼0
or equivalently as: y ¼ s
2 X
rk ak ðnÞ& x s þ r3 a3 ðnÞ& x a ;
y a ¼ r4 a 4 ðnÞ& xs þ r5 a5 ðnÞ& x a :
(1.72) ð1:72Þ
k¼0
The product of two N-operators is defined in the common way: Np ð nÞ Nq ðnÞ& Nr ð nÞ ¼
5 X
qk rm ak ðnÞ& am ð nÞ ¼
k;m¼0
6 X
pk ak ðnÞ:
ð1:8Þ
k¼0
With the use of multiplicative Table 1, the basic scalars pk for resulting operation are found from the fundamental equation: p0 ¼ q0 r0 ; p1 ¼ q0 r1 þ q1 r0 þ q1 r1 q3 r4 ; p2 ¼ q2 r0 þ q0 r2 þ 2=3q2 r2 p3 ¼ ðq0 þ q1 Þr3 þ q3 r5 ; p4 ¼ q4 ðr0 þ r1 Þ þ q5 r4 ; p5 ¼ q4 r3 þ q5 r5 ð1:9Þ
Table 1 Products of basis tensors ai aj j l
0
1
2
3
4
5
0 1 2 3 4 5
a0 a1 a2 0 a4 0
a1 a1 0 0 a4 0
a2 0 (2/3) a2 0 0 0
a3 a3 0 0 −a5 0
0 0 0 −a1 0 a4
0 0 0 a3 0 a5
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Even in the Onsager case, when Nq ð nÞ ¼ Noq ð nÞ, Nr ð nÞ ¼ Nor ð nÞ i.e. q4 =−q3 and r4 =−r3, generally p4 ≠−p3. It means that Np ðnÞ 6¼ Nop ð nÞ, i.e. the product of two ONoperators is not generally an ON-operator. We now show that the set of N+-operators Nr n , whose basis parameters r 2 Rþ 6 satisfy inequalities (1.31,3,5), constitute a non-commutative, multiplicative, sixparametric group N6, with the fundamental equations (1.9). Using the Table 1, one can define the unit N-operator I(n) and its basic properties as: Ið nÞ ¼ a0 þ a5 ðnÞ;
Nr ð nÞ IðnÞ ¼ Ið nÞ Nr ð nÞ ¼ Nr ðnÞ
ð1:10Þ
Because of (1.2, 1.21) the unit N-operator is positive, and therefore it is a N+operator. If Nr ðnÞ is N+-operator, its inverse, N1 r ðnÞ Nrˆ ðnÞ should satisfy the common condition, Nrˆ ð nÞ& Nr ð nÞ ¼ Nr ðnÞ& Nrˆ ð nÞ ¼ Ið nÞ which for parameters in (1.9) yields: p0 ¼ 1; p1 ¼ p2 ¼ p3 ¼ p4 ¼ 0; p5 ¼ 1:
ð1:11Þ
Then using (1.9 and 1.11) yields the expressions for basis parameters rˆk of inverse N-operator Nrˆ n as: 1 ðr3 r4 þ r1 r5 Þ=r0 r2 =r0 ; rˆ1 ¼ ; rˆ2 ¼ r0 r5 ðr0 þ r1 Þ þ r3 r4 r0 þ 2=3r2 r3 r4 r0 þ r1 rˆ3 ¼ ; rˆ4 ¼ ; rˆ5 ¼ r5 ðr0 þ r1 Þ þ r3 r4 r5 ðr0 þ r1 Þ þ r3 r4 r5 ðr0 þ r1 Þ þ r3 r4 ð1:12Þ rˆ0 ¼
+ Using inequalities (1.31,2,3,4 and 1.12) one can directly check that N1 r ð nÞ is a N 1 operator and if Nr ð nÞ is positive, Nr ð nÞ is positive too. Additionally, any positive ON operator has inverse one, which is a positive ON operator. Finally, we show that the product Np ð nÞ ¼ Nr ðnÞ& Nq ð nÞ of two N+ operators is + þ N operator, because 8r; q 2 Rþ 6 : p 2 R6 . This follows from the direct calculations with the use of (1.9):
p0 ¼ q0 r0 > 0; 3=2p0 þ p2 ¼ 3=2ðq0 þ 2=3q2 Þðr0 þ 2=3r2 Þ > 0 : p5 ðp0 þ p1 Þ þ p4 p3 ¼ ½q5 ðq0 þ q1 Þ þ q4 q3 ½r5 ðr0 þ r1 Þ þ r4 r3 > 0
ð1:13Þ
These properties prove that the N+ operators constitute a multiplicative group N6. Note that due to (1.9), p0 þ p1 ¼ ðq0 þ q1 Þðr0 þ r1 Þ q3 r4 ; p5 ¼ q5 r5 q4 r3 . Therefore the product of two positive N+-operators is positive only under additional constraints: q3r4, q4r3 <0. In particular, the product of two positive ON-operators is generally non-positive N+ operator, and is positive only under additional constraint, q3r3 >0. Since the N+-operators also constitute the additive group with the group operation: Nq ðnÞ þ Nr ðnÞ ¼ Nqþr ðnÞ, they form an associative ring Ń6 relative to both, additive and multiplicative operations. In the following we are interested only in the multiplicative properties of N-operators. Consider as an example the dual N-operations defined as: z ¼ Nq ðnÞ&y ¼ Nr ðnÞ&x. If both Nr ðnÞ and Nq ðnÞ are N+ operators, the dual operations uniquely
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determine the direct and reciprocal dependences, y ¼ Np ðnÞ&x and x ¼ Npˆ ðnÞ&y. Here Np ðnÞ ¼ N1 q ðnÞ&Nr ðnÞ;
1 Npˆ ðnÞ ¼ N1 p ðnÞ ¼ Nr ðnÞ&Nq ðnÞ:
ð1:14Þ
Formulae (1.9 and 1.12) express the basis scalars p and pˆ of dual N+-operators o o via given basis scalars r and q. E.g. in case Np ð nÞ ¼ No1 q ð nÞ&Nr ð nÞ, where Nr ðnÞ o and Nq ðnÞ are positive ON-operators, the parameters p are: r0 ðr0 þ r1 Þq5 r3 q3 r0 r2 q0 r0 q2 ; ; p1 ¼ ; p2 ¼ q0 q0 q0 ðq0 þ 2=3q2 Þ q5 ðq0 þ q1 Þ q23 r3 ðq0 þ q1 Þ q3 ðr0 þ r1 Þ r5 q3 r3 q5 r5 ðq 0 þ q 1 Þ r3 q 3 p3 ¼ ; p4 ¼ ; p5 ¼ 2 2 q5 ðq0 þ q1 Þ q3 q5 ðq0 þ q1 Þ q23 q5 ðq0 þ q1 Þ q3 p0 ¼
ð1:15Þ Due to (1.14) respective formulae for the basis parameters pˆ of inverse dual N+ operation are obtained from (1.15) by substitution r $ q. Note that N-operator Np ðnÞ with basis scalars defined in (1.15), is generally neither Onsager nor positive. The evident sufficient condition for Np ðnÞ to be positive is: r3q3 <0. We consider in the following the case of non-degenerating (NG) Nr ðnÞ operators, rk 6¼ 0
ðk ¼ 0; 1; 2; 3; 4; 5Þ:
ð1:16Þ
2.4 Spectral Properties of N-operators The spectral problem for a NG operator Nr ðnÞ is formulated in the standard way: Nr ð nÞ&x ¼ nx; Here Nr ðnÞ ¼
5 P k¼0
or
½Nr ðnÞ nIðnÞ&x ¼ 0:
(1.171) ð1:171Þ
rk ak ðnÞ, IðnÞ ¼ a0 þ a5 ðnÞ, ν being a generally complex ^
eigenvalue, and xðν Þ 2 X is a respective “eigentensor”. If an eigenvalue ν is known, it is convenient to search instead of eigentensor xðn Þ, the N-“eigenoperator” 5 P Qðn; nÞ ¼ qk ðn Þak from the equation: k¼0
Nr ðnÞ&Qðn; nÞ nQðn; nÞ ¼ Nr ðn; nÞ&Qðn; nÞ ¼ 0 Nr ðn; nÞ Nr ðnÞ nIðnÞ:
where
(1.172) ð1:172Þ
Evidently, the tensor x ¼ Qðn; nÞ&x 0 where x 0 is a given tensor, is the eigentensor, because it identically satisfies (1.172). A common analytical continuation n ! nˆ is used to find the eigenvalues of problem (1.171) for any NG operator Nr ðnÞ. Substituting r0 ! r0 nˆ and ˆ and using (1.12) results in formal finding the basis parameters r5 ! r5 n, ˆ rk Þ of operator N1 ˆ nÞ. The eigenvalues are then found as singular points rˆ k ðn; r ðn;
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ˆ rk Þ of the inverse operator N1 ˆ nÞ. Using this nˆ ¼ n for the basis parameters rˆk ðn; r ðn; procedure, the eigenvalues are expressed via basis parameters rk as: ν 1 ¼ r0 ;
ν 2 ¼ r0 þ 2=3r2 ; ν 3 ¼ 1=2ðr0 þ r1 þ r5 þ d Þ; ν 4 ¼ 1=2ðr0 þ r1 þ r5 d Þ d 2 ¼ ðr0 þ r1 þ r5 Þ2 4½r5 ðr0 þ r1 Þ þ r3 r4 ðr0 þ r1 r5 Þ2 4r3 r4 ;
ð1:18Þ To find the solution (1.172) one can use (1.9) with r0 ! r0 n, r5 ! r5 n. Demanding here due to (1.172) p ¼ 0 yields: ðr0 þ ν Þq0 ¼ 0; ðr0 þ r1 ν Þq1 þ r1 q0 r4 q3 ¼ 0; ðr0 þ 2=3r2 ν Þq2 þ r2 q0 ¼ 0 ðq0 þ q1 Þr3 þ q3 ðr5 ν Þ ¼ 0; ðr0 þ r1 ν Þq4 þ r4 q5 ¼ 0; r3 q4 þ ðr5 ν Þq5 ¼ 0
For each value of ν from (1.18), the parameters qðn Þ of eigenoperator Qðn; nÞ are found from the above equations as: qðν 1 Þ ¼ c1 ð1; 1; 3=2; 0; 0; 0Þ; qðν 2 Þ ¼ c2 ð0; 0; 1; 0; 0; 0Þ; qðν 3 Þ ¼ ð0; c3 ; 0; λ1 c3 ; c4 ; λ1 c4 Þ qðν 4 Þ ¼ ð0; c5 ; 0; λ2 c5 ; c6 ; λ2 c6 Þ fλ1 ¼ ðr0 þ r1 ν 3 Þ=r5 ; λ2 ¼ ðr0 þ r1 ν 4 Þ=r5 g
ð1:19Þ Here c1, c2,..., c6 are arbitrary constants. Additional solutions that use specific relations between parameters rk have been rejected because they are not robust. Using (1.19), the “eigenoperators” Qðnk ; nÞ are presented as: Q ν 1 ; n ¼ c1 ða0 a1 3=2a2 Þ; Q ν 2 ; n ¼ c2 a2 ; Q ν 3 ; n ¼ c3 ða1 λ1 a3 Þ þ c4 ða4 þ λ1 a5 Þ Q ν 4 ; n ¼ c5 ða1 λ2 a3 Þ þ c6 ða4 þ λ2 a5 Þ fλ1 ¼ ðr0 þ r1 ν 3 Þ=r4 ; λ2 ¼ ðr0 þ r1 ν 4 Þ=r4 g
ð1:20Þ Arbitrary parameters ck in (1.15) can be established from various additional conditions. One of possible physical condition is: 4 X
Qðn i ; nÞ ¼ IðnÞ ¼ a0 þ a5 ðnÞ:
ð1:21Þ
i¼1
Using (1.20 and 1.21) yields: c1 ¼ 1; c2 ¼ 3=2; c3 ¼ ðr0 þ r1 ν 4 Þ=d; c4 ¼ r4 =d; c5 ¼ ðν 3 r0 r1 Þ=d; c6 ¼ r4 =d
ð1:22Þ Here parameters rk, ν3, ν4, and d have been defined in (1.18). Easy analysis reveals various cases of behavior of eigenvalues νk in (1.18): (1) in general case of NG operators Nr ðnÞ, the eigenvalues ν1, ν2 are real but generally have arbitrary signs, the eigenvalues ν3, ν4 being generally complex and conjugated; (2) in case of N+ operators when Nr ðnÞ 2 N6 , the eigenvalues ν1, ν2 are positive, the eigenvalues ν3, ν4 being generally complex and conjugated; (3) in case of positive N+ operators, the eigenvalues ν1, ν2 are positive, the eigenvalues ν3, ν4 being generally complex and conjugated, with Re(ν3, ν4)>0, and (4) all eigenvalues νk in (1.18) are real positive if r3r4 <0, as in particular case of positive ON operators where r4 =−r3.
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There is also a remarkable feature of Onsager positive operators, which has an important application to the viscoelastic nematodynamics. As shown in “Appendix”, a dual, generally not a positive operator Np n constructed due to (1.14 and 1.15) from two ON positive operators has real positive eigenvalues. 2.5 Symmetric N-operators: Transversal Isotropy (TI) 2.5.1 General Properties ^
^
This section briefly describes the nematic operations on the additive subgroup X s X ^ of traceless second rank symmetric tensors xs 2 X s : trxs ¼ 0. Therefore the lower ^ index “s” is omitted here. The linear symmetric N-operator Sr n on X s is defined as: y ¼ Sr ðnÞ&x ¼
2 X
rk ak ðnÞ&x;
k¼0
or
S r ð nÞ ¼
2 X
r k a k ð nÞ
ð1:23Þ
k¼0
Here ak ðnÞ are the basis tensors defined in (1.41–1.46), and rk are the real-valued basic scalar ordered parameters {r}, characterizing operation. The common tensor presentation of symmetric operation (1.1), and corresponding quadratic form Ps are: h i . y ¼ r0 x þ r1 nn & x þ x & nn 2nn x : nn þ r2 nn δ 3 x : nn : ð1:24Þ 2 2 Ps x&Sr ðnÞ&x r0 xs þ2r1 nn : x2s þ ðr2 2r1 Þ nn : xs :
ð1:25Þ
Equations (1.23 and 1.24) could also be obtained from (1.1) when ya ≡0, using the normalizing procedure [11, 12]. In this case the second equation in (1.1) isused for expressing xa via xs and n. Substituting so found dependence xa ¼ xa xs ; n into (1) results in the (1.23 and 1.24). As seen, this procedure does not violate the nondegenerating conditions (1.16). Relations (1.23 and 1.24) show that symmetric Noperators are transversally isotropic. Therefore they are called TI-operators. 2.5.2 Multiplicative Group Due to the Table 1, the products of TI-operators are commutative: Sp ðnÞ ¼ Sr ðnÞ&Sq ðnÞ ¼ Sq ðnÞ&Sr ðnÞ: Here the basis parameters pk are found from the fundamental equation: p0 ¼ r 0 q0 ;
p1 ¼ r 0 q1 þ r 1 q0 þ r 1 q1 ;
p2 ¼ r0 q2 þ r2 q0 þ 2=3r2 q2 :
ð1:26Þ ð1:27Þ
^
A TI-operator is called positive if 8x 2 X the quadratic form Ps ¼ x & Sr ðnÞ & x > 0. Using the same approach as in the general case of N operators, one can proof that TI operator Sr n is positive iif r0 > 0; r0 þ r1 > 0; r0 þ 2=3r2 > 0: Direct calculations show that the unit TI operation is I=a0, so 8Sr n :I & Sr n ¼ Sr n & I ¼ Sr n Sr n :
ð1:28Þ
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Direct calculations also show that the product (1.26) of two positive TI-operators is 1 ˆ positive. The basis scalar parameters r of inverse positive TI-operator S k r n Srˆ n are found from the relation S1 n S n ¼ S n S n ¼ a n as: ˆ r r 0 r r r2 =r0 : ð1:29Þ r0 þ 2=3r2 Due to (1.28 and 1.29), any positive TI-operator Sr n has inverse positive. All these properties of positive TI operators show that they constitute commutative three parametric TI group S3 ⊂ N 6. The dual linear transformations z ¼ Sq n y ¼ Sr n x with positive TI operators Sq n and Sr n define the direct y ¼ Sp n x and inverse x ¼ Sbp n y linear relations, where Sp n ¼ S1 n & Sr n ; Spˆ n ¼ S1 n ¼ S1 n & Sq n ; ð1:30Þ q p r rˆ0 ¼
1 ; r0
rˆ1 ¼
r1 =r0 ; r0 þ r1
rˆ2 ¼
Here the parameters p are: p0 ¼
r0 ; q0
p1 ¼
r 1 q0 q1 r 0 ; q0 þ q1
p2 ¼
r 2 q0 q 2 r 0 : q0 þ 2=3q2
(1.311) ð1:311Þ
Parameters ˆp of inverse operation found by substitution q $ r are: pˆ0 ¼
q0 q1 r 0 r 1 q0 q2 r 0 r 2 q0 ; pˆ1 ¼ ; pˆ2 ¼ : r0 r0 þ r1 r0 þ 2=3r2
(1.31 ð1:312Þ 2)
2.5.3 Eigenvalue Problem The formulation of the eigenvalue problem, similar to (1.171) is: Sr ν; n & Q ν; n ¼ 0: ð1:32Þ Sr n νI x ¼ 0; or Sr n & Q ν; n νQ ν; n e 2 2 P P ^ Here Sr n ¼ rk ak n 2 S3 , I = a0, Q n; n ¼ qk ðnÞak n and x ¼ xðn Þ 2X s . k¼0 k¼0 Using the same methods as in “Section 2.4” it is easy to show that 8Sr n 2 S3 , the spectral points of problem (1.32) are:
ν 1 ¼ r0 ;
ν 2 ¼ r0 þ r1 ;
ν 3 ¼ r0 þ 2=3r2 :
ð1:33Þ
Here due to (1.28), vk >0. The corresponding eigentensors xðn k Þ are found as: xðν k Þ ¼Q ν k ; n & x0 with x0 being a given tensor, and the “eigenoperators” Q nk ; n given by: Q ν 1 ; n ¼ c1 ða0 a1 3=2a2 Þ;
Q ν 2 ; n ¼ c2 a1 ;
Q ν 3 ; n ¼ c3 a2 : ð1:34Þ
If the arbitrary parameters ck in (1.34) are found from the physical condition, 3 X Q n i ; n ¼ I n ¼ a0 ; i¼1
ð1:35Þ
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their values are: c1 ¼ c2 ¼ 1;
c3 ¼ 3=2:
ð1:36Þ
2.5.4 Singular TI Operators Consider now the limiting, marginal situation when some inequalities in (1.34) turn out to be equalities. If once again non-degeneration conditions rk 6¼ 0 ðk ¼ 0; 1; 2Þ with r0 >0 are valid, there might be only two independent marginal conditions: r0 þ r1 ¼ 0;
r0 þ 2=3r2 ¼ 0
ð1:371Þ (1.371)
When one of these conditions is satisfied, TI operator is called partially soft. When both of them are satisfied, TI operator is called completely soft. In both the partially or complete soft cases, the quadratic form Ps is positively semi-definite. The nearly marginal situations are defined as those that reduce (1.371) to: r0 þ r1 ¼ r0 δ;
r0 þ 2=3r2 ¼ 2=3r0 κ ð0 < δ; κ << 1Þ;
r ¼ r0 ð1; δ 1; κ 3=2Þ:
ð1:372Þ (1.372)
Formulae (1.372) mean that the corresponding TI operator Sr n Sd;k n is positive. Due to (1.37 1) there is one-parametric marginal family of completely soft TI operators, S n ¼ r0 a n ðr0 > 0Þ where: ! n ¼ a0 n a1 n 3=2a2 n ; ð1:38Þ The parameters of the marginal family are: r ¼ r0 ð1; 1; 3=2Þ. It is seen that the operator a n is singular, i.e. a1 n does not exist, and that: a n & a n ¼ a n : ð1:39Þ Consider now a pair Sr n ¼ Sd1 ;k1 n and Sq n ¼ Sd2 ;k2 n of positive, nearly marginal TI operators, with parameters: r ¼ ð1; δ1 1; κ1 3=2Þ and q ¼ ð1;δ2 1; κ2 3=2Þ, where 0 < δ 1 , κ 1 < <1, 0 < δ 2 , κ 2 << 1. Evidently Sδ1 ;κ1 n ! Sδ2 ;κ2 n ! a n when (δ1, κ1)→(δ2, κ2)→(0, 0). One can directly show that when the two independent limits exist: δ ¼ lim ðδ2 =δ1 Þ; δ1;2 !0
κ ¼ lim ðκ2 =κ1 Þ; ν 1;2 !0
ð0 < δ; κ < 1Þ
ð1:40Þ
there exist two positive limiting TI operators, aδ;κ n
lim
δ1;2 ;κ1;2 !0
S1 δ1 ;κ1 n & Sδ2 ;κ2 n ¼ a0 n þ ðδ 1Þa1 n þ 3 2ðκ 1Þa2 n
(1.411) ð1:411Þ bδ;κ n
lim
δ1;2 ;κ1;2 !0
1
S1 1 a1 n þ 3 2 κ1 1 a2 n : δ2 ;κ2 n & Sδ1 ;κ1 n ¼ a0 n þ δ
ð1:412Þ (1.412)
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Using direct calculations one can confirm that the TI operators defined in (1.411 and 1.412) have the following properties: bδ;κ n ¼ a1 n ¼ a1=δ;1=κ n ; ð1:421Þ δ;κ (1.421) aδ;κ n & a n ¼ bδ;κ n & a n ¼ a n ;
(1.422) ð1:422Þ
a1;1 n ¼ b1;1 n ¼ a0 n :
ð1:423Þ (1.423)
Using (1.421) it is easy to prove that when either δ→0 and/or . →0, or δ→∞ or/ and . →∞, there are only two non-trivial cases of singular limiting behavior of TI operators aδ;κ n and bδ;κ n : (i) α-type by singular TI operators a0;κ n or aδ;0 n , or described a0;0 n ¼ a n , when the operators bδ;κ n do not exist; and (ii) β-type b1;κ n or bδ;1 n , or described by singular TI operators b1;1 n ¼ a n , when the operators aδ;κ n do not exist. The cases a0;0 n ¼ a n and b1;1 n ¼ a n describe, respectively, the limiting super-soft behaviors of either α- or β-types. The eigenvalue problem for the singular operator S n ¼ r0 a n where a n is presented in (1.38), has an easy solution. Due to (1.33) the eigenvalues are: n 1 ¼ r0 ; n 1 ¼ n 2 ¼ 0; ð1:43Þ whereas the “eigenoperators” Q n k ; n are given by (1.34). Note that the formulae (1.35 and 1.36) are still valid in this case. 3 Linear Nematic Viscoelasticity 3.1 Linear Kinematical Relations Bearing in mind that the main objective of this part is establishing a simple case of nematic viscoelasticity, we omit discussing here the well-known general relations for the balance of moment of momentum and related rotational inertia effects. One can find their derivations in the texts [3, 9, 21]. We use in the following the linear viscoelastic kinematics based on the common decomposition of full infinitesimal strain gradient tensor g into elastic (transient) γ and inelastic γ parts. Each of these tensors consists of infinitesimal strain p e rotation tensors, 4b ; 4b ; 4b . Then decom “body” "; "e ; "p and infinitesimal e p b b b position γ; γ e ; γ p ¼ "; "e ; "p þ 4 ; 4e ; 4p is presented as γ ¼ γ e þ γ p . Differentiating this equation with respect to time and separating equations for deformation and rotations, yields the kinematical rate equations: ε: e þ ep ¼ e ð2:1Þ :b 4e þ ωbp ¼ ωb ð2:21Þ (2.21)
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Here overdots denote time derivatives, e ¼ ð1=2Þ rv þ rv
T
T ð1=2Þ rv rv
and, ωb ¼
are, respectfully, the full strain rate and “body” vorticity, and :b b : : b ω ¼ 4 v is the velocity vector. So + rv ¼ e þ ω , ep ¼ ε p and p p are, respectfully, the irreversible strain rate and vorticity. Following the papers [17, 18], we now introduce in (2.1 and 2.21) the additional rate equation describing internal rotations for nematic continuum: : 4ie þ ωip ¼ ωi : ð2:22Þ (2.22) i Here the total tensor decomposed into the rate of : i of internal vorticity w was reversible rotation 4e and irreversible vorticity wip. Extracting now (2.22) from (2.21) results in the equation describing the linear kinematics of relative rotations in nematic continuum:
: 4 e þ ωp ¼ ω
4e ¼ 4be 4ie ; ωp ¼ ωbp ωip ; ω ¼ ωb ωi :
ð2:23Þ (2.23)
The evolution of director, generally characterized by kinematical equation, : n ¼ wi n ¼ wi n, can be considered in the linear nematic theory as the evolution of director disturbance dn near the state with a known constant value of director, n ¼ const:
ð2:3Þ d δn dt ¼ ωi n: We finally introduce the kinematical equation and related kinematical variables, convenient for characterizing a combined effect of viscoelastic deformations and relative rotations, as follows:
: *e þ γ ¼ γ *e ¼ " þ 4e ; γ ¼ ep þ ωp ; γ ¼ e þ ω : ð2:4Þ p
p
Only incompressible case is analyzed below, when all the strains and their rates are traceless. 3.2 Thermodynamics and Constitutive Relations 3.2.1 Non-symmetric Theory To describe the quasi-equilibrium effects in weak nematic viscoelasticity we will use the de Gennes type [2, 11] of potential (Helmholtz’s free energy density): 2 2 f ¼ 1=2G0 " þ G1 nn : "2 þ G2 nn : " 2G3 nn : " Ωe G5 nn : 42e
ð2:5Þ
Here Gk are the elastic moduli, and the lower index in "e is missed for simplicity. The expression for the dissipation D in the system, D TPs jT ¼ σs : e þ σa : ω df jT =dt;
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can be represented with the aid of (2.5) in the form: D ¼ σsp : e þ σap : ω þ σse : ep þ σae : ωp : Here T is the temperature, Ps is the entropy production, s s and s a are the symmetric and asymmetric parts of the extra stress tensor, respectively, and . . σsp σs σse ; σap σa σae ; σse ¼ @f @"; σae ¼ @f @4e : ð2:6Þ Here s se s sp and s ae s ap are the equilibrium (non-equilibrium) symmetric and asymmetric parts of the extra stress. Keeping in mind possible applications to LCP we consider below only the simplest (Maxwell-like) liquid case s sp ¼ 0; s ap ¼ 0 with the dissipation presented as: D ¼ σ s : e p þ σ a : ωp :
ð2:7Þ
This case describes “instant” elastic response of LCP’s to the applied forces. Due to (2.5) the symmetric s s and asymmetric s a parts of extra stress traceless tensor are defined as: . h i σs ¼ @f @" ¼ G0 " þ G1 nn " þ " nn 2nn " : nn þ 2ðG1 þ G2 Þ . (2.81) nn δ 3 " : nn þ G3 nn 4e 4e nn ; ð2:81Þ
σa ¼ @f
.
@Ωe ¼ G4 nn " " nn þ G5 nn Ωe þ Ωe nn :
ðG4 G3 Þ ð2:82Þ (2.82)
Note that due to (2.5) the Onsager relation G4 ≡−G3 is automatically fulfilled. Among several sources of stress asymmetry, such as inertial effects of internal rotations, orientation (Frank) elasticity and the Cosserat/Born isotropic couples, the most important for LCP is the action of external magnetic field, H. The common assumption used below is that the magnetic field is potential, with the potential function < presented as: < ¼ 1=2χ n : H H. Here χ n ¼ χ? δ þ χa nn is the susceptibility tensor, χk and χk are the susceptibilities parallel and perpendicular to the director, with χa ¼ χk χ? being the magnetic anisotropy. Because of a typical weak magnetization of the considered diamagnetic liquid, the effect of magnetic field on constitutive parameters Gk in (2.5) can be neglected. In this case, the body couple (“effective magnetic field”) is defined as:
h ¼ @< @ n ¼ χ a H n HÞ: ð2:9Þ When the inertial effects of internal rotations are negligible, the equilibrium equation for internal torques in magnetic field is:
σa ¼ 1 2 h n nh ; Σ σa n ¼ 1 2 h n h n ¼ 1 2 χ a n H H n n H :
ð2:10Þ
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Due to (2.7), the constitutive relations between the irreversible kinematical variables and stress are of the LEP type: h i σs ¼ η0 ep þ η1 nn ep þ ep nn 2nn nn : ep þ 2ðη1 þ η2 Þ . nn δ 3 nn : ep þ η3 nn ωp ωp nn ;
ð2:111Þ (2.111)
σa ¼ η4 nn ep ep nn þ η5 nn ωp þ ωp nn ðη4 ¼ η3 Þ:
(2.112) ð2:112Þ
Here we used the Onsager relation: η4 =−η3. Under common assumptions, we consider the kinetic coefficients ηk, as well as the parameters Gk, being independent of H. Also, we further assume that Gk ≠0 and ηk ≠0 in order to avoid the degeneration of CE’s. Demanding the elastic potential (2.5) to be thermodynamically stable results in the necessary and sufficient stability conditions [11], imposed on the parameters Gk: G0 > 0 ;
G5 > 0 ; G0 þ G1 > 0 ;
3=4G0 þ G1 þ G2 > 0;
ðG0 þ G1 ÞG5 > G23
ð2:121Þ (2.121)
The thermodynamic stability conditions for dissipation in (2.81 and 2.82) in incompressible case are the same as in (2.121) with substitution Gk→ηk [12]: η0 > 0 ; η5 > 0;
η0 þ η1 > 0 ; 3=4η0 þ η1 þ η2 > 0 ;
ðη0 þ η1 Þη5 > η23
ð2:122Þ (2.122)
Substituting now CE’s (2.9 and 2.10) into the expression for dissipation (2.7) reduces it to the quadratic form: 2 2 D ¼ η0 ep þ2η1 nn : e2p þ 2η2 nn : ep 4η3 nn : ep ωp 2η5 nn : ω2p : ð2:13Þ Due to the stability constraints (2.121–2.123) the quadratic form (2.13) is positively definite. 3.2.2 Symmetric Theory As follows from (2.10), in the absence of magnetic field ðH ¼ 0Þ when Σ ¼ σa n ¼ 0 and s a ¼ 0, stress tensor is symmetric. So (82 and 92) yield the kinematical relations, 4e ¼ λ1 " nn nn " ; ðλ1 ¼ G3 =G5 ;
ωp ¼ λ2 ep nn nn ep λ2 ¼η3 =η5 Þ:
(2.141,2 ð2:141; 2Þ)
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Here 1 1 and 1 2 are sign indefinite parameters. Substituting (2.141,2) back into (82 and 92), and in (5) yields the reduced (or “renormalized”) formulation of these equations with symmetric stress as: 2 2 f ¼ 1=2G0 " þ Gr1 nn : "2 þ Gr2 nn : " ð2:15Þ r
G1 ¼ G1 G23 G5 ; Gr2 ¼ G2 þ G23 G5 . h i σ ¼ @f @" ¼ G0 " þ Gr1 nn " þ " nn 2nn ":nn þ 2 Gr1 þ Gr2 . " : nn nn 1 3δ
ð2:161Þ
(2.161)
h i 1 @D ¼ η0 ep þ ηr1 nn ep þ ep nn 2nn ep : nn þ 2 ηr1 þ ηr2 ð2:162Þ 2 @ep . ep : nn nn δ 3 (2.162)
σ¼
The dissipation D ¼ TPs jT ¼ s : ep is presented in the form: 2 D=2 ¼ η0 ep þηr1 nn : e2p 2
þ ηr2 nn : ep ηr1 ¼ η1 η23 η5 ;
ηr2 ¼ η2 þ η23 η5
ð2:17Þ
Equation (2.161) is the Ericksen CE. It is seen that D/2 there plays the role of Raleigh dissipative function. The (necessary and sufficient) conditions of thermodynamic stability [11] are: G0 > 0;
G0 þ Gr1 ¼ G0 þ G1 G23 G5 > 0;
3=4G0 þ Gr1 þ Gr2 ¼ 3=4G0 þ G1 þ G2 > 0 η0 > 0;
η0 þ ηr1 ¼ η0 þ η1 η23 η5 > 0;
3=4η0 þ ηr1 þ ηr2 ¼ 3=4η0 þ η1 þ η2 > 0:
(2.181) ð2:181Þ
(2.182) ð2:182Þ
One can see complete similarity in inequalities (2.181 and 2.182). When analyzing the symmetric case the notations for simplicity will be changed as: Grk $ Gk , hrk $ hk ðk ¼ 1; 2Þ. One can expect that excluding the variables wp and ep from the final formulation in non-symmetrical case will result in coupled equations for evolution of the hidden thermodynamic parameters Ωe and ". In symmetrical case only ep should be
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105
excluded to obtain evolution equation for ". These common awkward operations are easy with the use of N-operators. 3.3 N-operators in Viscoelastic Nematodynamics Table 2 establishes the correspondence between the continual equations in “Section 3.2” and algebraic relations in “Section 2.1”. The constitutive parameters Gˆ k and ηˆk in Table 2 are: Gˆ k ¼ Gk ðk ¼ 0; 1; 3; 4; 5Þ; Gˆ 2 ¼ 2G1 þ 2G2 ; ηˆk ¼ ηk ðk ¼ 0; 1; 3; 4; 5Þ; ηˆ2 ¼ 2η1 þ 2η2 ;
G4 ¼ G3 η4 ¼ η3 :
(2.191) ð2:191Þ (2.192) ð2:192Þ
Only non-degenerating conditions Gk ≠0, ηk ≠0 are considered below. Using Table 2 it is easy to establish direct and inverse relations for CE’s described by ONoperators. 3.3.1 N-operator Presentation of Non-symmetric Theory 1. N-operator presentation of “elastic” and “viscous” CE’s (2.181, 2.182 and 2.9): 5 2 P P Gˆ k ak n & *e , σs ¼ Gˆ k ak n & " þ G3 a3 n & 4e ; σ G n & *e ¼ k¼0
k¼0
σ ¼ G4 a4 n & "e þ G5 a5 n & Ωe (2.20 1,2) ð2:2012Þ 5 2 P P s hˆ k ak n & γ , σ ¼ hˆ k ak ðnÞ & ep þ η3 a3 n & ωp ; σ ¼ h n & γ ¼ a
p
k¼0
p
σa ¼ h4 a4 n & ep þ η5 a5 n & ωp
k¼0
Here G n RoG n and ) n Roη n are the ON-operators of moduli and viscosity. Table 2 Correspondence between algebraic and physical variables/parameters/equations Algebraic xs ……………….. (1.1) symmetric tensor (variable) xa ……………….. (1.1) asymmetric tensor variable y ………………. (1.1) s symmetric tensor, function y ………………. (1.1) a asymmetric tensor, function rk (k=0,.., 5) ……… (1.1) parameters of N-operators P ………………… (1.2) quadratic form
Elastic Nematic " ……………….. (2.81 and 2.82) elastic strain tensor We ……………… (2.81 and 2.82) elastic relative rotation s s ……………… (2.81) symmetric extra stress s a ……………… (2.82) asymmetric extra stress Gˆ k ……….. (2.81 and 2.82, 2.161) nematic elastic moduli 2 f ……………… (2.5) nematic free energy
Viscous Nematic ep …………….. (2.111 and 2.112) inelastic strain rate tensor wp …………….. (2.111 and 2.112) inelastic relative vorticity s s …………….. (2.111) symmetric extra stress s a …………… (2.112) asymmetric extra stress ηˆ k … (2.111 and 2.112, 2.162) nematic viscosities D ………………. (2.13) dissipation
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2. N-operator presentations of free energy and dissipation: 5 5 X
X ηˆk γ ak ðnÞ γ Gˆ k Γe ak n Γe ; D TPs jT ¼ f ¼1 2 k¼0
k¼0
p
p
(2.211,2 ð2:211; 2Þ)
3. Inverse relations expressing kinematic variables via stresses: *e ¼ G1 n &σ J n &σ;
5 P J n ¼ Jk ak n ;
γ ¼ ) n &σ F n &σ;
5 P F n ¼ 8k ak n :
(2.221,22Þ) ð2:221;
k¼0
1
k¼0
Here J n NoJ n and F n No8 n are the ON-operators of compliance and fluidity, respectively. Their basis scalars, compliances Jk (dimensionality of inverse modulus) in (2.221,2), and fluidities 8k (dimensionality of inverse viscosity) in (2.221,2), are: J0 ¼
1 G0
;
J1 ¼
J3 ¼ J4 ¼
ðG23 G1 G5 Þ=G0 G5 ðG0 þG1 ÞG23
G3 G5 ðG0 þG1 ÞG23
80 ¼
1 η0
81 ¼
83 ¼
η3 η5 ðη0 þη1 Þη23
;
;
;
ðG1 þG2 Þ=G0 J2 ¼ 3=2 3=4G0 þG1 þG2
;
J5 ¼
ðη23 η1 η5 Þ=η0 η5 ðη0 þη1 Þη23
G0 þG1 G5 ðG0 þG1 ÞG23
ð2:231Þ (2.231)
;
ðη1 þη2 Þ=η0 8 2 ¼ 3=2 3=4η þη þη
;
0
8 4 ¼ 8 3;
85 ¼
1
(2.232) ð2:232Þ
2
η0 þη1 η5 ðη0 þη1 Þη23
4. Expressions of g ¼ ep þ wp via Γe ¼ "e þ Ωe and inverse from the dual p equations (2.221,2): ! 5 X g ¼ sðnÞ&*e sðnÞ ¼ s k ak ð nÞ ; p
k¼0
*e ¼ EðnÞ& g
p
EðnÞ ¼
5 X
!
(2.241,22Þ ) ð2:241;
θ k ak ð nÞ
k¼0
Here sðnÞ ¼ )1 ðnÞ&GðnÞ ¼ FðnÞ&GðnÞ and EðnÞ ¼ G1 ðnÞ&)ðnÞ ¼ JðnÞ&)ðnÞ are the N-operators of relaxation frequencies and relaxation times, respectively. Using (2.231 and 2.232), their basis scalar parameters sk and qk are calculated as: s0 ¼
G0 η0
s3 ¼
θ0 ¼
η0 G0
θ3 ¼
η5 ðG1 η0 G0 η1 Þþη3 ðG0 η3 G3 η0 Þ 2 Þη0 G0 ðη1 þη2 Þ ; s2 ¼ 32 ðG1ηþG η0 ½η5 ðη0 þη1 Þη23 0 ð3=4η0 þη1 þη2 Þ G3 ðη0 þη1 Þη3 ðG0 þG1 Þ G3 η5 G5 ðη0 þη1 ÞG3 η3 ; s4 ¼ η Gð5ηη3þη 2 ; s5 ¼ η5 ðη0 þη1 Þη23 η5 ðη0 þη1 Þη23 5 0 1 Þη3
; s1 ¼
G5 ðη1 G0 η0 G1 ÞþG3 ðη0 G3 η3 G0 Þ 2 ÞG0 η0 ðG1 þG2 Þ ; θ2 ¼ 32 ðηG1 þη 0 ð3=4G0 þG1 þG2 Þ G0 ½G5 ðG0 þG1 ÞG23 η3 ðG0 þG1 ÞG3 ðη0 þη1 Þ η3 G5 η5 ðG0 þG1 Þη3 G3 ; θ4 ¼ G5ηð5GG03þG 2 ; θ5 ¼ G ðG þG ÞG2 G5 ðG0 þG1 ÞG23 1 ÞG3 5 0 1 3
; θ1 ¼
ð2:251Þ (2.251)
ð2:252Þ (2.252)
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107
Note that EðnÞ ¼ )ðnÞ G1 ðnÞ and sðnÞ ¼ GðnÞ & )1 ðnÞ ¼ E1 ðnÞ are not ONoperators. 5. Evolution equation for elastic (transient) strain " and elastic rotation Ωe , obtained upon substituting (2.241,2) in (2.4), is: X s k ak ð nÞ & " þ s 3 a3 ð nÞ & 4e *e þ sðnÞ & *e ¼ g , " þ 2
k¼0
¼ e; 4e þ s4 a4 ðnÞ & " þ s5 a5 ðnÞ & 4e ¼ ω
(2.261) ð2:261Þ
Equations (2.261) written in the common tensor form are presented as: h i . " þ s0 " þ s1 n n " þ " n n 2nn " : n n þ s2 n n δ 3 " : n n þs3 ðn n 4e 4e n nÞ ¼ e 4 e þ s 4 n n " " n n þ s 5 n n 4e þ 4e n n ¼ 4 (2.262) ð2:262Þ Here the basis scalars of N-operator of relaxation frequency sðnÞ are presented in (2.251). 6. Maxwell-like nematodynamic equations have the following equivalent forms: JðnÞ&σ þ 6ðnÞ&σ ¼ g ; σ þ sðnÞ&σ ¼ GðnÞ& g ;
(2.27 ð2:27123Þ 1,2,3)
EðnÞ&σ þ σ ¼ )ðnÞ& g
The basis scalar parameters Jk ; 8 k and sk ; qk are expressed via the given model parameters Gk and hk in 2.231, 2.232 and 2.241,2), respectively. The example of “split” expression for CE 2.221,2) is: 2 P σ s þ ak ðnÞ& sk σs Gk e þ a3 ðnÞ& s3 σa G3 w ¼ 0 k¼0 ð2:27 *2Þ (2.27 2) σa þ a ðnÞ& s σs G e þ a ðnÞ& s σa G w ¼ 0 4 4 3 5 5 5 7. The eigenvalues of N-operator of relaxation frequency sðnÞ due to (1.18) are: ν 1 ¼ s0 ;
ν 2 ¼ s0 þ 2=3s2 ;
ν 3;4 ¼ 1=2ðs0 þ s1 þ s5 d Þ;
ð2:28Þ
d 2 ¼ ðs0 þ s1 s5 Þ2 4s3 s4 : Here sk are the basis parameters (2.251) of the N-operator of relaxation frequency sðnÞ, Due to “Appendix,” all n k in (2.28) are positive, while the Noperator sðnÞ is not necessarily positive. 8. Basic representation theorem of non-symmetric linear nematic viscoelasticity: Maxwell-like nematodynamic (2.271,2,3) are always presented in the equivalent forms of LEP CE’s (2.111, 2.112 or 2.161, 2.162), where the parameters are changed for linear viscoelastic functionals.
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The N-operator formulation is: σ¼
5 X
n
o
0
ak ðnÞ& φk ðt Þ*γ ðt Þ @φk ðt Þ*γ ðt Þ
k¼0
σs ¼
2 X
Zt
1 φk ðt t1 Þγ ðt1 Þdt1A
1
ak ðnÞ& φk *e þ a3 ðnÞ& φ3 *ω ; σa ¼ a4 ðnÞ& φ4 *e þ a5 ðnÞ& φ5 *ω
ð2:291Þ (2.291)
k¼0
The common tensor formulation is: h i . σs ¼ φ0 *e þ φ1 * nn e þ e nn 2nn nn : e þ 2ðφ1 þ φ2 Þ* nn δ 3 nn : e þφ3 * nn w w nn ; σa ¼ φ4 * nn e e nn þ φ5 * nn w þ w nn
(2.292) ð2:292Þ Here φk ðt Þare: φ0 ðt Þ ¼ G0 eν1 t ; φ1 ðt Þ ¼ G0 eν1 t þ ðG0 þ G1 Þðκ2 eν3 t κ1 eν4 t Þ; φ2 ðt Þ ¼ 3=2G0 eν 1 t þ 2ð3=4G0 þ G1 þ G2 Þeν2 t φ3 ðt Þ ¼ ðs3 =d ÞðG0 þ G1 Þeν1 t þ ½ðs3 =d ÞðG0 þ G1 Þκ1 G3 s4 =d eν 3 t þ κ2 G3 eν 4 t φ4 ðt Þ ¼ ðκ2 G3 þ G5 s4 =d Þeν3 t þ ðκ1 G3 þ G5 s4 =d Þeν4 t φ5 ðt Þ ¼ G3 ðs3 =d Þeν 1 t þ ½G3 ðs3 =d Þ κ1 G5 s4 =deν3 t þ κ2 G5 eν 4 t fκ1 ¼ ðs0 þ s1 ν 3 Þ=d;
κ2 ¼ ðs0 þ s1 ν 4 Þ=d g ð2:30Þ
The derivation of the formulae (2.291, 2.292 and 2.30), which proves the theorem, is made in several steps. 1. Consider the first equation in (2.261), *e þ sðnÞ*e ¼ g . Searching for a b nt , reduces solution of the homogeneous equation in the form, Γhe ðt Þ ¼ Γe solution of this equation to the eigenvalue problem (1.172), where Nr ðn; nÞ ¼ sðnÞ nIðnÞand eigenvalues n k are exposed in (2.28). The eigenoperators Qðn k ; nÞare presented in (1.20), with substitutions rk ! sk . 2. Utilizing additionally the condition (1.21) yields the following solution of 4 P initial homogeneous problem: Γhe ðt Þ ¼ enk t Qðn k ; nÞ Γhe ð0Þ. Employing k¼1 then the standard technique yields the solution of the evolution (2.211,2) presented in N-operator form as a linear memory functional: Γe ðt Þ ¼ 4 Rt ν ðtt Þ P Qðν k ; nÞ e k 1 γ ðt1 Þdt1. Utilizing here the relations (1.19 and k¼1
1
1.21) with substitutions rk ! sk yields: 5 n o X Γ e ðt Þ ¼ ak ðnÞ& χk ðt Þ*γ ðt Þ
ð2:31Þ
k¼0
The following notations have been used in (2.31): χ0 ðt Þ ¼ eν1 t ; χ1 ðt Þ ¼ eν 1 t þ ðκ2 =d Þeν 3 t ðκ1 =d Þeν 4 t ; χ2 ðt Þ ¼ 3=2ðeν 2 t eν 1 t Þ; χ3 ðt Þ ¼ ðs3 =d Þðeν 3 t eν 1 t Þ; χ4 ðtÞ ¼ ðs4 =d Þðeν 4 t eν 3 t Þ;χ5 ðt Þ ¼ κ1 eν 3 t þ κ2 eν 4 t
ð2:32Þ
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3. Using (2.31) in the first relation of (2.161) yields (2.291, 2.292 and 2.30). Formulae (2.291, 2.292/2.30 and 2.31/2.32) correctly describe the two limiting cases: 1. The initial elastic “jump”, i.e. Γe ðþ0Þ ¼ Γ0 and σ ¼ GðnÞ Γ0 , when g ðt Þ ¼ Γ0 d ðt Þ, and 2. The case: Γ ð1Þ ¼ θðnÞ& γ ; σð1Þ ¼ )ðnÞ& γ , when g ðt Þ ¼ H ðt Þ g e 0 0 0 g ¼ constÞ. 0
Here δ (t) and H (t) are Dirac delta and Heaviside functions, and GðnÞ; EðnÞ, and )ðnÞare the N-operators of moduli, relaxation time, and viscosity, respectively. 3.3.2 TI-operator Presentation of Symmetric Theory 1. The TI-operator presentations of “elastic” and “viscous” CE’s (2.161 and 2.161) are: σ ¼ GðnÞ& " ¼ )ðnÞ&ep ; GðnÞ ¼
2 X
Gˆk ak ðnÞ;
ð2:3312Þ (2.33 1,2)
k¼0
)ð n Þ ¼
2 X ηˆk ak ðnÞ k¼0
Here GðnÞ SG ðnÞ and )ðnÞ Sη ðnÞ are the TI operators of moduli and viscosity. Also, Gˆ k and b hk are defined in (2.191 and 2.192) for k=0, 1, 2 with nodegenerating conditions Gk 6¼ 0, hk 6¼ 0. It is also worth reminding of simplifying notations, Grk $ Gk , hrk $ hk ðk ¼ 1; 2Þ, accepted in “Section 3.2.2” after renormalization procedure. 2. The TI-operator presentations of free energy and dissipation are: f ¼ 1=2"&GðnÞ&";
D ¼ ep &)ðnÞ ep :
ð2:3412Þ (2.34 1,2)
3. The inverse relations expressing kinematical variables via stresses are: " ¼ G1 ðnÞ&σ; G1 ðnÞ JðnÞ ¼ 6ðnÞ ¼
3 P
3 P
Jk ak ðnÞ; " ¼ )1 ðnÞ&σ; )1 ðnÞ
i¼0
8 k ak ð nÞ
i¼0
(2.351,2) ð2:3512Þ
Here JðnÞand FðnÞare the compliance and fluidity TI-operators, respectively. Due to (1.29) the expressions for their respective basis scalars are: J0 ¼ 1=G0 ; 80 ¼ 1=η0 ;
J1 ¼
81 ¼
G1 =G0 ; G0 þ G1
η1 =η0 ; η0 þ η1
J2 ¼
82 ¼
3=2ðG1 þ G2 Þ=G0 3=4G0 þ G1 þ G2
3=2ðη1 þ η2 Þ=η0 3=4η0 þ η1 þ η2
(2.36 ð2:361Þ 1) ð2:362Þ (2.362)
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4. The expressions ep via "e or vice versa using dual equations (2.331,2) are: ep ¼ EðnÞ&"; " ¼ sðnÞ&ep
ð2:371:2Þ (2.371,2)
Here sðnÞ = )1 ðnÞ&GðnÞ ¼ 6ðnÞ&GðnÞ and EðnÞ ¼ G1 ðnÞ&)ðnÞ ¼ JðnÞ )ðnÞ are the TI-operators of relaxation frequencies and relaxation times, respectively. Using (2.361 and 2.362), their basis scalar parameters sk and θk are calculated as: η0 η G 0 G 1 η0 3 G0 ðη1 þ η2 Þ η0 ðG1 þ G2 Þ ; θ2 ¼ ; θ1 ¼ 1 2 G0 ð3=4G0 þ G1 þ G2 Þ G0 G0 ðG0 þ G1 Þ
(2.381) ð2:381Þ
G0 G 1 η 0 η0 G 0 3 η ðG1 þ G2 Þ G0 ðη1 þ η2 Þ ; s2 ¼ 0 : ; s1 ¼ 2 η0 ð3=4η0 þ η1 þ η2 Þ η0 η0 ð η0 þ η1 Þ
(2.382) ð2:382Þ
θ0 ¼
s0 ¼
5. Evolution equation for elastic (transient) strain ", obtained upon substituting (2.371.2) in (2.4): 2 X : : " þ s n " ¼ e , " sk ak ðnÞ " ¼ e: þ
ð2:391Þ (2.39 1)
k¼0
It is written in the common tensor form as: h i . : " þ s0 " þ s1 nn " þ " nn 2nn " : nn þ s2 nn δ 3 " : nn ¼ e:
ð2:392Þ (2.39 2)
The basis scalars of TI-operator of relaxation frequency sðnÞ are presented in (2.382). 6. Maxwell-like nematodynamic equations have the following equivalent forms: :
:
:
σ σ σ JðnÞ þ FðnÞ σ ¼ e; þ sðnÞ σ ¼ GðnÞ e; EðnÞ þ σ ¼ )ðnÞ e
(2.401,2,3 ) ð2:401; 2; 3Þ
The basis scalar parameters Jk,8k and sk,Ek are expressed via the given model parameters Gk and ηk in (2.361, 2.362 and 2.381, 2.382), respectively. 7. The eigenvalues of TI-operator of relaxation frequency sðnÞ due to (1.39) are: ν 1 ¼ s0 ¼
G0 G0 þ G1 2 3=4G0 þ G1 þ G2 ; ν 2 ¼ s0 þ s1 ¼ ; ν 3 ¼ s0 þ s2 ¼ : η0 η0 þ η1 3=4η0 þ η1 þ η2 3
ð2:41Þ Due to the stability conditions (2.181 and 2.182) all the eigenvalues vk are positive and describe the relaxation frequencies, with the respective relaxation times θˆk ¼ 1=ν k .
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8. Basic representation theorem of symmetric linear nematic viscoelasticity: Maxwell-like nematodynamic equations (2.401,2,3) are always presented in the equivalent forms of Ericksen CE’s (2.162), where parameters are changed for linear viscoelastic functionals, as: 0 1 Zt 2 n o X (2.42 1) ð2:421Þ σ¼ ak ðnÞ& φk ðt Þ*eðt Þ @φk ðt Þ*eðt Þ φk ðt t1 Þeðt1 Þdt1A k¼0
1
or: h i . σ ¼ φ0 *e þ φ1 * n n e þ e n n 2n n n n : e þ 2ðφ1 þ φ2 Þ* n n δ 3 n n : e
ð2:422Þ (2.42 2) With eigenvalues vk given in (2.41), φk (t) are given as: φ0 ðt Þ ¼ G0 eν1 t ; φ1 ðt Þ ¼ G0 eν 1 t þ ðG0 þ G1 Þeν2 t ;
ð2:43Þ
φ2 ðt Þ ¼ 3=2G0 eν 1 t þ 2ð3=4G0 þ G1 þ G2 Þeν 3 t The proof is the same as in the non-symmetric case, and based on the solution of evolution (2.391 and 2.392) for elastic (transient) strain, "¼
2 X
n
ak ðnÞ χk ðt Þ*eðtÞ
o
χ0 ðt Þ ¼ e
ν 1 t
; χ 1 ðt Þ ¼ e
ν 2 t
e
ν 1 t
k¼0
3 ; χ2 ðt Þ ¼ ðeν3 t eν 1 t Þ 2
ð2:44Þ The same limiting cases as in non-symmetric linear nematic viscoelasticity are valid here. 3.3.3 Soft Modes in Linear Nematic Viscoelasticity Consider now non-generating TI operators, when the values of material parameters belong to the marginal stability boundaries in (2.181 and 2.182). There are four independent marginal stability conditions: (2.451)
G0 þ Gr1 ¼ G0 þ G1 G23 G5 ¼ 0; η0 þ ηr1 ¼ η0 þ η1 η23 η5 ¼ 0 ð2:451Þ 3=4G0 þ Gr1 þ Gr2 ¼ 3=4G0 þ G1 þ G2 ¼ 0; 3=4η0 þ ηr1 þ ηr2 ¼ 3=4η0 þ η1 þ η2 ¼ 0:
(2.452) ð2:452Þ The nearly marginal, still stable situations happen when instead (2.451 and 2.452) the four independent conditions are satisfied: G0 þ G1 ¼ d G G0 ; h0 þ h1 ¼ d h h0 ð0 << d G ; d h << 1Þ 3=4G0 þ G1 þ G2 ¼ 3=2kG G0 ; 3=4η0 þ η1 þ η2 ¼ 3=2kη η0
(2.461) ð2:461Þ
(2.462) 0 << kG ; kη << 1 ð2:462Þ
If one of (or both) the conditions (2.461) occurs, the behavior of viscoelastic (as the
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viscous or weakly elastic) nematics is sensitive to magnetic field, with great effects expected when the field is applied. On the contrary, the conditions (2.462) seem to be insensitive to magnetic field. When the magnetic field is absent, one could a use the marginal conditions (2.451 and 2.452) and employ the results of “Section 2.5.4”. We consider here for example the extreme case when both the elastic and viscous TI operators are completely soft. Then CE’s (2.331,2) take the form: σ ¼ G0 aðnÞ " ¼ η0 aðnÞ ep ; aðnÞ ¼ a0 ðnÞ a1 ðnÞ 3=2a2 ðnÞ:
ð2:47Þ
In this case, G1 ¼ G0 ; G2 ¼ G0 =4; η1 ¼ η0 ; η2 ¼ η0 =4; and free energy and dissipation are represented as: 2 2 f =G0 ¼ 1=2"&aðnÞ&" ¼ 1=2" n n:"2 þ1=4 n n:" 0 2 2 2D=η0 ¼ 1=2ep &aðnÞ&ep ¼ 1=2ep n n:e2p þ1=4 n n:ep 0:
ð2:48Þ
ð2:49Þ
The TI operator aðnÞ is singular, i.e. neither a1 ðnÞ exists nor the formulae (2.361, 2.362, 2.371,2 and generally, 2.381, 2.382). The evolution equation (2.391) for elastic (transient) strain ", for super-soft case due to limit in formulae (1.421, 1.422 and 1.423) is represented as: h . e þ s0 aðnÞ&e ¼ e , e þ s0 " nn" "nn þ 2nn ":n n 3=2 n n δ 3 ":n n ¼ e
ð2:50Þ Here the only parameter, s0 ¼ G0 =η0 , is the relaxation frequency. 4. The eigenvalues of super-soft TI-operator of relaxation frequency sðnÞ ¼ s0 aðnÞ due to (1.35) are: ν 1 ¼ s0 ¼ G0 =η0 ; ν 2 ¼ ν 3 ¼ 0:
ð2:51Þ
5. The basic representation theorem of symmetric linear nematic viscoelasticity in the super-soft case is presented by the limit singular case of (2.421) as: Zt σ ¼ G0 a n &Eðt Þ; E ðt Þ ¼ eðt1 Þes0 ðtt1 Þ dt1 1
h
. i σ ¼ G0 E nn E E nn þ 2nn E:nn 3=2 nn δ 3 E:nn
ð2:52Þ
The derivation is based on direct use of (2.51) and the marginal stability conditions (2.451 and 2.452).
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In this case, the form of convolution formula (2.44) for elastic (transient) strains " remains the same, but the functions χk (t) there change for:
# 0 ðt Þ ¼ es0 t ; # 1 ðt Þ ¼ 1 es0 t ; # 2 ðt Þ ¼ 3=2ð1 es0 t Þ:
ð2:53Þ
It means that the elastic strain tensor " can unrestrictedly grow in time. This is the asymptotic effect of the super-soft nematic viscoelastic behavior. Nevertheless, one should recall that the linear nematic viscoelasticity in the soft cases can only hold either for a restricted time with a given constant value of strain rate e, or for a small amplitude oscillatory flow.
4 Results and Discussions This paper consists of two parts. The first part analyzes the mathematical properties of linear nematic (N-) operators, which appeared in nematic viscous and elastic theories. The N-operators were first introduced and described in our preprint [22]. It is shown that N-operators are represented by a specific set of basis fourth rank tensors in both the general no-symmetric and particular symmetric cases. The analysis includes the group and spectral properties of N-operators in both nonsymmetric and symmetric cases, as well as their singular properties in symmetric case, with singular, semi-positively definite symmetric TI-operators. The main objective of this part is to introduce and develop a new theoretical tool for analyses of nematic theories, which highly simplifies theoretical description. The second part of the paper, developed a theory of linear viscoelastic nematodynamics applicable to LCP in the mono-domain limit, when the Frank elasticity effects are absent. The new theory of N-operators developed in the first part of paper, was applied to general analysis of the linear viscoelastic nematodynamics of Maxwell-type. Continuum approach employed in the paper is based on well-elaborated specific viscoelastic kinematics and non-equilibrium thermodynamics. The action of external fields is illustrated on example of magnetic field. Additionally, the concept of marginal stability introduced first in papers [11, 12] and mathematically developed for singular TI operators in the “Section 2.5.4”, is employed for analyses of soft viscoelastic modes in the “Section 3.3.3”. The main results are the reduction of the differential formulation to the LEP-like equations where the coefficients are changed for linear memory functionals whose parameters are calculated from original viscosities and moduli. Similar formulae have been obtained in paper [13] using statistical derivation. Keeping aside discussion about the derivation and concepts in [13], even a quick comparison of formulae for integral presentation of the theory in this paper with similar ones obtained in [13] (in the single relaxation mode Maxwell-type, or dumbbell limit) shows their qualitative difference, e.g. in the non-symmetric current approach, the anisotropic relaxation functions are presented via four independent exponential terms, whereas in [13] via three terms. In the symmetric case only two independent exponential terms describe the anisotropic relaxation functions in [13] instead of three terms in the present paper.
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It is remarkable that the complete soft shearing and soft elongation case in this paper is described when holding the complete anisotropy by formulae (2.52) with only one relaxation time as in the isotropic case. When comparing these results with real linear dynamic behavior of LCP’s one should involve the semi-soft description of relaxations, which are grouping near the basic soft relaxation frequency s0 in (2.52). The results of the present analysis could be directly applied to the description of linear dynamic behavior of rigid enough industrial thermotropic LCP’s, where the proposed Maxwell-like phenomenology seems to explain the major effect, the exact balance of the anisotropic viscous and elastic forces (including the magnetic torque), which cause extension, bending and twisting rigid enough macromolecules. For the more flexible, model types of LCP’s where the conformation effects should also be taken into account, the proposed approach could serve at least as the first approximation.
Appendix A prove of positive definiteness of eigenvalues for a dual operator Np ðnÞ in (1.14) represented via two positive Onsager operators o o o According to (1.14) Np ðnÞ ¼ No1 q ðnÞ&Nr ðnÞ; where Nr ðnÞ and Nq ðnÞ are positive, non- degenerating ON operators. The eigenvalues of operator Np ðnÞ are expressed in (1.18) via its basis parameters p with substitution r→p(r,q) and the use of (1.15), where the basis parameters rk and qk under condition r4 =−r3 satisfy inequalities (1.3). It is proved below that due to these inequalities, all eigenvalues vk of operator Np ðnÞ are positive. The first two eigenvalues, v1 and v2 defined in (1.18) with the substitution r→p(r, q) and (1.15), are positive because ν 1 ¼ p0 ¼ r0 =q0 > 0; ν 2 ¼ p0 þ 2=3p2 ¼ ðr0 þ 2=3r2 Þ=ðq0 þ 2=3q2 Þ > 0: Other two eigenvalues v3,4 due to (1.18) are presented as: ν 3;4 ¼ 1=2ðp0 þ p1 þ p5 d Þ; d 2 ¼ ðp0 þ p1 þ p5 Þ2 4½p5 ðp0 þ p1 Þ þ p3 p4 : ð3Þ Here p=p(r,q) is defined in (1.15). The proof that v3 and v4 are positive is given below in three steps. 1. We firstly show that p0 þ p1 þ p5 > 0. Due to (1.15) this sum is represented as: p0 þ p1 þ p5 ¼
ðr0 þ r1 Þq5 þ ðq0 þ q1 Þr5 2r3 q3 : q5 ðq0 þ q1 Þ q23
ð4Þ
The denominator in (4) is positive due to inequalities (1.3), and the numerator is positive, because ðr0 þ r1 Þq5 þ ðq0 þ q1 Þr5 2q3 r3 2½ðr0 þ r1 Þr5 ðq0 þ q1 Þq5 1=2 2q3 r3 > 2ðjq3 r3 j q3 r3 Þ > 0: 2
½ðr0 þr1 Þq5 ðq0 þq1 Þr5 þ4F 2 2 2. We secondly show that d >0. Presenting d ¼ , where F ¼ ½q5 ðq0 þq1 Þq23 2 r5 ðr0 þ r1 Þr3 gð xÞ; g ð xÞ ¼ ðx αÞðx β Þðx q3 =r3 Þ, α ¼ ðq0 þ q1 Þ=ðr0 þ r1 Þ; 2
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and β ¼ q5 =r5 , and establishing that min gð xÞ ¼ g ðxm Þ ¼ gf1=2ðα þ βÞg ¼ 1=4ðα β Þ2 , yields: 2 ðr0 þ rÞ1 q5 ðq0 þ q1 Þr5 þ4 min F 2 d 2 q5 ðq0 þ q1 Þ q23 ½ðr0 þ r1 Þq5 ðq0 þ q1 Þr5 2 q5 ðq0 þ q1 Þ q23 ¼ > 0: 2 r5 ðr0 þ r1 Þ q5 ðq0 þ q1 Þ q23 Using now (1.15), it is shown that p5 ðp0 þ p1 Þ þ p3 p4 ¼
r5 ðr0 þ r1 Þ r32 > 0: q5 ðq0 þ q1 Þ q23
ð5Þ
3. Finally, due to (3 and 4) it is seen that ν 3 ¼ 1=2ðp0 þ p1 þ p5 þ d Þ > 0, and due to (3 and 5) it is clear that ν 4 ¼ 1=2ðp0 þ p1 þ p5 d Þ ¼ ½p5 ðp0 þ p1 Þ þ p3 p4 =ν 3 > 0.
References 1. Beris, A.N., Edwards, B.J.: Thermodynamics of flowing systems. Oxford University Press, Oxford (1999) 2. de Gennes, P.G.: Weak nematic gels. In: Helfrich, W., Kleppke, G. (eds.) Liquid crystals in one and two dimensional order, pp. 231–237. Springer, Berlin (1980) 3. de Gennes, P.G., Prost, J.: The physics of liquid crystals, 2nd edn. Clarendon, Oxford (1993) 4. Doi, M., Edwards, S.F.: The theory of polymer dynamics. Clarendon, Oxford (1986) 5. Einaga, Y., Berry, G.C., Chu, S.-G.: Rheological properties of rod-like polymers in solution. 3. Transient and steady-state studies on nematic solutions. Polymer (Japan) 17, 239 (1985) 6. Feng, J.J., Sgalari, G., Leal, L.G.: A theory for flowing nematic polymers with orientational distortion. J. Rheol. 44, 1085–1101 (2000) 7. Golubovich, L., Lubensky, T.C.: Nonlinear elasticity of amorphous solids. Phys. Rev. Lett. 63, 1082– 1085 (1989) 8. Han, C.D., Ugaz, V.M., Burghardt, W.R.: Shear stress overshoots in flow inception of semi-flexible thermotropic liquid crystalline polymers: experimental test of a parameter-free model prediction. Macromolecules 34, 3642–3645 (2001) 9. Kleman, M.: Points, lines and walls. Wiley, New York (1983) 10. Larson, R.G.: The structure and rheology of complex fluids. Oxford Press, New York (1999) 11. Leonov, A.I., Volkov, V.S.: General analysis of linear nematic elasticity. J. Eng. Phys. Thermophys. 77, 717–726 (2004) 12. Leonov, A.I., Volkov, V.S.: Dissipative soft modes in viscous nematodynamics. Rheol. Acta 44, 331– 341 (2005) 13. Long, D., Morse, D.C.: A Rouse-like model of liquid crystalline polymer melts: director dynamics and linear viscoelasticity. J. Rheol. 46, 49–92 (2002) 14. Lubensky, T.C., Mukhopadya, R.: Symmetries and elasticity of nematic gels. Phys. Rev. E 66, 011702 (2002) 15. Marrucci, G., Greco, F.: Flow behavior of liquid crystalline polymers. Adv. Chem. Phys. 86, 331–404 (1993) 16. Odell, P.A., Unger, G., Feijo, J.L.: A rheological, optical and X-ray study of the relaxation and orientation of nematic PBZT. J. Polym. Sci.: Polym. Phys. 31, 141 (1993) 17. Pleiner, H., Brand, H.R.: Macroscopic dynamic equations for nematic liquid crystalline side-chain polymers. Mol. Cryst. Liq. Cryst. 199, 407–418 (1991)
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18. Pleiner, H., Brand, H.R.: Local rotational degrees of freedom in nematic liquid-crystalline side-chain polymers. Macromolecules 25, 895–901 (1992) 19. Ugaz, V.M., Burghardt, W.R.: In situ X-ray scattering study of a model thermotropic copolyester under shear: evidence and consequences of flow-aligning behavior. Macromolecules 31, 8474–8484 (1998) 20. Volkov, V.S., Kulichikhin, V.G.: Macromolecular dynamics in anisotropic viscoelastic liquids. Macromolec. Symposia 81, 45–53 (1994) 21. Warner, M., Terentjev, E.M.: Liquid crystal elastomers. Clarendon Press, Oxford (2003) 22. Leonov A.I.: Algebraic theory of linear viscoelastic nematodynamics. http://arxiv.org/e-print/cond-mat/ 0409274, http://arxiv.org/e-print/cond-mat/0409275
Math Phys Anal Geom (2008) 11:117–129 DOI 10.1007/s11040-008-9043-x
A Wegner-type Estimate for Correlated Potentials Victor Chulaevsky
Received: 19 February 2008 / Accepted: 5 May 2008 / Published online: 5 July 2008 © Springer Science + Business Media B.V. 2008
Abstract We propose a fairly simple and natural extension of Stollmann’s lemma to correlated random variables. This extension allows to obtain Wegner-type estimates even in various problems of spectral analysis of random operators where the original Wegner’s lemma is inapplicable, e.g., for correlated random potentials with singular marginal distributions and for multiparticle Hamiltonians. Keywords Wegner estimate · Stollmann’s lemma · Multi-scale analysis Mathematics Subject Classifications (2000) Primary 35P20 · Secondary 47F05
1 Introduction The regularity problem for the limiting distribution of eigenvalues of infinite dimensional self-adjoint operators appears in many problems of mathematical physics. Specifically, consider a lattice Schrödinger operator (LSO, for short) H: 2 (Zd ) → 2 (Zd ) given by (Hψ)(x) = ψ(y) + V(x)ψ(x); x, y ∈ Zd . y: |y−x|=1
V. Chulaevsky (B) Département de Mathématiques, Université de Reims, Moulin de la Housse, B.P. 1039 51687 Reims, France e-mail: [email protected]
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For each finite subset ⊂ Zd , let Ej , j = 1, . . . , ||, be eigenvalues of H with Dirichlet b.c. in . Consider the family of finite sets L = [−L, L]d ∩ Zd and define the following quantity (which does not necessarily exist for an arbitrary LSO): 1 L k(E) = lim card j: E E . j L→∞ (2L + 1)d If the above limit exists, k(E) is called the limiting distribution function (LDF) of e.v. of H. It is not difficult to construct various examples of a function V: Zd → R (called the potential of the operator H) for which the LDF does not exist. One can prove the existence of LDF for periodic potentials V, but even in this, relatively simple situation the existence of k(E) is not a trivial fact. However, the existence of k(E) can be proved for a large class of ergodic random potentials. Namely, consider an ergodic dynamical system (, F , P, {T x , x ∈ Zd }) with discrete time Zd and a measurable function (sometimes called a hull) v: → R. Then we can introduce a family of sample potentials V(x, ω) = v(T x ω), x ∈ Zd , labeled by ω ∈ . Under the assumption of ergodicity of {T x } (and, for example, boundedness of function v), the quantity 1 L k(E, ω) = lim card j: E (ω) E j L→∞ (2L + 1)d is well-defined P-a.s. Moreover, k(E, ω) is P-a.s. independent of ω, so its value for a.e. ω is natural to take as k(E). In such a context, k(E) is usually called the integrated density of states (IDS, for short). It admits an equivalent definition: k(E) = E ( f, (−∞,E] (H(ω)) f ) , where f ∈ 2 (Zd ) is any vector of unit norm, and (−∞,E] (H(ω)) is the spectral projection of H(ω) on (−∞, E]. The reader can find a detailed discussion of the existence problem of IDS in excellent monographs by Carmona and Lacroix [5] and by Pastur and Figotin [16]. See also articles [3, 4, 6, 10, 15]. It is not difficult to see that k(E) can be considered as the distribution function of a normalized measure, i.e. probability measure, on R. If this measure dk(E), called the measure of states, is absolutely continuous with respect to the Lebesgue measure dE, its density (or Radon–Nikodim derivative) dk(E)/dE is called the density of states (DoS). In physical literature, it is customary to neglect the problem of existence of such density, for if dk(E)/dE is not a function, then “it is simply a generalized function”. However, the real problem is not terminological. The actual, explicit estimates of the probabilities of the form P ∃ eigenvalue Ej L ∈ (a, a + ) for an LSO HL in a finite cube L of size L, for small , often depend essentially upon the existence and the regularity properties of the DoS dk(E)/dE.
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Apparently, the first fairly general result relative to the existence and boundedness of the DoS is due to Wegner [21]. Lemma 1.1 (Wegner) Assume that {V(x, ω), x ∈ Zd } are i.i.d. r.v. with a bounded density pV (u) of their common probability distribution: pV ∞ = C < ∞. Then the DoS dk(E)/dE exists and is bounded by the same constant C. The proof can be found, for example, in the monographs [5] and [16]. This estimate and some of its generalizations have been used in the multiscale analysis (MSA) developed in the works by Fröhlich and Spencer [13], Fröhlich et al. [12], von Dreifus and Klein [19, 20], and in a number of more recent works where the so-called Anderson Localization phenomenon has been observed. Namely, it has been proven that all eigenfunctions of random LSOs decay exponentially at infinity with probability one (for Pa.e. sample of random potential V(ω)). Von Dreifus and Klein [20] proved an analog of Wegner estimate and used it in their proof of localization for Gaussian and some other correlated (but non-deterministic) potentials. The author of this paper recently proved, in a joint work with Suhov [8], an analog of Wegner estimate for a system of two or more interacting quantum particles on the lattice under the assumption of analyticity of the probability density pV (u), using a rigorous path integral formula by Molchanov (see a detailed discussion of this formula in the monograph [5]). In order to relax the analyticity assumption in a multi-particle context, Chulaevsky and Suhov [9] used later a more general and flexible result: a lemma proved by Stollmann (cf. [17] and [18]) which we discuss below. In the present work, we propose a fairly simple and natural extension of Stollmann’s lemma to correlated, but still non-deterministic random fields generating random potentials. Our main motivation here is to lay out a way to interesting applications to localization problems for multi-particle systems.
2 Stollmann’s Lemma for Product Measures Recall the Stollmann’s lemma and its proof for independent random variables. Let m 1 be a positive integer, and J an abstract finite set with |J|(= cardJ) = m. Consider the Euclidean space R J ∼ = Rm with the standard basis (e1 , . . . , em ), and its positive orthant R+J = q ∈ R J: q j 0, j = 1, 2, . . . , m . Definition 2.1 Let J be a finite set with |J| = m. Consider a function : R J → R. It is called diagonally monotone (DM, for short) if it satisfies the following conditions: (1) for any r ∈ R+J and any q ∈ R J , (q + r) (q);
(1)
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(2) moreover, for e = e1 + · · · + em ∈ R J , for any q ∈ R J and for any t > 0 (q + t · e) − (q) t.
(2)
It is convenient to introduce the notion of DM operators considered as quadratic forms. In the following definition, we use the same notations as above. Definition 2.2 Let H be a Hilbert space. A family of self-adjoint operators B(q): H → H, q ∈ R J , is called DM if, ∀ q ∈ R J ∀ r ∈ R+J B(q + r) B(q), in the sense of quadratic forms, and for any vector f ∈ H with f = 1, the function f : R J → R defined by f (q) = (B(q) f, f ) is DM. In other words, the quadratic form Q B(q) ( f ):= (B(q) f, f ) as a function of q ∈ R J is non-decreasing in any q j, j = 1, . . . , |J|, and (B(q + t · e) f, f ) − (B(q) f, f ) t · f 2 . Remark 2.3 By virtue of the min-max principle for self-adjoint operators, if an operator family H(q) in a finite-dimensional Hilbert space H is DM, then each B(q) eigenvalue Ek of B(q) is a DM function. Remark 2.4 If H(q), q ∈ R J , is a DM operator family in a Hilbert space H, and H0 : H → H is an arbitrary self-adjoint operator, then the family H0 + H(q) is also DM. This explains why the notion of diagonal monotonicity is relevant to the spectral theory of random operators. Note also, that this property applies to physically interesting examples where dim H = +∞, but H(q) have, e.g., a compact resolvent, as in the case of Schrödinger operators in a finite cube with Dirichlet b.c. and with a bounded potential, so the respective spectrum is pure point, and even discrete. For any measure μ on R, we will denote by μ J the product measure μ × · · · × μ on R J . Furthermore, for any probability measure μ and for any > 0, define the following quantity: s(μ, ) = sup μ([a, a + ]) a∈R
the marginal probability distribution induced by μ J on We will denote by q = j = (q1 , . . . , q j−1 , q j+1 , . . . , qm ). μm−1 j
Lemma 2.5 (Stollmann [17]) Let J be a finite index set, |J| = m, μ be a probability measure on R, and μ J be the product measure on R J with marginal
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measures μ. If the function : R J → R is DM, then for any open interval I ⊂ R we have μ J { q: (q) ∈ I } m · s(μ, |I|). We provide below a proof of Stollmann’s lemma; this will allow to extend it to the case of correlated potentials. Proof Let I = (a, b ), b − a = > 0, and consider the set A = { q: (q) a }. Furthermore, define recursively sets Aj , j = 0, . . . , m, by setting A0 = A, Aj = Aj−1 + [0, ]e j := q + te j : q ∈ Aj−1 , t ∈ [0, ] . Obviously, the sequence of sets Aj , j = 1, 2, ..., is increasing with j. The DM property implies { q: (q) < b } ⊂ Am . Indeed, if (q) < b , then for the vector r := q − · e we have by (2): (r) (r + · e) − = (q) − b − a, meaning that r ∈ { a } = A and, therefore, q = r + · e ∈ Am . Now, we conclude that { q: (q) ∈ I } = { q: (q) ∈ (a, b ) } = { q: (q) < b } \ { q: (q) a } ⊂ Am \ A. Furthermore,
μm { q: (q) ∈ I } μm Am \ A ⎛ ⎞ m m
= μm ⎝ μm Aj \ Aj−1 . Aj \ Aj−1 ⎠ j=1
For
q =1
= (q2 , . . . , qm ) ∈ R
m−1
j=1
, set
I1 (q =1 ) = q1 ∈ R : (q1 , q =1 ) ∈ A1 \ A .
By definition of the set A1 , this is an interval of length not bigger than . Since μ J is a product measure, we have m m−1 μ (A1 \ A) = dμ (q =1 ) dμ(q1 ) s(μ, ). (3) I1
Similarly, we obtain for j = 2, . . . , m μm (Aj \ Aj−1 ) s(μ, ),
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yielding μm { q: (q) ∈ I }
m
μm (Aj \ Aj−1 ) m · c(μ, ).
j=1
Now, taking into account the above Remark 2.4, Lemma 2.5 yields immediately the following estimate. Lemma 2.6 Let H be an LSO with a random potential V(x; ω) in a finite box ⊂ Zd with Dirichlet b.c., and (H ) its spectrum, i.e. the collection of its eigenvalues E() j , j = 1, . . . , ||. Assume that r.v. V(x; ·) are i.i.d. with the marginal distribution μV . Then P { dist( (H (ω), E) } ||2 s(μV , 2).
This is an analogue of Wegner bound. One visible distinction is the form of its volume dependence: the factor ||2 instead of || in the conventional Wegner bound. One has to keep in mind, however, that • •
• •
•
Stollmann’s lemma on monotone functions is sharp (cf. [17]); eigenvalues, however, are particular monotone functions, which explains why conventional Wegner bound has the factor of ||1 ; while Wegner’s method requires the ensemble of random variables generating (or controlling) potential in a volume of cardinality || to have || degrees of freedom, correlated or not, the above approach works fine even for ensembles with one degree of freedom (the parameter m above may equal 1); in applications to the MSA, any upper bound of the form Const || N , or β even e|| , with β ∈ (0, 1) would be sufficient to make the MSA inductive scheme work; although the above version of Wegner bound does not allow to establish the existence of DoS even in models where the latter does exist (as can be shown with Wegner bound), the existence of the (limiting) DoS is not quite helpful per se for the finite-volume MSA, where upper bounds for the probability of high concentration of eigenvalues are vital; in applications to multi-particle localization problems with a short-range (or decaying) interaction between quantum particles, the existence of DoS for external potentials with regular marginal distributions can be proved by different methods. For example, Klopp and Zenk (2003, preprint) proved that the IDS for a multi-particle quantum system in Rd with a decaying particle interaction is the same as for the model without interaction. This result, quite natural from a physical point of view, was proved with the help of Helffer–Sjöstrand formula for almost analytic extensions. A similar result can be proved in a simpler way for lattice systems.
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3 Extension to Multi-particle Systems Results of this section have been obtained by the author and Suhov [9]. Let N > 1 and d 1 be two positive integers and consider a random LSO H = H(ω) which can be used, in the framework of tight-binding approximation, as the Hamiltonian of a system of N quantum particles in Zd with a random external potential V and an interaction potential U. Specifically, let x1 , . . . , x N ∈ Zd be the positions of quantum particles in the lattice Zd , and x = (x1 , . . . , x N ). Let {V(x; ω), x ∈ Zd } be a random field on Zd describing the external potential acting on all particles, and U: (x1 , . . . , x N ) → R be the interaction energy of the particles. In physics, U is usually to be a symmetric function of its N arguments x1 , . . . , x N ∈ Zd . We will assume in this section that the system in question obeys either Fermi or Bose quantum statistics, so it is convenient to assume U to be symmetric. Note, however, that the results of this section can be extended, with natural modifications, to more general interactions U. Further, U is assumed to be a finite-range interaction: supp U ⊂ {x : max(|x j − xk | r0 )}, r0 < ∞. Such an assumption is required in the proof of Anderson localization for multiparticle systems. However, it is irrelevant to the Wegner-type estimate we are going to discuss below. Now, let H be as follows: N
(H(ω) f )(x) =
( j ) + V(x j; ω) + U(x),
j=1 (j)
where
is the lattice Laplacian acting on the j-th particle, i.e. ( j ) = 1 ⊗ . . . ⊗ ⊗ . . . ⊗ 1 j
1
acting in the Hilbert space (Z
N
Nd
). For any finite “box”
(1)
× . . . × (N) ⊂ Z Nd
2
=
one can consider the restriction, H (ω), of H(ω) on with Dirichlet b.c. It is easy to see that the potential W(x) =
N
V(x j; ω) + U(x)
j=1
is no longer an i.i.d. random field on Z Nd , even if V is i.i.d. Therefore, neither version of the Wegner bound applies directly. But, in fact, Stollmann’s lemma does apply to multi-particle systems, virtually in the same way as to singleparticle ones. Lemma 3.1 Assume that r.v. V(x; ·) are i.i.d. with marginal distribution μV . Then P { dist( (H (ω), E) } || · M() · s(μV , 2),
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with M() =
N
card ( j ) .
j=1
A reader familiar with the MSA method may notice that, in fact, the latter requires two different kinds of Wegner-type bounds: for individual finite volumes and for couples of disjoint (or, more generally, distant) finite volumes , . In the conventional, single-particle MSA the two-volume bound can be deduced (under certain conditions) from its single-volume counterpart. This is far from obvious for multi-particle (even two-particle) systems with an interaction. A detailed discussion of the multi-particle MSA scheme is beyond the scope of this short note (for details, see [9]). Recently, Kirsch [14] proved an analog of Wegner bound for single volumes (but not for couples of volumes) under a more restrictive assumption of existence and boundedness of the marginal probability distribution of the potential of a multi-particle lattice Anderson model with interaction. 4 Extension to Correlated Random Variables Fix a positive integer m 1 and consider the Euclidean space Rm with coordinates q = (q1 , . . . , qm ). For a given point q ∈ Rm , set q = j := (q1 , . . . , q j−1 , q j+1 , . . . , qm ). Let μm be a probability measure on Rm with marginal dis(q = j) of order m − 1, and conditional distributions μ1j (q j | q = j) tributions μm−1 j of order 1; here j = 1, . . . , m. In the case where the measure μ1j (q j | q = j) is absolutely continuous, we denote by p(q j|q = j) its density with respect to the Lebesgue measure dq j. The measure μm (unlike the measure μ J in Section 2) is no longer assumed to be a product measure (we emphasize this fact by changing notation). For every > 0, define the following quantities, measuring in different ways continuity properties of μm : a+ m m−1 dμ C1 (μ , ) = max sup (q = j) dμ(q j|q = j), (4) j
a∈R
a
C2 (μ , ) = max ess m
j
sup
a+
sup
q = j ∈Rm−1 a∈R
a
dμ(q j|q = j),
(5)
and C3 (μm ) = ess
sup
q = j ∈Rm−1 , q j ∈R
p(q j|q = j).
(6)
Since μm is a finite (even probability) measure, the quantities C1 (μm , ) and C2 (μm , ) are always finite, and bounded by 1, while C3 (μm , ) may be infinite (in which case, naturally, it is useless). If the density p(q j|q = j) exists, we have dμ(q j|q = j) = p(q j|q = j)dq j
A Wegner-type estimate for correlated potentials
and if C3 (μm , ) < ∞, we can write C2 (μm , ) = max ess j
max ess j
sup
q = j ∈Rm−1
sup
q = j ∈Rm−1
125
a+
sup a∈R
a
a+
sup a∈R
p(q j|q = j)dq j C3 (μm ) dq j C3 (μm ) .
a
Also, it is easy to see that C1 (μm , ) = max sup
j
a∈R
dμm−1 (q = j)
a+
a
dμ(q j|q = j)
C2 (μm , ) dμm−1 (q = j) = C2 (μm , ),
since μm−1 is a probability measure. Therefore, C1 (μm , ) C2 (μm , ) C3 (μm , ) .
(7)
Remark 4.1 In applications to localization problems, the aforementioned continuity moduli C1 (μm , ), C2 (μm , ) need to decay not too slowly as → 0. A power decay of order O( β ) with β > 0 is certainly sufficient, but it can be essentially relaxed. For example, it suffices to have an upper bound of the form β C1 μm , e−L Const · L−B , uniformly for all sufficiently large L > 0 with some (arbitrarily small) β > 0 and with B > 0 which should sufficiently big, depending on the specific spectral problem. Using notations of the previous section, one can formulate the following generalization of Stollmann’s lemma. Lemma 4.1 Let J be a finite set with |J| = m, so that we can identify R J with Rm . Let : R J → R be a DM function and μm a probability measure on Rm ∼ R J with C1 (μm , ) < ∞. Then for any interval I ⊂ R of length |I| = > 0, we have μm { q: (q) ∈ I } m · C1 (μ, ). Proof We proceed as in the proof of Stollmann’s lemma and introduce in Rm the sets A = { q: (q) a } and Aj , j = 0, . . . , m. Here, again, we have { q: (q) ∈ I } =⊂ Am \ A and μm { q: (q) ∈ I }
m j=1
μm Aj \ Aj−1 .
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For q =1 ∈ Rm−1 , we set
I1 (q =1 ) = q1 ∈ R : (q1 , q =1 ) ∈ A1 \ A .
Furthermore, we come to the following upper bound which generalizes (3):
m m−1 μ A1 \ A = dμ (q ) dμ(q1 |q ) C1 (μ, ). (8) I1
Similarly, we obtain for j = 2, . . . , m μm Aj \ Aj−1 C1 (μ, ), yielding μm { q: (q) ∈ I }
m
μm Aj \ Aj−1 m · C1 (μ, ).
j=1
5 Application to Gibbs Fields with Continuous Spin There exists a large variety of correlated random lattice fields for which the hypothesis of Lemma 4.2 can be easily verified. For example, conditional distributions of Gibbs fields are given explicitly in terms of their respective interaction potentials. Gaussian fields can also be considered as a particular class of Gibbsian fields. The reader can find in the article by von Dreifus and Klein [20] a detailed discussion of such models and a proof of Wegner estimate for homogeneous non-deterministic Gaussian potentials. It suffices to notice, actually, that for such potentials the conditional density of a single-site value V(x0 ; ·) given all other values {V(y; ·), y = x0 } exists and is bounded. Therefore, Lemma 4.2 applies, but so does the traditional Wegner’s method. Anderson localization for Gibbsian potentials on the lattice was proved by von Dreifus and Klein [20] under a rather strong assumption of complete analyticity in the sense of Dobrushin and Shlosman (see, e.g., [11]) of the Gibbsian field V(x; ω), with continuous spins, generating the potential of the respective LSO. We show in this section how Lemma 4.2 allows to relax the complete analyticity hypothesis to a quite general, single-site condition on the Hamiltonian generating the respective Gibbs state for a model of the classical statistical mechanics with continuous spins. Indeed, original Dobrushin–Shlosman techniques are adapted to models with a finite number of spin values. Bourgain and Kenig [2] considered continuous Anderson models where amplitudes determining the random potential take two values. Recently, Aizenman et al. [1] extended this result to a quite general case, including random variables taking with positive probability any finite number n > 1 of values. Unfortunately, no analog of such techniques is known so far for the lattice models.
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It is worth mentioning that the condition described below can hold in some models where the marginal density does not exist, in which case more traditional methods do not apply. Consider a lattice Gibbs field Sx (ω) with bounded continuous spins, S: × Zd → S = [a, b ] ⊂ R generated by a short-range, bounded, two-body interaction potential u• (·, ·). The spin space is assumed to be equipped with a measure dS which may, in principle, be singular with respect to Lebesgue measure on [a, b ] (more general spin spaces S and measures dS can also be considered). In other words, consider the formal Hamiltonian H(S) = h(Sx ) + u|x−y| (Sx , S y ), x∈Zd |y−x| R
x∈Zd
where h: S → R is the self-energy of a given spin. Assume that the interaction potentials u|x−y| (Sx , S y ) vanish for |x − y| > R and are uniformly bounded: max sup |ul (S, S )| < ∞. l R S,S ∈S
Then for any lattice point x and any configuration S = S =x of spins outside {x}, the single-site conditional distribution of Sx given the external configuration S admits a bounded density with respect to measure dS, namely,
p(Sx | S =x ) =
e−βU(Sx |S ) e−βU(Sx |S ) = −βU(S |S ) dS
(β, S ) Se
with U(Sx |S ) :=
u|x−y| (Sx , S y )
y: |y−x| R
satisfying the upper bound |U(Sx |S )| (2R + 1)d sup |ul (S, S )| < ∞. S,S ∈S
A similar property is valid for sufficiently rapidly decaying long-range interaction potentials, for example, under the condition sup |u|y| (S, S )|
S,S ∈S
Const , δ > 0. |y|d+1+δ
(9)
as well as for more general, but still uniformly summable many-body interactions. Below we give one simple example of application of Wegner–Stollmanntype bound to such random potentials. Lemma 5.1 Let ⊂ Zd be a finite subset of the lattice, ⊂ Zd \ any subset disjoint with ( may be empty), and let Sx (ω) be a Gibbs field in with continuous spins S ∈ S = [a, b ] generated by a two-body interaction potential ul (S, S ) satisfying condition (9), with any b.c. on Zd \ . Consider a LSO
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H with the random potential V(x, ω) = Sx (ω). Then for any interval I ⊂ R of length > 0, we have P (H ) ∩ I = ∅ | V(y, ·), y ∈ C(V) ||2 , C(V) < ∞. In the case of unbounded spins and/or interaction potentials, the uniform boundedness of conditional single-spin distributions does not necessarily hold, since the energy of interaction of a given spin S0 with the external configuration S may be arbitrarily large (depending on a particular form of interaction) and even infinite, if S y → ∞ too fast. In such situations, our general condition (4) might still apply, provided that rapidly growing configurations S have sufficiently small probability, so that the outer integral in the r.h.s. of (4) converges. 6 Conclusion Wegner-type bounds of the IDS in finite volumes are a key ingredient of the MSA of spectra of random Schrödinger (and some other) operators. The proposed simple extension of Stollmann’s lemma shows that a very general assumption on correlated random fields generating the potential rules out an abnormal accumulation of eigenvalues in finite volumes. This extension applies also to multi-particle systems with interaction. Acknowledgements I would like to thank Senya Shlosman for numerous and very fruitful discussions of Dobrushin–Shlosman techniques. I also thank Lana Jitomirskaya and Abel Klein and the University of California at Irvine for their warm hospitality during my stay at UCI and stimulating discussions. I thank Anne Boutet de Monvel and Peter Stollmann for many fruitful discussions of Wegner-type estimates.
References 1. Aizenman, A., Germinet, F., Klein, A., Warzel, S.: On Bernoulli decompositions for random variables, conncenration bounds, and spectral localization. arXiv:0707.0095 (2007) 2. Bourgain, J., Kenig, C.: On localization in continuous Anderson–Bernoulli model in highre dimensions. Invent. Math 161, 389–426 (2005) 3. Bovier, A., Campanino, M., Klein, A., Perez, F.: Smoothness of the density of states in the Anderson model at high disorder. Comm. Math. Phys. 114, 439–461 (1988) 4. Campanino, M., Klein, A.: A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model. Comm. Math. Phys. 104, 227–241 (1986). 5. Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990) 6. Craig, W., Simon, B.: Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices. Comm. Math. Phys. 90, 207–218 (1983) 7. Chulaevsky, V.: A simple extension of Stollmann’s lemma to correlated potentials. Université de Reims. arXiv:0705:2873 May (2007) 8. Chulaevsky, V., Suhov, Y.: Anderson localisation for an interacting two-particle quantum system on Z. Université de Reims. arXiv:0705.0657 May (2007) 9. Chulaevsky, V., Suhov, Y.: Wegner bounds for a two-particle tight binding model. arXiv:0708:2056. Comm. Math. Phys. (2008, in press)
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10. Constantinescu, F., Fröhlich, J., Spencer, T.: Analyticity of the density of states and replica method for random Schrödinger operators on a lattice. J. Stat. Phys. 34, 571–596 (1983) 11. Dobrushin, R.L., Shlosman, S.: Completely analytical interactions: constructive description. J. Statist. Phys. 46, 983–1014 (1987) 12. Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: A constructive proof of localization in Anderson tight binding model. Comm. Math. Phys. 101, 21–46 (1985) 13. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Comm. Math. Phys. 88, 151–184 (1983) 14. Kirsch, W.: A Wegner estimate for multi-particle random Hamiltonians. arXiv:0704:2664 April (2007) 15. Pastur, L.A.: Spectral properties of disordered systems in one-body approximation. Comm. Math. Phys. 75, 179 (1980) 16. Pastur, L.A., Figotin, A.L.: Spectra of Random and Almost Periodic Operators. Springer, Berlin (1992) 17. Stollmann, P.: Wegner estimates and localization for continuous Anderson models with some singular distributions. Arch. Math. 75, 307–311 (2000) 18. Stollmann, P.: Caught by Disorder. Birkhäuser, Boston (2001) 19. Von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Comm. Math. Phys. 124, 285–299 (1989) 20. Von Dreifus, H., Klein, A.: Localization for Schrödinger operators with correlated potentials. Comm. Math. Phys. 140, 133–147 (1991) 21. Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B. Condens. Matter 44, 9–15 (1981)
Math Phys Anal Geom (2008) 11:131–154 DOI 10.1007/s11040-008-9044-9
The Two-Spectra Inverse Problem for Semi-infinite Jacobi Matrices in The Limit-Circle Case Luis O. Silva · Ricardo Weder
Received: 2 August 2007 / Accepted: 28 May 2008 / Published online: 12 July 2008 © Springer Science + Business Media B.V. 2008
Abstract We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be the spectra of two different self-adjoint extensions of a Jacobi operator in the limit circle case. Keywords Jacobi matrices · Two-spectra inverse problem · Limit circle case Mathematics Subject Classifications (2000) 47B36 · 49N45 · 81Q10 · 47A75 · 47B37 · 47B39
Research partially supported by CONACYT under Project P42553F. L. O. Silva partially supported by PAPIIT-UNAM through grant IN-111906. Ricardo Weder is a fellow of Sistema Nacional de Investigadores. L. O. Silva · R. Weder (B) Departamento de Métodos Matemáticos y Numéricos, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, México, D. F. C.P. 04510, Mexico e-mail: [email protected] L. O. Silva e-mail: [email protected]
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1 Introduction In the Hilbert space l2 (N), consider the operator J whose matrix representation with respect to the canonical basis in l2 (N) is the semi-infinite Jacobi matrix ⎞ ⎛ q1 b1 0 0 · · · ⎜b q b 0 ···⎟ ⎟ ⎜ 1 2 2 ⎟ ⎜ ⎟ ⎜ 0 b2 q3 b3 (1.1) ⎟, ⎜ ⎟ ⎜ . ⎜ 0 0 b3 q4 . . ⎟ ⎠ ⎝ .. .. .. .. . . . . where bn > 0 and qn ∈ R for n ∈ N. This operator is densely defined in l2 (N) and J ⊂ J ∗ (see Section 2 for details on how J is defined). It is well known that J can have either (1, 1) or (0, 0) as its deficiency indices [2, Sec. 1.2 Chap. 4], [31, Cor. 2.9]. By our definition (see Section 2), J is closed, so the case (0, 0) corresponds to J = J ∗ , while (1, 1) implies that J is a nontrivial restriction of J ∗ . The latter operator is always defined on the maximal domain in which the action of the matrix (1.1) makes sense [3, Sec. 47]. Throughout this work we assume that J has deficiency indices (1, 1). Jacobi operators of this kind are referred as being in the limit circle case and the moment problem associated with the corresponding Jacobi matrix is said to be indeterminate [2, 31]. In the limit circle case, all self-adjoint extensions of a Jacobi operator have discrete spectrum [31, Thm. 4.11]. The set of all selfadjoint extensions of a Jacobi operator can be characterized as a one parameter family of operators (see Section 2). The main results of the present work are Theorem 1 in Section 3 and Theorem 2 in Section 4. In Theorem 1 we show that a Jacobi matrix can be recovered uniquely from the spectra of two different self-adjoint extensions of the Jacobi operator J corresponding to that matrix. Moreover, these spectra also determine the parameters that define the self-adjoint extensions of J for which they are the spectra. The proof of Theorem 1 is constructive and it gives a method for the unique reconstruction. The uniqueness of this reconstruction in a more restricted setting has been announced in [13] without proof. In Theorem 2 we give necessary and sufficient conditions for two sequences to be the spectra of two self-adjoint extensions of a Jacobi operator in the limit circle case. This is a complete characterization of the spectral data for the twospectra inverse problem of a Jacobi operator in the limit circle case. In two spectra inverse problems, one may reconstruct a certain self-adjoint operator from the spectra of two different rank-one self-adjoint perturbations of the operator to be reconstructed. This is the case of recovering the potential of a Schrödinger differential expression in L2 (0, ∞), being regular at the origin and limit point at ∞, from the spectra of two operators defined by the differential expression with two different self-adjoint boundary conditions at the origin [4, 5, 9, 15, 17, 21, 24, 27, 28]. Necessary and sufficient conditions for this inverse problem are found in [27]. Characterization of spectral data of a
The two-spectra inverse problem
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related inverse problem was obtained in [10]. The inverse problem consisting in recovering a Jacobi matrix from the spectra of two rank-one self-adjoint perturbations, was studied in [8, 12, 19, 20, 32, 34]. A complete characterization of the spectral data for this two-spectra inverse problem is given in [29]. In the formulation of the inverse problem studied in the present work, the aim is to recover a symmetric non-self-adjoint operator from the spectra of its self-adjoint extensions, as well as the parameters that characterize the selfadjoint extensions. There are results for this setting of the two spectra inverse problem, for instance in [14, 23] for Sturm–Liouville operators, and in [13] for Jacobi matrices. It is well known that self-adjoint extensions of symmetric operators with deficiency indices (1, 1) can be treated within the rank-one perturbation theory (cf. [6, Sec. 1.1–1.3] and, in particular, [6, Thm. 1.3.3]). Thus, both settings may be regarded as particular cases of a general two-spectra inverse problem. A consideration similar to this is behind the treatment of inverse problems in [11]. For Jacobi operators, however, the type of rank-one perturbations in the referred formulations of the inverse spectral problem are different [6]. Indeed, in the setting studied in [29], one has the so-called bounded rankone perturbations [6, Sec. 1.1]. This means that all the family of rank-one perturbations share the same domain. In contrast the present work deals with singular rank-one perturbations [6, Sec. 1.3], meaning that every element of the family of rank-one perturbations has different domain. Note that for differential operators both settings involve a family of singular rank-one perturbations. The paper is organized as follows. In Section 2, we introduce Jacobi operators, in particular the class whose corresponding Jacobi matrix is in the limit circle case. Here we also present some preliminary results and lay down some notation used throughout the text. Section 3 contains the uniqueness result on the determination of a Jacobi matrix by the spectra of two self-adjoint extensions. The proof of this assertion yields a reconstruction algorithm. Finally in Section 4, we give a complete characterization of the spectral data for the two spectra inverse problem studied here.
2 Preliminaries Let l f in (N) be the linear space of sequences with a finite number of non-zero elements. In the Hilbert space l2 (N), consider the operator J defined for every f = { fk }∞ k=1 in l f in (N) by means of the recurrence relation (J f )k := bk−1 fk−1 + qk fk + bk fk+1 , (J f )1 := q1 f1 + b1 f2 ,
k ∈ N \ {1} ,
(2.1) (2.2)
where, for n ∈ N, bn is positive and qn is real. Clearly, J is symmetric since it is densely defined and Hermitian due to (2.1) and (2.2). Thus J is closable and henceforth we shall consider the closure of J and denote it by the same letter.
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We have defined the operator J so that the semi-infinite Jacobi matrix (1.1) is its matrix representation with respect to the canonical basis {en }∞ n=1 in l2 (N) (see [3, Sec. 47] for the definition of the matrix representation of an unbounded symmetric operator). Indeed, J is the minimal closed symmetric operator satisfying (Jen , en ) = qn , (Jen , en+1 ) = (Jen+1 , en ) = bn , (Jen , en+k ) = (Jen+k , en ) = 0 ,
n ∈ N,
k ∈ N \ {1} .
We shall refer to J as the Jacobi operator and to (1.1) as its associated matrix. The spectral analysis of J may be carried out by studying the following second order difference system bn−1 fn−1 + qn fn + bn fn+1 = ζ fn ,
n > 1,
ζ ∈ C,
(2.3)
with the “boundary condition” q1 f1 + b1 f2 = ζ f1 .
(2.4)
If one sets f1 = 1, then f2 is completely determined by (2.4). Having f1 and f2 , equation (2.3) gives all the other elements of a sequence { fn }∞ n=1 that formally satisfies (2.3) and (2.4). Clearly, fn is a polynomial of ζ of degree n − 1, so we denote fn =: Pn−1 (ζ ). The polynomials Pn (ζ ), n = 0, 1, 2, . . . , are referred to as the polynomials of the first kind associated with the matrix (1.1) [2, Sec. 2.1 Chap. 1]. The sequence P(ζ ) := {Pk−1 (ζ )}∞ k=1 is not in l f in (N), but it may happen that ∞
|Pk (ζ )|2 < ∞ ,
(2.5)
k=0
in which case P(ζ ) ∈ Ker(J ∗ − ζ I ). The polynomials of the second kind Q(ζ ) := {Qk−1 (ζ )}∞ k=1 associated with the matrix (1.1) are defined as the solutions of bn−1 fn−1 + qn fn + bn fn+1 = ζ fn ,
n ∈ N \ {1} ,
−1
under the assumption that f1 = 0 and f2 = b1 . Then Qn−1 (ζ ) := fn ,
∀n ∈ N .
Qn (ζ ) is a polynomial of degree n − 1. As pointed out in the introduction, J has either deficiency indices (1, 1) or (0, 0) [2, Sec. 1.2 Chap. 4] and [31, Cor. 2.9]. These cases correspond to the limit circle and limit point case, respectively. In terms of the polynomials of the first kind, J has deficiency indices (0, 0) if for one ζ ∈ C \ R the series in (2.5) diverges. In the limit circle case (2.5) holds for every ζ ∈ C [2, Thm. 1.3.2], [31, Thm. 3] and, therefore, P(ζ ) is always in Ker(J ∗ − ζ I ). Another peculiarity of the limit circle case is that every self-adjoint extension of J has purely discrete spectrum [31, Thm. 4.11]. Moreover, the resolvent of every self-adjoint extension is a Hilbert-Schmidt operator [33, Lem. 2.19].
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In what follows we always consider J to have deficiency indices (1, 1). The behavior of the polynomials of the first kind determines this class. There are various criteria for establishing whether a Jacobi operator is symmetric but non-self-adjoint. These criteria may be given in terms of the moments associated with the matrix, for instance the criterion [31, Prop. 1.7] due to Krein. A criterion in terms of the matrix entries is the following result which belongs to Berezans ki˘ı [2, Chap. 1], [7, Thm. 1.5 Chap. 7]. Proposition 1 Suppose that supn∈N |qn | < ∞ and that N ∈ N such that for n > N bn−1 bn+1 ≤ bn 2 ,
∞
1 n=1 b n
< ∞. If there is
then the Jacobi operator whose associated matrix is (1.1) is in the limit circle case. Jacobi operators in the limit circle case may be used to model physical processes. For instance Krein’s mechanical interpretation of Stieltjes continued fractions [22], in which one has a string carrying point masses with a certain distribution along the string, is modeled by an eigenvalue equation of a Jacobi operator [2, Appendix]. There are criteria in terms of the point masses and their distribution [2, Thm. 0.4 Thm. 0.5 Appendix] for the corresponding Jacobi operator to be in the limit circle case. In this work, all self-adjoint extensions of J are assumed to be restrictions of J ∗ . When dealing with all self-adjoint extensions of J, including those which imply an extension of the original Hilbert space, the self-adjoint restrictions of J ∗ are called von Neumann self-adjoint extensions of J (cf. [3, Appendix I], [31, Sec. 6]). There is also a well known result for J in the limit circle case, namely, that J is simple [2, Thm. 4.2.4]. In its turn this imply that the eigenvalues of any self-adjoint extension of J have multiplicity one [3, Thm.3 Sec. 81]. Let us now introduce a convenient way of parametrizing the self-adjoint extensions of J in the symmetric non-self-adjoint case. We first define the Wronskian associated with J for any pair of sequences ϕ = {ϕk }∞ k=1 and ψ = {ψk }∞ in l ( N ) as follows 2 k=1 Wk (ϕ, ψ) := bk (ϕk ψk+1 − ψk ϕk+1 ) ,
k ∈ N.
Now, consider the sequences v(τ ) = {vk (τ )}∞ k=1 such that, for k ∈ N, vk (τ ) := Pk−1 (0) + τ Qk−1 (0) ,
τ ∈ R,
(2.6)
and vk (∞) := Qk−1 (0) .
(2.7)
All the self-adjoint extensions J(τ ) of the symmetric non-self-adjoint operator J are restrictions of J ∗ to the set [33, Lem. 2.20]
∗ Dτ := f = { fk }∞ τ ∈ R ∪{∞} . (2.8) k=1 ∈ dom(J ) : lim Wn v(τ ), f = 0 , n→∞
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Different values of τ imply different self-adjoint extensions, so J(τ ) is a selfadjoint extension of J uniquely determined by τ [33, Lem. 2.20]. Observe that the domains Dτ are defined by a boundary condition at infinity given by τ . We also remark that given two sequences ϕ and ψ in dom(J ∗ ) the following limit always exists [33, Sec. 2.6] lim Wn (ϕ, ψ) =: W∞ (ϕ, ψ) .
n→∞
It follows from [31, Thm. 3] that, in the limit circle case, P(ζ ) and Q(ζ ) are in dom(J ∗ ) for every ζ ∈ C. From what has just been said, one can consider the functions (see also [2, Sec. 2.4 Chap. 1, Sec. 4.2 Chap. 2]) W∞ (P(0), P(ζ )) =: D(ζ ) , W∞ (Q(0), P(ζ )) =: B(ζ ) .
(2.9)
The notation for these limits has not been chosen arbitrarily; they are the elements of the second row of the Nevanlinna matrix associated with the matrix (1.1) and they are usually denoted by these letters [2, Sec. 4.2 Chap. 2], [31, Eq. 4.17]. It is well known that the functions D(ζ ) and B(ζ ) are entire of at most minimal type of order one [2, Thm. 2.4.3], [31, Thm. 4.8], that is, for each > 0 there exist constants C1 (), C2 () such that |D(ζ )| C1 ()e|ζ | ,
|B(ζ )| C2 ()e|ζ | .
If P(ζ ) is in Dτ the following holds D(ζ ) + τ B(ζ ) 0 = W∞ (v(τ ), P(ζ )) = B(ζ ) Thus, the zeros of the function Rτ (ζ ) :=
D(ζ ) + τ B(ζ ) B(ζ )
if τ ∈ R if τ = ∞ .
if τ ∈ R if τ = ∞
(2.10)
constitute the spectrum of the self-adjoint extension J(τ ) of J. A Jacobi matrix of the form (1.1) determines, in a unique way, the sequence P(t) = {Pn−1 (t)}∞ n=1 , t ∈ R. This sequence is orthonormal in any space L2 (R, dρ), where ρ is a solution of the moment problem associated with the Jacobi matrix (1.1) [2, Sec. 2.1 Chap. 2]. The elements of the sequence {Pn−1 (t)}∞ n=1 form a basis in L2 (R, dρ) if ρ is an N-extremal solution of the moment problem [2, Def. 2.3.3] or, in other words, if ρ can be written as (2.11) ρ(t) = E(t)e1 , e1 , t ∈ R, where E(t) is the spectral resolution of the identity for some von Neumann selfadjoint extension of the Jacobi operator J associated with (1.1) [2, Thm. 2.3.3, Thm. 4.1.4].
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Let ρ be given by (2.11), then we can consider the linear isometric operator U which maps the canonical basis {en }∞ n=1 in l2 (N) into the orthonormal basis in L ( R , dρ) as follows {Pn (t)}∞ 2 n=0 Uen = Pn−1 ,
n ∈ N.
(2.12)
By linearity, one extends U to the span of {en }∞ n=1 and by continuity, to all l2 (N). Clearly, the range of U is all L2 (R, dρ). The Jacobi operator J given by the matrix (1.1) is transformed by U into the operator of multiplication by the independent variable in L2 (R, dρ) if J = J ∗ , and into a symmetric restriction of the operator of multiplication if J = J ∗ . Following the terminology used in [11], we call the operator U JU −1 in L2 (R, dρ) the canonical representation of J. By virtue of the discreteness of σ (J(τ )) in the limit circle case (here and in the sequel, σ (A) stands for the spectrum of operator A), the function ρτ given by (2.11), with E(t) being the resolution of the identity of J(τ ), can be written as follows a(λk )−1 , λk ∈ σ (J(τ )) , ρτ (t) = λk t
where the positive constant a(λk ) is the so-called normalizing constant of J(τ ) corresponding to λk . In the limit circle case it is easy to obtain the following formula for the normalizing constants [2, Sec. 4.1 Chap 3], [31, Thm. 4.11] a(λk ) = P(λk ) l22 (N) ,
λk ∈ σ (J(τ )) .
(2.13)
Formula (2.13), which gives the jump of the spectral function at λk , also holds true in the limit point case, when λk is an eigenvalue of J [7, Thm. 1.17 Chap. 7]. It turns out that the spectral function ρτ uniquely determines J(τ ). Indeed, there are two ways of recovering the matrix from the spectral function. One method, developed in [16] (see also [32]), makes use of the asymptotic behaviour of the Weyl m-function mτ (ζ ) :=
R
ρτ (t) t−ζ
and the Ricatti equation [16, Eq. 2.15], [32, Eq. 2.23], b 2n m(n) τ (ζ ) = qn − ζ −
1 m(n−1) (ζ ) τ
,
n ∈ N,
(2.14)
where m(n) τ (ζ ) is the Weyl m-function of the Jacobi operator associated with the matrix (1.1) with the first n columns and n rows removed. The other method for the reconstruction of the matrix is more straightforward (see [7, Sec. 1.5 Chap. 7 and, particularly, Thm. 1.11]). The starting
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point is the sequence {tk }∞ k=0 , t ∈ R. From what we discussed above, all the elements of the sequence {tk }∞ k=0 are in L2 (R, dρτ ) and one can apply, in this Hilbert space, the Gram-Schmidt procedure of orthonormalization to the ∞ sequence {tk }∞ k=0 . One, thus, obtains a sequence of polynomials {Pk (t)}k=0 normalized and orthogonal in L2 (R, dρτ ). These polynomials satisfy a three term recurrence equation [7, Sec. 1.5 Chap. 7], [31, Sec. 1] t Pk−1 (t) = bk−1 Pk−2 (t) + qk Pk−1 (t) + bk Pk (t) , t P0 (t) = q1 P0 (t) + b1 P1 (t) ,
k ∈ N \ {1} ,
(2.15) (2.16)
where all the coefficients bk (k ∈ N) turn out to be positive and qk (k ∈ N) are real numbers. The system (2.15) and (2.16) defines a matrix which is the matrix representation of J. After obtaining the matrix associated with J, if it turns out to be non-selfadjoint, one can easily obtain the boundary condition at infinity which defines the domain of J(τ ). The recipe is based on the fact that the spectra of different self-adjoint extensions are disjoint [2, Sec. 2.4 Chap. 4]. Take an eigenvalue, λ, of J(τ ), i. e., λ is a point of discontinuity of ρτ or a pole of mτ . Since the corresponding eigenvector P(λ) = {Pk−1 (λ)}∞ k=1 is in dom(J(τ )), it must be that W∞ v(τ ), P(λ) = 0 . This implies that either W∞ Q(0), P(λ) = 0, which means that τ = ∞, or W∞ P(0), P(λ) . τ =− W∞ Q(0), P(λ) Notation We conclude this section with a remark on the notation. The elements of the unbounded set σ (J(τ )), τ ∈ R ∪ ∞, may be enumerated in different ways. Let σ (J(τ )) = {λk }k∈K , where K is a countable set through which the subscript k runs. If σ (J(τ )) is either bounded from above or below, one may take K = N. If σ (J(τ )) is unbounded below and above, one may set K = Z. Of course, other choices of K are possible. Since the particular choice of K is not important in our formulae, we shall drop K from the notation and simple write {λk }k . All our formulae will be written so that they are independent of the way the elements of a sequence are enumerated, so our convention for denoting sequences should not lead to misunderstanding. Similarly, we write y instead of k k k∈K yk , and the convergence of the series to a number c means that for any sequence of sets {K j}∞ j=1 , with K j ⊂ K j+1 ⊂ K, such that ∞ ∞ j=1 K j = K, the sequence { k∈K j yk } j=1 tends to c whenever j → ∞. 3 Unique Reconstruction of the Matrix In this section we show that, given the spectra of two different self-adjoint extensions J(τ1 ), J(τ2 ) of the Jacobi operator J in the limit circle case, one can
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always recover the matrix, being the matrix representation of J with respect to the canonical basis in l2 (N), and the two parameters τ1 , τ2 that define the self-adjoint extensions. It has already been announced [13, Thm. 1] that, when τ1 , τ2 ∈ R and τ1 = τ2 , the spectra σ (J(τ1 )) and σ (J(τ2 )) uniquely determine the matrix of J and the numbers τ1 and τ2 . A similar result, but in a more general setting can be found in [11, Thm. 7]. Consider the following expression which follows from the Christoffel– Darboux formula [2, Eq. 1.17]: n−1
Pk2 (ζ ) = bn Pn−1 (ζ )Pn (ζ ) − Pn (ζ )Pn−1 (ζ ) = Wn (P(ζ ), P (ζ )) .
k=0
It is easy to verify, taking into account the analogue of the Liouville– Ostrogradskii formula [2, Eq. 1.15], that Wn (P(ζ ), P (ζ )) = Wn (P(0), P(ζ ))Wn (Q(0), P (ζ )) −Wn (Q(0), P(ζ ))Wn (P(0), P (ζ )) . Thus,
∞
Pk2 (ζ ) = W∞ (P(ζ ), P (ζ ))
k=0
= D(ζ )B (ζ ) − B(ζ )D (ζ ) . Indeed, due to the uniform convergence of the limits in (2.9) [2, Sec. 4.2 Chap. 2], the following is valid B (ζ ) = W∞ (Q(0), P (ζ )) D (ζ ) = W∞ (P(0), P (ζ )) . Now, a straightforward computation yields (τ1 , τ2 ∈ R, τ1 = τ2 ) Rτ1 (ζ )Rτ2 (ζ ) − Rτ1 (ζ )Rτ2 (ζ ) = (τ2 − τ1 ) D(ζ )B (ζ ) − B(ζ )D (ζ ) . On the other hand one clearly has Rτ1 (ζ )R∞ (ζ ) − Rτ1 (ζ )R∞ (ζ ) = D(ζ )B (ζ ) − B(ζ )D (ζ ) ,
Hence, a(ζ ) :=
∞ k=0
τ1 ∈ R .
⎧ ⎪ ⎨ Rτ1 (ζ )Rτ2 (ζ ) − Rτ1 (ζ )Rτ2 (ζ ) τ1 = τ2 , τ1 , τ2 ∈ R τ2 − τ1 Pk2 (ζ ) = ⎪ ⎩ Rτ1 (ζ )R∞ (ζ ) − Rτ1 (ζ )R∞ (ζ ) τ1 ∈ R .
(3.1)
It follows from (2.13) that the values of the function a(ζ ) evaluated at the points of the spectrum of some self-adjoint extension of J are the corresponding normalizing constants of that extension.
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The analogue of (3.1) with τ1 = τ2 and τ1 , τ2 ∈ R, for the Schrödinger operator in L2 (0, ∞) being in the limit circle case is [14, Eq. 1.20]. Formula [14, Eq. 1.20] plays a central rôle in proving the unique reconstruction theorem for that operator [14, Thm. 1.1]. The discrete counterpart of [14, Thm. 1.1] is [13, Thm. 1]. It is worth mentioning that the reconstruction technique we present below is also based on (3.1). It is well known that the spectra of any two different self-adjoint extensions of J are disjoint [2, Sec. 2.4 Chap. 4]. One can easily conclude this from (3.1). Moreover, the following assertion holds true. Proposition 2 The eigenvalues of two different self-adjoint extensions of a Jacobi operator interlace, that is, there is only one eigenvalue of a self-adjoint extension between two eigenvalues of any other self-adjoint extension. Remark 1 One may arrive at this assertion via rank-one perturbation theory, in particular by recurring to the Aronzajn–Krein formula [30, Eq. 1.13]. Nonetheless, we provide below a simple proof to illustrate the use of (3.1). The proof of this statement for regular simple symmetric operators can be found in [18, Prop. 3.4 Chap. 1]. Proof The proof of this assertion follows from the expression (3.1). It is similar to the proof of [2, Thm. 1.2.2]. Note that (2.10) implies that the entire function Rτ (ζ ), τ ∈ R ∪ {∞}, is real, i. e., it takes real values when evaluated on the real line. Let λk < λk+1 be two neighboring eigenvalues of the self-adjoint extension J(τ2 ) of J, with τ2 ∈ R ∪ {∞}. So λk , λk+1 are zeros of Rτ2 and by (3.1) these zeros are simple. Since Rτ2 (λk ) and Rτ2 (λk+1 ) have different signs, it follows from (3.1) that Rτ1 (λk ) and Rτ1 (λk+1 ) (τ1 ∈ R ∪ {∞}, τ1 = τ2 ) have also opposite signs. From the continuity of Rτ1 on the interval [λk , λk+1 ], there is at least one zero of Rτ1 in (λk , λk+1 ). Now, suppose that in this interval there is more than one zero of Rτ1 , so one can take two neighboring zeros of Rτ1 in (λk , λk+1 ). By reproducing the argumentation above with τ1 and τ2 interchanged, one obtains that there is at least one zero of Rτ2 somewhere in (λk , λk+1 ). This contradicts
the assumption that λk and λk+1 are neighbors. The assertion of the following proposition is a well established fact (see, for instance [23, Thm. 1]). We, nevertheless, provide the proof for the reader’s convenience and because we introduce in it notation for later use. Note that a non-constant entire function of at most minimal type of order one must have zeros, otherwise, by Weierstrass theorem on the representation of entire functions by infinite products [25, Thm. 3 Chap. 1], it would be a function of at least normal type. Before stating the proposition we remind the definition of convergence exponent of a sequence of complex numbers (see [25, Sec. 4 Chap. 1]). The
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convergence exponent ρ1 of a sequence {νk }k of non-zero complex numbers accumulating only at infinity is given by ⎧ ⎫ ⎨ ⎬ 1 ρ1 := inf γ ∈ R : lim < ∞ . (3.2) r→∞ ⎩ ⎭ |νk |γ |νk |r
We also remark that, as it is customary, whenever we say that an infinite product is convergent we mean that at most a finite number of factors may be zero and the partial product formed by the non-vanishing factors tends to a number different from zero [1, Sec. 2.2 Chap. 5]. Proposition 3 Let f (ζ ) be an entire function of at most minimal type of order one with an infinite number of zeros. Let the elements of the sequence {νk }k , which accumulate only at infinity, be the non-zero roots of f , where {νk }k contains as many elements for each zero as its multiplicity. Assume that m ∈ N ∪ {0} is the order of the zero of f at the origin. Then there exists a complex constant C such that ζ f (ζ ) = Cζ m lim 1− , (3.3) r→∞ νk |ν |r k
where the limit converges uniformly on compacts of C. Proof The convergence exponent ρ1 of the zeros of an arbitrary entire function does not exceed its order [25, Thm. 6 Chap. 1]. Then, for a function of at most minimal type of order one, ρ1 1. According to Hadamard’s theorem [25, Thm. 13 Chap. 1], the expansion of f in an infinite product has either the form: ζ f (ζ ) = ζ m eaζ +b lim G ;0 , a, b ∈ C (3.4) r→∞ νk |ν |r k
if the limit lim
r→∞
|νk |r
converges, or m cζ +d
f (ζ ) = ζ e
lim
r→∞
|νk
1 |νk |
ζ G ;1 νk |r
(3.5)
,
c, d ∈ C
(3.6)
if (3.5) diverges. We have used here the Weierstrass primary factors G (for details see [25, Sec. 3 Chap. 1]). Let us suppose that the order is one and (3.5) diverges, then, in view of the fact that f is of minimal type, by a theorem due to Lindelöf [25, Thm. 15 a Chap. 1], we have in particular that lim νk−1 = −c . r→∞
|νk |r
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This implies the uniform convergence of the series limr→∞ compacts of C. In its turn, since ρ1 = 1, this yields that ζ lim 1− r→∞ νk |ν |r
ζ |νk |r νk
on
k
is uniformly convergent on any compact of C. Therefore, ζ ζ −cζ lim G ;1 = e lim 1− . r→∞ r→∞ νk νk |ν |r |ν |r k
k
Thus, (3.6) can be written as (3.3). Suppose now that the limit (3.5) converges. If the order of the function is less than one, then, by [25, Thm. 13 Chap. 1], one may write (3.4) as (3.3). If the order of the function is one, by [25, Thm. 12, Thm. 15 b Chap. 1], one concludes again that (3.4) can be written as (3.3) (cf. Thm 15 in the Russian version of [25] or, alternatively, [26, Lect. 5]).
Let {λn (τ )}n be the eigenvalues of J(τ ). In view of the fact that Rτ (ζ ) is an entire function of at most minimal type of order one, by Proposition 3, one can always write ζ Rτ (ζ ) = Cτ ζ δτ lim 1− , τ ∈ R ∪ {∞} , (3.7) r→∞ λk (τ ) 0<|λ (τ )|r k
where Cτ ∈ R \ {0} and δτ is the Kronecker delta, i. e., δτ = 1 if τ = 0, and δτ = 0 otherwise. The limits in (3.7) converge uniformly on compacts of C. Note that when τ = 0 we have naturally excluded λk (0) = 0 from the infinite product. When writing (3.7), we have taken into account, on the one hand, that R0 (0) = 0, which follows from (2.10) and the definition of the function D, and on the other, that different self-adjoint extensions have disjoint spectra (see Section 2). Now, let us consider the following expressions derived from the Green’s formula [2, Eqs. 1.23, 2.28] D(ζ ) = ζ
∞
Pk (0)Pk (ζ ) ,
(3.8)
k=0
B(ζ ) = −1 + ζ
∞
Qk (0)Pk (ζ ) .
(3.9)
k=0
Again we verify from (3.8) that D(0) = 0, while from (3.9) we have B(0) = −1. Therefore Rτ (0) = −τ for every τ ∈ R, and R∞ (0) = −1. Thus, Cτ = −τ provided that τ ∈ R and τ = 0, and C∞ = −1.
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143
To simplify the writing of some of the formulae below, let us introduce ) Rτ (ζ ) := RCτ (ζ , that is, τ ζ 1− Rτ (ζ ) := ζ δτ lim , τ ∈ R ∪ {∞} , (3.10) r→∞ λk (τ ) 0<|λk (τ )|r
where δτ is defined as in (3.7). Due to the uniform convergence of the expression ⎡ ⎤ ζ d ⎣ ⎦ , as r → ∞ , 1− dζ 0<|λ (τ )|r λk (τ ) k
one has Rτ (λ j(τ )) =
⎧ ⎪ ⎪ ⎨ −[λ j(τ )]δτ −1 lim ⎪ ⎪ ⎩
r→∞ # # 0 < #λk (τ )# r k = j
1−
λ j(τ ) , λ j(τ ) = 0 λk (τ )
1
(3.11)
λ j(τ ) = 0
By (3.1) and (3.11), one obtains C0 = a(0).
(3.12)
Theorem 1 Let τ1 , τ2 ∈ R ∪ {∞} with τ1 = τ2 . The spectra {λk (τ1 )}k , {λk (τ2 )}k of two different self-adjoint extensions J(τ1 ), J(τ2 ) of a Jacobi operator J in the limit circle case uniquely determine the matrix associated with J, and the numbers τ1 and τ2 . Proof For definiteness assume that τ1 = 0, in other words that the sequence {λk (τ1 )}k does not contain any zero element. By (3.1), we have a(λk (τ1 )) = MRτ2 (λk (τ1 ))Rτ1 (λk (τ1 )) , where
M=
⎧ τ1 τ2 if τ1 , τ2 ∈ R \ {0} ⎪ τ1 −τ2 ⎪ ⎪ ⎪ ⎨ τ2 if τ1 = ∞ , τ2 = 0 ⎪ −τ1 if τ2 = ∞ ⎪ ⎪ ⎪ ⎩ −C0 if τ2 = 0
(3.13)
(3.14)
Now, since {a(λk (τ1 ))}k are the normalizing constants of J(τ1 ) we must have 1 1 1 1= = . a(λk (τ1 )) M Rτ2 (λk (τ1 ))Rτ1 (λk (τ1 )) k
k
Therefore M=
k
1 . Rτ2 (λk (τ1 ))Rτ1 (λk (τ1 ))
(3.15)
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Thus, M is completely determined by the sequences {λk (τ2 )}k and {λk (τ1 )}k . Inserting the obtained value of M into (3.13) one obtains the normalizing constants. Having the normalizing constants allows us to construct the spectral measure for J(τ1 ). Then, by standard methods (see Section 2), one reconstructs the matrix associated with J and the boundary condition at infinity τ1 . From the value of M and τ1 one obtains τ2 , by using the first three cases in (3.14). When 0 ∈ {λk (τ2 )}k , one does not use (3.14), since it is already known that τ2 = 0. Remark 2 Note that the proof of Theorem 1 gives a reconstruction method of the Jacobi matrix. Although mentioned earlier, we also remark here that the assertion of Theorem 1, for the case of τ1 , τ2 ∈ R, was announced without proof in [13, Thm. 1].
4 Necessary and Sufficient Conditions In this section we give a complete characterization of our two-spectra inverse problem. We remind the reader about the remark on the notation at the end of Section 2. First we prove the following simple proposition related to the converse of Proposition 3. Proposition 4 Let {νk }k be an infinite sequence of non-vanishing complex numbers accumulating only at ∞, and whose convergence exponent ρ1 does not exceed one. Suppose that the infinite product ζ 1− (4.1) lim r→∞ νk |ν |r k
converges uniformly on any compact of C. Then this product is an entire function of at most minimal type of order one if either (3.5) converges or if (3.5) diverges but the following holds n(r) = 0, r where n(r) is the number of elements of {νk }k in the circle |ζ | < r. lim
r→∞
(4.2)
Proof Clearly, by the conditions of the theorem, one can express (4.1) in terms of canonical products [25, Sec. 3 Chap. 1] either in the form ζ ζ lim 1− = lim G ;0 (4.3) r→∞ r→∞ νk νk |ν |r |ν |r k
k
whenever (3.5) converges, or in the form ζ ζ cζ 1− = e lim lim G ;1 r→∞ r→∞ νk νk |ν |r |ν |r k
k
(4.4)
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145
otherwise, where lim
r→∞
νk−1 = −c.
(4.5)
|νk |r
In the case (4.3), in which the genus of the product is less than the convergence exponent of {νk }k , it is clear that (4.1) does not grow faster than an entire function of minimal type of order one. Indeed, by [25, Thm. 7 Chap. 1], the order of a canonical product is equal to the convergence exponent, so when ρ1 < 1 the assertion is obvious. For ρ1 = 1 the statement follows from [25, Thm. 15 b Chap. 1]. If we have the representation (4.4), then ρ1 = 1. By [25, Thm. 7 Chap. 1], the canonical product has order one. Since the product of functions of the same order is of that same order, the order of (4.1) is one. Then, the assertion follows from [25, Thm. 15 a Chap. 1] due to (4.2) and (4.5).
Before passing on to the main results of this section, we establish an auxiliary result which is related to part of the proof of Theorem 1 in the Addenda and Problems of [2, Chap. 4]. Lemma 1 Consider an infinite real sequence {κ j} j and a sequence {α j} j of positive numbers such that κ 2m j j
αj
<∞
for all m = 0, 1, . . .
Let F be an entire function of at most minimal type of order one whose zeros, {κ j} j, are simple, and such that |F(it)| → ∞ as t → ±∞ ,
t ∈ R.
(4.6)
If j
αj $ % 2 < ∞ , 2 1 + κ j F (κ j)
then κm j j
F (κ j)
(4.7)
is absolutely convergent for m = 0, 1, . . . , and the absolutely convergent expansion 1 1 = (κ )(ζ − κ ) F(ζ ) F j j j holds true for all ζ ∈ C \ {κ j} j.
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Proof The absolutely convergence of (4.7) follows from # # ## & # # # # # κm 1 + κ2 # √ # κm # # # # # α j j j # j #= #& # ## √ # F (κ ) # # # # # αj 2 j # j j # 1 + κ j F (κ j ) # # ' ' ( 2m ( κ j + κ 2m+2 ( αj j ( ) $ % ) <∞ 2 α 2 j 1 + κ [F (κ )] j j j
j
Construct the function h(ζ ) := 1 − F(ζ )
j
F (κ
1 , j )(ζ − κ j )
where the series is absolutely convergent in compact subsets of C \ {κ j} j because of (4.7). Clearly, h(κ j) = 0 for any j. Moreover, it turns out that h is an entire function of at most minimal type of order one. To show this, first ## ##−1 < ∞. Here, by what we have discussed in consider the case when j κj the proof of Proposition 3, the function F(ζ )/(ζ − κ j) can be expressed by a canonical product of genus zero. It follows from [25, Lem. 3 Chap. 1] (see also the proof of [25, Thm. 4 Chap. 1]) that, on the one hand, # # # F(ζ ) # # # < exp (C(α)rα ) , max # ρ1 < α < 1 , |ζ |=r (ζ − κ j ) # for any r > 0 provided that ρ1 < 1. If, on the other hand, ρ1 = 1, then for any > 0, there exists R0 > 0 such that # # # F(ζ ) # # < exp (r) max ## (4.8) |ζ |=r (ζ − κ j ) # for all r > R0 . Hence, in any case, we have the uniform, with respect to j, # #−1 asymptotic estimation (4.8) when j #κ j# < ∞. # #−1 Suppose now that j #κ j# = ∞. In this case, as was shown in the proof of Proposition 3, F(ζ ) 1 m (c+κ −1 ζ )ζ +d j =− ζ e lim G ;1 , r→∞ # # (ζ − κ j) κj κk # # κk r k = j
where lim
r→∞
|κk |r
κk−1 = −c .
(4.9)
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147
On the basis of the estimates found in the proof of [25, Thm. 15 Chap. 1] (see in particular the inequality next to [25, Eq. 1.43]), one can find R1 , independent of j, such that # # F(ζ ) max ## |ζ |=r (ζ − κ
# ⎡ ⎛# ⎞⎤ # # # # # # n(r) 1 # < exp ⎣r ⎝#c + ⎠⎦ κk−1 ## + C lim sup + +O # # r r r→∞ j) # |κk |r #
for all r > R1 and > 0 (see # # the definition of n(r) in the statement of inequality follows directly Proposition 4). Note that if #κ j# r, then the# above # from the inequality next to [25, Eq. 1.43]. If #κ j# > r, the same inequality holds due to # # # # # # # # # # # 1 # −1 # −1 # # #c + κ −1 + κk # #c + κk # + . j # # # # # r |κk |r |κk |r Since F does not grow faster than a function of minimal type of order one, by [25, Thm. 15 a Chap. 1], one again verifies that, for any > 0, (4.8) holds for all r greater than a certain R2 depending only on the velocity of convergence in the limits (4.9) and (4.2). Thus, one concludes that, for any > 0, there is R > 0 such that # # # # # # # # # F(ζ ) # 1 1 # # # < exp(r) # # # max max #F(ζ ) (κ )(ζ − κ ) # #F (κ j)# |ζ |=r # (ζ − κ j) # |ζ |=r # F j j # j j for all r > R, which shows that h is an entire function of at most minimal type of order one. Now, the function h/F is also an entire function of at most minimal type of order one [25, Cor. Sec. 9 Chap. 1]. By the hypothesis (4.6), lim
t → ±∞ t∈R
h(it) = 0, F(it)
which implies that h/F ≡ 0 (see Corollary of [25, Sec. 14 Chap. 1]).
Theorem 2 Let {λk }k and {μk }k be two infinite sequences of real numbers such that a) {λk }k ∩ {μk }k = ∅. For definiteness we assume that 0 ∈ {λk }k b) the sequences accumulate only at the point at infinity. c) λk = λ j, μk = μ j for k = j.
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Then there exist unique τ1 , τ2 ∈ R ∪ {∞}, with τ1 = 0, τ1 = τ2 , and a unique Jacobi operator J = J ∗ such that {λk }k = σ (J(τ1 )) and {μk }k = σ (J(τ2 )) if and only if the following conditions are satisfied. 1. The convergence exponents of the sequence {λk }k , and of the non-zero elements of {μk }k do not exceed one. Additionally, if 1 nλ (r) lim = ∞ , require that lim = 0, r→∞ r→∞ |λk | r |λk |r
and if
lim
r→∞
0<|μk |r
1 = ∞, |μk |
require that
lim
r→∞
nμ (r) = 0, r
where nλ (r) and nμ (r) are the number of elements of {λk }k and {μk }k , respectively, in the circle |ζ | < r. 2. The limits ζ ζ lim 1− lim 1− , r→∞ r→∞ λk μk |λ |r 0<|μ |r k
k
converge uniformly on compact subsets of C, and they define the functions ζ Rλ (ζ ) := lim 1− (4.10) r→∞ λk |λk |r ζ Rμ (ζ ) := ζ δ lim 1− , (4.11) r→∞ μk 0<|μ |r k
where δ = 1 if 0 ∈ {μk }k , and δ = 0 otherwise. 3. All numbers Rμ (λ j)Rλ (λ j) have the same sign for all j. The same is true for the numbers Rλ (μ j)Rμ (μ j). 4. For every m = 0, 1, 2, . . . the series below are convergent and the following equalities hold λmj μmj = − Rμ (λ j)Rλ (λ j) Rλ (μ j)Rμ (μ j) j j 5. The series
Rμ (λ j) j
Rλ (λ j)
diverge either to −∞ or +∞. 6. The series Rμ (λ j) $ % 1 + λ2j Rλ (λ j) j are convergent.
and
and
Rλ (μ j) Rμ (μ j) j
j
Rλ (μ j) $ % 1 + μ2j Rμ (μ j)
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Proof We begin by proving that if {λk }k and {μk }k are, respectively, the spectra of the self-adjoint extensions J(τ1 ) and J(τ2 ) of a Jacobi operator J, then conditions 1–6 hold true. Since {λk }k = σ (J(τ1 )) and {μk }k = σ (J(τ2 )), the functions Rτ1 and Rτ2 , given by (2.10), have the sequences {λk }k and {μk }k , respectively, as their sets of zeros. These functions do not grow faster than an entire function of minimal type of order one. By Proposition 3 [see (3.7)], the limits ζ ζ lim 1− , lim 1− r→∞ r→∞ λk μk |λ |r 0<|μ |r k
k
converge uniformly on compacts of C. This is condition 2. Moreover, by (3.6) and [25, Thm. 15 a Chap. 1], condition 1 holds. The functions Rλ and Rμ , given by (4.10) and (4.11), coincide with Rτ , given by (3.10), with τ = τ1 and τ = τ2 , respectively. Thus (3.13) is rewritten as follows a(λ j) = MRμ (λ j)Rλ (λ j) ,
(4.12)
where M is given by (3.14). Analogously, a(μ j) = −MRλ (μ j)Rμ (μ j) .
(4.13)
On the basis of the positiveness of the normalizing constants, from (4.12) and (4.13), we obtain condition 3. From what we discussed in Section 2 all the moments exist for the spectral functions of J(τ1 ) and J(τ2 ), which are, respectively, 1 1 and . (4.14) a(λk ) a(μk ) λ t μ t k
k
Hence the series in both sides of condition 4 are convergent for m ∈ N ∪ {0}. Moreover, the spectral functions (4.14) are solutions of the same moment problem associated with J [see the paragraph surrounding (2.11)], therefore the equality of condition 4 holds. Theorem 1 in the Addenda and Problems of [2, Chap. 4] tells us that a(λ j) (4.15) 2 = +∞ Rλ (λ j) j is a necessary condition for the sequences {λ j} j and {a(λ j)} j to be the spectrum of J(τ1 ) and its corresponding normalizing constants. Thus, substituting (4.12) into (4.15), one establishes the divergence of the first series in condition 5. Similarly, a(μ j) 2 = +∞ Rμ (μ j) j must hold, which, by (4.13), implies the divergence of the second series in condition 5.
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By the same theorem in [2] mentioned above, and taking into account (4.12) and (4.13), one obtains the convergence of the series in condition 6. Let us now prove that the conditions 1–6 are sufficient. Using condition 2 and the convergence of the series in the left hand side of condition 4 with m = 0, we define the real constant M :=
j
1 Rμ (λ j)Rλ (λ j)
(4.16)
and the sequence of numbers a j := MRμ (λ j)Rλ (λ j)
(4.17)
By condition 3 and (4.16), it follows that a j > 0 for all j. Moreover, (4.16) and (4.17) imply that j a−1 j = 1. With the aid of the sequences {λk } and {ak } define the function ρ : R → R+ as follows ρ(t) := a−1 (4.18) k . λk t
Consider the self-adjoint operator of multiplication Aρ by the independent variable in L2 (R, dρ). We show below that this operator is the canonical representation (see Section 2) of a self-adjoint extension of a Jacobi matrix in the limit circle case. The proof of this fact is similar to the proof of Theorem 2 in Addenda and Problems of [2, Chap. 4]. Note, however, that our conditions are slightly different. Consider a function θk (t) ∈ L2 (R, dρ) such that √ θk (λ j) = ak δkj . (4.19) Clearly, θk (t) is the normalized eigenvector of Aρ corresponding to λk . Let ϕ(t) ∈ L2 (R, dρ) be such that √ aj ϕ, θ j L2 (R,dρ) = . (4.20) (λ j − i)Rλ (λ j) Taking into account (4.17), it is clear that the convergence of the first series in condition 6 ensures that ϕ(t) is indeed an element of L2 (R, dρ). Define + * D := ξ ∈ L2 (R, dρ) : ξ = (Aρ + iI )−1 ψ, ψ ∈ L2 (R, dρ), ψ ⊥ ϕ . (4.21) Since D ⊂ dom(Aρ ), we can consider the restriction of Aρ to the linear set D. Let us show that this restriction is a symmetric operator with deficiency indices (1, 1). First we verify that D is dense in L2 (R, dρ). Suppose that a non-zero η ∈ L2 (R, dρ) is orthogonal to D. This would imply that there is a non-zero constant C ∈ C such that η = C(Aρ − iI )ϕ .
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151
Therefore, η, θ j L2 (R,dρ) = C
√ aj , Rλ (λ j)
whence we easily conclude that η ∈ L2 (R, dρ) would contradict condition 5. Consider now the restriction of Aρ to the set D, denoted henceforth by Aρ D , and let us find the dimension of ker((Aρ D )∗ − iI ) which is characterized as the set of all ω ∈ L2 (R, dρ) for which the equation (Aρ + iI )ξ, ω L2 (R,dρ) = 0 is satisfied for any ξ ∈ D. It is not difficult to show that any such ω can be written as follows , , ω = Cϕ
, ∈ C. 0 = C
Hence dim ker((Aρ D )∗ −iI ) = 1. Analogously, it can be shown that the dimension of ker((Aρ D )∗ +iI ) also equals one. Indeed, if ω ∈ ker((Aρ D )∗+ iI ) then, up to a complex constant ω = (Aρ − iI )(Aρ + iI )−1 ϕ . Now we show that Aρ D is the canonical representation of a Jacobi operator in the limit circle case and Aρ is the canonical representation of a self-adjoint extension of this Jacobi operator. We proceed stepwise. 1. We orthonormalize the sequence of functions {tn }∞ n=0 with respect to the inner product of L2 (R, dρ). Note that condition 4 guarantees that all elements of the sequence {tn }∞ n=0 are in L2 (R, dρ). We obtain thus a sequence of polynomials {Pn−1 (t)}∞ n=1 which satisfy the three term recurrence equation (2.15) and (2.16), where all the coefficients bk (k ∈ N) turn out to be positive and qk (k ∈ N) are real numbers. 2. We verify that the polynomials are dense in L2 (R, dρ), so the sequence we have constructed is a basis in L2 (R, dρ). Note first that the function Rλ (ζ ) is entire of at most minimal type of order one. Indeed, this follows from Proposition 4, in view of conditions 1 and 2. Now, for any element λk0 of the sequence {λk }k , we clearly have # # # t2 ## # |Rλ (it)| #1 + 2 # , t ∈ R. # λk # 0
This implies that Rλ satisfies (4.6). Hence the function Rλ and the sequences {λ j} j, {a j} j satisfy the condition of Lemma 1. Thus, we have shown the convergence of the series
λmj
j
Rλ (λ j)
(4.22)
for all m = 0, 1, 2, . . . , and that 1 1 = . Rλ (ζ ) R (λ )(ζ − λ j) j λ j
(4.23)
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Taking into account Definitions 1 and 2 of the Addenda and Problems of [2, Chap. 4], one obtains from Corollary 2 of [2, Addenda and Problems Chap. 4], together with conditions 5 and 6, that {λk } is a canonical sequence of nodes and {a−1 k } the corresponding sequence of masses for the moment problem given by {sm }∞ m=0 with sm :=
λmj 1 . M j Rμ (λ j)Rλ (λ j)
(4.24)
Hence, for this moment problem, ρ is a canonical solution [2, Def. 3.4.1]. By definition, a canonical solution is N-extremal and by [2, Thm. 2.3.3], the polynomials are dense in L2 (R, dρ). 3. We prove that the elements of the basis {Pn−1 (t)}∞ n=1 are in D. From (4.22) and (4.23), by Lemma 1 of the Addenda and Problems of [2, Chap. 4], one has for m = 0, 1, 2, . . . λmj = 0. Rλ (λ j) j Then, if S(t) is a polynomial S(λ j) = 0. (Aρ + iI )S, ϕ L2 (R,dρ) = Rλ (λ j) j Whence it follows that S ∈ D. Now, by (2.15) and (2.16), it is straightforward to show that U −1 Aρ D U (see (2.12)) is a Jacobi operator in the limit circle case. Denote by J the Jacobi operator U −1 Aρ D U. On the basis of what was discussed in Section 2 one can find τ1 ∈ (R ∪ {∞}) \ {0} such that the selfadjoint operator of multiplication in L2 (R, dρ) is the canonical representation of J(τ1 ). τ1 cannot be zero since then {λk } should contain the zero. If 0 ∈ {μk }k , we define ⎧ ⎨ M if τ1 = ∞ if τ1 = −M (4.25) τ2 := ∞ ⎩ Mτ1 in all other cases, τ1 + M and if 0 ∈ {μk }k simply assign τ2 := 0. For the proof to be complete it remains to show that {μk }k are the eigenvalues of J(τ2 ). To this end we first show that {μk }k are the eigenvalues of some - = −M and define self-adjoint extension of J. Let M - λ (μ j)Rμ (μ j) . , a j := MR
−1 From condition 4 with m = 0, it follows that , a j > 0 for any j and j , a j = 1, −1 and that the function ρ ,(t) := μk t , ak is a solution of the moment problem with s given by (4.24). Moreover, taking into account conditions 1 {sk }∞ k k=0 and 6, one easily verifies as before that the sequences {μ j} j and {, a j} j, and the function Rμ satisfy the conditions of Lemma 1. Therefore, by Definitions 1,
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2 and Corollary 2 of the Addenda and Problems of [2, Chap. 4], as well as conditions 5 and 6, it turns out that the sequence {μ j} j is a canonical sequence of nodes and {, a j} j the corresponding sequence of masses for the moment problem given by {sk }∞ , is a canonical k=0 with sk satisfying (4.24). Hence ρ solution of this moment problem. Denote by J(, τ2 ) the self-adjoint extension of J having ρ , as its spectral function. Let us consider now the functions Rτ2 and Rτ,2 corresponding to J(τ2 ) and J(, τ2 ), respectively (see (3.10)). It is straightforward to verify that Rτ2 (λ j) =
aj = Rτ,2 (λ j) , M Rτ1 (λ j)
(4.26)
where the first equality follows from (3.13), while the second follows from (3.11) and (4.17). By (3.14), (3.15), and (4.25), one easily concludes from (4.26) that τ2 = τ,2 .
Remark 3 When 0 ∈ {μk }k , the signs of the real numbers Rμ (λ j)Rλ (λ j) and Rλ (μ j)Rμ (μ j) are known. Thus, we can write condition 3 as follows Rμ (λ j)Rλ (λ j) < 0
Rλ (μ j)Rμ (μ j) > 0
for all j .
This is a consequence of (3.12) and (3.14) by which we know that in equation (4.12) M = −a(0) < 0. Remark 4 Note that, by Proposition 2, conditions 1–6 imply the interlacing of the sequences {λk }k and {μk }k . Acknowledgements We thank A. Osipov for drawing our attention to [14] and the anonymous referees whose comments led to an improved presentation of our work.
References 1. Ahlfors, L.V.: Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable. McGraw-Hill Book Co., New York (1966) 2. Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Hafner Publishing Co., New York (1965) 3. Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover Publications Inc., New York (1993) 4. Aktosun, T., Weder, R.: Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrödinger equation. Inverse Problems 22, 89–114 (2006) 5. Aktosun, T., Weder, R.: The Borg–Marchenko theorem with a continuous spectrum. In: Recent Advances in Differential Equations and Mathematical Physics. Contemp. Math. vol. 412, pp. 15–30. Amer. Math. Soc., Providence, RI (2006) 6. Albeverio, S., Kurasov, P.: Singular perturbations of differential operators. London Mathematical Society Lecture Note Series, vol. 271. Cambridge University Press, Cambridge (2000) 7. Berezans ki˘ı, J.M.: Expansions in eigenfunctions of selfadjoint operators. Translations of Mathematical Monographs, vol. 17. American Mathematical Society, Providence, RI (1968) 8. Brown, B.M., Naboko, S., Weikard, R.: The inverse resonance problem for Jacobi operators. Bull. London Math. Soc. 37, 727–737 (2005)
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9. Borg, G.: Uniqueness theorems in the spectral theory of y + (λ − q(x))y = 0. In: Proc. 11th Scandinavian Congress of Mathematicians, pp. 276–287. Johan Grundt Tanums Forlag, Oslo (1952) 10. Chelkak, D., Korotyaev, E.: The inverse problem for perturbed harmonic oscillator on the half-line with a Dirichlet boundary condition. Ann. Henri Poincaré 8(6), 1115–1150 (2007) 11. Donoghue, W.F., Jr.: On the perturbation of spectra. Comm. Pure Appl. Math. 18, 559–579 (1965) 12. Fu, L., Hochstadt, H.: Inverse theorems for Jacobi matrices. J. Math. Anal. Appl. 47, 162–168 (1974) 13. Gasymov, M.G., Guse˘ınov, G.S.: On inverse problems of spectral analysis for infinite Jacobi matrices in the limit-circle case. Dokl. Akad. Nauk SSSR 309(6), 1293–1296 (1989). In Russian. Translation in Soviet Math. Dokl. 40(3), 627–630 (1990) 14. Gasymov, M.G., Guse˘ınov, G.S.: Uniqueness theorems in inverse problems of spectral analysis for Sturm–Liouville operators in the case of the Weyl limit circle. Differentsial nye Uravneniya 25(4), 588–599 (1989). In Russian. Translation in Differential Equations 25(4), 394–402 (1989) 15. Gesztesy, F., Simon, B.: Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators. Trans. Amer. Math. Soc. 348, 349–373 (1996) 16. Gesztesy, F., Simon, B.: m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices. J. Anal. Math. 73, 267–297 (1997) 17. Gesztesy, F., Simon, B.: On local Borg-Marchenko uniqueness results. Comm. Math. Phys. 211, 273–287 (2000) 18. Gorbachuk, M.L., Gorbachuk, V.I.: M. G. Krein’s lectures on entire operators. Operator Theory: Advances and Applications, vol. 97. Birkhaüser Verlag, Basel (1997) 19. Guse˘ınov, G.Š.: The determination of the infinite Jacobi matrix from two spectra. Mat. Zametki 23(5), 709–720 (1978) 20. Halilova, R.Z.: An inverse problem. Izv. Akad. Nauk Azerba˘ıdžan. SSR Ser. Fiz.-Tehn. Mat. Nauk 1967(3–4), 169–175 (1967) (In Russian) 21. Kre˘ın, M.G.: Solution of the inverse Sturm–Liouville problem. Dokl. Akad. Nauk SSSR (N.S.) 76, 21–24 (1951) (In Russian) 22. Kre˘ın, M.G.: On a generalization of investigations of Stieltjes. Dokl. Akad. Nauk SSSR (N.S.) 87, 881–884 (1952) (In Russian) 23. Kre˘ın, M.G.: On the indeterminate case of the Sturm-Liouville boundary problem in the interval (0, ∞). Izvestiya Akad. Nauk SSSR. Ser. Mat. 16, 293–324 (1952) (In Russian) 24. Kre˘ın, M.: On a method of effective solution of an inverse boundary problem. Dokl. Akad. Nauk SSSR (N.S.) 94, 987–990 (1954) (In Russian) 25. Levin, B.Ja.: Distribution of zeros of entire functions. Translations of Mathematical Monographs, vol. 5. American Mathematical Society, Providence, R.I. (1980) 26. Levin, B.Ja.: Lectures on entire functions. Translations of Mathematical Monographs, vol. 150. American Mathematical Society, Providence, R.I. (1996) 27. Levitan, B.M., Gasymov, M.G.: Determination of a differential equation by two spectra. Uspehi Mat. Nauk 19(2(116)), 3–63 (1964) 28. Marˇcenko, V.A.: Some questions of the theory of one-dimensional linear differential operators of the second order. I. Tr. Mosk. Mat. Obs. 1, 327–420 (1952). In Russian. Translation in Amer. Math. Soc. Transl. Ser. 2 101, 1–104 (1973) 29. Silva, L.O., Weder, R.: On the two spectra inverse problem for semi-infinite Jacobi matrices. Math. Phys. Anal. Geom. 3(9), 263–290 (2006) 30. Simon, B.: Spectral analysis of rank one perturbations and applications. In: Mathematical Quantum Theory. II. Schrödinger Operators (Vancouver, BC, 1993), CRM Proc. Lecture Notes, vol. 8, pp. 109–149. Amer. Math. Soc., Providence, RI (1995) 31. Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137(1), 82–203 (1998) 32. Teschl, G.: Trace formulas and inverse spectral theory for Jacobi operators. Comm. Math. Phys. 196(1), 175–202 (1998) 33. Teschl, G.: Jacobi operators and completely integrable nonlinear lattices. Mathematical Surveys and Monographs, vol. 72. American Mathematical Society, Providence, RI (2000) 34. Weikard, R.: A local Borg-Marchenko theorem for difference equations with complex coefficients. In: Partial Differential Equations and Inverse Problems, Contemp. Math., vol. 362, pp. 403–410. Amer. Math. Soc., Providence, RI (2004)
Math Phys Anal Geom (2008) 11:155–173 DOI 10.1007/s11040-008-9045-8
Anisotropic Lavine’s Formula and Symmetrised Time Delay in Scattering Theory Rafael Tiedra de Aldecoa
Received: 30 September 2007 / Accepted: 27 June 2008 / Published online: 6 August 2008 © Springer Science + Business Media B.V. 2008
Abstract We consider, in quantum scattering theory, symmetrised time delay defined in terms of sojourn times in arbitrary spatial regions symmetric with respect to the origin. For potentials decaying more rapidly than |x|−4 at infinity, we show the existence of symmetrised time delay, and prove that it satisfies an anisotropic version of Lavine’s formula. The importance of an anisotropic dilations-type operator is revealed in our study. Keywords Time delay · Lavine’s formula · Scattering theory Mathematics Subject Classifications (2000) 46N50 · 81Q10 · 35J10
1 Introduction and Main Results It is known for quite some time that the definition of time delay (in terms of sojourn times) in scattering theory has to be symmetrised in the case of multichannel-type scattering processes (see e.g. [3, 4, 12, 13, 21, 22]). More recently [6] it has been shown that symmetrised time delay does exist, in two-body scattering processes, for arbitrary dilated spatial regions symmetric with respect to the origin (the usual time delay does exist only for spherical spatial regions [20]). This leads to a generalised formula for time delay, which reduces to the usual one in the case of spherical spatial regions. The aim of the present paper is to provide a reasonable interpretation of this formula
R. Tiedra de Aldecoa (B) CNRS (UMR 8088) and Department of Mathematics, University of Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France e-mail: [email protected]
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for potential scattering by proving its identity with an anisotropic version of Lavine’s formula [11]. Let us recall the definition of symmetrised time delay for a two-body scattering process in Rd , d ≥ 1. Consider a bounded open set in Rd containing the origin and the dilated spatial regions r := {rx | x ∈ }, r > 0. Let H0 := − 12 be the kinetic energy operator in H := L2 (Rd ) (endowed with the norm · and scalar product ·, ·). Let H be a selfadjoint perturbation of H0 such that the wave operators W± := s- limt→±∞ eitH e−itH0 exist and are complete (so that the scattering operator S := W+∗ W− is unitary). Then one defines for some states ϕ ∈ H and r > 0 two sojourn times, namely: ∞ 2 T 0 (ϕ) := dt dd x e−itH0 ϕ (x) r
−∞
and
Tr (ϕ) :=
∞ −∞
x∈r
dt x∈r
2 dd x e−itH W− ϕ (x) .
If the state ϕ is normalized to one the first number is interpreted as the time spent by the freely evolving state e−itH0 ϕ inside the set r , whereas the second one is interpreted as the time spent by the associated scattering state e−itH W− ϕ within the same region. The usual time delay of the scattering process for r with incoming state ϕ is defined as τrin (ϕ) := Tr (ϕ) − Tr0 (ϕ), and the corresponding symmetrised time delay for r is given by τr (ϕ) := Tr (ϕ) −
1 0 T (ϕ) + Tr0 (Sϕ) . 2 r
If is spherical and some abstract assumptions are verified, the limits of τrin (ϕ) and τr (ϕ) as r → ∞ exist and satisfy [6, Sec. 4.3] lim τr (ϕ) = lim τrin (ϕ) = −
r→∞
r→∞
1 −1/2 −1/2 H ϕ, S∗ [D, S]H0 ϕ , 2 0
(1.1)
where D is the generator of dilations. If is not spherical the limit of τrin (ϕ) as r → ∞ does not exist anymore [20], but the limit of τr (ϕ) as r → ∞ still exists, provided that is symmetric with respect to the origin [6, Rem. 4.8]. In this paper we study τr (ϕ) in the setting of potential scattering. For potentials decaying more rapidly than |x|−4 at infinity, we prove the existence of limr→∞ τr (ϕ) by using the results of [6]. In a first step we show that the limit satisfies the equation lim τr (ϕ) = − f (H0 )−1/2 ϕ, S∗ [D , S] f (H0 )−1/2 ϕ , (1.2) r→∞
where f is a real symbol of degree 1 and D ≡ D ( f ) is an operator acting as an anisotropic generator of dilations. Then we prove that formula (1.2) can be rewritten as an anisotropic Lavine’s formula. Namely, one has (see Theorem
Anisotropic Lavine’s formula and symmetrised time delay
157
4.5 for a precise statement) ∞ ds e−isH W− f (H0 )−1/2 ϕ, V, f e−isH W− f (H0 )−1/2 ϕ , (1.3) lim τr (ϕ) = r→∞
−∞
where the operator V, f = f (H) − f (H0 ) − i[V, D ]
:= 2V − i[V, D]. Formula (1.3) provides an interesting generalises the virial V relation between the potential V and symmetrised time delay, which we discuss. Let us give a description of this paper. In Section 2 we introduce the condition on the set (see Assumption 2.1) under which our results are proved. We also define the anisotropic generator of dilations D and establish some of its properties. Section 3 is devoted to symmetrised time delay in potential scattering; the existence of symmetrised time delay for potentials decaying more rapidly than |x|−4 at infinity is established in Theorem 3.5. In Theorem 4.5 of Section 4 we prove the anisotropic Lavine’s formula (1.3) for the same class of potentials. Remarks and examples are to be found at the end of Section 4. We emphasize that the extension of Lavine’s formula to non spherical sets is not straightforward due, among other things, to the appearance of a singularity in the space of momenta not present in the isotropic case (see Eq. 2.7 and the paragraphs that follow). The adjunction of the symbol f in the definition of the operator D (see Definition 2.2) is made to circumvent the difficulty. Finally we refer to [9] (see also [8, 11, 15–17]) for a related work on Lavine’s formula for time delay.
2 Anisotropic Dilations In this section we define the operator D and establish some of its properties in relation with the generator of dilations D and the shape of . We start by recalling some notations. Given two Hilbert spaces H1 and H2 , we write B (H1 , H2 ) for the set of bounded operators from H1 to H2 with norm · H1 →H2 , and put B (H1 ) := B (H1 , H1 ). We set Q := (Q1 , Q2 , . . . , Qd ) and P := (P1 , P2 , . . . , Pd ), where Q j (resp. P j) stands for the j-th component of the position (resp. momentum) operator in H. N := {0, 1, 2, . . .} is the set of natural numbers. Hk , k ∈ N, are the usual Sobolev spaces over Rd , and Hts , s, t ∈ R, are the weighted Sobolev spaces over Rd [1, Sec. 4.1], with the convention that Hs := H0s and Ht := Ht0 . Given a set M ⊂ Rd we write 1lM for the characteristic function for M. We always assume that is a bounded open set in Rd containing 0, with boundary ∂ of class C4 . Often we even suppose that satisfies the following stronger assumption (see [6, Sec. 2]).
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Assumption 2.1 is a bounded open set in Rd containing 0, with boundary ∂ of class C4 . Furthermore satisfies ∞ dμ 1l (μx) − 1l (−μx) = 0, ∀x ∈ Rd . 0
∞ If p ∈ Rd , then the number 0 dt 1l (tp) is the sojourn time in of a free classical particle moving along the trajectory t → x(t) := tp, t ≥ 0. Obviously satisfies Assumption 2.1 if is symmetric with respect to 0 (i.e. = −). Moreover if is star-shaped with respect to 0 and satisfies Assumption 2.1, then = −. We recall from [6, Lemma 2.2] that the limit +∞
dμ R (x) := lim 1l (μx) + ln ε (2.4) ε0 μ ε exists for each x ∈ Rd \ {0}, and we define the function G : Rd \ {0} → R by G (x) :=
1 [R (x) + R (−x)] . 2
(2.5)
The function G : Rd \ {0} → R is of class C4 since ∂ is of class C4 . Let x ∈ Rd \ {0} and t > 0, then formulas (2.4) and (2.5) imply that G (tx) = G (x) − ln(t). From this one easily gets the following identities for the derivatives of G : x · (∇G )(x) = −1, t|α| ∂ α G (tx) = ∂ α G (x),
(2.6) (2.7) ∂1α1
· · · ∂dαd .
The where α is a d-dimensional multi-index with |α| ≥ 1 and ∂ α := second identity suggests a way of regularizing the functions ∂ j G which partly motivates the following definition. We use the notation Sμ (R; R), μ ∈ R, for the vector space of real symbols of degree μ on R (see [1, Sec. 1.1]). Definition 2.2 Let f ∈ S1 (R; R) be such that (1) f (0) = 0 and f (u) > 0 for each u > 0, (2) for each j = 1, 2, . . . , d, the function x → (∂ j G )(x) f (x2 /2) (a priori only defined for x ∈ Rd \ {0}) belongs to C3 (Rd ; R). Then we define F : Rd → Rd by F (x) := −(∇G )(x) f (x2 /2). Given a set there are many appropriate choices for the function f . For instance if γ > 0 one can always take f (u) = 2(u2 + γ )−1 u3 , u ∈ R. But when is equal to the open unit ball B := {x ∈ Rd | |x| < 1} one can obviously make a simpler choice. Indeed in this case one has [6, Rem. 2.3.(b)] (∂ j GB )(x) = −x j x−2 , and the choice f (u) = 2u, u ∈ R, leads to the C∞ -function F (x) = x.
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symmetric and starRemark 2.3 One can associate to each set a unique set shaped with respect to 0 such that G = G
[6, Rem. 2.3.(c)]. The boundary
of
satisfies ∂
= eG (x) x | x ∈ Rd \ {0} , ∂
r = rx | x ∈
, r > 0. Thus the vector field F = F and
is orthogonal to
r in the following sense: if v belongs to the tangent space the hypersurfaces ∂
r , then F (y) is orthogonal to v. To see this let s → y(s) ≡
r at y ∈ ∂ of ∂
r . Then d y(s) belongs to the r eG (x(s)) x(s) be any differentiable curve on ∂ ds
r at y(s), and a direct calculation using Eqs. 2.6–2.7 gives tangent space of ∂ d y(s) = 0. F (y(s)) · ds In the rest of the section we give a meaning to the expression D :=
1 [F (P) · Q + Q · F (P)], 2
and we establish some properties of D in relation with the generator of dilations 1 D := (P · Q + Q · P). 2 in the domain For the next lemma we emphasize that H2 is contained D f (H0 ) of f (H0 ). The notation · stands for 1 + | · |2 , and S is the Schwartz space on Rd . Lemma 2.4 Let be a bounded open set in Rd containing 0, with boundary ∂ of class C4 . Then on (a) The operator D is essentially selfadjoint S . As a bounded operator, s−1 for each s ∈ R, t ∈ [−2, 0] ∪ D extends to an element of B Hts , Ht−1 [1, 3]. (b) One has for each t ∈ R and ϕ ∈ D(D ) ∩ D f (H0 ) e−itH0 D eitH0 ϕ = [D − t f (H0 )]ϕ.
(2.8)
In particular one has the equality i[H0 , D ] = f (H0 )
(2.9)
as sesquilinear forms on D(D ) ∩ H2 . The second claim of point (a) is sufficient for our purposes, even if it is only a particular case of a more general result. Proof (a) The essential seladjointness of D on S follows from the fact that F is of class C3 (see e.g. [1, Prop. 7.6.3.(a)]).
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Due to the hypotheses on F one has for each ϕ ∈ S the bound α (∂ F j)(P)ϕ Const. P ϕ ,
(2.10)
where F j is the j-th component of F and α is a d-dimensional multiindex with |α| 3. Furthermore s−1 P Q2 F j(P)Q j + i (∂ j F j)(P) ϕ D Hs3 →Hs−1 sup 2 2 s jd ϕ∈S ,ϕH3 =1
for each s ∈ R. Since Q2 acts as the operator 1 − after a Fourier transform, the inequalities above imply that D extends to an element of B (H3s , H2s−1 ). A similar argument shows that D extends to an element of B (H1s , Hs−1 ) for each s ∈ R. The second part of the claim follows then by using interpolation and duality. (b) Let ϕ ∈ e−itH0 S . Since e−itH0 Q j eitH0 ϕ = (Q j − t P j)ϕ, it follows by formula (2.6) that e−itH0 D eitH0 ϕ = [D + t P · (∇G )(P) f (H0 )]ϕ = [D − t f (H0 )]ϕ. This together with the essential selfajointness of e−itH0 D eitH0 on e−itH0 S implies the first part of the claim. Relation (2.9) follows by taking the derivative of (2.8) w.r.t. t in the form sense and then setting t = 0. Remark 2.5 If = B and f (u) = 2u, then F (x) = x for each x ∈ Rd , and the operators D and D coincide. If is not spherical it is still possible to determine part of the behaviour of the group Wt := eitD . Indeed let R × Rd (t, x) → ξt (x) ∈ Rd be the flow associated to the vector field −F , that is, the solution of the differential equation d ξ0 (x) = x. (2.11) ξt (x) = (∇G )(ξt (x)) f ξt (x)2 /2 , dt Then it is known (see e.g. the proof of [1, Prop. 7.6.3.(a)]) that the group Wt acts in the Fourier space as t ϕ (x) := ηt (x)ϕ(ξt (x)), W (2.12) where ηt (x) ≡ det(∇ξt (x)) is the Jacobian at x of the mapping x → ξt (x). Taking the scalar product of (2.11) with ξt (x) and then using formula (2.6) leads to the equation d ξt (x)2 = −2 f ξt (x)2 /2 , dt
ξ0 (x) = x.
If t < 0 and x = 0, then ξt (x)2 x2 > 0, and ξt (x)2 is given by the implicit formula ξt (x)2 du f (u/2)−1 = 0. 2t + x2
This, together with the facts that x → f (x2 /2) belongs to S2 (R; R) and f (u) > 0 for u > 0, implies the estimate ξt (x) e−ct x for some constant c > 0.
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Since ξt (x) x for each t 0 it follows that ξt (x) (1 + e−ct ) x
(2.13)
for all t ∈ R and x ∈ Rd (the case x = 0 is covered since ξt (0) = 0 for all t ∈ R). Equation (2.13) implies that the domain H2 of H0 is left invariant by the group Wt . The results of Remarks 2.3 and 2.5 suggest that Wt may be interpreted as an anisotropic version of the dilation group, which reduces to the usual dilation group in the case = B and f (u) = 2u. In the next lemma we show some properties of the mollified resolvent Rλ := iλ(D + iλ)−1 ,
λ ∈ R \ {0}.
We refer to [18, Lemma 6.2] for the same results for the usual generator of the dilation group D, that is, when = B and f (u) = 2u. See also [5, Lemma 4.5] for a general result. Lemma 2.6 Let be a bounded open set in Rd containing 0, with boundary ∂ of class C4 . Then (a) One has for each t ∈ R and ϕ ∈ D ξt (P)2 eitD H0 e−itD ϕ =
1 ξt (P)2 ϕ. 2
(2.14)
(b) For each λ ∈ R with |λ| large enough, Rλ belongs to B (H2 ), and Rλ extends to an element of B (H−2 ). Furthermore we have for each ϕ ∈ H2 and each ψ ∈ H−2 lim (1 − Rλ )ϕH2 = 0
|λ|→∞
and
lim (1 − Rλ )ψH−2 = 0.
|λ|→∞
Proof (a) Let ϕ ∈ eitD S . A direct calculation using formula (2.12) gives 1 F eitD H0 e−itD ϕ (k) = ξt (k)2 (F ϕ)(k), 2 where F is the Fourier transformation. This together with the essential selfajointness of eitD H0 e−itD on eitD S implies the claim. (b) Let ϕ ∈ H2 and take λ ∈ R with |λ| > c, where c is the constant in the inequality (2.13). Using the (strong) integral formula −1
(D + iλ)
∓∞
=i 0
dt eλt e−itD ,
sgn(λ) = ±1,
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and Relation (2.14) we get the equalities (D + iλ)−1 ϕ = (H0 + 1)−1 (D + iλ)−1 (H0 + 1)ϕ + ∓∞ +i dt eλt e−itD , (H0 + 1)−1 (H0 + 1)ϕ 0
= (H0 + 1)−1 (D + iλ)−1 (H0 + 1)ϕ − ∓∞ 1 −i dt eλt (H0 + 1)−1 e−itD H0 − ξt (P)2 ϕ 2 0 = (H0 + 1)−1 (D + iλ)−1 ϕ + ∓∞ i dt eλt e−itD ξt (P)2 ϕ. + (H0 + 1)−1 2 0 It follows that H0 Rλ ϕ = −
λ 2
∓∞
dt eλt e−itD ξt (P)2 ϕ,
sgn(λ) = ±1.
0
Now |λ| > c, and ξt (P)2 ϕ (1 + e−ct )ϕH2 due to the bound (2.13). Thus |λ| ∞ H0 Rλ ϕ dt e−|λ|t ξ− sgn(λ)t (P)2 ϕ 2 0 |λ| ∞ −|λ|t dt e + e(sgn(λ)c−|λ|)t ϕH2 2 0 Const. ϕH2 .
(2.15)
Using the estimate (2.15) and a duality argument one gets the bounds Rλ H2 →H2 Const.
and
Rλ H−2 →H−2 Const.,
(2.16)
which imply the first part of the claim. For the second part we remark that 1 − Rλ = (iλ)−1 D Rλ on H. Using this together with the bounds (2.16) one easily shows that lim|λ|→∞ (1 − Rλ )ϕH2 = 0 for each ϕ ∈ H2 and that lim|λ|→∞ (1 − Rλ )ψ H−2 = 0 for each ψ ∈ H−2 .
3 Symmetrised Time Delay In this section we collect some facts on short-range scattering theory in connection with the existence of symmetrised time delay. We always assume that the potential V satisfies the usual Agmon-type condition: Assumption 3.1 V is a multiplication operator by a real-valued function such that V defines a compact operator from H2 to Hκ for some κ > 1.
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By using duality, interpolation and the fact that V commutes with the operator Qt , t ∈ R, one shows that V also defines a bounded operator from 2(s−1) Ht2s to Ht+κ for any s ∈ [0, 1], t ∈ R. Furthermore the operator sum H := H0 + V is selfadjoint on D(H) = H2 , the wave operators W± exist and are complete, and the projections 1lr (Q) are locally H-smooth on (0, ∞) \ σpp (H) (see e.g. [7, Sec. 3] and [19, Sec. XIII.8]). Since the first two lemmas are somehow standard, we give their proofs in Appendix. Lemma 3.2 Let V satisfy Assumption 3.1 with κ > 1, and take z ∈ C \ {σ (H0 ) ∪ σ (H)}. Then the operator (H − z)−1 extends to an element of B Ht−2s , Ht2(1−s) for each s ∈ [0, 1], t ∈ R. Alternate formulations of the next lemma can be found in [7, Lemma 4.6] and [22, Lemma 3.9]. For each s 0 we define the dense set Ds := ϕ ∈ D(Qs ) | η(H0 )ϕ = ϕ for some η ∈ C0∞ ((0, ∞) \ σpp (H)) . Lemma 3.3 Let V satisfy Assumption 3.1 with κ > 2. Then one has for each ϕ ∈ Ds with s > 2 (W− − 1) e−itH0 ϕ ∈ L1 (R− , dt) (3.17) and (W+ − 1) e−itH0 ϕ ∈ L1 (R+ , dt).
(3.18)
Lemma 3.4 Let V satisfy Assumption 3.1 with κ > 4, and let ϕ ∈ Ds for some s > 2. Then there exists s > 2 such that Sϕ ∈ Ds , and the following conditions are satisfied: (W− − 1) e−itH0 ϕ ∈ L1 (R− , dt) and (W+ − 1) e−itH0 Sϕ ∈ L1 (R+ , dt). Proof The first part of the claim follows by [10, Thm. 1.4.(ii)]. Since ϕ ∈ Ds and Sϕ ∈ Ds with s, s > 2, the second part of the claim follows by Lemma 3.3. Theorem 3.5 Let satisfy Assumption 2.1. Suppose that V satisfies Assumption 3.1 with κ > 4. Let ϕ ∈ Ds with s > 2. Then the limit of τr (ϕ) as r → ∞ exists, and one has lim τr (ϕ) = − f (H0 )−1/2 ϕ, S∗ [D , S] f (H0 )−1/2 ϕ . (3.19) r→∞
Proof Due to Lemma 3.4 all the assumptions for the existence of limr→∞ τr (ϕ) are verified (see [6, Sec. 4]), and we know by Theorem [6, Thm. 4.6] that lim τr (ϕ) = −
r→∞
1 ϕ, S∗ i[Q2 , G (P)], S ϕ . 2
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It follows that 1 ϕ, S∗ [Q · (∇G )(P) + (∇G )(P) · Q, S]ϕ 2 1 f (H0 )−1/2 ϕ, S∗ f (H0 )1/2 Q · (∇G )(P) + = 2 + (∇G )(P) · Q f (H0 )1/2 , S f (H0 )−1/2 ϕ = − f (H0 )−1/2 ϕ, S∗ [D , S] f (H0 )−1/2 ϕ .
lim τr (ϕ) =
r→∞
Note that Theorem 3.5 can be proved with the function f (u) = 2u, even if is not spherical. Indeed, in such a case, point (2) of Definition 2.2 is the only assumption not satisfied by f , and a direct inspection shows that this assumption does not play any role in the proof of Theorem 3.5. Remark 3.6 Some results of the literature suggest that Theorem 3.5 may be proved under a less restrictive decay assumption on V if one modifies some of the previous definitions. Typically one proves the existence of (usual) time delay for potentials decaying more rapidly than |x|−2 (or even |x|−1 ) at infinity by using a smooth cutoff in configuration space and by considering particular potentials. The reader is referred to [2, 14, 15, 23, 24] for more information on this issue.
4 Anisotropic Lavine’s Formula In this section we prove the anisotropic Lavine’s formula (1.3). We first give a precise meaning to some commutators. Lemma 4.1 Let be a bounded open set in Rd containing 0 with boundary ∂ of class C4 . Let V satisfy Assumption 3.1 with κ > 1. Then (a) The commutator [V, D ], defined as a sesquilinear form on D(D ) ∩ H2 , extends uniquely to an element of B (H2 , H−2 ). (b) For each t ∈ R the commutator [D , e−itH ], defined as a sesquilinear form on D(D ) ∩ H2 , extends uniquely to an element [D , e−itH ]a of B (H2 , H−2 ) which satisfies [D , e−itH ]a 2 Const. |t|. H →H−2 (c) For each η ∈ C0∞ (R) the commutator [D , η(H)], defined as a sesquilinear form on D(D ) ∩ H2 , extends uniquely to an element of B (H). In particular, the operator η(H) leaves D(D ) invariant.
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Proof Point (a) follows easily from Lemma 2.4.(a) and the hypotheses on V. Given point (a) and Lemma 2.6.(b), one shows points (b) and (c) as in [18, Lemma 7.4]. If V satisfies Assumption 3.1 with κ > 2, then the result of Lemma 4.1.(a) can be improved by using Lemma 2.4.(a). Namely, there exists δ > 1/2 such form on D(D ) ∩ H2 , that the commutator [V, D ], defined as a sesquilinear 2 −2 a extends uniquely to an element [V, D ] of B H−δ , Hδ . The next Lemma is a generalisation of [9, Lemmas 2.5 & 2.7]. It is proved under the following assumption on the function f . that the Assumption 4.2 For each t ∈ R there exists ρ > 1 such operator f (H) − f (H0 ), defined on H2 , extends to an element of B Ht2 , Ht+ρ . We refer to Remark 4.4 for examples of admissible functions f . Here we only note that the operator V, f := f (H) − i[H, D ]a = f (H) − f (H0 ) − i[V, D ]a . 2 belongs to B (H−δ , Hδ−2 ) for some δ > 1/2 as soon as f satisfies Assumption 4.2.
Lemma 4.3 Let be a bounded open set in Rd containing 0, with boundary ∂ of class C4 . Let V satisfy Assumption 3.1 with κ > 2. Suppose that Assumption 4.2 is verified. Then (a) One has for each η ∈ C0∞ ((0, ∞) \ σpp (H)) and each t ∈ R the inequality (D + i)−1 e−itH η(H)(D + i)−1 Const. t−1 . (b) For each η ∈ C0∞ ((0, ∞) \ σpp (H)) the operators [D , W± η(H0 )] and [D , W±∗ η(H)], defined as sesquilinear forms on D(D ), extend uniquely to elements of B (H). In particular, the operators W± η(H0 ) and W±∗ η(H) leave D(D ) invariant. Proof (a) Since the case t = 0 is trivial, we can suppose t = 0. Let ϕ, ψ ∈ D(D ) ∩ H2 , then
D ϕ, e−itH ψ − ϕ, e−itH D ψ t = lim ds ϕ, ei(s−t)H i[H, D Rλ ] e−isH ψ λ→∞ 0
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due to Lemma 2.6.(b). By using Lemma 2.4.(b) and Lemma 4.1.(b) we get in B (H2 , H−2 ) the equalities
D , e−itH
a
= e−itH
t
ds eisH i[H, D ]a e−isH
0
= t e−itH f (H) − e−itH
t
ds eisH V, f e−isH . (4.20)
0
Take η, ϑ ∈ C0∞ ((0, ∞) \ σpp (H)) with ϑ identically one on the support of η, and let ζ ∈ C0∞ ((0, ∞) \ σpp (H)) be defined by ζ (u) := f (u)−1 ϑ(u). Then η(H) = f (H)ζ (H)η(H) and 1 ζ (H)t e−itH f (H)η(H) t t 1 = ζ (H) e−itH ds eisH V, f e−isH η(H) + t 0 a 1 + ζ (H) D , e−itH η(H). t
e−itH η(H) =
2 Since V, f belongs to B (H−δ , Hδ−2 ) for some δ > 1/2, a local Hsmoothness argument shows that the first term is bounded by Const.|t|−1 in H. Furthermore by using Lemma 4.1.(c) one shows that (D + i)−1 ζ (H)[D , e−itH ]a η(H)(D + i)−1 is bounded in H by a constant independent of t. Thus
(D + i)−1 e−itH η(H)(D + i)−1 Const. |t|−1 , and the claim follows. (b) Consider first [D , W+ η(H0 )]. Given η ∈ C0∞ ((0, ∞) \ σpp (H)) let ζ ∈ C0∞ ((0, ∞) \ σpp (H)) be identically one on the support of η. Due to Lemma 4.1.(c) one has on D(D )
D , ζ (H) eitH η(H) e−itH0 ζ (H0 ) = ζ (H) D , eitH η(H) e−itH0 ζ (H0 )+[D , ζ (H)] eitH η(H) e−itH0 ζ (H0 ) + + ζ (H) eitH η(H) e−itH0 [D , ζ (H0 )],
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and the last two operators belong to B (H) with norm uniformly bounded in t. Let ϕ, ψ ∈ D(D ). Using Lemma 2.4.(b) and Lemma 2.6.(b) one gets for the first operator the following equalities ϕ, ζ (H) D , eitH η(H) e−itH0 ζ (H0 )ψ = ϕ, ζ (H) D , eitH η(H) e−itH0 ζ (H0 )ψ + + ϕ, ζ (H) eitH [D , η(H)] e−itH0 ζ (H0 )ψ + + ϕ, ζ (H) eitH η(H) D , e−itH0 ζ (H0 )ψ t ds ϕ, ζ (H) ei(t−s)H i[H, D ]a eisH η(H) e−itH0 ζ (H0 ) + =− 0
+ ϕ, ζ (H) eitH [D , η(H)] e−itH0 ζ (H0 )ψ + + t ϕ, ζ (H) eitH η(H) e−itH0 f (H0 )ζ (H0 )ψ t = ds ϕ, ζ (H) ei(t−s)H V, f eisH η(H) e−itH0 ζ (H0 ) + 0
+ ϕ, ζ (H) eitH [D , η(H)] e−itH0 ζ (H0 )ψ − − t ϕ, η(H) eitH { f (H) − f (H0 )} e−itH0 ζ (H0 )ψ . The first two terms are bounded by c ϕ · ψ with c > 0 independent of ϕ, ψ and t (use the local H-smoothness of V, f for the first term). Furthermore, due to the local H- and H0 -smoothness of f (H) − f (H0 ) one can find a sequence tn → ∞ as n → ∞ such that lim tn ϕ, η(H) eitn H { f (H) − f (H0 )} e−itn H0 ζ (H0 )ψ = 0. n→∞
This together with the previous remarks implies that lim ϕ, [D , ζ (H) eitn H η(H) e−itn H0 ζ (H0 )]ψ c ϕ · ψ, n→∞
with c > 0 independent of ϕ, ψ and t. Thus using the intertwining relation and the identity η(H0 ) = ζ (H0 )η(H0 )ζ (H0 ) one finds that D ϕ, W+ η(H0 )ψ − ϕ, W+ η(H0 )ψ = lim ϕ, [D , ζ (H) eitn H η(H) e−itn H0 ζ (H0 )]ψ n→∞
c ϕ · ψ.
This proves the result for [D , W+ η(H0 )]. A similar proof holds for [D , W− η(H0 )]. Since the wave operators are complete, one has W±∗ η(H) = s- limt→±∞ eitH0 e−itH η(H), and an analogous proof can be given for the operators [D , W±∗ η(H)].
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Remark 4.4 In the case = B the requirements of Definition 2.2 and Assumption 4.2 are satisfied by many functions f . A natural choice is f (u) = 2u, u ∈ R, since in such a case f (H) − f (H0 ) = 2V ∈ B Ht2 , Ht+κ , t ∈ R, κ > 1. If is not spherical there are still many appropriate choices for f . For instance if γ > 0, then the function f (u) = 2(u2 + γ )−1 u3 , u ∈ R, satisfies all the desired requirements. Indeed in such a case one has on H2 the following equalities f (H) − f (H0 ) −1 H0 = 2V − 2γ (H 2 + γ )−1 H − H02 + γ −1 = 2V − 2γ (H 2 + γ )−1 V + 2γ (H 2 + γ )−1 H0 V + V H0 + V 2 H02 + γ H0 , 2 and thus f (H) − f (H0 ) also extends to an element of B Ht , Ht+κ , t ∈ R, κ > 1, due to Lemma 3.2 and the assumptions on V. The next Theorem provides a rigorous meaning to the anisotropic Lavine’s formula (1.3). Theorem 4.5 Let satisfy Assumption 2.1. Let V satisfy Assumption 3.1 with κ > 4. Suppose that Assumption 4.2 is verified. Then one has for each ϕ ∈ Ds with s > 2 ∞ ds e−isH W− f (H0 )−1/2 ϕ, V, f e−isH W− f (H0 )−1/2 ϕ 2,−2 , lim τr (ϕ) = r→∞
−∞
(4.21)
where · , · 2,−2 : H2 × H−2 → C is the anti-duality map between H2 and H−2 . Proof
, where η ∈ C0∞ ((0, ∞) \ (1) Set W(t) := eitH e−itH0 , and let ψ := η(H)ψ
σpp (H)) and ψ ∈ D(D ). We shall prove that D W(t)∗ ψ c, with c independent of t. Due to Lemma 2.4.(b) and Lemma 4.1.(c) one has D W(t)∗ ψ = e−itH0 D eitH0 e−itH η(H)(D + i)−1 ψ1 |t|{ f (H) − f (H0 )} e−itH η(H)(D + i)−1 ψ1 + + {D − t f (H)} e−itH η(H)(D + i)−1 ψ1 , (4.22) where ψ ≡ η(H)(D + i)−1 ψ1 . Let z ∈ C \ {σ (H0 ) ∪ σ (H)} and set
η(H) := (H − z)2 η(H). Then Lemmas 2.4.(a), 3.2, and 4.3.(a) imply that |t|{ f (H) − f (H0 )} e−itH η(H)(D + i)−1 ψ1 |t|{ f (H) − f (H0 )}(H − z)−2 (D + i) · · (D + i)−1 e−itH
η(H)(D + i)−1 Const.
Anisotropic Lavine’s formula and symmetrised time delay
169
Calculations similar to those of Lemma 4.3.(a) show that the second term of (4.22) is also bounded uniformly in t. (2) Let W(t) and ψ be as in point (1). Lemma 2.4.(b), Lemma 4.1.(c), and commutator calculations as in (4.20) lead to W(t)∗ ψ, D W(t)∗ ψ = ψ, eitH D e−itH ψ − t ψ, eitH f (H0 ) e−itH ψ t ds e−isH ψ, V, f e−isH ψ 2,−2 + = ψ, D ψ − 0
+ t ψ, eitH { f (H) − f (H0 )} e−itH ψ .
The local H-smoothness of f (H) − f (H0 ) implies the existence of a sequence tn → ∞ as n → ∞ such that lim tn ψ, eitn H { f (H) − f (H0 )} e−itn H ψ = 0. n→∞
This together with point (1) and the local H-smoothness of V, f implies that ∞ ∗ ∗ ψ, D ψ − ds e−isH ψ, V, f e−isH ψ 2,−2 . W+ ψ, D W+ ψ = 0
Similarly, one finds
W−∗ ψ, D W−∗ ψ = ψ, D ψ +
0 −∞
ds e−isH ψ, V, f e−isH ψ 2,−2 ,
and thus
W+∗ ψ, D W+∗ ψ − W−∗ ψ, D W−∗ ψ = −
∞
ds e−isH ψ, V, f e−isH ψ 2,−2 .
−∞
(4.23) Let ϕ ∈ Ds with s > 2. Due to Lemma 4.3.(b) the vector W− f (H0 )−1/2 ϕ is
, with η ∈ C0∞ ((0, ∞) \ σpp (H)) and ψ
∈ D(D ). Thus of the form η(H)ψ one can set ψ = W− f (H0 )−1/2 ϕ in formula (4.23). This gives Sf (H0 )−1/2 ϕ, D Sf (H0 )−1/2 ϕ − f (H0 )−1/2 ϕ, D f (H0 )−1/2 ϕ ∞ =− ds e−isH W− f (H0 )−1/2 ϕ, V, f e−isH W− f (H0 )−1/2 ϕ 2,−2 , −∞
and the claim follows by Theorem 3.5.
Remark 4.6 Symmetrised time delay and usual time delay are equal when is spherical (see formula (1.1)). Therefore in such a case formula (4.21) must reduce to the usual Lavine’s formula. This turns out to be true. Indeed
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:= if = B and f (u) = 2u, then f (H0 ) = 2H0 , V, f is equal to the virial V a 2V − i[V, D] , and formula (4.21) takes the usual form lim τr (ϕ) =
r→∞
1 2
∞ −∞
−1/2
e−isH W− H −1/2 ϕ ds e−isH W− H0 ϕ, V . 0 2,−2
In the following remark we give some insight into the meaning of formula (4.21) when is not spherical. Then we present two simple examples as an illustration. Remark 4.7 Let V satisfy Assumption 3.1 with κ > 4, and choose a set = B satisfying Assumption 2.1. In such a case the function fγ (u) := 2(u2 + γ )−1 u3 , u ∈ R, fulfills the requirements of Definition 2.2 and Assumption 4.2 (see Remark 4.4). Thus Theorem 4.5 applies, and one has for ϕ ∈ Ds with s > 2 ∞ lim τr (ϕ) = lim ds e−isH W− fγ (H0 )−1/2 ϕ, V, fγ e−isH W− fγ (H0 )−1/2 ϕ 2,−2 . r→∞
γ 0 −∞
Now fγ (H0 )ϕ converges in norm to 2H0 ϕ as γ 0, so formally one gets the identity 1 ∞ −isH −1/2 −1/2 ds e W− H0 ϕ, V e−isH W− H0 ϕ 2,−2 , (4.24) lim τr (ϕ) = r→∞ 2 −∞ where V := 2V − i[V, D ]a = 2V −
i V, F j(P) · Q j + Q j · V, F j(P) , 2 jd
and F j(P) = −(∂ j G )(P)P2 .
(4.25)
of the isotropic The pseudodifferential operator V generalises the virial V case. It furnishes a measure of the variation of the potential V along the Fig. 1 The vector field FE and the sets ∂Er
Anisotropic Lavine’s formula and symmetrised time delay
171
Fig. 2 The vector field FS and the sets ∂Sr
vector field −F , which is orthogonal to the hypersurfaces ∂r due to Remark 2.3. Therefore formula (4.24) establishes a relation between symmetrised time delay and the variation of V along −F . Moreover one can rewrite V as
+ i[V, D − D ]a V = V
+ i V, P j − F j(P) · Q j + Q j · V, P j − F j(P) , =V 2 jd
where P − F (P) is orthogonal to P due to formulas (4.25) and (2.6). Consequently there are two distinct contributions to symmetrised time delay. The
and it is due to the first one is standard; it is associated with the term V, variation of the potential V along the radial coordinate (see [11, Sec. 6] for details). The second one is new; it is associated with the term i[V, D − D ]a and it is due to the variation of V along the vector field x → x − F (x). Example 4.8 (Examples in R2 ) Set d = 2, supposethat V satisfies Assumption 3.1 with κ > 4, and let be the superellipse E := (x1 , x2 ) ∈ R2 | x41 + x42 < 1 . −1 Then one has GE (x) = − 41 ln x41 + x42 and (∂ j GE )(x) = −x3j x41 + x42 . Thus, due to Remark 4.7 the symmetrised time delay associated with E is (formally) characterised by the pseudodifferential operator VE = 2V −
i V, FE j(P) · Q j + Q j · V, FE j(P) , 2 jd
−1 where FE j(P) = P3j P2 P14 + P24 (see Fig. 1). When is equal to the star-type set −1/2 , S := (θ) eiθ ∈ R2 | θ ∈ [0, 2π ), (θ) < cos(2θ)8 + sin(2θ)8
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one has GS (x) = 72 ln(x21 + x22 ) − 12 ln (x21 − x22 )8 + 28 (x1 x2 )8 , and a direct calculation using formula (4.25) gives the vector field FS . The result is plotted in Fig. 2. Acknowledgements The author thanks the Swiss National Science Foundation and the Department of Mathematics of the University of Cergy-Pontoise for financial support.
Appendix Proof 3.2 We first prove that (H − z)−1 extends to an element of −2of Lemma B Ht , Ht for each t 0. This clearly holds for t = 0. Since (H0 − z)−1 P2 = 2 + (1 + 2z)(H0 − z)−1 one has by virtue of the second resolvent equation Qt (H − z)−1 P2 Q−t = 2 + (1 + 2z) Qt (H0 − z)−1 Q−t − − Qt (H0 − z)−1 (Q V) Q−t · Qt−1 (H − z)−1 P2 Q−t . (4.26) If we take t = 1 we find that each term on the r.h.s. of (4.26) is in B (H) due to [2, Lemmas 1 & 2]. Hence, by interpolation, Qt (H − z)−1 P2 Q−t ∈ B (H) for each t ∈ [0, 1]. Next we choose t ∈ (1, 2] and obtain, by using the preceding result and (4.26), that Qt (H − z)−1 P2 Q−t ∈ B (H) for these values of t. By iteration (take t ∈ (2, 3], then t ∈ (3, 4], etc.) one obtains that Qt (H − −t 2 −1 z)−1 P −2Q ∈ B (H) for each t > 0. Thus (H − z) extends to an element of B Ht , Ht for each t ≥ 0. A similar argument shows that (H − z)−1 also extends to an element of B Ht−2 , Ht for each t < 0. The claim follows then by using duality and interpolation. Proof of Lemma 3.3 For ϕ ∈ Ds and t ∈ R, we have (see the proof of [7, Lemma 4.6]) t dτ eiτ H V e−iτ H0 ϕ, (W− − 1) e−itH0 ϕ = −i e−itH −∞
where the integral is strongly convergent. Hence to prove (3.17) it is enough to show that −δ t dt dτ V e−iτ H0 ϕ < ∞ (4.27) −∞
−∞
for some δ > 0. If ζ := min{κ, s}, then Qζ ϕ < ∞, and V P−2 Qζ belongs to B (H) due to Assumption 3.1. Since η(H0 )ϕ = ϕ for some η ∈ C0∞ ((0, ∞) \ σpp (H)), this implies that −iτ H 0 V e ϕ Const. Q−ζ P2 η(H0 ) e−iτ H0 Q−ζ .
Anisotropic Lavine’s formula and symmetrised time delay
173
For each ε> 0, it follows from [2, Lemma 9] that there exists a constant c > 0 such that V e−iτ H0 ϕ c (1 + |τ |)−ζ +ε . Since ζ > 2, this implies (3.17). The proof of (3.18) is similar.
References 1. Amrein, W.O., Boutet de Monvel, A., Georgescu, V.: C0 -groups, commutator methods and spectral theory of N-body Hamiltonians. In: Progress in Math, vol. 135. Birkhäuser, Basel (1996) 2. Amrein, W.O., Cibils, M.B., Sinha, K.B.: Configuration space properties of the S-matrix and time delay in potential scattering. Ann. Inst. Henri Poincaré 47, 367–382 (1987) 3. Amrein, W.O., Jacquet, Ph.: Time delay for one-dimensional quantum systems with steplike potentials. Phys. Rev. A 022106 (2008) 4. Bollé, D., Osborn, T.A.: Time delay in N-body scattering. J. Math. Phys. 20, 1121–1134 (1979) 5. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin (1987) 6. Gérard, C., Tiedra de Aldecoa, R.: Time-delay and Lavine’s formula. J. Math. Phys. 48, 122101 (2007) 7. Jensen, A.: Time-delay in potential scattering theory. Comm. Math. Phys. 82, 435–456 (1981) 8. Jensen, A.: A stationary proof of Lavine’s formula for time-delay. Lett. Math. Phys. 7(2), 137–143 (1983) 9. Jensen, A.: On Lavine’s formula for time-delay. Math. Scand. 54(2), 253–261 (1984) 10. Jensen, A., Nakamura, S.: Mapping properties of wave and scattering operators for two-body Schrödinger operators. Lett. Math. Phys. 24, 295–305 (1992) 11. Lavine, R.: Commutators and local decay. In: Lavita, J.A., Marchand, J.P. (eds.) Scattering Theory in Mathematical Physics, pp. 141–156. D. Reidel, Dordrecht (1974) 12. Martin, P.A.: Scattering theory with dissipative interactions and time delay. Nuovo Cimento B 30, 217–238 (1975) 13. Martin, P.A.: Time delay in quantum scattering processes. Acta Phys. Austriaca Suppl., XXIII 157–208 (1981) 14. Mohapatra, A., Sinha, K.B., Amrein, W.O.: Configuration space properties of the scattering operator and time delay for potentials decaying like |x|−α , α > 1. Ann. Inst. H. Poincaré Phys. Théor. 57(1), 89–113 (1992) 15. Nakamura, S.: Time-delay and Lavine’s formula. Comm. Math. Phys. 109(3), 397–415 (1987) 16. Narnhofer, H.: Another definition for time delay. Phys. Rev. D 22(10), 2387–2390 (1980) 17. Narnhofer, H.: Time delay and dilation properties in scattering theory. J. Math. Phys. 25(4), 987–991 (1984) 18. Perry, P., Sigal, I.M., Simon, B.: Spectral analysis of N-body Schrödinger operators. Ann. of Math. (2) 114(3), 519–567 (1981) 19. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978) 20. Sassoli de Bianchi, M., Martin, P.A.: On the definition of time delay in scattering theory. Helv. Phys. Acta 65(8), 1119–1126 (1992) 21. Smith, F.T.: Lifetime matrix in collision theory. Phys. Rev. 118, 349–356 (1960) 22. Tiedra de Aldecoa, R.: Time delay and short-range scattering in quantum waveguides. Ann. Henri Poincaré 7(1), 105–124 (2006) 23. Wang, X.P.: Time-delay operator for a class of singular potentials. Helv. Phys. Acta 60(4), 501–509 (1987) 24. Wang, X.P.: Phase-space description of time-delay in scattering theory. Comm. Parttial Differential Equations 13(2), 223–259 (1988)
Math Phys Anal Geom (2008) 11:175–186 DOI 10.1007/s11040-008-9046-7
Estimates for Entries of Matrix Valued Functions of Infinite Matrices M. I. Gil’
Received: 5 March 2008 / Accepted: 15 July 2008 / Published online: 14 August 2008 © Springer Science + Business Media B.V. 2008
Abstract Sharp upper estimates for the absolute values of entries of matrix valued functions of infinite matrices, as well as two sided estimates for the entries of matrix valued functions of infinite M-matrices (monotone matrices) are derived. They give us bounds for the lattice norms of matrix valued functions and positivity conditions for functions of M-matrices. In addition, some results on perturbations and comparison of matrix functions are proved. Applications of the obtained estimates to the Hille-Tamarkin matrices and differential equations are also discussed. Keywords Infinite matrices · Matrix valued functions · Monotone matrices · Positivity · Hille-Tamarkin matrices · Norm estimates · Perturbations · Comparison Mathematics Subject Classifications (2000) 47A56 · 47A60 1 Introduction and Statement of the Main Result In the book [8], I.M. Gel’fand and G.E. Shilov have established an estimate for the norm of a regular matrix valued function in connection with their investigations of partial differential equations. However that estimate is not sharp, it is not attained for any matrix. The problem of obtaining a precise estimate for the norm of a matrix function has been repeatedly discussed in
This research was supported by the Kamea fund of the Israel M. I. Gil’ (B) Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel e-mail: [email protected]
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the literature, cf. [5]. In the paper [10] (see also [13]) the author has derived a precise estimate for the Euclidean norm which is attained in the case of normal matrices. But that estimate requires bounds for the eigenvalues. In this paper we derive sharp estimates for the absolute values of entries of matrix valued functions of infinite matrices. In addition, two sided estimates for the entries of functions of infinite monotone matrices are derived. These estimates give us bounds for the lattice norms of matrix valued functions, and enable us to investigate perturbations of matrix valued functions and compare them. Besides, bounds for the eigenvalues are not required. Applications of the obtained estimates to differential equations are also discussed. Our results supplement the very interesting recent investigations of matrix valued functions [7, 17, 24], M-matrices [1, 2, 23], infinite matrices and their applications [3, 11, 16, 19, 20, 25]. Of course we cannot survey the whole subject here and refer the reader to the above pointed papers and references given therein. Everywhere below, X is a complex Banach space of scalar sequences h = {hk } with a norm .. In particular, X = l p ( p 1) with the norm 1/ p ∞ p hl p = |hk | (1 p < ∞) and hl∞ = sup |hk |. k
k=1
The unit operator in X is denoted by I. Let σ (A) be the spectrum of a linear operator A and Rz (A) = (A − zI)−1 (z ∈ σ (A)) be the resolvent of A; rs (A) denotes the spectral radius of A. Everywhere below, A = (a jk )∞ j,k=1 is a matrix representing a bounded linear operator in the standard basis of X whose diagonal part is D = diag [a11 , a22 , ... ] and off diagonal is V := A − D. That is, the entries v jk of V are v jk = a jk ( j = k) and v jj = 0 ( j, k = 1, 2, ...). Clearly, rs (D) = sup |a jj|. j=1,2,....
Denote by co(D) the closed convex hull of the diagonal entries a11 , a22 , ... and put (A) := {z ∈ C : |z| rs (D) + V}. Clearly co(D) ⊂ (A). In the sequel f (λ) is a scalar function holomorphic on a neighborhood of (A). The matrix valued function f (A) is defined by 1 f (A) = − f (λ)Rλ (A)dλ, (1.1) 2πi where ⊂ (A) is a closed contour surrounding σ (A). We put |A| = (|a jk |)∞ j,l=1 , i.e. |A| is the matrix whose entries are absolute values of A in the standard basis. We also write C 0 if all the entries of a matrix C are nonnegative. If C and B are two matrices, then we write C B if C − B 0. The same sense have the symbols |h|, h 0 and h g for sequences h, g ∈ X.
Estimates for entries of matrix valued functions of infinite matrices
177
Moreover, it is assumed that any considered number series converges and any operator series strongly converges. Now we are in a position to formulate our main result. Theorem 1.1 With the notation γk (A) :=
| f (k) (z)| (k = 0, 1, 2, ...), k! z∈co (D) sup
the inequality | f (A)|
∞
γk |V|k
(1.2)
k=0
holds. This theorem is proved in the next section. It generalizes the main result from [14]. In the sequel the norm in X is a lattice norm. That is, f h whenever | f | |h| for f, h ∈ X, cf. [18, p. 6]. By A the operator norm of A is denoted: A := supx∈X Ax/x. So h = |h| and A |A|. Theorem 1.1 implies Corollary 1.2 The inequality f (A)
∞
γk |V|k
(1.3)
k=0
is valid. Theorem 1.1 and Corollary 1.2 are sharp: inequalities (1.2) and (1.3) become equalities, provided A is diagonal: V = 0 and the set of the diagonal entries {a11 , a22 , ...} is convex. For instance, let X = l p for some finite p > 1, and V be a Hille-Tamarkin matrix, cf. [22]. Namely, ⎛ ⎡ ⎤ p/q ⎞1/ p ∞ ∞ ⎣ N p (V) := ⎝ |a jk |q ⎦ ⎠ < ∞ (1.4) j=1
k=1, k= j
with 1/ p + 1/q = 1. So under (1.4), A = (a jk ) represents a linear operator in l p which is bounded, provided D is bounded. As it is well-known and |V| N p (V), cf. [22]. Now (1.3) yields Corollary 1.3 Let X = l p and condition (1.4) hold. Then f (A)l p
∞ k=0
γk N kp (V).
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Note that in the paper [12], estimates for the norm of the powers of HilleTamarkin quasinilpotent matrices have been established.
2 Proof of Theorem 1.1 First assume that A is a finite matrix. By the equality A = D + V we get Rλ (A) ≡ (A − Iλ)−1 = (D + V − λI)−1 = (I + Rλ (D)V)−1 Rλ (D), provided the norm of Rλ (D)V is less than one. But Rλ (D)V
V V <1 infk |λ − akk | |λ| − rs (D)
for any λ with |λ| > r A := V + rs (D). Thus, ∞ 1 f (A) = − f (λ)Rλ (A)dλ = Ck (r = r˜ A + , > 0), 2πi |λ|=r
(2.1)
k=0
where Ck = (−1)
k+1
1 2πi
|λ|=r
f (λ)(Rλ (D)V)k Rλ (D)dλ.
Since D is a diagonal matrix with respect to basis {ek }, we can write out Rλ (D) =
n j=1
Qj (λ j = a jj), λj − λ
where Qk = (., ek )ek . We thus have Ck =
n
Q j1 V
n
j1 =1
j2 =1
Here I j1 ... jk+1 =
Q j2 V . . . V
(−1)k+1 2πi
n
Q jk+1 I j1 j2 ... jk+1 .
jk =1
|λ|=r
f (λ)dλ . (λ j1 − λ) . . . (λ jk+1 − λ)
Lemma 1.5.1 from [13] gives us the inequalities |I j1 ... jk+1 | γk ( j1 , j2 , ..., jk+1 = 1, ..., n). Hence, by (2.2) |Ck | γk
n
Q j1 |V|
j1 =1
n
Q j2 |V| . . . |V|
j2 =1
n
Q jk+1 .
jk =1
But n j1 =1
Q j1 |V|
n j2 =1
Q j2 |V| . . . |V|
n jk =1
Q jk+1 = |V|k .
(2.2)
Estimates for entries of matrix valued functions of infinite matrices
179
Thus Ck γk |V|k . Now (2.1) implies | f (A)|
∞
|Ck |
k=0
∞
γk |V|k .
k=0
So in the finite dimensional case the theorem is proved. Now let A be infinite dimensional and Pn the projection onto subspace generated by the first n elements of the standard basis. Then the finite dimensional matrices An = Pn APn strongly converge to A. Let Dn = Pn D and Vn = Pn V Pn be the diagonal and off-diagonal parts of An , respectively. Then as it is above proved, the required relations hold with A = An . But f (An ) → f (A) in the strong topology [6]. So each entry of f (An ) converges to the corresponding entry of f (A). This proves the result.
3 Functions of M-Matrices A real matrix A in X is said to be an M-matrix (a monotone matrix) if its off diagonal part V is nonnegative, cf. [4]. Put a=
inf
j=1,2,...
a jj,
b = sup
a jj.
j=1,2,...
In this section A is an M-matrix, f (λ) is holomorphic on a neighborhood of (A), as above, and, in addition, it is real on [a, b ]. Theorem 3.1 With the notations αk := inf
a x b
f (k) (x) , k!
βk = sup
a x b
f (k) (x) (k = 0, 1, 2, ...), k!
the inequalities f (A)
∞
αk V k ,
(3.1)
k=0
and f (A)
∞
βk V k
(3.2)
k=0
are valid. In particular, if αk 0 (k = 0, 1, 2, ...), then f (A) 0. Proof First let A be n-dimensional. Again use relations (2.1), (2.2). Lemma 1.5.2 from [13] gives us the equality I j1 ... jk+1 =
f (k) (θ) (a θ b ). k!
(3.3)
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So αk I j1 ... jk+1 βk . Since V 0, Q j V Qk = a jk (., ek )e j, and n
Q j1 V
j1 =1
n
Q j2 V . . . V
j2 =1
n
Q jk = V k ,
jk =1
we have Ck αk
n
Q j1 V
j1 =1
n
Q j2 V . . . V
j2 =1
n
Q jk = αk V k .
jk =1
Similarly, Ck βk V k . Hence (2.1) proves the theorem in the finite dimensional case. Now let A be infinite dimensional and Pn the projection onto subspace generated by the first n elements of the standard basis. Then the finite dimensional matrices An = Pn APn strongly converge to A. Let Dn = Pn D and Vn = Pn V Pn be the diagonal and off-diagonal parts of An , respectively. Now taking into account the above proved result and that f (An ) → f (A) in the strong topology [6] we can assert that each entry of f (An ) converges to the corresponding entry of f (A). This proves the result.
Recall that the norm in X is assumed to be lattice. Theorem 3.1 yields the following result. Corollary 3.2 The inequalities | f (A)|
∞
νk V k (νk := max{|αk |, |βk |})
(3.4)
k=0
and f (A)
∞
νk V k
k=0
are true. Denote by R+ (l 1 ) the cone of vectors from l 1 , whose coordinates in the standard basis are nonnegative. Thanks to (3.1) we get Corollary 3.3 Let X = l 1 and αk 0, k = 0, 1, 2, . . . . Then for any h ∈ R+ (l 1 ) we have the inequality f (A)hl1
∞ k=0
Furthermore, inequality (3.4) yields
αk V k hl1 .
Estimates for entries of matrix valued functions of infinite matrices
181
Corollary 3.4 Let X = l p (1 < p < ∞) and V satisfy condition (1.4). Then f (A)l p
∞
νk N kp (V).
k=0
4 Perturbations of Entire Functions of Hille-Tamarkin Matrices We need the following result. Lemma 4.1 Let Z and Y be complex normed spaces with norms . Z and .Y , respectively, and h a Y-valued function defined on Z . Assume that h(C + λB) (λ ∈ C) is an entire function for all C, B ∈ Z . That is, for any φ from the space adjoint to Y, the functional < φ, h(C + λB) > is an entire function. In addition, let there be a monotone non-decreasing function G : [0, ∞) → [0, ∞), such that h(C)Y G(C Z ) (C ∈ Z ). Then h(C)−h(B)Y C− B Z G(1+1/2C+ B Z +1/2C − B Z ) (C, B ∈ Z ). For the proof see [15]. Let A = (a jk ) and A˜ = (˜a jk ) be matrices representing bounded operators in a Banach space X with a lattice norm .. Thanks to Corollary 1.2, for any entire f , f (A)
∞
γk (A)|V|k
k=0
∞
γk (A)|A|k .
(4.1)
k=0
Now let X = l p for some finite p > 1, and A be a Hille-Tamarkin matrix: ⎛ ∞ p/q ⎞1/ p ∞ ⎠ < ∞. |a jk |q N p (A) = ⎝ j=1
k=1
Then |A|l p N p (A) and γk (A)
| f (k) (z)| (k = 0, 1, 2, ...). k! |z| N p (A) sup
The previous lemma and (4.1) imply Theorem 4.2 Let A and A˜ be Hille-Tamarkin matrices in l p , 1 < p < ∞. Assume that f (λ) (λ ∈ C) is an entire function. Then ˜ l p N p (A− A) ˜ f (A)− f ( A)
∞ k=0
k ˜ ˜ ˜ ηk (A, A)(1+ N p (A+ A)/2+ N p (A− A)/2)
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M. I. Gil’
where ˜ := ηk (A, A)
1 sup | f (k) (z)|. k! |z|1+N p (A+ A)/2+N ˜ ˜ p (A− A)/2
5 Comparison of Functions of M-Matrices ˜ + V, ˜ where In this section A = (a jk ) and A˜ = (˜a jk ) are M-matrices. So A˜ = D ˜ respectively. Recall that ˜ and V˜ are the diagonal and off-diagonal parts of A, D D and V are the diagonal and off-diagonal parts of A, respectively. Lemma 5.1 Let a akk = a˜ kk b but a jk a˜ jk ( j = k, j, k = 1, 2, ...).
(5.1)
In addition, let f be holomorphic on a neighborhood of (A) and positive on ˜ 0. Moreover, [a, b ]: αk 0, k = 1, 2, . . . . Then f (A) f ( A) ∞
˜ αk (V k − V˜ k ) f (A) − f ( A)
k=0
∞
βk (V k − V˜ k ).
k=0
Proof First, let A, A˜ be n-dimensional (n < ∞). We have by (2.1), ˜ = f (A) − f ( A)
∞
C˜ k ,
k=0
where C˜ k =
n
(Q j1 V Q j2 V . . . V Q jk+1 − Q j1 V˜ Q j2 V˜ . . . V˜ Q jk+1 )I j1 j2 ... jk+1
j1 , j2 ,... jk+1 =1
Under assumption (5.1), according to (3.3) we have I j1 j2 ... jk+1 0. Thus C˜ k 0. ˜ Moreover, since V V, C˜ k βk
n
˜ j2 V˜ . . . VQ ˜ jk+1 ) = βk (V k − V˜ k ), (Q j1 VQ j2 V . . . VQ jk+1 − Q j1 VQ
j1 , j2 ,... jk+1 =1
and C˜ k αk (V k − V˜ k ). So in the finite dimensional case, the lemma is proved. Taking into account that any bounded operator is a strong limit of finite dimensional operators, we arrive at the required result.
Furthermore, for real constants a˜ and b, let b akk a˜ kk a˜ but a jk = a˜ jk ( j = k, j, k = 1, 2, ...). ˜ + V and Put rˆ := max{rs (D), rs ( D)} dk := max a jj − a˜ jj. 1 jk
(5.2)
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183
Lemma 5.2 Under conditions (5.2), let f be holomorphic on a neighborhood of {z ∈ C : |z| rˆ} and positive on [˜a, b ]: f (k+1) (x) 0 (k = 0, 1, 2, ...). k!
αˆ k := inf
a˜ xb
˜ 0 and Then f (A) f ( A) ∞
˜ αˆ k dk V k f (A) − f ( A)
k=0
∞
dk βˆk V k
k=0
where βˆk := sup
a˜ xb
f (k+1) (x) (k = 0, 1, 2, ...). k!
Proof First, let A, A˜ be n-dimensional. We have by (2.1), ˜ = f (A) − f ( A)
∞
Tk ,
k=0
where Tk :=
n
Q j1 V Q j2 V . . . V Q jk+1 (I j1 j2 ... jk+1 − I˜ j1 j2 ... jk+1 ).
j1 , j2 ,... jk+1 =1
Here (−1)k I˜ j1 , j2 ,..., jk+1 = 2πi
|λ|=r
(˜a j1 j1
f (λ)dλ (r > rˆ). − λ) . . . (˜a jk+1 jk+1 − λ)
As it is well-known [9, Section I.4.3], I j1 , j2 ,..., jk+1 is the k-order divided difference of f in the points a j1 j1 , . . . , a˜ jk+1 jk+1 . By the Herimit integral representation [21, p. 4], we have 1 t1 tk ... f (k) [(1 − t1 )a j1 j1 + (t1 − t2 )a j2 j2 + ... I j1 , j2 ,..., jk+1 = 0
0
0
+ a jk+1 jk+1 tk+1 ]dtk+1 ... dt1 .
(5.3)
The same representation has I˜ j1 , j2 ,..., jk+1 with a˜ kk instead akk . By (5.3) we obtain I j1 , j2 ,..., jk+1 I˜ j1 , j2 ,..., jk+1 , provided f n+1 (x) 0, x ∈ [˜a, b ]. So Tk 0 and thus f (A) f˜(A). Furthermore, for real points x1 , ..., xk , y1 , ..., yk with x j y j, j k, we can write out f (k) [(1 − t1 )x1 + (t1 − t2 )x2 + ... + xk tk ] − f (k) [(1 − t1 )y1 + (t1 − t2 )y2 + ... ... + yk tk ] = f (k+1) (θ) (min y j θ max x j), jk
jk
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where = (1 − t1 )(x1 − y1 ) + (t1 − t2 )(x2 − y2 ) + ... + (xk − yk )tk dk (t j 1). Hence, taking into account that 1 t1 0
...
0
tk−1
dtk ... dt1 =
0
1 k!
we get αˆ k dk I j1 , j2 ,..., jk+1 − I˜ j1 , j2 ,..., jk+1 dk βˆk . This implies the required result in the finite dimensional case. Reducing the obtained result to the infinite dimensional case as in the previous lemma, we get the required result.
6 Examples Example 6.1 Let f (A) = e At (t 0). With the notations of Section 1 we can write out f (k) (λ) = tk eλt ; γk =
tk α(D)t e k!
where α(D) = maxk Re akk . So by Theorem 1.1, |e At | eα(D)t
∞ k t |V|k = e(α(D)I+|V|)t (t 0). k! k=0
Now let A be an M-matrix. Then with the notations of Section 3, αk =
tk at e , k!
βk =
tk b t e , k!
k = 0, 1, 2, ....
So by Theorem 3.1, ∞ k ∞ k t k t k V e At ebt V (t 0). k! k! k=0
k=0
Or e(aI+V)t e At e(bI+V)t (t 0). Example 6.2 Let f (A) = sin (At) (t 0). Then
f (2k) (λ) = t2k (−1)k sin (λt); f (2k+1) (λ) = t2k+1 (−1)k cos (λt) (k = 0, 1, 2, ...).
Estimates for entries of matrix valued functions of infinite matrices
Let the diagonal matrix D be real. Then γk |sin (At)|
1 k t . k!
185
So by Theorem 1.1,
∞ 1 k k t |V| = et|V| . k!
(6.1)
k=0
Consider the second order nonlinear differential equation d2 x + A2 x(t) = F(x(t)) (t > 0) dt2
(6.2)
dx(0) = x(0) = 0, dt
(6.3)
where A is a real matrix, F : Rn → Rn is a continuous function, satisfying F(h) v + qh (v, q = const; h ∈ X) with some lattice norm. Problem (6.2), (6.3) is equivalent to the following equation: t x(t) = sin A(t − s)F(x(s))ds. 0
So
x(t)
t
sin A(t − s)(qx(s) + v)ds
0
with an arbitrary ideal norm. Put y(t) = x(t). Then by (6.1), t y(t) v f (t) + q e|V|(t−s) y(s)ds 0
where
f (t) =
t
e|V|s ds.
0
Hence taking into account that f monotonically increases, we get by the Gronwall inequality, y(t) v f (t)exp [q f (t)]. Such estimates are important, in particular, in the theory of oscillations, cf. [5].
References 1. Ameur, Y., Kaijser, S., Silvestrov, S.: Interpolation classes and matrix monotone functions. J. Operator Theory 57(2), 409–427 (2007) 2. Bapat, R.B., Catral, M., Neumann, M.: On functions that preserve M-matrices and inverse M-matrices. Linear and Multilinear Algebra 53(3), 193–201 (2005) 3. Candan, M., Solak, I.: On some difference sequence spaces generated by infinite matrices. Int. J. Pure Appl. Math. 25(1), 79–85 (2005) 4. Collatz, L.: Functional Analysis and Numerical Mathematics. Academic, New York (1966)
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5. Daleckii, Y.L., Krein, M.G.: Stability of solutions of differential equations in Banach space. American Mathematical Society, Providence (1974) 6. Dunford, N., Schwartz, J.T.: Linear Operators, part I. Interscience, New York (1966) 7. Fritzsche, B., Kirstein, B., Lasarow, A.: Orthogonal rational matrix-valued functions on the unit circle: recurrence relations and a Favard-type theorem. Math. Nachr. 279(5–6), 513–542 (2006) 8. Gel’fand, I.M., Shilov, G.E.: Some Questions of Theory of Differential Equations. Nauka, Moscow (in Russian) (1958) 9. Gel’fond, A.O.: Calculations of Finite Differences. Nauka, Moscow (in Russian) (1967) 10. Gil’, M.I.: Estimates for norm of matrix-valued functions. Linear and Multilinear Algebra 35, 65–73 (1993) 11. Gil’, M.I.: Spectrum localization of infinite matrices. Math. Phys. Anal. Geom. 4(4), 379–394 (2001) 12. Gil’, M.I.: Invertibility and spectrum of Hille-Tamarkin matrices. Math. Nachr. 244, 1–11 (2002) 13. Gil’, M.I.: Operator functions and localization of spectra. In: Lectures Notes In Mathematics vol. 1830. Springer, Berlin (2003) 14. Gil’, M.I.: Estimates for absolute values of matrix functions. Electron. J. Linear Algebra 16, 444–450 (2007) 15. Gil’, M.I.: Inequalities of the Carleman type for Schatten-von Neumann operators. AsianEuropean J. Math. 1(2), 1–11 (2008) 16. Golinskii, L.: On the spectra of infinite Hessenberg and Jacobi matrices. Mat. Fiz. Anal. Geom. 7(3), 284–298 (2000) 17. Lasarow, A.: Dual Szego˝ pairs of sequences of rational matrix-valued functions. Int. J. Math. Math. Sci. 2006(5), 37 (2006) 18. Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991) 19. Mittal, M.L., Rhoades, B.E., Mishra, V.N., Singh, U.: Using infinite matrices to approximate functions of class Lip(α, p) using trigonometric polynomials. J. Math. Anal. Appl. 326(1), 667– 676 (2007) 20. Nagar, D.K., Tamayo-Acevedo, A.C.: Integrals involving functions of Hermitian matrices. Far East J. Appl. Math. 27(3), 461–471 (2007) 21. Ostrowski, A.M.: Solution of Equations in Euclidean and Banach Spaces. Academic, New York (1973) 22. Pietsch, A.: Eigenvalues and s-Numbers. Cambridge University Press, Cambridge (1987) 23. Singh, M., Vasudeva, H.L.: Monotone matrix functions of two variables. Linear Algebra Appl. 328(1–3), 131–152 (2001) 24. Werpachowski, R.: On the approximation of real powers of sparse, infinite, bounded and Hermitian matrices. Linear Algebra Appl. 428(1), 316–323 (2008) 25. Zhao, X., Wang, T.: The algebraic properties of a type of infinite lower triangular matrices related to derivatives. J. Math. Res. Exposition 22(4), 549–554 (2002)
Math Phys Anal Geom (2008) 11:187–364 DOI 10.1007/s11040-008-9042-y
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal with Respect to Varying Exponential Weights: I K. T.-R. McLaughlin · A. H. Vartanian · X. Zhou
Received: 19 January 2008 / Accepted: 10 April 2008 / Published online: 18 October 2008 © Springer Science + Business Media B.V. 2008
Abstract Orthogonal rational functions are characterized in terms of a family of matrix Riemann–Hilbert problems on R, and a related family of energy minimisation problems is presented. Existence, uniqueness, and regularity properties of the equilibrium measures which solve the energy minimisation problems are established. These measures are used to derive a family of ‘model’ matrix Riemann–Hilbert problems which are amenable to asymptotic analysis via the Deift–Zhou non-linear steepest-descent method. Keywords Asymptotics · Equilibrium measures · Orthogonal rational functions · Riemann–Hilbert problems · Variational problems Mathematics Subject Classifications (2000) Primary 42C05; Secondary 30E20 · 30E25 · 30C15
K. T.-R. McLaughlin Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., P. O. Box 210089, Tucson, AZ 85721-0089, USA e-mail: [email protected] A. H. Vartanian (B) Department of Mathematics, College of Charleston, 66 George Street, Charleston, SC 29424, USA e-mail: [email protected] X. Zhou Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, USA e-mail: [email protected]
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1 Introduction, Background, and Summary of Results 1.1 Introduction This manuscript considers a number of questions related to general orthogonal rational functions (ORFs). These may be thought of as a natural generalisation of orthogonal polynomials. Just as orthogonal polynomials are obtained via the Gram-Schmidt orthogonalisation procedure (with respect to a given measure) applied to the sequence {1, z, z2 , . . . }, ORFs may be obtained via the GramSchmidt procedure, but applied to a more general, pre-determined sequence of simple rational functions. A basic example follows. Using the measure defined on R, −2
−2
−2
dψ(z) = e−z e−z e−(z−1) e−(z−2) dz, 2
one may apply the Gram-Schmidt procedure to a sequence of rational functions with poles at 0, 1, and 2. Clearly, the order of this initial sequence of rational functions matters. This infinite sequence of rational functions will be described by first specifying a finite pole sequence, {0, 0, 1, 0, 1, 2}, which yields the following seven rational functions: {1, z−1 , z−2 , (z−1)−1 , z−3 , (z−1)−2 , (z−2)−1 }. The infinite sequence of rational functions is specified by repeating this pole sequence, and augmenting the order of the pole at 0, 1, and 2 every time that pole is encountered. Thus, after the first seven rational functions specified above, the next six members of the infinite sequence are {z−4 , z−5 , (z−1)−3 , z−6 , (z−1)−4 , (z−2)−2 }, and this continues, with the rational functions appearing in six-tuples. The result of applying the Gram-Schmidt procedure to this sequence of functions, using the aforementioned measure, is a sequence of ORFs. More precisely, the nth term in the new sequence is in the linear span of the first n members of the original sequence of simple rational functions, and is orthogonal to all ‘previous’ members of the simple sequence. The general definition of a sequence of ORFs is quite cumbersome. The reason is that one must specify the pole sequence, and if a pole should happen to repeat, then the associated ‘simple rational function’ must be linearly independent from all previous members of the sequence; so typically, the order of the pole increases with each occurrence of that pole. Although one might merely describe the sequence of rational functions abstractly, it is useful, and necessary for a Riemann–Hilbert characterization, to explicitly annotate this level of detail. This manuscript only deals with the case of a finite sequence of poles, which must necessarily repeat: the more general situation in which the pole sequence is infinite, is not considered. In Subsection 1.2, then, the reader will find a description of the pole sequence that requires effort to absorb. It is important to note that the Riemann–Hilbert problems described here,
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which characterize the ORFs, depend on this level of detail regarding the pole sequence. The uninitiated reader might wonder why one would consider such a generalisation. Although such questions may be delegated to a matter of taste, we describe here one application [1], and refer the reader to the monograph [2] for a complete list of applications. Suppose one would like to find a multi-point rational approximant to a function which is given as the Stieltjes transform of the measure defined above: dψ(ξ ) Fψ (z) = . R z−ξ Now, in a vicinity of each of the points z = 0, z = 1, and z = 2 the function Fψ (z) possesses a complete asymptotic description. The (n−1, n) 3-point Padé approximation to Fψ (z) is a rational function of the form U n (z)/Vn (z), with deg(U n ) n−1 and deg(Vn ) = n, which matches each of the asymptotic expansions to specified degrees at each of the points 0, 1, and 2. A challenging question is to describe the asymptotic behaviour of the sequence of Padé approximants as n → ∞. The amazing connection is this: the approximants themselves, as well as the error in approximation, are described explicitly in terms of the ORFs! For a precise description of these (and more) connections, we refer the reader to [1]. The goal of describing the asymptotic behaviour of the sequence of Padé approximants may be achieved if one has a complete asymptotic description (as n → ∞) of the ORFs. Toward the ultimate goal of a complete asymptotic description of ORFs with respect to varying exponential weights, the purpose of this manuscript is to lay the foundation: we describe families of Riemann–Hilbert problems which characterize the ORFs. In addition, we derive variational problems which are central to the subsequent asymptotic analysis of the Riemann– Hilbert problems. Since the variational problems contain external fields with singular points, we also establish existence, uniqueness, and regularity of the associated equilibrium measures. Historically, the analysis of ORFs played a major rôle in a variety of socalled moment problems, in which one is given a list of p pole locations, and p infinite sequences of real numbers, and one asks for the existence of a measure whose moments relative to each pole are prescribed by one of the infinite sequences of real numbers. Two examples are as follows: (i) the extended Hamburger moment problem (EHMP) [3, 4]: given p distinct (fixed) real numbers a˜ 1 , a˜ 2 , . . . , a˜ p and p sequences of finite real numbers
EH(i)
ck
k∈N
, i = 1, 2, . . . , p, find necessary and sufficient condi-
on (−∞, +∞) tions for the existence of a distribution function1 μEH MP
1A
real-valued, bounded, non-decreasing function with infinitely many points of increase on its domain of definition.
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+∞ +∞ EH ˜ i )−k dμEH such that −∞ dμEH (τ ) = 1 and c (i) (τ ), k ∈ N, i = MP MP k = −∞ (τ − a 1, 2, . . . , p; (ii) the extended Stieltjes moment problem (ESMP) [5]: given q distinct (fixed) real numbers aˆ 1 , aˆ 2 , . . . , aˆ q ordered by size (e.g., aˆ 1 < aˆ 2 < · · · < aˆ q ), agree to call the real interval [c, d] a Stieltjes interval for the finite point set {ˆa1 , aˆ 2 , . . . , aˆ q } if (c,d) ∩ {ˆa1 , aˆ 2 , . . . , aˆ q } = ∅. Given q sequences of finite real numbers c (r) j ES
j∈N
, r = 1, 2, . . . , q, and a finite real
ES
number c 0 , find necessary and sufficient conditions for the existence of a distribution function μES , with all its points of increase on a given Stieltjes MP d d ES ES ˆ r )− j dμES interval [c, d], such that c 0 = c dμES (τ ) and c (r) (τ ), MP MP j = c (τ − a j∈ N, r = 1, 2, . . . , q. Two other, related problems of analysis are the Pick–Nevanlinna problem and the frequency analysis problem, which are now described: (iii) consider a sequence of complex points {zn }n∈N such that zn coalesce into a finite number, p, say, of distinct real points aˇ 1 , aˇ 2 , . . . , aˇ p according to the prescription z pq+1 = aˇ 1 , z pq+2 = aˇ 2 , . . . , z pq+ p = aˇ p , q ∈ Z+ 0 := N ∪ {0}. The corresponding Pick–Nevanlinna problem [6] can be formulated (m) thus: given the p sequences of numbers γ˜l , m = 1, 2, . . . , p, + l∈Z0
find a Nevanlinna function2 X(z) which has the asymptotic expansions X(z) = Rm,δ z→ˇam l∈Z+0 γ˜l(m) (z− aˇ m )l , m = 1, 2, . . . , p, where Rm,δ := {z ∈ C; δ < Arg(z− aˇ m ) < π −δ}, δ > 0. (As shown in [6], this Pick–Nevanlinna problem is related to the EHMP and certain (weak) multi-point Padé approximation problems.); (iv) the so-called frequency analysis problem (see, e.g., the review article [7]) consists of determining the unknown frequencies ω j, j= 1, 2, . . . , J, N−1 J ∈ N, from a discrete time signal x N (m) = Jj=−J α jeimω j of obm=0 served values, where (N ) N is the ‘sample size’, α0 0, 0 = α− j = α j ∈ C, 0 =: ω0 < ω1 < ω2 < · · · < ω J < π , and ω− j = −ω j ∈ R, j= 1, 2, . . . , J. The frequency analysis problem has been dealt with by exploiting the fact that, under various conditions, certain roots of the Szegö polynomials [8] on the unit circle (T := {z ∈ C; |z| = 1}) converge, as N → ∞, to the ‘frequency points’ eiω j and 1, j= ±1, ±2, . . . , ±J, and the ‘remaining— uninteresting—roots’ are bounded away from T as N → ∞. Recently, however, the question of the generalisation, or extension, of this theory, where, in lieu of Szegö polynomials, certain rational functions replace polynomials, has been raised [9–11].
function X(z) which is analytic for z ∈ C+ := {z ∈ C; Im(z) > 0} with Im(X(z)) 0 is called a Nevanlinna function.
2A
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As shown in [1, 3–6, 9–11], the key ingredient subsumed in the analysis of the multi-point rational approximation problem and the above-mentioned problems (i)–(iv) (by no means an exhaustive list!) is to consider, in lieu of (Szegö) orthogonal polynomials, suitably orthogonalised rational functions with no poles in the extended complex plane (C := C ∪ {∞}) outside of a prescribed (fixed) pole set, called, generically, ORFs. Historically, and to the best of the authors’ knowledge as at the time of the presents, M. M. Djrbashian instigated the study of ORFs; in particular, the investigation of ORFs on T (see, e.g., [12–16]). Since then, a monumental, systematic and comprehensive study of ORFs has been undertaken by A. Bultheel, P. González-Vera, E. Hendriksen, and O. Njåstad (see the recent monograph [2] and the plethora of references therein; see, also, [17–23]). There has also been concomitant recent progress in the matrix generalisation of the scalar ORF theory on T (see, e.g., [24, 25]). ORFs have applications, and potential applications, to generalised problems in moment theory (see, e.g., [26, 27]), interpolation theory and numerical analysis (see, e.g., [28–35]), control theory [36], multi-point Padé-type approximants [37–39], the theory of Cauchy and Cauchy-type integrals and their derivatives with meromorphic density (see, e.g., [40–47]), an extended Toda lattice [48], uniform approximation of sgn(x) (see, e.g., [49]), and spectral theory (see, e.g., [50]). Thus far, the bulk of the analyses, asymptotic or otherwise, of ORFs on T [2, 51, 52] assume that the poles of the ORFs lie in the interior of the open unit disc (O := {z ∈ C; |z| < 1}) (see, however, Chapter 11 of [2], and [17, 19]). For the scarce number of analyses, asymptotic or otherwise, of ORFs on the extended real line (R := R ∪ {±∞}), the poles of the ORFs are assumed to be real and disjoint from the support of the orthogonality measure [18, 53], to lie in C \ R, or, due to so-called ‘special technical considerations’, some ‘forbidden value’ of a pole must be excluded from the analyses. Since the generalisation of orthogonal polynomials on T and on R requires the poles to be in the exterior of the closed unit disc (C \ (O ∪ T)) or in R, respectively, the lacunae described above must be addressed in order to have an exhaustive understanding of the general ORF theory. For the case of ORFs on T with poles in C \ (O ∪ T), and for the case of ORFs on R with poles in the open lower half-plane (C− := {z ∈ C; Im(z) < 0}), Velázquez [50] has recently presented an operator-theoretic approach to address non-asymptotic aspects of these issues. Most poignantly, for the case of ORFs on R, there is a dearth of analysis. In addition, a technically challenging component of the analysis, asymptotic or otherwise, is where the poles of the ORFs lie on the support of the orthogonality measure. In fact, the present paper, which is Part I (the ‘finitepole case’, FPC) of a two-part study of ORFs on R, considers the case where the finite-in-number and not necessarily distinct real poles of the ORFs are bounded and lie in R. Part II (the ‘mixed-pole case’, MPC) considers the more general case where at least one, but not all, of the finite-in-number and not necessarily distinct poles of the ORFs is the point at infinity. Hereafter, only the FPC ORFs will be considered.
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1.2 Background and Notation There is a plethora of definitions which must be presented in this Subsection 1.2 in order to completely describe the FPC ORFs. Since these definitions may appear quite daunting, we have chosen to intersperse, at various points, a basic example of an FPC ORF real pole sequence. This is intended to parse the definitions in a digestible manner, and to elucidate the ideas presented. One starts with the real pole sequence denoted by α1 , α2 , . . . , α K . The sequence is assumed to be bounded; but, the individual pole locations are not necessarily distinct. The example which we will return to frequently in this Subsection 1.2 is the following real pole sequence of ‘length’ K = 6: {α1 , α2 , α3 , α4 , α5 , α6 } = {0, 0, 1, 0, 1, 2}. Notational Remark 1.2.1 The reader may mistakenly interpret the notation {α1 , α2 , . . . , α K } to be standard notation from set theory. In this work, however, such notation shall be thought of as a lexicographic listing of elements which may, therefore, repeat. The next ingredient is the measure of orthogonality, which will be taken to be a probability measure μ of the form dμ(z) = w (z) dz,
(1)
with varying exponential weight function w (z) = exp(−N V(z)),
N ∈ N,
(2)
where the external field V : R \ {α1 , α2 , . . . , α K } → R satisfies the following conditions: V(z) is real analytic on R \ {α1 , α2 , . . . , α K }; V(x) lim = +∞; |x|→+∞ ln(x2 +1) V(x) lim = +∞, k = 1, 2, . . . , K. x→αk ln((x−αk )−2 +1)
(3) (4)
(5)
Next, the definition of a collection of spaces of rational functions R n,k is needed. For n ∈ N and k = 1, 2, . . . , K, R n,k is defined to be the set of all rational functions with poles restricted to the real pole sequence α1 , α2 , . . . , α K ; more precisely, ⎧ K n−1 ⎨
di, j R
n,k := f : C \ {α1 , α2 , . . . , α K } → C; f (z) = d0 + ⎩ (z−α j)κij i=1 j=1 +
k
r=1
dn,r
, d , d , d ∈R , (z−αr )κnr 0 i, j n,r
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
with the convention
0
m=1 ∗ := 0,
193
where
κnk : N × {1, 2, . . . , K} → N, (n, k) → κnk := (n−1)γk +ρk
denotes the multiplicity of the pole αk , k = 1, 2, . . . , K, in the repeated real pole sequence n−1
1
n
{α1 , α2 , . . . , α K } ∪ · · · ∪ {α1 , α2 , . . . , α K } ∪ {α1 , α2 , . . . , αk }, with γk the ‘repeating number’ of pole αk in the set {α1 , α2 , . . . , α K }, and ρk the ‘repeating index’ of pole αk in the set {α1 , α2 , . . . , α K } up to, and including, ‘position k’. As an illustration, for the pole set {α1 , α2 , α3 , α4 , α5 , α6 } = {0, 0, 1, 0, 1, 2}, γ1 = γ2 = γ4 = 3, γ3 = γ5 = 2, γ6 = 1 and ρ1 = 1, ρ2 = 2, ρ3 = 1, ρ4 = 3, ρ5 = 2, ρ6 = 1, and, for n ∈ N, κn1 = (n−1)γ1 +ρ1 = 3(n−1)+1 = 3n−2, κn2 = (n−1)γ2 +ρ2 = 3(n−1)+2 = 3n−1, κn3 = (n−1)γ3 +ρ3 = 2(n−1)+1 = 2n−1, κn4 = (n−1)γ4 +ρ4 = 3(n−1)+3 = 3n, κn5 = (n−1)γ5 +ρ5 = 2(n−1)+2 = 2n, κn6 = (n−1)γ6 +ρ6 = (n−1)+1 = n.
Denote the linear space over R spanned by a constant and the rational functions {(z−αk )−κnk } n∈N by R ; more precisely, k=1,2,...,K
R :=
K
R n,k .
n∈N k=1
A function element 0 = f ∈ R is called a rational function corresponding to the real pole set {α1 , α2 , . . . , α K }. The ordered base of rational functions for R is ⎧ n=1 ⎪ ⎨ −κ11 B ∼ const., (z−α1 ) , (z−α2 )−κ12 , . . . , (z−α K )−κ1K , ⎪ ⎩ k=1,2,...,K
n=2
(z−α1 )−κ21 , (z−α2 )−κ22 , . . . , (z−α K )−κ2K , . . . . . . k=1,2,...,K
⎫ n=m ⎪ ⎬ −κm1 −κm2 −κmK . . . . . . , (z−α1 ) , (z−α2 ) , . . . , (z−α K ) , ...... , ⎪ ⎭ k=1,2,...,K
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corresponding, respectively, to the cyclically repeated real pole sequence ⎧ ⎫ n=1 n=2 n=m ⎨ ⎬ P ∼ no pole, α1 , α2 , . . . , α K , α1 , α2 , . . . , α K , . . . . . . , α1 , α2 , . . . , α K , . . . . . . . ⎩ ⎭ k=1,2,...,K
k=1,2,...,K
k=1,2,...,K
R
For 0 = f ∈ , there exists a unique pair (n, k) ∈ N × {1, 2, . . . , K} such R that f ∈ R n,k . For (n, k) ∈ N × {1, 2, . . . , K} and 0 = f ∈ n,k , define the leading coefficient of f , symbolically LC( f ), as
LC( f ) := dn,k . For (n, k) ∈ N × {1, 2, . . . , K}, 0 = f ∈ R n,k is called monic if LC( f ) = 1. A linear functional L will now be defined; however, in order for the integrals appearing in its definition to be well defined, it will be assumed that the probability measure μ ∈ M1 (R), where M1 (R) := μ; dμ(ξ ) = 1, ξ m dμ(ξ ) < ∞, (ξ −αk )−m dμ(ξ ) < ∞, R
R
R
m ∈ N, k = 1, 2, . . . , K denotes the set of all non-negative unit Borel measures on R for which all moments at αk , k = 1, 2, . . . , K, and at the point at infinity exist. Define, for (n, k) ∈ N × {1, 2, . . . , K}, the linear functional L by its action on the (rational) basis elements of R as follows:
L : R → R, f = d0 +
K n−1
i=1 j=1
di, j
k
dn,r + (z−α j)κij r=1 (z−αr )κnr
→ L( f ) := d0 +
K n−1
di, jci, j +
−κij
ci, j = L((z−α j)
) :=
R
(ξ −α j)−κij dμ(ξ ),
(i, j) ∈ N × {1, 2, . . . , K}.
(Of course, since M1 (R) μ, c0,0 := L(1) = R dμ(ξ ) = 1.) Define the real bilinear form ··, · L as follows: R
R
d n,r cn,r ,
r=1
i=1 j=1
where
k
··, · L : × → R, ( f, g) → f, g L := L( f (z)g(z)) =
R
f (ξ )g(ξ ) dμ(ξ ).
It is a fact that the bilinear form ··, · L thus defined is an inner product (see Section 2, the proof of Lemma 2.1); and this fact is used, with little or no further reference, throughout this work. If f ∈ R , then || f (··)||L := ( f, f )1/2 is called the norm of f with respect to L (note that || f (··)||L 0 for all f ∈ R , and || f (··)||L > 0 if 0 = f ∈ R ). A
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sequence of rational functions will now be defined. {φˆ kn }
n∈N
195
is called a—
k=1,2,...,K
real—orthonormal rational function sequence with respect to L if, for n ∈ N and k = 1, 2, . . . , K: (i) φˆ kn ∈ R n,k ; n n (ii) φˆ k , φˆ k L = δnn δkk , where δij is the Kronecker delta; (iii) φˆ kn , φˆ kn L =: ||φˆ kn (··)||2L = 1. (For consistency of notation, set φˆ 00 (z) ≡ 1.) In order to elucidate the precise structure of the orthogonality conditions for the FPC ORFs, and to state the results of this work (see Subsection 1.3), the following notational prelude is requisite. What follows next is an ordered partitioning of {1, 2, . . . , K} and the real pole set {α1 , α2 , . . . , α K }. In order to proceed, though, the appearance of a parameter, designated s, must first be explained. For the real pole set {α1 , α2 , . . . , α K } (described above), let s denote the number of distinct poles; e.g., for {0, 0, 1, 0, 1, 2}, s = 3. For k = 1, 2, . . . , K, a decomposition of the index set corresponding to poles distinct from αk will be needed, that is: {k ∈ {1, 2, . . . , K}; αk = αk }. In order to decompose this set, one needs to consider the (possibly smaller) collection of distinct poles, ordered consistently with the original pole sequence, and with the pole αk excised: the size of this set is s −1. For the jth member of the reduced collection of poles (also referred to as the ‘residual’ pole set), the number of times that that pole appears in the original pole sequence shall be denoted k j, j= 1, 2, . . . , s −1. (Note that this integer should be thought of as a function of k; but, for simplicity of notation, this dependence is suppressed.) One then decomposes the set of integers written above into a disjoint union so that the first subset is the collection of all integers (from 1 to K) corresponding to the first pole in this reduced collection of poles, the second subset is the collection of all integers corresponding to the second pole in this reduced collection of poles, etc. The precise definition of this decomposition is as follows. Write the ordered disjoint integer partition {k ∈ {1, 2, . . . , K}; αk = αk } := i(1)1 , i(1)2 , . . . , i(1)k1 k1
∪ i(2)1 , i(2)2 , . . . , i(2)k2 ∪ · · ·
k2
· · · ∪ i(s −1)1 , i(s −1)2 , . . . , i(s −1)ks−1 ks−1
:=
s −1
q=1
i(q)1 , i(q)2 , . . . , i(q)kq , kq
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where, for q ∈ {1, 2, . . . , s −1}, 1 i(q)1 < i(q)2 < · · · < i(q)kq K, and {i( j )1 , i( j)2 , . . . , i( j)k j } ∩ {i(l)1 , i(l)2 , . . . , i(l)kl } = ∅ ∀ l = j∈ {1, 2, . . . , s −1}, which induces, on the real pole set {α1 , α2 , . . . , α K }, the following disjoint ordering, {αk ; k ∈ {1, 2, . . . , K}, αk = αk } := αi(1)1 , αi(1)2 , . . . , αi(1)k1 k1
∪ αi(2)1 , αi(2)2 , . . . , αi(2)k2 ∪ · · ·
k2
· · · ∪ αi(s−1)1 , αi(s−1)2 , . . . , αi(s−1)ks−1
ks−1
:=
s −1
q=1
αi(q)1 , αi(q)2 , . . . , αi(q)kq , kq
where, for q ∈ {1, 2, . . . , s −1}, αi(q)1 ≺ αi(q)2 ≺ · · · ≺ αi(q)kq , where the notation ‘a ≺ b ’ means “a precedes b ” or “a is to the left of b ”, {αi( j)1 , αi( j)2 , . . . , αi( j)k j } ∩ {αi(l)1 , αi(l)2 , . . . , αi(l)kl } = ∅ ∀ l = j∈ {1, 2, . . . , s −1}, such that αi(q)1 = αi(q)2 = · · · = αi(q)kq , # αi(q)1 , αi(q)2 , . . . , αi(q)kq = kq ,
q = 1, 2, . . . , s −1, q = 1, 2, . . . , s −1.
s−1 (Note, also, the interesting relation #{k ∈ {1, 2, . . . , K}; αk = αk } = q=1 kq = K−γk , k = 1, 2, . . . , K.)3 In order to illustrate the above notation, consider the real pole sequence (of ‘length’ K = 6) {α1 , α2 , α3 , α4 , α5 , α6 } = {0, 0, 1, 0, 1, 2}, for which s = 3: (i)
k=1 {k ∈ {1, 2, . . . , 6}; αk = α1 = 0} = {3, 5, 6} = {3, 5} ∪ {6} := {i(1)1 , i(1)k1} ∪ {i(2)k2 }
⇒
k1 = 2, i(1)1 = 3, i(1)k1 =2 = 5, k2 = 1, i(2)k2 =1 = 6,
all the real poles are distinct, that is, αi = α j ∀ i = j∈ {1, 2, . . . , K}, then, for k = 1, 2, . . . , K, K−1 {k ∈ {1, 2, . . . , K}; αk = αk } is the ordered disjoint union of singeltons, that is, ∪q=1 {i(q)kq }, with kq = 1, q = 1, 2, . . . , K−1, 1 i(1)1 < i(2)1 < · · · < i(K−1)1 K, and {i(q)kq } ∩ {i(r)kr } = ∅ ∀ q = r ∈ {1, 2, . . . , K−1}, which induces, on the real pole set {α1 , α2 , . . . , α K }, the followK−1 ing disjoint ordering, {αk ; k ∈ {1, 2, . . . , K}, αk = αk } := ∪q=1 {αi(q)kq }, with αi(1)1 ≺ αi(2)1 ≺ · · · ≺ αi(K−1)1 , #{αi(q)kq } = kq = 1, q = 1, 2, . . . , K−1, {αi(q)kq } ∩ {αi(r)kr } = ∅ ∀ q = r ∈ {1, 2, . . . , K−1}, and K−1 kq = K−1. #{αk ; k ∈ {1, 2, . . . , K}, αk = αk } = q=1 3 If
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which induces the ordering (on the ‘residual’ pole set) αk ; k ∈ {1, 2, . . . , 6}, αk = α1 = 0 := αi(1)1 , αi(1)k1 ∪ αi(2)k2 = {α3 , α5 } ∪ {α6 } = {1, 1} ∪ {2}. (ii)
k=2 {k ∈ {1, 2, . . . , 6}; αk = α2 = 0} = {3, 5, 6} = {3, 5} ∪ {6} := {i(1)1 , i(1)k1 } ∪ {i(2)k2 }
⇒
k1 = 2, i(1)1 = 3, i(1)k1 =2 = 5, k2 = 1, i(2)k2 =1 = 6, which induces the ordering (on the ‘residual’ pole set) αk ; k ∈ {1, 2, . . . , 6}, αk = α2 = 0 := αi(1)1 , αi(1)k1 ∪ αi(2)k2 = {α3 , α5 } ∪ {α6 } = {1, 1} ∪ {2}. (iii)
k=3 {k ∈ {1, 2, . . . , 6}; αk = α3 = 1} = {1, 2, 4, 6} = {1, 2, 4} ∪ {6} := {i(1)1 , i(1)2 , i(1)k1 } ∪ {i(2)k2 }
⇒
k1 = 3, i(1)1 = 1, i(1)2 = 2, i(1)k1 =3 = 4, k2 = 1, i(2)k2 =1 = 6, which induces the ordering (on the ‘residual’ pole set) αk ; k ∈ {1, 2, . . . , 6}, αk = α3 = 1 := αi(1)1 , αi(1)2 , αi(1)k1 ∪ αi(2)k2 = {α1 , α2 , α4 } ∪ {α6 } = {0, 0, 0} ∪ {2}. (iv)
k=4 {k ∈ {1, 2, . . . , 6}; αk = α4 = 0} = {3, 5, 6} = {3, 5} ∪ {6} := {i(1)1 , i(1)k1 } ∪ {i(2)k2 }
⇒
k1 = 2, i(1)1 = 3, i(1)k1 =2 = 5, k2 = 1, i(2)k2 =1 = 6, which induces the ordering (on the ‘residual’ pole set) αk ; k ∈ {1, 2, . . . , 6}, αk = α4 = 0 := αi(1)1 , αi(1)k1 ∪ αi(2)k2 = {α3 , α5 } ∪ {α6 } = {1, 1} ∪ {2}. (v)
k=5 {k ∈ {1, 2, . . . , 6}; αk = α5 = 1} = {1, 2, 4, 6} = {1, 2, 4} ∪ {6} := i(1)1 , i(1)2 , i(1)k1 } ∪ {i(2)k2 k1 = 3, i(1)1 = 1, i(1)2 = 2, i(1)k1 =3 = 4, k2 = 1, i(2)k2 =1 = 6,
⇒
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which induces the ordering (on the ‘residual’ pole set) αk ; k ∈ {1, 2, . . . , 6}, αk = α5 = 1 := αi(1)1 , αi(1)2 , αi(1)k1 ∪ αi(2)k2 = {α1 , α2 , α4 } ∪ {α6 } = {0, 0, 0} ∪ {2}. (vi)
k=6 k ∈ {1, 2, . . . , 6}; αk = α6 = 2 = {1, 2, 3, 4, 5} = {1, 2, 4} ∪ {3, 5} := i(1)1 , i(1)2 , i(1)k1 ∪ i(2)1 , i(2)k2 ⇒ k1 = 3, i(1)1 = 1, i(1)2 = 2, i(1)k1 =3 = 4, k2 = 2, i(2)1 = 3, i(2)k2 =2 = 5, which induces the ordering (on the ‘residual’ pole set) {αk ; k ∈ {1, 2, . . . , 6}, αk = α6 = 2} := αi(1)1 , αi(1)2 , αi(1)k1 ∪ αi(2)1 , αi(2)k2 = {α1 , α2 , α4 } ∪ {α3 , α5 } = {0, 0, 0} ∪ {1, 1}.
This concludes the example. In order to introduce the notion of the residual multiplicity (see below), a notational remark is requisite. For k = 1, 2, . . . , K and a given set of positive integers i1 , i2 , . . . , i M , let j= ind{i1 , i2 , . . . , i M | k} denote the largest positive integer j, if it exists, from the collection i1 , i2 , . . . , i M that is less than k; e.g., j= ind{1, 3, 7, 11| 8} ⇒ j= 7
and
j= ind{5, 6, 9| 3} ⇒ no such j exists.
Recall that we have defined, for each member of the reduced collection of real poles, the index set {i(q)1 , i(q)2 , . . . , i(q)kq }, q = 1, 2, . . . , s −1. It will be important to know if, within each of these index sets, there is a positive integer less than k. Towards this end, define, for each choice of k = 1, 2, . . . , K, the following set: Jq (k) := j= ind i(q)1 , i(q)2 , . . . , i(q)kq | k , q = 1, 2, . . . , s −1, and denote by q (k), m
q = 1, 2, . . . , s −1,
the unique element of the set Jq (k), if it is not empty. For n ∈ N and k = 1, 2, . . . , K, define, with the above orderings and definitions, (n−1)γi(q)kq , Jq (k) = ∅, q ∈ {1, 2, . . . , s −1}, κnkkq := q ∈ {1, 2, . . . , s −1}, (n−1)γm q (k) +ρm q (k) , Jq (k) = ∅,
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where κnkkq : N × {1, 2, . . . , K} → Z+ 0 , q = 1, 2, . . . , s −1, is the residual multiplicity of pole αi(q)kq in the repeated real pole sequence 1 n−1 n Pn,k := α1 , α2 , . . . , α K ∪ · · · ∪ α1 , α2 , . . . , α K ∪ α1 , α2 , . . . , αk . K
K
k
More precisely, for n ∈ N and k = 1, 2, . . . , K, as all occurrences of the real pole αk , k = 1, 2, . . . , K, are excised from the repeated real pole sequence Pn,k , where the multiplicity, or number of occurrences, of the real pole αk is κnk = (n−1)γk +ρk , one is left with the ‘residual’ real pole set (via the above induced ordering on the real poles) s −1 Pn,k \ αk , αk , . . . , αk := αi(q)kq , αi(q)kq , . . . , αi(q)kq q=1 κnk
=
κnkkq
αi(1)k1 , αi(1)k1 , . . . , αi(1)k1
κnkk
1
∪ αi(2)k2 , αi(2)k2 , . . . , αi(2)k2 ∪ · · ·
κnkk
2
· · · ∪ αi(s−1)ks−1 , αi(s−1)ks−1 , . . . , αi(s−1)ks−1 , κnkk
s−1
where the number of times the real pole αi(q)kq (= αk ) occurs is (its multiplicity) κnkkq , q = 1, 2, . . . , s −1. It can occur that, for n = 1, κ1kkq = 0, for some values of q ∈ {1, 2, . . . , s −1}: in such cases, one defines, {αi(q)kq , αi(q)kq , . . . , αi(q)kq } := ∅, q ∈ {1, 2, . . . , s −1}; however, for (N ) n 2 and k = 1, 2, . . . , K, κnkkq 1, q ∈ {1, 2, . . . , s −1}. For n ∈ N and k = 1, 2, . . . , K, a counting-of-residual-multiplicities argument gives rise to the following ordered sum formula: s−1
κnkkq := κnkk1 + κnkk2 +· · ·+ κnkks−1 = (n−1)K+k− κnk .
q=1
In order to illustrate the latter notation, consider, again, the real pole sequence (of ‘length’ K = 6) {α1 , α2 , α3 , α4 , α5 , α6 } = {0, 0, 1, 0, 1, 2}, for which s = 3 (recall, also, the above example): (i)
k=1 J1 (1) := j= ind i(1)1 , i(1)k1 | 1 = j= ind{3, 5| 1} = ∅, J2 (1) := j= ind i(2)k2 | 1 = j= ind{6| 1} = ∅,
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hence κn1k1 = (n−1)γi(1)k1 = (n−1)γ5 = 2(n−1), κn1k2 = (n−1)γi(2)k2 = (n−1)γ6 = n−1,
that is, as one moves from left to right across the repeated real pole sequence n 1 n−1 Pn,1 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 6
1
6
n−1 n = 0, 0, 1, 0, 1, 2 ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0 1 6
1
6
and removes all occurrences of the real pole α1 = 0, which occurs κn1 = (n−1)γ1 +ρ1 = 3(n−1)+1 = 3n−2 times, one is left with the residual real pole set (via the above induced ordering) Pn,1\{α1 , α1 , . . . , α1 } = Pn,1\{0, 0, . . . , 0} := κn1
3n−2
2 q=1
αi(q)kq , αi(q)kq , . . . , αi(q)kq κn1kq
= αi(1)k1, αi(1)k1, . . . , αi(1)k1 κn1k
1
∪ αi(2)k2, αi(2)k2, . . . , αi(2)k2 κn1k
2
= {1, 1, . . . , 1}∪{2, 2, . . . , 2}, 2(n−1)
n−1
where the number of times the real pole αi(1)k1 = 1 (= 0 = α1 ) occurs is κn1k1 = 2(n−1), and the number of times the real pole αi(2)k2 = 2 (= 0 = α1 ) occurs is κn1k2 = n−1. For n = 1, since κ11k1 = κ11k2 = 0, one sets, as per the convention above, {αi(q)kq , αi(q)kq , . . . , αi(q)kq } := ∅, q = 1, 2, in which case, as κ11 = ρ1 = 1, P1,1 \ {α1 } = P1,1 \ {0} = ∅ ∪ ∅ = ∅. In this case, the ordered sum formula reads 2
κn1kq := κn1k1 + κn1k2 = 2(n−1)+(n−1) = 3(n−1).
q=1
(ii)
k=2 J1 (2) := j= ind i(1)1 , i(1)k1 | 2 = j= ind{3, 5| 2} = ∅, J2 (2) := j= ind i(2)k2 | 2 = j= ind{6| 2} = ∅,
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hence κn2k1 = (n−1)γi(1)k1 = (n−1)γ5 = 2(n−1), κn2k2 = (n−1)γi(2)k2 = (n−1)γ6 = n−1,
that is, as one moves from left to right across the repeated real pole sequence n 1 n−1 Pn,2 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 , α2 6
2
6
n−1 n = 0, 0, 1, 0, 1, 2 ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0, 0 1 6
6
2
and removes all occurrences of the real pole α2 = 0, which occurs κn2 = (n−1)γ2 +ρ2 = 3(n−1)+2 = 3n−1 times, one is left with the residual real pole set (via the above induced ordering) Pn,2\{α2 , α2 , . . . , α2 } = Pn,2\{0, 0, . . . , 0} := κn2
3n−1
2 q=1
αi(q)kq , αi(q)kq , . . . , αi(q)kq κn2kq
= αi(1)k1, αi(1)k1, . . . , αi(1)k1
κn2k
1
∪ αi(2)k2, αi(2)k2, . . . , αi(2)k2 κn2k
2
= {1, 1, . . . , 1}∪{2, 2, . . . , 2}, 2(n−1)
n−1
where the number of times the real pole αi(1)k1 = 1 (= 0 = α2 ) occurs is κn2k1 = 2(n−1), and the number of times the real pole αi(2)k2 = 2 (= 0 = α2 ) occurs is κn2k2 = n−1. For n = 1, since κ12k1 = κ12k2 = 0, one sets, as per the convention above, {αi(q)kq , αi(q)kq , . . . , αi(q)kq } := ∅, q = 1, 2, in which case, as κ12 = ρ2 = 2, P1,2 \ {α2 , α2 } = P1,2 \ {0, 0} = ∅ ∪ ∅ = ∅. In this case, the ordered sum formula reads 2
κn2kq := κn2k1 + κn2k2 = 2(n−1)+(n−1) = 3(n−1).
q=1
(iii)
k=3 J1 (3) := { j= ind{i(1)1 , i(1)2 , i(1)k1 | 3}} = { j= ind{1, 2, 4| 3}}
1 (3) = 2, = {2} ⇒ m J2 (3) := { j= ind{i(2)k2 | 3}} = { j= ind{6| 3}} = ∅,
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hence κn3k1 = (n−1)γm 1 (3) +ρm 1 (3) = (n−1)γ2 +ρ2 = 3(n−1)+2 = 3n−1, κn3k2 = (n−1)γi(2)k2 = (n−1)γ6 = n−1,
that is, as one moves from left to right across the repeated real pole sequence 1 n−1 n Pn,3 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 , α2 , α3 6
6
3
1 n−1 n = 0, 0, 1, 0, 1, 2 ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0, 0, 1 6
6
3
and removes all occurrences of the real pole α3 = 1, which occurs κn3 = (n−1)γ3 +ρ3 = 2(n−1)+1 = 2n−1 times, one is left with the residual real pole set (via the above induced ordering) Pn,3\{α3 , α3 , . . . , α3 } = Pn,3 \{1, 1, . . . , 1} := κn3
2n−1
2 q=1
αi(q)kq , αi(q)kq , . . . , αi(q)kq κn3 kq
= αi(1)k1, αi(1)k1, . . . , αi(1)k1 κn3 k
1
∪ αi(2)k2, αi(2)k2, . . . , αi(2)k2
κn3 k
2
= {0, 0, . . . , 0} ∪ {2, 2, . . . , 2}, 3n−1
n−1
where the number of times the real pole αi(1)k1 = 0 (= 1 = α3 ) occurs is κn3k1 = 3n−1, and the number of times the real pole αi(2)k2 = 2 (= 1 = α3 ) occurs is κn3k2 = n−1. For n = 1, since κ13k2 = 0, one sets, as per the convention above, {αi(2)k2 , αi(2)k2 , . . . , αi(2)k2 } := ∅, in which case, as κ13 = ρ3 = 1, P1,3 \ {α3 } = P1,3 \ {1} = {0, 0} ∪ ∅ = {0, 0}. In this case, the ordered sum formula reads 2
κn3kq := κn3k1 + κn3k2 = (3n−1)+(n−1) = 4n−2.
q=1
(iv)
k=4
J1 (4) := j= ind i(1)1 , i(1)k1 | 4 = { j= ind{3, 5| 4}} 1 (4) = 3, = {3} ⇒ m J2 (4) := j= ind i(2)k2 | 4 = { j= ind{6| 4}} = ∅,
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hence κn4k1 = (n−1)γm 1 (4) +ρm 1 (4) = (n−1)γ3 +ρ3 = 2(n−1)+1 = 2n−1, κn4k2 = (n−1)γi(2)k2 = (n−1)γ6 = n−1,
that is, as one moves from left to right across the repeated real pole sequence 1 n−1 n Pn,4 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 , α2 , α3 , α4 6
6
4
n−1 n = 0, 0, 1, 0, 1, 2 ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0, 0, 1, 0 1 6
6
4
and removes all occurrences of the real pole α4 = 0, which occurs κn4 = (n−1)γ4 +ρ4 = 3(n−1)+3 = 3n times, one is left with the residual real pole set (via the above induced ordering) Pn,4\{α4 , α4 , . . . , α4 } = Pn,4\{0, 0, . . . , 0} := κn4
3n
2 q=1
αi(q)kq , αi(q)kq , . . . , αi(q)kq κn4 kq
= αi(1)k1, αi(1)k1, . . . , αi(1)k1 κn4 k
1
∪ αi(2)k2, αi(2)k2, . . . , αi(2)k2 κn4 k
2
= {1, 1, . . . , 1}∪{2, 2, . . . , 2}, 2n−1
n−1
where the number of times the real pole αi(1)k1 = 1 (= 0 = α4 ) occurs is κn4k1 = 2n−1, and the number of times the real pole αi(2)k2 = 2 (= 0 = α4 ) occurs is κn4k2 = n−1. For n = 1, since κ14k2 = 0, one sets, as per the convention above, {αi(2)k2 , αi(2)k2 , . . . , αi(2)k2 } := ∅, in which case, as κ14 = ρ4 = 3, P1,4 \ {α4 , α4 , α4 } = P1,4 \ {0, 0, 0} = {1} ∪ ∅ = {1}. In this case, the ordered sum formula reads 2
κn4kq := κn4k1 + κn4k2 = (2n−1)+(n−1) = 3n−2.
q=1
(v)
k=5
J1 (5) := j= ind i(1)1 , i(1)2 , i(1)k1 | 5 = j= ind{1, 2, 4| 5} 1 (5) = 4, = {4} ⇒ m J2 (5) := j= ind i(2)k2 | 5 = j= ind{6| 5} = ∅,
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hence κn5k1 = (n−1)γm 1 (5) +ρm 1 (5) = (n−1)γ4 +ρ4 = 3(n−1)+3 = 3n, κn5k2 = (n−1)γi(2)k2 = (n−1)γ6 = n−1,
that is, as one moves from left to right across the repeated real pole sequence 1 n−1 n Pn,5 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 , α2 , α3 , α4 , α5 6
6
5
1 n−1 n = {0, 0, 1, 0, 1, 2} ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0, 0, 1, 0, 1 6
6
5
and removes all occurrences of the real pole α5 = 1, which occurs κn5 = (n−1)γ5 +ρ5 = 2(n−1)+2 = 2n times, one is left with the residual real pole set (via the above induced ordering) Pn,5\{α5 , α5 , . . . , α5 } = Pn,5\{1, 1, . . . , 1} := κn5
2n
2 q=1
αi(q)kq , αi(q)kq , . . . , αi(q)kq κn5 kq
= αi(1)k1, αi(1)k1, . . . , αi(1)k1 κn5 k
1
∪ αi(2)k2, αi(2)k2, . . . , αi(2)k2
κn5 k
2
= {0, 0, . . . , 0}∪{2, 2, . . . , 2}, 3n
n−1
where the number of times the real pole αi(1)k1 = 0 (= 1 = α5 ) occurs is κn5k1 = 3n, and the number of times the real pole αi(2)k2 = 2 (= 1 = α5 ) occurs is κn5k2 = n−1. For n = 1, since κ15k2 = 0, one sets, as per the convention above, {αi(2)k2 , αi(2)k2 , . . . , αi(2)k2 } := ∅, in which case, as κ15 = ρ5 = 2, P1,5 \ {α5 , α5 } = P1,5 \ {1, 1} = {0, 0, 0} ∪ ∅ = {0, 0, 0}. In this case, the ordered sum formula reads 2
κn5kq := κn5k1 + κn5k2 = 3n+(n−1) = 4n−1.
q=1
(vi)
k=6
J1 (6) := j= ind i(1)1 , i(1)2 , i(1)k1 | 6 = { j= ind{1, 2, 4| 6}} 1 (6) = 4, = {4} ⇒ m J2 (6) := j= ind i(2)1 , i(2)k2 | 6 = { j= ind{3, 5| 6}} 2 (6) = 5, = {5} ⇒ m
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hence κn6k1 = (n−1)γm 1 (6) +ρm 1 (6) = (n−1)γ4 +ρ4 = 3(n−1)+3 = 3n, κn6k2 = (n−1)γm 2 (6) +ρm 2 (6) = (n−1)γ5 +ρ5 = 2(n−1)+2 = 2n,
that is, as one moves from left to right across the cyclically repeated real pole sequence 1 n−1 n Pn,6 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 , α2 , α3 , α4 , α5 , α6 6
6
6
1 n−1 n = 0, 0, 1, 0, 1, 2 ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0, 0, 1, 0, 1, 2 6
6
6
and removes all occurrences of the real pole α6 = 2, which occurs κn6 = (n−1)γ6 +ρ6 = (n−1)+1 = n times, one is left with the residual real pole set (via the above induced ordering) Pn,6\{α6 , α6 , . . . , α6 } = Pn,6\{2, 2, . . . , 2} := κn6
n
2 q=1
αi(q)kq , αi(q)kq , . . . , αi(q)kq κn6 kq
= αi(1)k1, αi(1)k1, . . . , αi(1)k1 κn6 k
1
∪ αi(2)k2, αi(2)k2, . . . , αi(2)k2
κn6 k
2
= {0, 0, . . . , 0}∪{1, 1, . . . , 1}, 3n
2n
where the number of times the real pole αi(1)k1 = 0 (= 2 = α6 ) occurs is κn6k1 = 3n, and the number of times the real pole αi(2)k2 = 1 (= 2 = α6 ) occurs is κn6k2 = 2n. For n = 1, since κ16 = ρ6 = 1, it follows that P1,6 \ {α6 } = P1,6 \ {2} = {0, 0, 0} ∪ {1, 1}. In this case, the ordered sum formula reads 2
κn6kq := κn6k1 + κn6k2 = 3n+2n = 5n.
q=1
This concludes the example. For simplicity of notation, set, hereafter, αi(q)kq := α pq ,
q = 1, 2, . . . , s −1.
Recall that, for each choice of k (from 1 to K), we have previously decomposed the index set corresponding to poles distinct from αk into an ordered disjoint union of subsets so that the jth subset (for j running from 1 to s −1) is the collection of all integers (from 1 to K) corresponding to the j th pole in the reduced collection of poles. Now, we will define the next subset in this ordering
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containing all integers (now from 1 to k) corresponding to the pole αk . Thus, for each choice of k from 1 to K, write the ordered integer partition {{1, 2, . . . , K} k k; αk = αk } := i(s)1 , i(s)2 , . . . , i(s)ks , ks
with i(s)ks := k, 1 i(s)1 < i(s)2 < · · · < i(s)ks K, #{i(s)1 , i(s)2 , . . . , i(s)ks } = ks (= κ1i(s)ks = κ1k ) = ρi(s)ks = ρk , and {i( j )1 , i( j )2 , . . . , i( j )k j } ∩ {i(s)1 , i(s)2 , . . . , i(s)ks } = ∅, j= 1, 2, . . . , s −1, which induces, by the definition above, the following real pole ordering, {αk ; k ∈ {1, 2, . . . , K}, k k, αk = αk } := αi(s)1 , αi(s)2 , . . . , αi(s)ks , with αi(s)ks := αk , αi(s)1 ≺ αi(s)2 ≺ · · · ≺ αi(s)ks , {αi( j )1 , αi( j )2 , . . . , αi( j )k j } ∩ {αi(s)1 , αi(s)2 , . . . , αi(s)ks } = ∅, j= 1, 2, . . . , s −1, such that αi(s)1 = αi(s)2 = · · · = αi(s)ks := α ps := αk , # αi(s)1 , αi(s)2 , . . . , αi(s)ks = ks = κ1i(s)ks = κ1k = ρk . For (N ) n 2 and k = 1, 2, . . . , K, # αi(s)ks , αi(s)ks , . . . , αi(s)ks = κni(s)ks = κnk = (n−1)γk +ρk . In order to illustrate this latter notation, consider, again, the real pole sequence (of ‘length’ K = 6) {α1 , α2 , α3 , α4 , α5 , α6 } = {0, 0, 1, 0, 1, 2}, for which s = 3: (i)
k=1
{{1, 2, . . . , 6} k 1; αk = α1 = 0} = {1} := i(3)k3 ⇒ k3 = 1, i(3)k3 =1 = 1,
which induces the real pole ordering {αk ; k ∈ {1, 2, . . . , 6}, k 1, αk = α1 = 0} := {αi(3)k3 } = {α1 } = {0}, hence κni(s)ks = κni(3)1 = κn1 = (n−1)γ1 +ρ1 = 3(n−1)+1 = 3n−2,
that is, as one moves from left to right across the repeated real pole sequence 1
n−1
n
6
6
1
1
n−1
n
6
6
1
Pn,1 = {α1 , α2 , . . . , α6 } ∪ · · · ∪ {α1 , α2 , . . . , α6 } ∪ { α1 }
0 } = {0, 0, 1, 0, 1, 2} ∪ · · · ∪ {0, 0, 1, 0, 1, 2} ∪ { and removes the residual pole set (recall the examples above) {αk ; k ∈ {1, 2, . . . , 6}, αk = α1 = 0}, one is left with the set (via the above-induced
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
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ordering) that consists of all occurrences of the real pole α1 = 0, which occurs κn1 = 3n−2 times, Pn,1 \
2
{α pq , α pq , . . . , α pq } := αi(s)ks , αi(s)ks , . . . , αi(s)ks = {0, 0, . . . , 0}. q=1 κni(s)k =κn1
κn1kq
(ii)
3n−2
s
k=2 {{1, 2, . . . , 6} k 2; αk = α2 = 0} = {1, 2} := {i(3)1 , i(3)k3 } ⇒ k3 = 2, i(3)1 = 1, i(3)k3 =2 = 2, which induces the real pole ordering
{αk ; k∈{1, 2, . . . , 6}, k 2, αk = α2 = 0} := αi(3)1 , αi(3)k3 = {α1 , α2 } = {0, 0},
hence κni(s)ks = κni(3)2 = κn2 = (n−1)γ2 +ρ2 = 3(n−1)+2 = 3n−1,
that is, as one moves from left to right across the repeated real pole sequence n 1 n−1 Pn,2 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 , α2 6
2
6
1 n−1 n = 0, 0, 1, 0, 1, 2 ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0, 0 6
6
2
and removes the residual pole set (recall the examples above) {αk ; k ∈ {1, 2, . . . , 6}, αk = α2 = 0}, one is left with the set (via the above-induced ordering) that consists of all occurrences of the real pole α2 = 0, which occurs κn2 = 3n−1 times, Pn,2 \
2 q=1
(iii)
α pq , α pq , . . . , α pq := αi(s)ks , αi(s)ks , . . . , αi(s)ks = {0, 0, . . . , 0}. κn2 kq
κni(s)k =κn2
3n−1
s
k=3 {{1, 2, . . . , 6} k 3; αk = α3 = 1} = {3} := {i(3)k3 } ⇒ k3 = 1, i(3)k3 =1 = 3, which induces the real pole ordering {αk ; k ∈ {1, 2, . . . , 6}, k 3, αk = α3 = 1} := {αi(3)k3 } = {α3 } = {1}, hence κni(s)ks = κni(3)1 = κn3 = (n−1)γ3 +ρ3 = 2(n−1)+1 = 2n−1,
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that is, as one moves from left to right across the repeated real pole sequence 1 n−1 n Pn,3 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 , α2 , α3 } 6
6
3
1 n−1 n = 0, 0, 1, 0, 1, 2 ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0, 0, 1 6
6
3
and removes the residual pole set (recall the examples above) {αk ; k ∈ {1, 2, . . . , 6}, αk = α3 = 1}, one is left with the set (via the above-induced ordering) that consists of all occurrences of the real pole α3 = 1, which occurs κn3 = 2n−1 times, Pn,3 \
2 q=1
(iv)
α pq , α pq , . . . , α pq := αi(s)ks , αi(s)ks , . . . , αi(s)ks = {1, 1, . . . , 1}. κn3 kq
κni(s)k =κn3
2n−1
s
k=4 {{1, 2, . . . , 6} k 4; αk = α4 = 0} = {1, 2, 4} := {i(3)1 , i(3)2 , i(3)k3 } ⇒ k3 = 3, i(3)1 = 1, i(3)2 = 2, i(3)k3 =3 = 4, which induces the real pole ordering
{αk ; k ∈ {1, 2, . . . , 6}, k 4, αk = α4 = 0} := αi(3)1 , αi(3)2 , αi(3)k3 = {α1 , α2 , α4 } = {0, 0, 0},
hence κni(s)ks = κni(3)3 = κn4 = (n−1)γ4 +ρ4 = 3(n−1)+3 = 3n,
that is, as one moves from left to right across the repeated real pole sequence 1 n−1 n Pn,4 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 , α2 , α3 , α4 6
6
4
1 n−1 n = 0, 0, 1, 0, 1, 2 ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0, 0, 1, 0 6
6
4
and removes the residual pole set (recall the examples above) {αk ; k ∈ {1, 2, . . . , 6}, αk = α4 = 0}, one is left with the set (via the above-induced ordering) that consists of all occurrences of the real pole α4 = 0, which occurs κn4 = 3n times, Pn,4 \
2 q=1
α pq , α pq , . . . , α pq := αi(s)ks , αi(s)ks , . . . , αi(s)ks = {0, 0, . . . , 0}. κn4kq
κni(s)k =κn4 s
3n
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(v)
k=5
209
{1, 2, . . . , 6} k 5; αk = α5 = 1 = {3, 5} := i(3)1 , i(3)k3 ⇒ k3 = 2, i(3)1 = 3, i(3)k3 =2 = 5,
which induces the real pole ordering αk ; k∈{1, 2, . . . , 6}, k 5, αk = α5 = 1 := αi(3)1, αi(3)k3 = {α3, α5 } = {1, 1}, hence κni(s)ks = κni(3)2 = κn5 = (n−1)γ5 +ρ5 = 2(n−1)+2 = 2n,
that is, as one moves from left to right across the repeated real pole sequence 1 n−1 n Pn,5 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 , α2 , α3 , α4 , α5 6
6
5
1 n−1 n = 0, 0, 1, 0, 1, 2 ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0, 0, 1, 0, 1 6
6
5
and removes the residual pole set (recall the examples above) {αk ; k ∈ {1, 2, . . . , 6}, αk = α5 = 1}, one is left with the set (via the above-induced ordering) that consists of all occurrences of the real pole α5 = 1, which occurs κn5 = 2n times, Pn,5 \
2 q=1
(vi)
α pq , α pq , . . . , α pq := αi(s)ks , αi(s)ks , . . . , αi(s)ks = 1, 1, . . . , 1 . κni(s)k =κn5
κn5kq
2n
s
k=6 {{1, 2, . . . , 6} k 6; αk = α6 = 2} = {6} := {i(3)k3 } ⇒ k3 = 1, i(3)k3 =1 = 6, which induces the real pole ordering {αk ; k ∈ {1, 2, . . . , 6}, k 6, αk = α6 = 2} := {αi(3)k3 } = {α6 } = {2}, hence κni(s)ks = κni(3)1 = κn6 = (n−1)γ6 +ρ6 = (n−1)+1 = n,
that is, as one moves from left to right across the repeated real pole sequence 1 n−1 n Pn,6 = α1 , α2 , . . . , α6 ∪ · · · ∪ α1 , α2 , . . . , α6 ∪ α1 , α2 , α3 , α4 , α5 , α6 6
6
6
1 n−1 n = 0, 0, 1, 0, 1, 2 ∪ · · · ∪ 0, 0, 1, 0, 1, 2 ∪ 0, 0, 1, 0, 1, 2 6
6
6
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and removes the residual pole set (recall the examples above) {αk ; k ∈ {1, 2, . . . , 6}, αk = α6 = 2}, one is left with the set (via the above-induced ordering) that consists of all occurrences of the real pole α6 = 2, which occurs κn6 = n times, Pn,6 \
2 q=1
α pq , α pq , . . . , α pq := αi(s)ks , αi(s)ks , . . . , αi(s)ks = {2, 2, . . . , 2}. n
κni(s)k =κn6
κn6kq
s
This concludes the example. With the above conventions and ordered disjoint partitions, one writes, for n ∈ N and k = 1, 2, . . . , K, the repeated real pole sequence Pn,k as the following ordered disjoint partition: s −1
q=1
:=
αi(q)kq , αi(q)kq , . . . , αi(q)kq ∪ αi(s)ks , αi(s)ks , . . . , αi(s)ks
s −1
q=1
κni(s)k
κnkkq
s
α pq , α pq , . . . , α pq ∪ αk , αk , . . . , αk , κnk
κnkkq
where, by convention, the set {αk , αk , . . . , αk } is written as the right-most set. With the above notational preamble concluded, one now returns to the precise formulation of the orthogonality conditions for the FPC ORFs. We have a nested sequence of (rational) base sets. Fixing n ∈ N and k = 1, 2, . . . , K, we determine one member of this nested sequence of (rational) base sets: ⎧ ⎨ 1 const., (z−α1 )−κ11 , (z−α2 )−κ12 , . . . , (z−α K )−κ1K , ⎩ K 2 −κ22
(z−α1 )−κ21 , (z−α2 ) , . . . , (z−α K )−κ2K , . . . . . . K n −κn2
. . . . . . , (z−α1 )−κn1 , (z−α2 ) := {const.}
⎫ ⎬
, . . . , (z−αk )−κnk ⎭
k
s−1 κnkkq ∪q=1 ∪r=1
κnk (z−α pq )−r ∪m=1 (z−αk )−m
s−1 (z−α pq )−1 , (z−α pq )−2 , . . . , (z−α pq )−κnkkq := {const.} ∪q=1 (z−αk )−1 , (z−αk )−2 , . . . , (z−αk )−κnk ,
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211
corresponding, respectively, to the ordered repeated real pole sequence ⎧ ⎫ ⎨ ⎬ 1 2 n no pole, α1 , α2 , . . . , α K , α1 , α2 , . . . , α K , . . . , α1 , α2 , . . . , αk ⎭ ⎩ K
:= {no pole}
K
s−1 ∪q=1 {α pq , α pq , . . . , α pq }
k
{αk , αk , . . . , αk }. κnk
κnkkq
Orthonormalisation with respect to ··, · L , via the Gram-Schmidt orthogonalisation method, leads to the FPC orthonormal rational functions, {φkn (z)} n∈N k=1,2,...,K
(for consistency of notation, set φ00 (z) ≡ 1), which can be written as n−1 K
νm, j(n, k)
φkn (z) = φ0 (n, k)+
m=1 j=1
k
μn,r (n, k) + . (z−α j)κmj r=1 (z−αr )κnr
Using the above orderings, it is convenient to express the FPC orthonormal rational functions as follows: κ
φkn (z) := φ0 (n, k)+
nkkq κnk s−1
νr,q (n, k) μn,m (n, k) + . r (z−α ) (z−αk )m p q q=1 r=1 m=1
The φkn ’s are normalised so that they all have real coefficients; in particular, for n ∈ N and k = 1, 2, . . . , K,
LC(φkn ) = μn,k (n, k) = μn,κnk (n, k) > 0.
(For consistency of notation, set φ0 (0, 0) = φ0 (0, 0) ≡ 1.) Furthermore, note that, for n ∈ N and k = 1, 2, . . . , K, by construction: j= 0, 1, . . . , κnk −1; (1) φkn , (z−αk )− j L = φkn (ξ )(ξ −αk )− j dμ(ξ ) = 0, R
(2) (3)
n φk , μn,κnk (n, k)(z−αk )−κnk L = μn,κnk (n, k) φkn (ξ )(ξ −αk )−κnk dμ(ξ ) = 1; n φk , (z−α pq )−r L =
R
R
φkn (ξ )(ξ −α pq )−r dμ(ξ ) = 0,
q = 1, 2, . . . , s −1, r = 1, 2, . . . , κnkkq . (Note: if, for k = 1, 2, . . . , K, the residual real pole set {αk ; k ∈ {1, 2, . . . , K}, αk = αk } = ∅, then the corresponding orthogonality conditions (3) above are vacuous; actually, this can only occur for n = 1.)
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For n ∈ N and k = 1, 2, . . . , K, the orthogonality conditions (1)–(3) above give rise to a total of s−1
κnk +1+
κnkkq
= (n−1)K+k+1
q=1
= (n−1)K+k−κnk
(linear) equations determining the (n−1)K+k+1 real (n- and k-dependent) coefficients. It is convenient to introduce, at this stage, the main object of study of π nk (z)} n∈N (for consistency of this work, namely, the monic FPC ORFs, {π k=1,2,...,K
notation, set π 00 (z) ≡ 1). For n ∈ N and k = 1, 2, . . . , K: π nk (z) :=
n−1 K
νm, j(n, k) φkn (z) φ0 (n, k) 1 = + LC(φkn ) μn,k (n, k) μn,k (n, k) m=1 j=1 (z−α j)κmj k−1
μn,r (n, k) 1 + + κnr κ (z−α ) (z−α r k ) nk μn,k (n, k) r=1
1
κ
nkkq s−1
νr,q (n, k) 1 φ0 (n, k) + := μn,κnk (n, k) μn,κnk (n, k) q=1 r=1 (z−α pq )r
κ nk −1 1 1 μn,m (n, k) + + . μn,κnk (n, k) m=1 (z−αk )m (z−αk )κnk
(Recall that LC(φkn ) denotes the leading coefficient of the FPC orthonormal π nk (z)} n∈N , possess the following rational functions.) The monic FPC ORFs, {π k=1,2,...,K
orthogonality properties: (1 )
(2 )
n π k , (z−αk )− j L =
R
n π k , (z−αk )−κnk L =
π nk (ξ )(ξ −αk )− j dμ(ξ ) = 0,
R
π nk (ξ )(ξ −αk )−κnk dμ(ξ ) = (LC(φkn ))−2
= (μn,κnk (n, k))−2 ;
(3 )
n π k , (z−α pq )−r L =
R
j= 0, 1, . . . , κnk −1;
π nk (ξ )(ξ −α pq )−r dμ(ξ ) = 0,
q = 1, 2, . . . , s −1, r = 1, 2, . . . , κnkkq .
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(Note: if, for k = 1, 2, . . . , K, the residual real pole set {αk ; k ∈ {1, 2, . . . , K}, αk = αk } = ∅, then the corresponding orthogonality conditions (3 ) above are vacuous; actually, this can occur only for n = 1.) For n ∈ N and k = 1, 2, . . . , K, it follows from the monic FPC ORF orthogonality conditions (1 )–(3 ) above that n n π nk (··)||2L = (LC(φkn ))−2 = (μn,κnk (n, k))−2 , π k , π k L =: ||π π nk (··)||L = (μn,κnk (n, k))−1 > 0. whence ||π Remark 1.2.1 There is a circuitous connection between FPC ORFs and the more standard orthogonal polynomials sequence(s). For n ∈ N and k = 1, 2, . . . , K, an FPC ORF has numerator which is a polynomial of degree (n−1)K+k that is orthogonal to lower-degree polynomials (of degrees 0, 1, . . . , (n−1)K+k−1), but with respect to the non-standard (exponentially s−1 (z−α pq )−2κnkkq (z−αk )−2κnk varying) measure of orthogonality dμ(z) = q=1 ·(z−αk ) exp(−N V(z)) dz, N ∈ N, which changes signs (due to the factor z−αk ). (Note: the latter measure represents, in fact, a doubly-indexed family of measures.) Intuition from weighted approximation and this doubly-indexed family of measures provides an alternative understanding for the importance of the associated family of variational (minimisation) problems, which are summarised in Subsection 1.3, (9) (see, also, Section 3, Lemma 3.8). The abovementioned connection (to orthogonal polynomials) also has the potential to yield an alternative approach to the asymptotic analysis (in the double-scaling limit N, n → ∞ such that N/n = 1+o(1)) of the FPC ORFs; however, in the opinion of the author’s, it is more convenient to present RHPs that are directly associated to the FPC ORFs. 1.3 Summary of Results Having defined, heretofore, and in considerable detail, the principal objects π nk (z)} n∈N , the ‘norming of this study, that is, the monic FPC ORFs, {π k=1,2,...,K
constant’, μn,κnk (n, k), (n, k) ∈ N × {1, 2, . . . , K}, and the FPC orthonormal π nk (z), (n, k) ∈ N × {1, 2, . . . , K}, it must rational functions, φkn (z) := μn,κnk (n, k)π be mentioned that the ultimate goal of this multi-fold study of ORFs (FPC and MPC) is to obtain precise, and uniform, asymptotics, in the doublescaling limit N, n → ∞ such that N/n = 1+o(1), of π nk (z), z ∈ C, μn,κnk (n, k), and, subsequently, φkn (z), z ∈ C. In order to follow through with the above-mentioned asymptotic programme, however, a correct formulation of the ORF problem, for an a priori prescribed, not necessarily distinct, real pole set {α1 , α2 , . . . , α K } lying on the support of the orthogonality measure, is a seminal necessity. In fact, the present work, which is the first installment of a multi-fold series of works dedicated to a detailed study of the above-described ORFs, serves a dual purpose, namely: (i) to address the above-mentioned ‘formulation problem’ for the FPC
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ORFs; and (ii) to prepare the groundwork for subsequent asymptotic analyses, in the double-scaling limit N, n → ∞ such that N/n = 1+o(1). The genesis of our ORF studies (FPC and MPC) consists in reformulating, in the spirit of Fokas et al. [54, 55] (see, also, [56]), the ORF problem as K families of matrix Riemann–Hilbert problems (RHPs) on R, and then to study the large-n (as N, n → ∞ such that N/n = 1+o(1)) behaviour of the corresponding K solution families, wherein the latter family of K asymptotic analyses consists of a union of the Deift–Zhou (DZ) non-linear steepest-descent method for undulatory—matrix—RHPs [57, 58] and the extension of Deift et al. [59]. Given the (N ) K arbitrary, bounded and not necessarily distinct real poles α1 , α2 , . . . , α K lying on the support of the orthogonality measure (with varying exponential weight) dμ(z) = exp(−N V(z)) dz, N ∈ N, where V: R \ {α1 , α2 , . . . , α K } → R is characterised by conditions (3)–(5), the first set of results of this work (Part I) can be summarised thus; for n ∈ N and k = 1, 2, . . . , K: • an equivalent reformulation of the monic FPC ORF problem as a family of K matrix RHPs on R (see Section 2, Lemma RHPFPC ); • explicit solution formulae for each of these K matrix RHPs, constructed explicitly from the monic FPC ORFs and their Cauchy transforms (see Section 2, Lemma 2.1); • the subsequent establishment of the existence and the uniqueness of the monic FPC ORFs via a detailed analysis of a novel family of K generalised Hankel determinants associated with rational moments of the orthogonality measure with respect to the given real pole set {α1 , α2 , . . . , α K } (see Section 2, Lemma 2.1); • explicit multi-integral representation for the ‘norming constant’, μn,κnk (n, k) (see Section 2, Corollary 2.1); • explicit multi-integral representation for the monic FPC ORFs, π nk (z), π nk (z), explicit multi-integral z ∈ C, and, via the relation φkn (z) := μn,κnk (n, k)π representation for the FPC orthonormal rational functions, φkn (z), z ∈ C (see Section 2, Lemma 2.2, and Remark 2.3, respectively). The ultimate goal of this multi-fold study of ORFs is to prepare the foundation for the asymptotic analysis (in the double-scaling limit N, n → ∞ such that N/n = 1+o(1)) of π nk (z), z ∈ C, and μn,κnk (n, k) (subsequently, π nk (z), z ∈ C). The present work (Part I) deals exclusively φkn (z) := μn,κnk (n, k)π with the FPC ORFs (see the follow-up work, Part II, for the MPC ORFs). The proceeding discussion, which summarises the remaining results of this work, while valid in its own right for finite n (∈ N), is germane, principally, to transforming the family of K matrix RHPs on R into a family of K equivalent (‘model’) matrix RHPs on R suitable for asymptotic analysis. It is a well-established mathematical fact that variational conditions for minimisation problems in logarithmic potential theory, via the equilibrium measure (see, e.g., [56, 60–63]), play an absolutely crucial rôle in asymptotic analyses of (matrix) RHPs associated with (continuous and discrete)
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orthogonal polynomials, their roots, and corresponding recurrence relation coefficients (see, e.g., [64–67]). The situation with respect to the large-n asymptotic analysis for the monic FPC ORFs is analogous; however, unlike asymptotic analyses for the orthogonal polynomials case, the asymptotic analysis for the monic FPC ORFs requires the consideration of K different families of matrix RHPs on R, one for each k = 1, 2, . . . , K (see Section 2, Lemmata RHPFPC and 2.1). Thus, one must consider, for n ∈ N, K sets of variational conditions for K suitably posed minimisation problems. Remark 1.3.1 Before proceeding, a minor notational preamble is requisite. Write (cf. (1) and (2)) dμ(z) = exp(−N V(z)) dz = exp(−nV(z)) dz =: d μ(z), n ∈ N, where = zo V(z), V(z) with zo : N × N → R+ ,
(N, n) → zo := N/n,
R\ where R+ := {x ∈ R; x > 0}, and where the ‘scaled’ external field V: {α1 , α2 , . . . , α K } → R satisfies the following conditions: is real analytic on R \ {α1 , α2 , . . . , α K }; V(z)
lim
|x|→+∞
lim
x→αk
! V(x) = +∞; ln(x2 +1)
! V(x) = +∞, ln((x−αk )−2 +1)
k = 1, 2, . . . , K.
(6)
(7)
(8)
R \ {α1 , α2 , . . . , α K } z → K (E.g., a rational function of the form V: k=1 −1 2m∞ q q ς ˜ (z−α ) + ς ˜ z , where, for k = 1, 2, . . . , K, m ∈ N , m ∈ N, q,k k q,∞ k ∞ q=0 q=−2mk 4 ς˜−2mk ,k > 0, and ς˜2m∞ ,∞ > 0, would satisfy conditions (6)–(8).) The following discussion summarises, succinctly, the remaining, principal results of this work, all of which are seminal ingredients for the subsequent asymptotic analysis (in the double-scaling limit N, n → ∞ such that zo = 1+
4 Since the double-scaling limit of interest is N, n → ∞ such that z = 1+o(1), the monic FPC ORFs o are now orthogonal with respect to the varying exponential measure d μ(z) = exp(−nV(z)) dz, (= zo V(z)) satisfying conditions (6)–(8), where the large parameter, n, enters n ∈ N, with V(z) simultaneously into the order (= (n−1)K+k) of the monic FPC ORFs and the (varying exponential) weight; thus, asymptotics of the monic FPC ORFs are studied along a ‘diagonal strip’ of a doubly-indexed sequence.
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o(1)) of the monic FPC ORFs, π nk (z), z ∈ C, the ‘norming constant’, μn,κnk (n, k), π nk (z), z ∈ C. and the FPC orthonormal rational functions, φkn (z) := μn,κnk (n, k)π •
R \ {α1 , α2 , . . . , α K } → R satisfy conditions (6)–(8). Let IV : N × Let V: {1, 2, . . . , K} × M1 (R) → R denote the energy functional ⎛
IV [n, k; μ ] := IV [μ ] = EQ
1 ln ⎝|ξ −τ | n
EQ
×
R2
|ξ −τ | |ξ −αk ||τ −αk |
κnkn −1
⎞ κnknkq −1 |ξ −τ | ⎠ dμEQ (ξ ) dμEQ (τ ) |ξ −α pq ||τ −α pq |
s −1 $
q=1
+2
R
) dμEQ (ξ ), V(ξ
and consider, for n ∈ N and k = 1, 2, . . . , K, the associated minimisation problem EV (n, k) := EV = inf IV [μEQ ]; μEQ ∈ M1 (R) . For n ∈ N and k = 1, 2, . . . , K, the infimum is finite, and there exists a unique measure μV (n, k) := μV , called the equilibrium measure, achieving this minimum, that is, M1 (R) μV = inf{IV [μEQ ]; μEQ ∈ M1 (R)} (see Section 3, Lemmata 3.1, 3.2, and 3.3). For n ∈ N and k = 1, 2, . . . , K, the equilibrium measure, μV , has the following ‘regularity properties’: •
the equilibrium measure has support which consists of the disjoint union of a finite number, N+1 (∈ N), of bounded real (compact) intervals. In fact, as shown in Section 3, Lemma 3.7, item (1) (1), supp(μV ) =: J = ∪ N+1 j=1 [bj−1 , a j ] (⊂ R \ {α1 , α2 , . . . , α K }). The end-points of the support of μV , that is, {bj−1 , a j} N+1 j=1 , as well as the non-negative integer N, depend on n and k; e.g., bj−1 = bj−1 (n, k) and a j = a j(n, k), j= 1, 2, . . . , N+1. It is instructive to note that the real poles do not lie within the support of the equilibrium measure: [bj−1 , a j] ∩ {α1 , α2 , . . . , α K } = ∅, j= 1, 2, . . . , N+1. The compact real intervals have been enumerated so that −∞ < b0 < a1 < b1 < a2 < · · · < bN < a N+1 < +∞. Note: all of the quantities above also depend on zo := N/n; but, for notational simplicity, this dependence is suppressed;
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•
217
the end-points, {bj−1 , a j} N+1 j=1 , satisfy the locally solvable system of 2(N+1) real moment equations (transcendental equations) ⎛ ⎛ ⎞ ⎞ s−1 j
κ ξ nkkq ⎝2 ⎝(κnk −1) + ⎠+ V (ξ )⎠dξ= 0, 1/2 iπ n(ξ −α ) n(ξ −α ) iπ k p J (R(ξ ))+ q q=1 j= 0,1, . . . ,N, ⎛
ξ N+1 J
1/2
(R(ξ ))+
⎛
⎞ ⎞ s−1
κ 2 ( V κ −1) (ξ ) nk k q ⎝ ⎝ nk ⎠+ ⎠ dξ + iπ n(ξ −αk ) q=1 n(ξ −α pq ) iπ
(n−1)K+k , n ⎛ ⎛ ⎛ ⎞ bj s−1
κnkkq 2 ( κ −1) nk −1/2 1/2 ⎝(R(ς)) ⎠ (R(ξ ))+ ⎝ ⎝ + iπ n(ξ −αk ) q=1 n(ξ −α pq ) aj J
= −2
(ξ ) V + iπ
•
!
dξ ξ −ς
!
' ' ' ' s−1
κnkkq ' bj −α pq ' ' bj −αk ' κnk −1 ' ' ' ' dς = 2 ln ' +2 ln ' ' n a j −αk ' n a −α j p q q=1
j)), j)− V(a j= 1, 2, . . . , N, + (V(b ( N+1 )1/2 1/2 , with (R(z))± := limε↓0 (R where (R(z))1/2 := j=1 (z−bj−1 )(z−a j ) (z±iε))1/2 , and the branch of the square root is chosen so that z−(N+1) (R(z))1/2 ∼C± z→∞ ±1 (see Section 3, Lemma 3.7, item (1) (1)). In fact, in the double-scaling limit N, n → ∞ such that zo = 1+o(1), the end-points, {bj−1 , a j} N+1 j=1 , are real-analytic functions of zo ; the density of the equilibrium measure is given by dμV (x) := ψV (x) dx =
1 1/2 (R(x))+ hV (x)11 J (x) dx, 2π i
where hV (z) =
1 2
(n−1)K+k n
−1 *
⎛
⎛
s−1
κnkkq
⎞
⎝ 2i ⎝ (κnk −1) + ⎠ π n(ξ −αk ) q=1 n(ξ −α pq ) CV
⎞ (ξ ) (R(ξ ))−1/2 iV ⎠ dξ + π ξ −z (real analytic for z ∈ R \ {α1 , α2 , . . . , α K }), with CV (⊂ C \ {α1 , α2 , . . . , α K }) the disjoint union of s +1 circular contours, one outer one of large radius
218
•
K. T.-R. McLaughlin et al.
R traversed clockwise, and s inner ones, one about each real pole α pq , q = 1, 2, . . . , s, of small radii rq , q = 1, 2, . . . , s, traversed counter-clockwise, with the numbers 0 0) for x ∈ J (resp., x ∈ int(J)); in fact, ψV (x) behaves like a square root at the end-points of the support of the equilibrium measure, that is, ψV (x) =x↓bj−1 O ((x−bj−1 )1/2 ) and ψV (x) =x↑a j O ((a j −x)1/2 ), j= 1, 2, . . . , N+1 (see Section 3, Lemma 3.7, item (2) (2)); the equilibrium measure and its—compact—support are uniquely characterised by the following Euler–Lagrange variational equations: there exists (n, k) := ∈ R, the Lagrange multiplier, such that
(n−1)K+k 2 n −2
s−1
κnkkq
q=1
2
s−1
κnkkq
q=1
n
J
' ' z−ξ ln '' ξ −α
k
ln|z−α pq | + 2
' ' ' dμ (ξ )−2 κnk −1 ln|z−αk | V ' n
s−1
κnkkq
n
q=1
(n−1)K+k n
−2
•
n
J
' ' z−ξ ln '' ξ −α
k
ln|z−αpq |+2
' ' ' dμ (ξ )−2 κnk −1 ln|z−αk | V ' n
s−1
κnkkq
n
q=1
ln|α pq −αk | − V(z)− = 0, z ∈ J, (9)
ln|αpq −αk |− V(z)− 0, z ∈ R \ J
(see Section 3, Lemma 3.8); the Euler–Lagrange variational conditions can be conveniently recast in terms of the complex potential (the ‘g-function’) g: N × {1, 2, . . . , K} × C \ (−∞, max{max j=1,2,...,K {α j}, max{supp(μV )}}) → C (of μV ): ⎛
1 ln ⎝(z−ξ ) n
g(n, k; z) := g(z) = J
×
s −1 $
q=1
(z−ξ ) (z−αk )(ξ −αk )
(z−ξ ) (z−α pq )(ξ −α pq )
κnknkq
κnkn −1
⎞ ⎠ dμV (ξ )
(see Section 3, Lemma 3.4). For n ∈ N and k = 1, 2, . . . , K, g(z) satisfies: (G1) g(z) is analytic for z∈C\(−∞, max{maxj=1,2,...,K {αj}, max{supp(μV )}}) (see Section 3, Lemma 3.4);
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
( κnk −1 )
(G2) g(z) = − where P± 0 :=
n
ln(z−αk )+ P± 0 + O (z−αk ) as z → αk , with z ∈ C± ,
⎛
J
⎜ ln ⎝|ξ −αk |
1 n
s$ −1 q=1
− iπ
κnk −1 n
|ξ −αk | |ξ −α pq ||α pq −αk |
J∩{x∈R; x<αk }
dμV (ξ ) − iπ
κ
nk kq n
dμV (ξ )±iπ
(n−1)K+k n
⎞ ⎟ ⎠ dμV (ξ )
s−1 κ
nk kq q=1
J∩{x∈R; x<α pq }
∓ iπ
!
×
219
n
J∩{x∈R; x>αk }
dμV (ξ )
κnk kq
q∈{ j∈{1,2,...,s−1}; α p j >αk }
n
(see Section 3, Lemma 3.4); − (G3) g+ (z)+g− (z)− P+ 0 − P0 − V(z)− = 0, z ∈ J, where g± (z) := limε↓0 g(z±iε) (see Section 3, Lemma 3.8); − (G4) g+ (z) + g− (z) − P+ 0 − P0 − V(z) − 0, z ∈ R \ J, where equality holds for at most a finite number of points (see Section 3, Lemma 3.8); + R R (G5) g+ (z) − g− (z) + P− 0 − P0 = i fg (z), z ∈ R, where fg (z) is a piecewise-continuous, real-valued bounded function (see Section 3, Lemma 3.8); κnkkq + (G6) i(g+ (z) – g− (z) + P− + q∈{ j∈{1,2,...,s−1}; α p j >z} n 0 – P0 + 2π i (κnk −1) 2π i n 1 {x∈R; x<αk } (z)) = (n, k; z), with 2π (n, k; z) = 0,
-
(n−1)K+k n
.
ψV (z) 0,
z ∈ J, z ∈ R \ J,
where equality holds for at most a finite number of points (see Section 3, Lemma 3.8); •
for k = 1, 2, . . . , K, an equivalent family of K asymptotic, ‘model’ matrix RHPs on R is derived (see Section 4, Lemma 4.3). From these K model matrix RHPs, asymptotics (in the double-scaling limit N, n → ∞ such that π nk (z), zo = 1+o(1)) of π nk (z), z ∈ C, μn,κnk (n, k), and φkn (z) := μn,κnk (n, k)π z ∈ C, will be derived, in a future publication. Note: the equilibrium measure and the corresponding variational problems emerge naturally in the asymptotic analysis of this family of K equivalent ‘model’ matrix RHPs on R.
In this work on the characterisation and asymptotics (in the double-scaling limit N, n → ∞ such that zo = 1+o(1)) of the FPC ORFs and related quantities,
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the so-called ‘regular case’ is studied. For n ∈ N and k = 1, 2, . . . , K, one says R \ {α1 , α2 , . . . , α K } → R, is regular if: that dμV , or V: (i) hV (x) = 0, x ∈ J; '. . -' ( ) s−1κnkkq ' z−ξ ' (ii) 2 (n−1)K+k ln dμV (ξ )−2 κnkn−1 ln|z−αk |−2 q=1 ln|z−αpq| ' J n ξ −αk ' n s−1 κnkkq < 0, z ∈ R \ J; +2 q=1 n ln|α pq −αk |− V(z)− (iii) inequalities (G4) and (G6) are strict, that is, (in (G4)) and (in (G6)) are replaced by < and >, respectively.5 Remark 1.3.2 The following correspondence should also be noted: for n ∈ N and k = 1, 2, . . . , K, g(z) solves the phase conditions (G1)–(G6) ⇔ M1 (R) μV solves the variational conditions (9). Remark 1.3.3 In the next section, the RHP formulation for the monic FPC ORFs is presented. The reader may wonder whence came the RHP. For various values of n (∈ N) and different choices of the pole set, FPC ORFs were constructed; and, based on these calculations, and in the spirit of the RHP formulation for the orthogonal polynomials problem, an integral representation for the ‘solution matrix’ was conjectured, but with the Cauchy kernels normalised at the poles. Then, the following question was posed: “what kind of RHP do the FPC ORFs solve?” The ‘full’ RHP formulation for the FPC ORFs (more precisely, for the monic FPC ORFs) then follows from a careful analysis of the asymptotic behaviour of the ‘solution matrix’ in open neighbourhoods of the poles, supplemented with a computation (which uses the Sokhotski– Plemelj formula) of the corresponding jump, or discontinuity, matrix. Remark 1.3.4 A heuristic explanation for the origin of the (doubly-indexed family of) energy—minimisation—problems appearing in (9) is as follows. Starting with the RHP for the monic FPC ORFs (see Section 2, Lemma RHPFPC ), one makes a transformation involving a so-called g-function (or complex logarithmic potential), whose properties are determined at a later stage: the result of this transformation is a ‘new’ RHP (see Section 3, Lemma 3.4). Then, one asks if there exists a choice of this g-function so that
n ∈ N and k = 1, 2, . . . , K, there are three distinct situations in which these conditions (i)– . κ nk −1 z˜ −ξ ln| z ˜ −α (iii) may fail: (1) for at least one z˜ ∈ R \ J, 2( (n−1)K+k ) ln(| |) dμ (ξ )−2 k |− V J n ξ −αk n s −1 κ nkkq s −1 κ nkkq z˜ )− = 0, that is, equality is attained for at 2 q=1 n ln|z˜ −α pq |+2 q=1 n ln|α pq −αk |− V( least one point z˜ in the complement of the support of the equilibrium measure, which corresponds to the situation in which a ‘band’ has just closed, or is about to open, about z˜ ; (2) for at least one ˆ ) = 0, that is, the function hV ˆ within the support of the zˆ , hV (z (z) vanishes for at least one point z equilibrium measure, which corresponds to the situation in which a ‘gap’ is about to open, or close, about zˆ ; and (3) there exists at least one j∈ {1, 2, . . . , N+1}, denoted j∗ , such that hV (bj∗ −1 ) = 0 or hV (a j∗ ) = 0. Each of these three cases can occur only a finite number of times due to the fact that R \ {α1 , α2 , . . . , α K } → R satisfies conditions (6)–(8) [63, 66]. V: 5 For
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221
the ‘new’, or transformed, RHP is in a form suitable for subsequent asymptotic analysis (in the double-scaling limit N, n → ∞ such that N/n = 1+o(1)). Based on experience from other (asymptotic) RHP analyses, one arrives at a collection of equations and inequalities, which, if satisfied, achieve the desired result of obtaining an RHP in so-called ‘standard form’. These equations and inequalities, in turn, are shown to be equivalent to Euler–Lagrange variational conditions associated with the energy—minimisation—problems. Remark 1.3.5 A general energy minimisation problem for all n ∈ N and k = 1, 2, . . . , K has been presented. As has been mentioned heretofore, the principal interest of this multi-fold study of ORFs is asymptotics as n → ∞. In this limit, the family of energy minimisation problems stabilizes in the sense that one may write the energy functional as a small perturbation (o(1)) of the following ‘core’ energy functional: !−1 ∞ |ξ −τ | K EQ IV [μ ] := ln s dμEQ (ξ ) dμEQ (τ ) γi(q)k q 2 (|ξ −α ||τ −α |) R pq pq q=1 +2
R
V(ξ ) dμEQ (ξ ),
s where (cf. Subsection 1.2) α pq := αi(q)kq , q = 1, 2, . . . , s, and the factor q=1 (|ξ − γi(q)k q α pq ||τ −α pq |) is independent of, or invariant with respect to, k (= 1, 2, . . . , K); e.g., for the real pole sequence (of ‘length’ K = 6) {α1 , α2 , α3 , α4 , α5 , α6 } = {0, 0, 1, 0, 1, 2}, for which s = 3, the above formula reads −1 ∞ |ξ −τ |6 IV [μEQ ] = ln dμEQ (ξ ) dμEQ (τ ) (|ξ ||τ |)3 (|ξ −1||τ −1|)2 (|ξ −2||τ −2|)1 R2 + 2 V(ξ ) dμEQ (ξ ). R
Note: if all the poles in the sequence {α1 , α2 , . . . , α K } are distinct, that is, αi = α j ∀ i = j∈ {1, 2, . . . , K}, then the ‘core’ energy functional is given by !−1 ∞ |ξ −τ | K EQ IV [μ ] = ln K dμEQ (ξ )dμEQ (τ ) |ξ −α ||τ −α | R2 j j j=1 +2
R
V(ξ ) dμEQ (ξ ).
Remark 1.3.6 The energy minimisation problem described herein requires a new existence and regularity theory because of the presence of the real at each αk , k = poles α1 , α2 , . . . , α K , as well as the singular behaviour of V 1, 2, . . . , K.
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2 The Monic FPC ORF Family of Riemann–Hilbert Problems: Existence and Uniqueness In this section, the family of K matrix Riemann–Hilbert problems (RHPs) π nk (z)} n∈N , z ∈ C, is stated, on R characterising the monic FPC ORFs, {π k=1,2,...,K
whence the existence and the uniqueness of the monic FPC ORFs and the associated norming constants, μn,κnk (n, k), is established via a generalised Hankel determinant analysis (see Lemmata RHPFPC and 2.1), and explicit multi-integral representations for the norming constant (see Corollary 2.1) and the monic FPC ORFs (see Lemma 2.2) are obtained. Before launching into the theory proper, however, and in order to prune the usual foliage attendant upon this topic, it is convenient to summarise the notation used throughout this work. Notational Conventions ) ( ) ( ) ( ( 0 ) and σ3 = 10 −1 (1) I = 10 01 is the 2 × 2 identity matrix, σ1 = 01 10 , σ2 = 0i −i 0 ( ) ( ) are the Pauli matrices, σ+ = 00 10 and σ− = 01 00 are, respectively, the ≷
raising and lowering matrices, R± := {x ∈ R; ±x > 0}, Rx0 := {x ∈ R; x ≷ x0 }, C± := {z ∈ C; ± Im(z) > 0}, C∗ := C \ {0}, C := C ∪ {∞}, and sgn(x) := 0 if x = 0 and x|x|−1 if x = 0; (2) for a scalar ω and a 2×2 matrix ϒ, ωad(σ3 ) ϒ := ωσ3 ϒω−σ3 ; (3) a contour D which is the finite union of piecewise-smooth, simple curves (as closed sets) is said to be orientable if its complement C \ D can always be divided into two, possibly disconnected, disjoint open sets + and − , either of which has finitely many components, such that D admits an orientation so that it can either be viewed as a positively oriented boundary D + for + or as a negatively oriented boundary D − for − [68], that is, the (possibly disconnected) components of C \ D can be coloured by + or by − in such a way that the + regions do not share boundary with the − regions, except, possibly, at finitely many points [69]; (4) for each segment of an oriented contour D , according to the given orientation, the “+” side is to the left and the “–” side is to the right as one traverses the contour in the direction of orientation, that is, for a matrix Aij(··), i, j= 1, 2, (Aij(··))± denote the non-tangential limits (Aij(z))± := lim z → z Aij(z ); z ∈ ± side of D
(5) for 1 p< ∞ and D some point set, p LM2 (C) (D ) :=
f: D → M2 (C); || f (··)||LMp (C) (D ) := 2
1p | f (z)| |dz| <∞ , p
D
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( 2 )1/2 where, for A (··) ∈ M2 (C), |A (··)| := (the Hilberti, j=1 Aij (·· ) Aij (·· ) Schmidt norm), with ∗ denoting complex conjugation of ∗ , for p= ∞, LM∞2 (C) (D ) := g : D → M2 (C); ||g(··)||LM∞ (C) (D ) := max sup |gij(z)| < ∞ , 2
i, j=1,2 z∈D
and, for f ∈ I+ LM2 2 (C) (D ) := {I+h; h ∈ LM2 2 (C) (D )}, || f (··)||I+LM2 (C) (D ) := || f (∞)||2LM∞ (C) (D ) +|| f (··)− f (∞)||2L 2 2
M2 (C) (D )
2
. 12
;
(6) for a matrix Aij(··), i, j= 1, 2, to have boundary values in the LM2 2 (C) (D ) sense on an oriented contour D , it is meant that 2 lim z → z D |A (z )−(A (z))± | |dz| = 0; e.g., if D = R is oriented z ∈ ± side of D
from +∞ to −∞, then A (··) has LM2 2 (C) (D ) boundary values on D means that limε↓0 R |A (x∓iε)−(A (x))± |2 dx = 0; p (7) ||F(··)||∩ p∈J LM (C) (∗) := p∈J ||F(··)||LMp (C) (∗) , with card(J) < ∞; 2 2 (8) M1 (R) denotes the set of all non-negative unit Borel measures on R for which all moments at αk , k = 1, 2, . . . , K, and at the point at infinity exist, that is, m M1 (R) := μ; dμ(ξ ) = 1, ξ dμ(ξ ) < ∞, (ξ −αk )−m dμ(ξ ) < ∞, R
R
R
m ∈ N, k = 1, 2, . . . , K ; (9) for (ν1 , ν2 ) ∈ R × R, the function (•−ν1 )iν2 : C \ (−∞, ν1 ) → C, • → exp(iν2 ln(• − ν1 )), with the branch cut taken along (−∞, ν1 ), and with the principal branch of the logarithm chosen; (10) for a 2 × 2 matrix-valued function T(z), the notation T(z) =z→z0 O (∗) (resp., o(∗)) means Tij(z) =z→z0 O (∗ij) (resp., o(∗ij)), i, j= 1, 2. Remark 2.1 The superscripts ± , and sometimes the subscripts ± , in this section should not be confused with the subscripts ± appearing in the various RHPs (this is a general comment which applies throughout the entire text, unless stated otherwise). Although C := C ∪ {∞} (resp., R := R ∪ {−∞} ∪ {+∞}) is the standard notation for the (closed) Riemann sphere (resp., closed real line), the simplified, and somewhat abusive, notation C (resp., R) is used to denote both the (closed) Riemann sphere, C (resp., closed real line, R), and the (open) complex field, C (resp., open real line, R), and the context(s) should make clear which object(s) the notation C (resp., R) represents. Remark 2.2 Throughout the remainder of this work, the notations of Subsection 1.2 will be used extensively, with little, or no, explanation.
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The family of K matrix RHPs characterising the monic FPC ORFs is now stated. RHPFPC . Let V: R \ {α1 , α2 , . . . , α K } → R satisfy conditions (3)–(5). For n ∈ N and k = 1, 2, . . . , K, find Y : N × {1, 2, . . . , K} × C \ R → SL2 (C) solving: (i) Y (n, k; z) := Y (z) is holomorphic for z ∈ C \ R; (ii) the boundary values Y± (z) := limε↓0 Y (z±iε) satisfy the jump condition Y+ (z) = Y− (z)υ(z),
z ∈ R,
where υ: R → GL2 (R), z → υ(z) := I+exp(−N V(z)) σ+ ,
N ∈ N;
(iii) Y (z)(z−αk )(κnk −1)σ3
=
C\R z→αk
I+O (z−αk );
(iv) Y (z)z−σ3
=
C\R z→∞
O (1);
(v) for q ∈ {1, 2, . . . , s −1}, Y (z)(z−α pq )κnkkq σ3
=
C\R z→α pq
O (1).
Lemma 2.1 Let Y : N ×{1, 2, . . . , K}× C \ R → SL2 (C) solve RHPFPC . For n ∈ N and k = 1, 2, . . . , K, RHPFPC possesses a unique solution given by ⎞ ⎛ ((ξ −αk )π π nk (ξ )) exp(−N V(ξ )) dξ n π k (z) (z−αk ) R ⎜(z−αk )π (ξ − αk )(ξ − z) 2π i ⎟ ⎟, z ∈ C \ R, Y (z) = ⎜ ⎝ ⎠ Y21 (ξ ) exp(−N V(ξ )) dξ Y21 (z) (z−αk ) R (ξ − αk )(ξ − z) 2π i where Y21 : N × {1, 2, . . . , K} × C \ {α1 , α2 , . . . , α K } → C denotes the (2 1)element of Y , and π nk : N × {1, 2, . . . , K} × C \ {α1 , α2 , . . . , α K } → C is the monic FPC ORF defined in Subsection 1.2. Proof Set w (z) := exp(−N V(z)), N ∈ N, where V: R \ {α1 , α2 , . . . , α K } → R satisfies conditions (3)–(5). If, for n ∈ N and k = 1, 2, . . . , K, (recall that Y (n, k; z) := Y (z)) Y : C \ R → SL2 (C) solves RHPFPC , then, from the jump condition (ii) of RHPFPC , it follows that, for the elements of the first column of Y (z), (Y j1 (z))+ = (Y j1 (z))− := Y j1 (z),
j= 1, 2,
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and, for the elements of the second column of Y (z), (Y j 2 (z))+ −(Y j 2 (z))− = Y j1 (z) w (z),
j= 1, 2.
(10)
Via the normalisation condition (iii) of RHPFPC , and the boundedness conditions (iv) and (v) of RHPFPC , it follows that, for n ∈ N and k = 1, 2, . . . , K, Y11 (z)(z−αk )κnk −1
=
C\R z→αk
Y12 (z)(z−αk )−(κnk −1)
=
C\R z→αk
Y22 (z)(z−αk )−(κnk −1) Y11 (z)z−1 Y21 (z)z−1
=
C\R z→αk
=
C\R z→αk
O (1),
Y12 (z)z
=
O (1),
Y22 (z)z
C\R z→∞
O (z−αk),
1 + O (z−αk ),
=
C\R z→∞
O (z−αk ),
(11)
κnk −1
Y21 (z)(z−αk )
1 + O (z−αk ),
=
O (1),
=
O (1),
C\R z→∞
C\R z→∞
(12)
and, for q ∈ {1, 2, . . . , s −1}, Y11 (z)(z−α pq )κnkkq Y21 (z)(z−α pq )κnkkq
=
O (1),
Y12 (z)(z−α pq )−κnkkq
=
O (1),
Y22 (z)(z−α pq )−κnkkq
C\R z→α pq
C\R z→α pq
=
O (1),
=
O (1),
C\R z→α pq
C\R z→α pq
(13) whence, temporarily re-inserting explicit n- and k-dependencies (Yij(n, k; z) := Yij(z), i, j= 1, 2), one notes that, for m = 1, 2, Ym1 : N × {1, 2, . . . , K} × C \ {α1 , α2 , . . . , α K } → C and Ym2 : N × {1, 2, . . . , K} × C \ R → C; in particular, for n ∈ N and k = 1, 2, . . . , K, Y11 (z) and Y21 (z) have no jumps throughout the z-plane, Y11 (z) is a monic (that is, coeff((z−αk )−κnk ) = 1) meromorphic function with poles at α1 , α2 , . . . , α K , and Y21 (z) is a meromorphic function with poles at α1 , α2 , . . . , α K . Application of the Sokhotski–Plemelj formula to the jump condition (10), with the Cauchy kernel normalised at αk , k = 1, 2, . . . , K, gives rise to the following Cauchy-type integral representation: (z−αk )Y j1 (ξ ) w (ξ ) dξ Y j 2 (z) = , z ∈ C \ R, j= 1, 2; (14) (ξ −α )(ξ −z) 2π i k R hence, for n ∈ N and k = 1, 2, . . . , K, Y : C \ R → SL2 (C) has the integral representation ⎛ ⎞ (z − αk )Y11 (ξ ) w (ξ ) dξ Y11 (z) ⎜ (ξ − αk )(ξ − z) 2π i ⎟ R ⎟, Y (z) = ⎜ z ∈ C \ R. ⎝ (z − αk )Y21 (ξ ) w (ξ ) dξ ⎠ Y21 (z) (ξ − αk )(ξ − z) 2π i R
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A detailed analysis for the elements of the first row of Y (z), that is, Y1m (z), m = 1, 2, is presented first; then, the corresponding analysis for the elements of the second row of Y (z), that is, Y2m (z), m = 1, 2, is presented. Recalling that (cf. Subsection 1.2) R ξ m w (ξ ) dξ < ∞ and R (ξ −α j)−(m+1) w (ξ ) dξ < ∞, m ∈ Z+ 0, zl+1 zi2 l 1 2 j= 1, 2, . . . , K, it follows via the expansion z1 −z2 = i=0 zi+1 + zl+1 (z −z ) , l ∈ Z+ 0, 1 2 1 (z−αk )Y111 (ξ ) w(ξ ) dξ the integral representation (cf. (14)) Y12 (z) = R (ξ −αk )(ξ −z) 2π i , z ∈ C \ R, and the first of each of the asymptotic conditions (11), (12), and (13) that, for n ∈ N and k = 1, 2, . . . , K, w (ξ ) Y11 (ξ ) dξ = 0, p= 0, 1, . . . , κnk −1, (15) ξ −α (ξ −αk ) p k R Y11 (ξ ) w (ξ ) dξ = 0, (16) κ ξ −α (ξ −α k k ) nk R Y11 (ξ ) w (ξ ) dξ = 0, q = 1, 2, . . . , s −1, r = 1, 2, . . . , κnkkq . (17) r ξ −α (ξ −α k pq ) R (Note: if, for n ∈ N and k = 1, 2, . . . , K, the set {αk ; k ∈ {1, 2, . . . , K}, αk = αk } = ∅, then the corresponding (17) are vacuous; actually, this can only occur if n = 1.) Recalling from the analysis preceding the integral representation (14) that, for n ∈ N and k = 1, 2, . . . , K, Y11 : N × {1, 2, . . . , K} × C \ {α1 , α2 , . . . , α K } → C is a monic (coeff((z−αk )−κnk ) = 1) meromorphic function with pole set {α1 , α2 , . . . , α K } and with no jumps throughout the z-plane, and that, for n ∈ N and k = 1, 2, . . . , K, the monic FPC ORFs, π nk (z), satisfy the orthogonality conditions stated in Subsection 1.2, it follows from these latter two observations and (15), (16), and (17) that, for n ∈ N and k = 1, 2, . . . , K, π nk (z) =
Y11 (z) . z−αk
Via (15) and (17), and this latter formula, one writes, for n ∈ N and k = 1, 2, . . . , K, (16) in a more transparent form:
w (ξ ) Y11 (ξ ) dξ κ ξ −α (ξ −α k k ) nk R Y11 (ξ ) μn,κ nk (n, k) w (ξ ) Y11 (ξ ) dξ = = ξ −αk (ξ −αk )κ nk μn,κ nk (n, k) ξ −αk R R ⎛ ⎞ n−1 k−1 K ν
(n, k) μn,κ nk (n, k) μ (n, k) m, j n,r ⎠ × ⎝φ0 (n, k)+ + + (ξ −α j )κ mj (ξ −αr )κ nr (ξ −αk )κ nk m=1 j=1 r=1 ×
w (ξ ) dξ = μn,κ nk (n, k)
= (μn,κ nk (n, k))−2
= φkn (ξ )
R
R
π nk (ξ )
φkn (ξ ) w(ξ ) dξ μn,κ nk (n, k)
= (μn,κnk (n,k))−1 φkn (ξ )
(φkn (ξ ))2 w (ξ ) dξ = (μn,κ nk (n, k))−2 ; =1
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hence, via this latter relation, (15)–(17) are written in the following, more convenient, form: for n ∈ N and k = 1, 2, . . . , K, R
π nk (ξ )(ξ −αk )− p w (ξ ) dξ = 0, R
R
p= 0, 1, . . . , κnk −1,
π nk (ξ )(ξ −αk )−κnk w (ξ ) dξ = (μn,κnk (n, k))−2 ,
(18)
(19)
π nk (ξ )(ξ −α pq )−r w (ξ ) dξ = 0, q = 1, 2, . . . , s −1, r = 1, 2, . . . , κnkkq . (20)
(Note: if, for n ∈ N and k = 1, 2, . . . , K, the set {αk ; k ∈ {1, 2, . . . , K}, αk = αk } = ∅, then the corresponding (20) are vacuous; in fact, this can only occur if n = 1.) For n ∈ N and k = 1, 2, . . . , K, (18) gives to κnk conditions, (19) srise −1 κnkkq = (n−1)K+k− κnk gives rise to 1 condition, and (20) give rise to q=1 conditions, for a total of (n−1)K+k+1 conditions, which is precisely the necessary count in order to determine, uniquely (see below), the (n- and kdependent) ‘norming constant’, μn,κnk (n, k), and the (n- and k-dependent) coefficients of the partial fraction expansion of π nk (z). One now examines, for n ∈ N and k = 1, 2, . . . , K, (18)–(20) in detail. Proceeding as per the detailed discussion of Subsection 1.2, write, for n ∈ N and k = 1, 2, . . . , K, the ordered disjoint partition for the repeated real pole sequence (with the convention ∪0m=1 {∗∗, ∗ , . . . , ∗ } := ∅):
1
n−1
n
K
K
k
{α1 , α2 , . . . , α K } ∪ · · · ∪ {α1 , α2 , . . . , α K } ∪ {α1 , α2 , . . . , αk }
:=
s −1
q=1
:=
s −1
q=1
{αi(q)kq , αi(q)kq , . . . , αi(q)kq } ∪ {αi(s)ks , αi(s)ks , . . . , αi(s)ks } ls =κni(s)k
lq =κnkkq
s
{α pq , α pq , . . . , α pq } ∪ {αk , αk , . . . , αk }, lq =κnkkq
ls =κnk
s−1 s−1 s lq = q=1 lq +ls = q=1 κnkkq + κnk = (n−1)K+k. Hence, via this where q=1 notational preamble, and the analysis leading up to the orthogonality
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conditions (18)–(20), one writes, in the indicated order, for n ∈ N and k = 1, 2, . . . , K, (cf. Subsection 1.2) κ
π nk (z)
nkkq s−1
νr,q (n, k) 1 φ0 (n, k) + = μn,κnk (n, k) μn,κnk (n, k) q=1 r=1 (z−αi(q)kq )r
1 + μn,κnk (n, k)
κni(s)k −1
s
m=1
μn,m (n, k) 1 + m (z−αi(s)ks ) (z−αk )κnk κ
nkkq s−1
νr,q (n, k) φ0 (n, k) 1 = + μn,κnk (n, k) μn,κnk (n, k) q=1 r=1 (z−α pq )r
+
κ nk −1 μn,m (n, k) 1 1 + m μn,κnk (n, k) m=1 (z−αk ) (z−αk )κnk κ
nk k1
φ0 (n, k) 1 1 νr,1 (n, k) = +· · · + + r μn,κnk (n, k) μn,κnk (n, k) r=1 (z−α p1 ) μn,κnk (n, k)
κnkk
×
κ s−1 nk −1
1 1 νr,s−1 (n, k) μn,m (n, k) + + r m (z−α ps−1 ) μn,κnk (n, k) m=1 (z−αk ) (z−αk )κnk r=1
κnkkm l =κ s−1 lm =
νm,q (n, k) s nk νs,r (n, k) 0 (n, k)+ := φ + , q (z−α pm ) (z−αk )r m=1 q=1 r=1
νs,ls (n, k) ≡ 1.
Substituting the latter partial fraction expansion of π nk (z) into the orthogonality conditions (18)–(20), one arrives at, for n ∈ N and k = 1, 2, . . . , K, the orthogonality conditions (recall that νs,ls (n, k) ≡ 1) ⎛ ⎞ κnkkm l =κ s−1 lm =
νs, j(n, k) νm,q (n, k) s nk (ξ ) ⎝ ⎠ w + dξ = 0, φ0 (n, k)+ q j (ξ −α ) (ξ −α ) (ξ −αk )r pm k R m=1 q=1 j=1 ⎛
R
⎝ φ0 (n, k)+
r = 0, 1, . . . , κnk −1, κnkkm s−1 lm =
m=1
⎛
q=1
⎞ l =κ νs, j(n, k) νm,q (n, k) s nk (ξ ) ⎠ w + dξ j κ (ξ −α pm )q (ξ −α ) (ξ −α k k ) nk j=1
= (μn,κnk (n, k))−2 ,
⎞ κnkkm l =κ s−1 lm =
νs, j(n, k) νm,q (n, k) s nk (ξ ) ⎝ ⎠ w φ0 (n, k)+ + dξ = 0, q j (ξ −α pm ) (ξ −αk ) (ξ −α pq )r R m=1 q=1 j=1 q = 1, 2, . . . , s −1,
r = 1, 2, . . . , li .
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For n ∈ N and k = 1, 2, . . . , K, the above orthogonality conditions give rise to a total of (n−1)K+k+1 linear inhomogeneous algebraic equations 0 (n, k), ν1,1 (n, k), . . . , ν1,l1 (n, k), . . . , for the (n−1)K+k+1 real unknowns φ νs−1,1 (n, k), . . . , νs−1,ls−1 (n, k), νs,1 (n, k), . . . , νs,ls −1 (n, k), (μn,κnk (n, k))−2 , that is, with dμ(z) := w (z) dz:
⎛
dμ(ξ1 ) R (ξ −α )
⎜ 1 p1 ⎜ dμ(ξ ) ⎜ R (ξ2 −α p21 )2 ⎜ ⎜ .. ⎜ . ⎜ dμ(ξ ⎜ l) ⎜ R (ξl −α p11 )l1 1 ⎜ ⎜ dμ(ξl1 +1 ) ⎜ R (ξl +1 −α p2 ) 1 ⎜ ⎜ .. ⎜ ⎜ . ⎜ dμ(ξl1 +l2 ) ⎜ R ⎜ (ξl1 +l2 −α p2 )l2 ⎜ .. ⎜ ⎜ . ⎜ dμ(ξ ⎜ R (ξ −α0 ))0 0 k ⎜ ⎜ .. ⎜ . ⎜ ⎜ dμ(ξm1 ) ⎜ R (ξm −α )κnk −1 k 1 ⎝
.. .
..
···
.. .
.. .
(ξl +l −α p2 )−l2 dμ(ξl +l ) 1 2 1 2 R (ξl +l −α p2 )l2 1 2
.. .
(ξ0 −αk )0 dμ(ξ0 ) R (ξ0 −α p2 )l2
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −α p2 )l2 (ξm2 −αk )−κnk dμ(ξm2 ) R (ξm2 −α p2 )l2
···
···
···
···
···
···
.. .. .. . . . ···
···
···
···
···
···
.. .. .. . . . ···
···
···
R
R
···
··· ···
··· ···
··· ···
(ξ0 −αk )0 dμ(ξ0 ) R (ξ0 −α p1 )l1
···
(ξ2 −α p1 )−2 dμ(ξ2 ) (ξ2 −αk )
···
.. . .. .
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −αk )
(ξm2 −αk )−κnk dμ(ξm2 ) R (ξm2 −αk )
..
.
··· ···
..
.
···
.. ..
.
··· ···
···
.. .
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −αk )κnk −1 1 (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) R (ξl +1 −αk )κnk −1 1 R
.. .
(ξl +l −α p2 )−l2 dμ(ξl +l ) 1 2 1 2 R (ξl +l −αk )κnk −1 1 2
.. .
(ξ0 −αk )0 dμ(ξ0 ) R (ξ −α )κnk −1 0 k
.. .
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −αk )κnk −1 (ξm2 −αk )−κnk dμ(ξm2 ) R (ξm2 −αk )κnk −1 R
.
···
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −αk )κnk −1 (ξ2 −α p1 )−2 dμ(ξ2 ) R (ξ −α )κnk −1 2 k
···
..
(ξm2 −αk )−κnk dμ(ξm2 ) R (ξm2 −α p2 )
R
.
.
(ξm1 −αk )−κnk +1 dμ(ξm1 ) R (ξm1 −α p2 )
.
···
···
···
..
.. .
(ξm1 −αk )−κnk +1 dμ(ξm1 ) R (ξm1 −α p1 )l1 (ξm2 −αk )−κnk dμ(ξm2 ) R (ξm2 −α p1 )l1
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −αk )
.. .
.
···
..
(ξ0 −αk )0 dμ(ξ0 ) R (ξ0 −α p2 )
.. .
(ξ0 −αk )0 dμ(ξ0 ) R (ξ0 −αk )
R
.. .
.. .
.. .
..
(ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) R (ξl +l −α p2 ) 1 2
(ξl +l −α p2 )−l2 dμ(ξl +l ) 1 2 1 2 R (ξl +l −αk ) 1 2
.. .
(ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) R (ξl +l −α p1 )l1 1 2
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −αk ) 1 (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) R (ξl +1 −αk ) 1
···
.. .
R
···
.. .. .. . . .
.. .
.. .
.. .. .. . . . ···
···
···
(ξ2 −α p1 )−2 dμ(ξ2 ) R (ξ2 −α p2 )
(ξl −α p1 )−l1 dμ(ξl ) 1 1 R (ξl −α p2 ) 1 (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) R (ξl +1 −α p2 ) 1
. .
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p2 )
R
(ξl −α p1 )−l1 dμ(ξl ) 1 1 R (ξl −α p1 )l1 1 (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) R (ξl +1 −α p1 )l1 1
···
(ξm2 −αk )−κnk dμ(ξm2 ) R (ξm2 −α p1 )
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −α p2 )l2 1 (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) R (ξl +1 −α p2 )l2 1
R
..
R
.. . .. .
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p2 )l2 (ξ2 −α p1 )−2 dμ(ξ2 ) R (ξ2 −α p2 )l2
.
···
(ξm1 −αk )−κnk +1 dμ(ξm1 ) R (ξm1 −α p1 )
R
.. .
..
(ξ0 −αk )0 dμ(ξ0 ) R (ξ0 −α p1 )
···
(ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) R (ξl +l −α p1 ) 1 2
.
···
.. .
dμ(ξm2 ) R (ξm −α )κnk k 2
..
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p1 )l1 (ξ2 −α p1 )−2 dμ(ξ2 ) R (ξ2 −α p1 )l1 R
···
(ξl −α p1 )−l1 dμ(ξl ) 1 1 R (ξl −α p1 ) 1 (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) R (ξl +1 −α p1 ) 1
···
(ξ2 −α p1 )−2 dμ(ξ2 ) R (ξ2 −α p1 )
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p1 )
R
⎞ 0 0
.. . 0 0
.. . 0
.. . 0
.. . 0 −1
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
230
K. T.-R. McLaughlin et al.
⎛
⎞
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1−αk )κnk (ξ −α p )−2 dμ(ξ ) − R 2 (ξ −α1 )κnk 2 2 k
−
R
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ .. ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (ξl1 −α p1 )−l1 dμ(ξl1 ) ⎟ ⎜ − R (ξl −αk )κnk ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 (n, k) ν 1,l 1 ⎟ ⎜ (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) ⎟ ⎜ ⎟ ⎜ − R ⎟ ⎜ κ ν (n, k) nk 2,1 (ξl +1−αk ) ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ .. . ⎟ ⎜ ⎟ ⎜ .. . ⎟=⎜ ⎟, ×⎜ −l2 dμ(ξ ⎟ ⎜ ⎟ ⎜ (ξ −α ) ) ⎜ ν2,l2 (n, k) ⎟ ⎜ − R l1 +l2(ξ p2 −α )κnkl1 +l2 ⎟ l1 +l2 k ⎟ ⎜ ⎟ ⎜ . .. ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ν (n, k) s,1 (ξ0 −αk )0 dμ(ξ0 ) ⎟ ⎜ ⎟ ⎜ − R (ξ −α )κnk ⎟ ⎜ ⎟ ⎜ . 0 k .. ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ .. ⎜ ⎟ ⎠ ⎝ νs,ls −1 (n, k) ⎜ (ξm −α )−. κnk +1 dμ(ξm ) ⎟ k 1 1 −2 ⎜ ⎟ (μn,κnk (n, k)) (ξm1 −αk )κnk ⎝− R ⎠ ⎛
0 (n, k) φ ν1,1 (n, k) .. .
⎞
−
R
(21)
(ξm2 −αk )−κnk dμ(ξm2 ) (ξm2 −αk )κnk
where m1 := (n−1)K+k−1
m2 := (n−1)K+k.
and
For n ∈ N and k = 1, 2, . . . , K, the linear system (21) of (n−1)K+k+1 inhomogeneous algebraic equations for the (n−1)K+k+1 real unknowns 0 (n, k), ν1,1 (n, k), . . . , ν1,l1 (n, k), . . . , νs−1,1 (n, k), . . . , νs−1,ls−1 (n, k), νs,1 (n, φ k), . . . , νs,ls −1 (n, k), (μn,κnk (n, k))−2 admits a unique solution if, and only if, the determinant of the coefficient matrix is non-zero: this fact will now be established; and, en route, an explicit multi-integral representation for the (n- and k-dependent) ‘norming constant’, (μn,κnk (n, k))−2 , will be derived. For n ∈ N and k = 1, 2, . . . , K, one uses Cramer’s rule to show that (μn,κnk (n, k))−2 =
cˆ N , cˆ D
where, by the multi-linearity property of the determinant, cˆ N =
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k+1
0
··· dμ(ξ
) ··· dμ(ξ
)
m2 −κnk +1 m2 κ l l 0 1 1 1 k ) (ξ1 −α p1 ) ···(ξl1 −α p1 ) 1 (ξl1 +1 −α p2 ) ···(ξl1 +l2 −α p2 ) 2 ···(ξm2 −κnk +1 −αk ) ···(ξm2 −αk ) nk
× (ξ −α
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
'1 ' ' '1 ' '. '. '. '1 ' ' '1 ' × '. '. '. '1 ' ' ' .. '. '1 ' ' '1
1 (ξ0 −α p1 )
···
1 (ξ1 −α p1 )
···
.. .
..
.
···
1 (ξl −α p1 ) 1 1 (ξl +1 −α p1 ) 1
···
.. .
..
.. .
..
1 (ξm1 −α p1 )
···
1 (ξm2 −α p1 )
···
.
···
1 (ξl +l −α p1 ) 1 2
.
1 (ξ0 −α p1 )l1 1 (ξ1 −α p1 )l1
1 (ξ0 −α p2 )
···
1 (ξ1 −α p2 )
···
1 (ξl −α p1 )l1 1 1 (ξl +1 −α p1 )l1 1
1 (ξl −α p2 ) 1 1 (ξl +1 −α p2 ) 1
1 (ξl +l −α p1 )l1 1 2
1 (ξl +l −α p2 ) 1 2
1 (ξm1 −α p1 )l1 1 (ξm2 −α p1 )l1
.. .
.. . .. .
.. .
..
1 (ξ0 −α p2 )l2 1 (ξ1 −α p2 )l2
.
··· ···
.. .
..
.. .
..
1 (ξm1 −α p2 )
···
1 (ξm2 −α p2 )
···
.
···
.
.. .
1 (ξl −α p2 )l2 1 1 (ξl +1 −α p2 )l2 1
···
1 (ξ0 −αk )
···
1 (ξ0 −αk )κnk
···
1 (ξ1 −αk )
···
..
..
···
.. .
1 (ξ1 −αk )κnk
1 (ξl −αk ) 1
···
1 (ξl −αk )κnk 1
···
1 (ξl +1 −αk ) 1
···
1 (ξl +1 −αk )κnk 1
.. .
..
.. .
..
1 (ξl +l −α p2 )l2 1 2
231
.
.
···
..
.. .
..
1 (ξl +l −αk ) 1 2
.
···
1 (ξm1 −α p2 )l2 1 (ξm2 −α p2 )l2
.. .
1 (ξm1 −αk ) 1 (ξm2 −αk )
···
.
.
···
.
.. .
.. .
1 (ξl +l −αk )κnk 1 2
.. .
···
1 (ξm1 −αk )κnk
···
1 (ξm2 −αk )κnk
' ' ' ' ' ' ' ' ' ' ' ' ' ', ' ' ' ' ' ' ' ' ' ' '
and cˆ D =
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R R (n−1)K+k
··· dμ(ξm2 −κnk +1 ) ··· dμ(ξm1 ) κ −1 l l 0 1 1 1 0 −αk ) (ξ1 −α p1 ) ···(ξl1 −α p1 ) 1 (ξl1 +1 −α p2 ) ···(ξl1 +l2 −α p2 ) 2 ···(ξm2 −κnk +1 −αk ) ···(ξm1 −αk ) nk
× (ξ
'1 ' (ξ0 −α1 p1 ) ··· (ξ0 −α1p )l1 1 ' 1 ' 1 (ξ1 −α1 p ) ··· (ξ1 −α p1 )l1 1 ' '. .. .. .. '. . '. . . 1 ' 1 (ξ −α1 ) ··· p1 ' l1 (ξl −α p1 )l1 1 ' 1 ' 1 (ξl +11−α p1 ) ··· (ξ −α l p1 ) 1 l1 +1 ' 1 ×' . .. .. .. '. . . . '. 1 ' 1 (ξ 1−α p ) ··· ' l1 +l2 1 (ξl +l −α p1 )l1 1 2 ' .. .. ' .. .. '. . . . '1 1 1 ' (ξm1 −1 −α p1 ) ··· (ξm −1 −α p1 )l1 1 ' 1 1 ' 1 (ξm −α ··· l1 p ) 1
1
(ξm1 −α p1 )
1 (ξ0 −α p2 )
···
1 (ξ1 −α p2 )
···
.. .
1 (ξl −α p2 ) 1 1 (ξl +1 −α p2 ) 1
..
··· ···
.. .
..
.. .
..
1 (ξl +l −α p2 ) 1 2
.
.
···
.
1 (ξm −1 −α p2 ) 1
···
1 (ξm1 −α p2 )
···
1 (ξ0 −α p2 )l2 1 (ξ1 −α p2 )l2
.. .
1 (ξl −α p2 )l2 1 1 (ξl +1 −α p2 )l2 1
.. .
1 (ξl +l −α p2 )l2 1 2
.. .
1 (ξm −1 −α p2 )l2 1 1 (ξm1 −α p2 )l2
' ' ' 1 ' ··· (ξ −α ··· 1 k) ' ' .. .. .. .. ' . . ' . . 1 1 ' ··· (ξ −α ) ··· l1 k (ξl −αk )κnk −1 ' 1 ' 1 ··· (ξ 1 −α ) ··· ' l1 +1 k (ξl +1 −αk )κnk −1 ' 1 '. .. .. .. .. ' . . . . ' 1 1 ' ··· (ξ ··· l1 +l2 −αk ) (ξl +l −αk )κnk −1 ' 1 2 ' .. .. ' .. .. ' . . . . ' 1 1 ··· (ξ ··· −1 κ ' −α ) m1 −1 k (ξm −1 −αk ) nk 1 ' 1 1 ' ··· (ξm −α ) ··· κnk −1 ···
1 (ξ0 −αk )
1
···
k
1 (ξ0 −αk )κnk −1 1 (ξ1 −αk )κnk −1
(ξm1 −αk )
For n ∈ N and k = 1, 2, . . . , K, cˆ N is studied first; then, cˆ D . For n ∈ N and k = 1, 2, . . . , K, introduce the following notation (cf. Subsection 1.2; recall that lm := κnkkm , m = 1, 2, . . . , s −1, ls := κnk , and 0j=1 ∗ := 0):
ϕ0 (z) :=
s −1 $
m=1
(z−α pm )lm (z−αk )κnk =:
(n−1)K+k
j=0
a j,0 z j,
232
K. T.-R. McLaughlin et al.
r−1 r−1 and, for r = 1, 2, . . . , s, q(r) = r−1 i=1 li +1, i=1 li +2, . . . , i=1 li +lr , and m(r) = 1, 2, . . . , lr , ⎧ ⎫ (n−1)K+k ⎨ ⎬
ϕq(r) (z) := ϕ0 (z)(z−α pr )−m(r) =: a j,q(r) z j ; ⎩ ⎭ j=0
e.g., for r = 1, the notation ϕq(1) (z) := {ϕ0 (z)(z−α p1 )−m(1) =: (n−1)K+k a j,q(1) z j}, j=0 q(1) = 1, 2, . . . , l1 , m(1) = 1, 2, . . . , l1 , encapsulates the l1 := κnkk1 functions ϕ0 (z) =: z−α p1
(n−1)K+k
ϕ0 (z) =: (z−α p1 )2
(n−1)K+k
ϕ1 (z) =
ϕ2 (z) =
. . . , ϕl1 (z) =
a j,1 z j,
j=0
a j,2 z j, . . .
j=0
ϕ0 (z) =: (z−α p1 )l1
(n−1)K+k
a j,l1 z j,
j=0
for r = 2, the notation ϕq(2) (z) := {ϕ0 (z)(z−α p2 )−m(2) =: (n−1)K+k a j,q(2) z j}, j=0 q(2) =l1 +1, l1 +2, . . . , l1 +l2 , m(2) = 1, 2, . . . , l2 , encapsulates the l2 := κnkk2 functions ϕl1 +1 (z) =
ϕl1 +2 (z) =
ϕ0 (z) =: z−α p2
ϕ0 (z) =: (z−α p2 )2
. . . , ϕl1 +l2 (z) =
(n−1)K+k
a j,l1 +1 z j,
j=0
(n−1)K+k
a j,l1 +2 z j, . . .
j=0
ϕ0 (z) =: (z−α p2 )l2
(n−1)K+k
a j,l1 +l2 z j,
j=0
s−1 etc., and, for r = s (cf. Subsection 1.2; recall that α ps := αk and i=1 li = (n − 1) (n−1)K+k −m(s) ·K + k− κnk ), the notation ϕq(s) (z) := {ϕ0 (z)(z−αk ) =: j=0 a j,q(s) z j}, q(s) = (n−1)K+k− κnk + 1, (n−1)K+k− κnk +2, . . . , (n− 1)K + k, m(s) = 1, 2, . . . , ls , encapsulates the ls := κnk functions ϕ0 (z) =: z−αk
(n−1)K+k
ϕ0 (z) =: (z−αk )2
(n−1)K+k
ϕ0 (z) =: (z−αk )κnk
(n−1)K+k
ϕ(n−1)K+k−κnk +1 (z) =
ϕ(n−1)K+k−κnk +2 (z) =
. . . , ϕ(n−1)K+k (z) =
a j,(n−1)K+k−κnk +1 z j,
j=0
a j,(n−1)K+k−κnk +2 z j, . . .
j=0
j=0
a j,(n−1)K+k z j.
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
233
s (Note: #{ϕ0 (z)(z − α pr )−m(r) } = lr , r = 1, 2, . . . , s, and # ∪r=1 {ϕ0 (z)(z − s −m(r) α pr ) } = r=1 lr = (n−1)K+k.) One notes that, for n ∈ N and k = 1, 2, . . . , K, the l1 +l2 +· · ·+ls +1 = (n−1)K+k+1 functions ϕ0 (z), ϕ1 (z), . . . , ϕl1 (z), ϕl1 +1 (z),. . . , ϕl1 +l2 (z),. . . , ϕ(n−1)K+k−κnk +1 (z),. . . , ϕ(n−1)K+k (z) are linear(n−1)K+k ly independent on R, that is, for z ∈ R, c jϕ j(z) = 0 ⇒ (via a j=0 Vandermonde-type argument) c j = 0, j= 0, 1, . . . , (n−1)K+k (see the ((n−1)K+k+1) × ((n−1)K+k+1) non-zero determinant D in (22) below). For n ∈ N and k = 1, 2, . . . , K, let S(n−1)K+k+1 denote the ((n−1)K+k+1)! permutations of {0, 1, . . . , (n−1)K+k}. Using the above notation and the multi-linearity property of the determinant, one studies, thus, for n ∈ N and k = 1, 2, . . . , K, cˆ N (see, also, [70]):
cˆ N =
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k+1
× ×
··· dμ(ξm2 −κnk +1 ) ··· dμ(ξm2 )
(ξ0 −αk )0 (ξ1 −α p1 )1 ···(ξl1 −α p1 )l1 (ξl1 +1 −α p2 )1 ···(ξl1 +l2 −α p2 )l2 ···(ξm2 −κnk +1 −αk )1 ···(ξm2 −αk )κnk
1 ϕ0 (ξ0 )ϕ0 (ξ1 ) · · · ϕ0 (ξl1 )ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξm2 )
' ϕ0 (ξ0 ) ϕ1 (ξ0 ) ··· ϕl (ξ0 ) ϕl +1 (ξ0 ) ··· ϕl +l (ξ0 ) ··· ϕm −κ +1 (ξ0 ) ··· ϕ(n−1)K+k (ξ0 ) ' 2 ' ' nk 1 1 1 2 ' ϕ0 (ξ1 ) ϕ1 (ξ1 ) ··· ϕl1 (ξ1 ) ϕl1 +1 (ξ1 ) ··· ϕl1 +l2 (ξ1 ) ··· ϕm2 −κnk +1 (ξ1 ) ··· ϕ(n−1)K+k (ξ1 ) ' ' ' ' ' .. .. .. .. .. .. .. .. .. .. .. ' ' . . . . ' ' . . . . . . . ' ϕ0 (ξl1 ) ϕ1 (ξl1 ) ··· ϕl1 (ξl1 ) ϕl1 +1 (ξl1 ) ··· ϕl1 +l2 (ξl1 ) ··· ϕm2 −κnk +1 (ξl1 ) ··· ϕ(n−1)K+k (ξl1 ) ' ' ' ' ϕ0 (ξl1 +1 ) ϕ1 (ξl1 +1 ) ··· ϕl1 (ξl1 +1 ) ϕl1 +1 (ξl1 +1 ) ··· ϕl1 +l2 (ξl1 +1 ) ··· ϕm2 −κnk +1 (ξl1 +1 ) ··· ϕ(n−1)K+k (ξl1 +1 ) ' ' ' ×' ' .. .. .. .. .. .. .. .. .. .. .. ' ' . . . . . . . . . . . ' ' ' ϕ0 (ξl1 +l2 )ϕ1 (ξl1 +l2 ) ··· ϕl1 (ξl1 +l2 )ϕl1 +1 (ξl1 +l2 ) ··· ϕl1 +l2 (ξl1 +l2 ) ··· ϕm2 −κnk +1 (ξl1 +l2 ) ··· ϕ(n−1)K+k (ξl1 +l2 ) ' ' ' ' ' .. .. .. .. .. .. .. .. .. .. .. ' ' . . . . . . . . . . . ' ' ' ϕ0 (ξm1 ) ϕ1 (ξm1 ) ··· ϕl1 (ξm1 ) ϕl1 +1 (ξm1 ) ··· ϕl1 +l2 (ξm1 ) ··· ϕm2 −κnk +1 (ξm1 ) ··· ϕ(n−1)K+k (ξm1 ) ' ' ' ϕ0 (ξm2 ) ϕ1 (ξm2 ) ··· ϕl1 (ξm2 ) ϕl1 +1 (ξm2 ) ··· ϕl1 +l2 (ξm2 ) ··· ϕm2 −κnk +1 (ξm2 ) ··· ϕ(n−1)K+k (ξm2 ) =: G(ξ0 ,ξ1 ,...,ξl1 ,ξl1 +1 ,...,ξl1 +l2 ,...,ξm2 −κnk +1 ,...,ξ(n−1)K+k )
=
1 (m2 +1)!
σ ∈S m2 +1 R
· · · dμ(ξσ (0) ) dμ(ξσ (1) ) · · · dμ(ξσ (l1 ) ) dμ(ξσ (l1 +1) ) · · · R R
(n−1)K+k+1
× ×
···dμ(ξσ (l1 +l2 ) ) ···dμ(ξσ (m2 −κnk +1) )···dμ(ξσ (m2 ) )
(ξσ (0) −αk )0 (ξσ (1) −α p1 )1 ···(ξσ (l1 ) −α p1 )l1 (ξσ (l1 +1) −α p2 )1 ···(ξσ (l1 +l2 ) −α p2 )l2 ···(ξσ (m2 −κnk+1) −αk )1 ···(ξσ (m2 ) −αk )κnk
1 ϕ0 (ξσ (0) )ϕ0 (ξσ (1) ) · · · ϕ0 (ξσ (l1 ) )ϕ0 (ξσ (l1 +1) ) · · · ϕ0 (ξσ (l1 +l2 ) ) · · · ϕ0 (ξσ (m2 −κ nk +1) ) · · · ϕ0 (ξσ (m2 ) )
×G(ξσ (0) , ξσ (1) ,. . . ,ξσ (l1 ) , ξσ (l1+1) ,. . . ,ξσ (l1 +l2 ) ,. . . ,ξσ (m2 −κ nk +1) ,. . . ,ξσ ((n−1)K+k) ) =
1 (m2 +1)!
R
· · · dμ(ξ0 )dμ(ξ1 ) · · · dμ(ξl1 )dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · dμ(ξm2 −κ nk +1 ) · · · R R
(n−1)K+k+1
234
K. T.-R. McLaughlin et al. ×· · · dμ(ξm2 )G(ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξm2 −κ nk +1 , . . . , ξ(n−1)K+k )
σ) sgn(σ
σ ∈S m2 +1
×
1 (ξσ (0) −αk )0 (ξσ (1) −α p1 )1 ···(ξσ (l1 ) −α p1 )l1 (ξσ (l1 +1) −α p2 )1 ···(ξσ (l1 +l2 ) −α p2 )l2 ···(ξσ (m2 −κnk +1) −αk )1 ···(ξσ (m2 ) −αk )κnk
1 ϕ0 (ξσ (0) )ϕ0 (ξσ (1) )· · ·ϕ0 (ξσ (l1 ) )ϕ0 (ξσ (l1 +1) )· · ·ϕ0 (ξσ (l1 +l2 ) )· · ·ϕ0 (ξσ (m2 −κ nk +1) )· · ·ϕ0 (ξσ (m2 ) ) 1 = · · · dμ(ξ0 ) dμ(ξ1 )· · ·dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 )· · ·dμ(ξm2 −κ nk +1 )· · · (m2 +1)! R R R
×
(n−1)K+k+1
×· · · dμ(ξm2 ) G(ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξm2 −κ nk +1 , . . . , ξ(n−1)K+k ) ' (ϕ (ξ ))−1 (ϕ0 (ξ0 ))−1 (ϕ0 (ξ0 ))−1 (ϕ0 (ξ0 ))−1 (ϕ0 (ξ0 ))−1 (ϕ0 (ξ0 ))−1 (ϕ0 (ξ0 ))−1 ' 0 0 ··· ··· ··· ··· (ξ0 −α p1 ) (ξ0 −α p2 ) (ξ0 −αk ) ' (ξ0 −αk )0 (ξ0 −αk )κnk (ξ0 −α p1 )l1 (ξ0 −α p2 )l2 ' ' (ϕ0 (ξ1 ))−1 (ϕ0 (ξ1 ))−1 (ϕ0 (ξ1 ))−1 (ϕ0 (ξ1 ))−1 (ϕ0 (ξ1 ))−1 (ϕ0 (ξ1 ))−1 (ϕ0 (ξ1 ))−1 ··· ··· ··· ··· ' (ξ −α )0 (ξ1 −α p1 ) (ξ1 −α p2 ) (ξ1 −αk ) (ξ1 −αk )κnk (ξ1 −α p1 )l1 (ξ1 −α p2 )l2 1 k ' ' .. .. .. .. .. .. .. ' .. .. .. .. ' . . . . . . . . . . . ' (ϕ0 (ξl ))−1 (ϕ0 (ξl ))−1 (ϕ0 (ξl ))−1 (ϕ0 (ξl ))−1 (ϕ0 (ξl ))−1 ' (ϕ0 (ξl1 ))−1 (ϕ0 (ξl1 ))−1 1 1 1 1 1 ' (ξ −α )0 ··· ··· ··· ··· κ (ξl −α p1 ) (ξ (ξ −α ) −α ) l l p (ξ −α l1 l1 k (ξl −α p1 ) 1 (ξl −α p2 ) 2 2 ' l1 k l1 k ) nk 1 1 1 ' (ϕ0 (ξl +1 ))−1 (ϕ0 (ξl +1 ))−1 (ϕ0 (ξl +1 ))−1 (ϕ0 (ξl +1 ))−1 (ϕ0 (ξl +1 ))−1 ' (ϕ0 (ξl1 +1 ))−1 (ϕ0 (ξl1 +1 ))−1 1 1 1 1 ··· ··· (ξ 1 −α ) ··· ' (ξ −α )0 (ξl +1 −α p ) ··· (ξl +1 −αk )κnk l1 +1 k (ξl +1 −α p1 )l1 (ξl1 +1 −α p2 ) (ξl +1 −α p2 )l2 1 ' l +1 k 1 1 1 1 ×' 1 ' .. .. .. .. .. .. .. .. .. .. .. ' . . . . . . . . . . . ' −1 −1 −1 −1 −1 −1 ' (ϕ0 (ξl +l )) (ϕ0 (ξl +l )) (ϕ0 (ξl +l )) (ϕ0 (ξl +l )) (ϕ0 (ξl +l )) (ϕ0 (ξl +l ))−1 (ϕ0 (ξl +l )) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ' ··· ··· ··· ··· (ξl +l −αk ) (ξl +l −αk )κnk ' (ξl1 +l2 −αk )0 (ξl1 +l2 −α p1 ) (ξl +l −α p1 )l1 (ξl1 +l2 −α p2 ) (ξl +l −α p2 )l2 1 2 1 2 1 2 1 2 ' ' .. .. .. .. .. .. .. .. .. .. .. ' . . . . ' . . . . . ' (ϕ0 (ξm. ))−1 (ϕ0 (ξm. ))−1 −1 −1 −1 −1 (ϕ0 (ξm1 )) (ϕ0 (ξm1 )) (ϕ0 (ξm1 )) (ϕ0 (ξm1 ))−1 (ϕ0 (ξm1 )) 1 1 ' ' (ξm1 −αk )0 (ξm1 −α p1 ) ··· (ξm1 −α p1 )l1 (ξm1 −α p2 ) ··· (ξm1 −α p2 )l2 ··· (ξm1 −αk ) ··· (ξm1 −αk )κnk ' ' (ϕ0 (ξm2 ))−1 (ϕ0 (ξm2 ))−1 (ϕ (ξm2 ))−1 (ϕ0 (ξm2 ))−1 (ϕ (ξm2 ))−1 (ϕ0 (ξm2 ))−1 (ϕ0 (ξm2 ))−1 ··· ··· (ξ0m −α ··· ' (ξm −α )0 (ξm −α p ) ··· (ξ 0 −α (ξm −α p ) (ξm −α )κnk k) )l1 (ξ −α )l2 2
=
k
1 (m2 +1)!
2
R
1
m2
2
p1
2
m2
2
p2
2
k
' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R
R
(n−1)K+k+1
×
· · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξm2 )G(ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξm2 −κ nk +1 , . . . , ξ(n−1)K+k ) ϕ02 (ξ0 )ϕ02 (ξ1 ) · · · ϕ02 (ξl1 )ϕ02 (ξl1 +1 ) · · · ϕ02 (ξl1 +l2 ) · · · ϕ02 (ξm2 −κ nk +1 ) · · · ϕ02 (ξm2 )
' ϕ0 (ξ0 ) ϕ1 (ξ0 ) ' ' ϕ0 (ξ1 ) ϕ1 (ξ1 ) ' ' . .. ' .. ' . ' ϕ0 (ξl1 ) ϕ1 (ξl1 ) ' ' ϕ0 (ξl1 +1 ) ϕ1 (ξl1 +1 ) ' ×' .. .. ' . . ' ' ϕ0 (ξl1 +l2 ) ϕ1 (ξl1 +l2 ) ' ' .. .. ' . . ' ' ϕ0 (ξm1 ) ϕ1 (ξm1 ) ' ϕ0 (ξm2 ) ϕ1 (ξm2 )
··· ···
..
ϕl1 (ξ0 ) ϕl1 (ξ1 )
.
.. .
.
.. .
ϕl1 +1 (ξ0 ) ϕl1 +1 (ξ1 )
··· ···
.. .
..
.. .
..
.. .
..
ϕl1 +l2 (ξ0 ) ϕl1 +l2 (ξ1 )
··· ···
.
.. .
..
.
.. .
..
.. .
..
ϕm2 −κnk +1 (ξ0 ) ϕm2 −κnk +1 (ξ1 )
··· ···
.
.. .
..
.
.. .
..
.. .
..
..
1 (m2 +1)!
R
.
··· ϕl1 (ξl1 +l2 ) ϕl1 +1 (ξl1 +l2 ) ··· ϕl1 +l2 (ξl1 +l2 ) ··· ϕm2 −κnk +1 (ξl1 +l2 ) ···
..
.
··· ···
.. .
ϕl1 (ξm1 ) ϕl1 (ξm2 )
ϕl1 +1 (ξm1 ) ϕl1 +1 (ξm2 )
.
··· ···
ϕl1 +l2 (ξm1 ) ϕl1 +l2 (ξm2 )
.
··· ···
ϕm2 −κnk +1 (ξm1 ) ϕm2 −κnk +1 (ξm2 )
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k+1
.
··· ···
= G(ξ0 ,ξ1 ,...,ξl1 ,ξl1 +1 ,...,ξl1 +l2 ,...,ξm2 −κnk +1 ,...,ξ(n−1)K+k )
=
.
··· ϕl1 (ξl1 ) ϕl1 +1 (ξl1 ) ··· ϕl1 +l2 (ξl1 ) ··· ϕm2 −κnk +1 (ξl1 ) ··· ··· ϕl1 (ξl1 +1 ) ϕl1 +1 (ξl1 +1 ) ··· ϕl1 +l2 (ξl1 +1 ) ··· ϕm2 −κnk +1 (ξl1 +1 ) ···
' ' ' ' ' .. ' ' . ϕ(n−1)K+k (ξl1 ) ' ' ϕ(n−1)K+k (ξl1 +1 ) ' ' ' .. ' . ' ϕ(n−1)K+k (ξl1 +l2 ) ' ' ' .. ' . ' ϕ(n−1)K+k (ξm1 ) ' ' ϕ(n−1)K+k (ξm2 ) ϕ(n−1)K+k (ξ0 ) ϕ(n−1)K+k (ξ1 )
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
235
×· · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξm2 ) ×
G(ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξm2 −κ nk +1 , . . . , ξ(n−1)K+k ) ϕ0 (ξ0 )ϕ0 (ξ1 ) · · · ϕ0 (ξl1 )ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξm2 )
2 ;
but, noting the determinantal factorisation G(ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξm2 −κnk +1 , . . . , ξ(n−1)K+k )
= D V1 (ξ0 , ξ1 , . . . , ξ(n−1)K+k ), where
' a0,0 ' a ' 0,1 ' ' .. ' . ' a ' 0,l1 ' a0,l1 +1 ' D := '' . ' .. ' a0,l1 +l2 ' ' . ' . ' . ' a0,m1 ' a
a1,0 a1,1
··· ···
.. .
..
a1,l1
a1,l1 +1
al1 ,0 al1 ,1
.
.. .
.
.. .
··· ···
al1 +1,0 al1 +1,1
.. .
..
.. .
..
.. .
..
al1 +l2 ,0 al1 +l2 ,1
··· ···
.
.. .
..
.
.. .
..
.. .
..
··· ···
am2 −κnk +1,0 am2 −κnk +1,1
.
.. .
..
.
.. .
..
.. .
..
.
··· al1 ,l1 al1 +1,l1 ··· al1 +l2 ,l1 ··· am2 −κnk +1,l1 ··· ··· al1 ,l1 +1 al1 +1,l1 +1 ··· al1 +l2 ,l1 +1 ··· am2 −κnk +1,l1 +1 ···
.. .
..
.. .
..
.
a1,l1 +l2 ··· al1 ,l1 +l2 al1 +1,l1 +l2 ··· al1 +l2 ,l1 +l2 ··· am2 −κnk +1,l1 +l2 ···
0,m2
.. .
.
··· ···
a1,m1 a1,m2
al1 ,m1 al1 ,m2
.
··· ···
al1 +1,m1 al1 +1,m2
al1 +l2 ,m1 al1 +l2 ,m2
.
··· ···
am2 −κnk +1,m1 am2 −κnk +1,m2
.
··· ···
' ' ' ' .. ' ' . ' a(n−1)K+k,l1 ' a(n−1)K+k,l1 +1 ' ' ' ( = 0), .. ' ' . a(n−1)K+k,l1 +l2 ' ' ' .. ' ' . a(n−1)K+k,m1 ' ' a a(n−1)K+k,0 a(n−1)K+k,1
(n−1)K+k,m2
(22) and V1 (ξ0 , ξ1 , . . . , ξ(n−1)K+k ) := ' 1 ' ξ0 ' ' . ' . ' . ' l1 ' ξ0 ' ' ξ l1 +1 ' 0 ' ' . ' . ' . ' ξ l1 +l2 ' 0 ' ' . ' .. ' m ' ξ0 1 ' ' ξ m2 0
1 ξ1
··· ···
.. .
..
l
···
l +1 ξ11
···
.. .
..
l +l2
.. .
m ξ1 1 m ξ1 2
.. .
.
ξ11
ξ11
1 ξl1
l
ξl 1
1 l +1 ξl 1 1
.
.. .
.
··· ···
l
.. .
..
.
.. .
l
..
.
l
ξl 1+1 ··· ξl 1+l
··· ξm1
···
··· ξm1
1 l +1 ξl 1+1 1
.. .
..
.
.. .
..
m1 ξl +1 1 m2 ξl +1 1
.. .
l +l
.
m ξl 1 1 m ξl 2 1
1
1
..
.
··· ···
.. .
..
..
.
··· ···
.
··· ···
.. .
..
.. .
..
l +l2 2 −κnk +1
2
m1 ξl +l 1 2 m2 ξl +l 1 2
.. .
2 −κnk +1 l +1 2 −κnk +1
ξl 1+12 ··· ξl 1+l 2 ··· ξm1
.. .
l +l
1 2 l +1 ξl 1+l 1 2
l +l2 1
··· ξl 1
..
' 1 ··· 1 ··· 1 ··· 1 ξl1 +1 ··· ξl1 +l2 ··· ξm2 −κnk +1 ··· ξ(n−1)K+k ''
.
···
m ξm 1−κ +1 2 nk m ξm 2−κ +1 2 nk
.
··· ···
' ' ' ' l1 ξ(n−1)K+k ' ' l1 +1 ' ξ(n−1)K+k ' ' (n−1)K+k '= .. (ξi −ξ j ), i, j=0 ' . ' j
(n−1)K+k
it follows that, for n ∈ N and k = 1, 2, . . . , K, cˆ N =
1 (m2 +1)!
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k+1
· · · dμ(ξm2 −κnk +1 ) · · · dμ(ξm2 )D2 ×
=
(n−1)K+k i, j=0
(ξi −ξ j)2
j
(ϕ0 (ξ0 )ϕ0 (ξ1 )· · ·ϕ0 (ξl1 )ϕ0 (ξl1+1 )· · ·ϕ0 (ξl1+l2 ) · · · ϕ0 (ξm2−κnk+1 )· · ·ϕ0 (ξ(n−1)K+k ))2 D2 ((n−1)K+k+1)!
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · R R
(n−1)K+k+1
236
K. T.-R. McLaughlin et al.
× · · · dμ(ξl1 +l2 ) · · · dμ(ξ(n−1)K+k−κnk +1 ) · · · dμ(ξ(n−1)K+k )
(n−1)K+k $
×
⎛
(n−1)K+k $
(ξi −ξ j)2 ⎝
⎞−2 ϕ0 (ξm )⎠
m=0
i, j=0 j
=
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · R R
D2 ((n−1)K+k+1)!
R
(n−1)K+k+1
× · · · dμ(ξl1 +l2 ) · · · dμ(ξ(n−1)K+k−κnk +1 ) · · · dμ(ξ(n−1)K+k )
(n−1)K+k $
(ξi −ξ j)2
i, j=0 j
⎛ ×⎝
(n−1)K+k −1 $ s$
(ξl −α pq )
(ξl −αk )κnk ⎠
⇒
q=1
l=0
cˆ N =
⎞−2 κnk kq
D2 ((n−1)K+k+1)!
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξ(n−1)K+k ) R R
R
(n−1)K+k+1
×
(n−1)K+k $
⎛
(n−1)K+k −1 $ s$
(ξi −ξ j)2 ⎝
m=0
i, j=0
⎞−2 (ξm −α pq )
κnk kq
(ξm −αk )κnk⎠
(> 0).
(23)
q=1
j
Now, cˆ D is studied. For n ∈ N and k = 1, 2, . . . , K, introduce the following notation:
ϕ0 (z) :=
s$ −1
(z−α pm )lm (z−αk )κnk −1 =:
(n−1)K+k−1
m=1
and, for r = 1, 2, . . . , s, q(r) = m(r) = 1, 2, . . . , lr −δrs ,
a j,0 z j,
j=0
r−1
i=1 li +1,
r−1
i=1 li +2, . . . ,
r−1
i=1 li +lr −δrs ,
⎧ ⎫ (n−1)K+k−1 ⎨ ⎬
−m(r) j ϕq(r) (z) := ϕ0 (z)(z−α pr ) =: a j,q(r) z ; ⎩ ⎭ j=0
and
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
237
(n−1)K+k−1 e.g., for r = 1, the notation ϕq(1) (z) := { ϕ0 (z)(z−α p1 )−m(1) =: j=0 a j,q(1) z j }, q(1) = 1, 2, . . . , l1 , m(1) = 1, 2, . . . , l1 , encapsulates the l1 := κnkk1 functions
ϕ0 (z) =: z−α p1
(n−1)K+k−1
ϕ0 (z) =: (z−α p1 )2
(n−1)K+k−1
ϕ0 (z) =: (z−α p1 )l1
(n−1)K+k−1
ϕ1 (z) =
ϕ2 (z) =
..., ϕl1 (z) =
a j,1 z j ,
j=0
a j,2 z j , . . .
j=0
a j,l1 z j ,
j=0
for r = 2, the notation ϕq(2) (z) := { ϕ0 (z)(z−α p2 )−m(2) =: (n−1)K+k−1 a j,q(2) z j}, j=0 q(2) =l1 +1, l1 +2, . . . , l1 +l2 , m(2) = 1, 2, . . . , l2 , encapsulates the l2 := κnkk2 functions ϕl1 +1 (z) =
ϕl1 +2 (z) =
. . ., ϕl1 +l2 (z) =
ϕ0 (z) =: z−α p2
(n−1)K+k−1
a j,l1 +1 z j ,
j=0
ϕ0 (z) =: (z−α p2 )2
(n−1)K+k−1
ϕ0 (z) =: (z−α p2 )l2
(n−1)K+k−1
a j,l1 +2 z j , . . .
j=0
a j,l1 +l2 z j ,
j=0
r = s, the notation ϕq(s) (z) := { ϕ0 (z)(z−αk )−m(s) =: q(s) = (n−1)K+k− κnk +1, (n−1)K+k− κnk +2, . . . , j=0 (n−1)K+k−1, m(s) = 1, 2, . . . , ls −1, encapsulates the ls −1 = κnk −1 functions
etc., and, (n−1)K+k−1
for
a j,q(s) z j},
ϕ(n−1)K+k−κ nk +1 (z) =
ϕ(n−1)K+k−κ nk +2 (z) =
..., ϕ(n−1)K+k−1(z) =
ϕ0 (z) =: z−αk
(n−1)K+k−1
a j,(n−1)K+k−κ
nk +1
z j,
j=0 (n−1)K+k−1
ϕ0 (z) =: (z−αk )2
a j,(n−1)K+k−κ
nk +2
z j, . . .
j=0
ϕ0 (z) =: (z−αk )κ nk −1
(n−1)K+k−1
a j,(n−1)K+k−1 z j .
j=0
s (Note: #{ ϕ0 (z)(z−α pr )−m(r) } =lr −δrs , r = 1, 2, . . . , s, and # ∪r=1 { ϕ0 (z)(z− s −m(r) } = r=1 lr − 1 = (n−1)K+k−1.) One notes that, for n ∈ N and α pr ) k = 1, 2, . . . , K, the l1 +l2 +· · ·+ (ls −1)+1 = (n−1)K+k functions ϕ0 (z), ϕ1 (z), ..., ϕl1 (z), ϕl1 +1 (z), . . . , ϕl1 +l2 (z), . . . , ϕ(n − 1)K + k − κnk + 1 (z), . . . , ϕ(n − 1)K + k − 1 (z) are linearly independent on R, that is, for z ∈ R, (n−1)K+k−1 c j ϕ j(z) = 0 ⇒ j=0
238
K. T.-R. McLaughlin et al.
(via a Vandermonde-type argument) c j = 0, j= 0, 1, . . . , (n−1)K+k−1 (see the ((n−1)K+k) × ((n−1)K+k) non-zero determinant D in (24) below). For n ∈ N and k = 1, 2, . . . , K, let S(n−1)K+k denote the ((n−1)K+k)! permutations of {0, 1, . . . , (n−1)K+k−1}. Using the above notation and the multilinearity property of the determinant, one proceeds to study, for n ∈ N and k = 1, 2, . . . , K, cˆ D :
cˆ D =
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k
× ×
··· dμ(ξm2 −κnk +1 ) ··· dμ(ξm1 )
(ξ0 −αk )0 (ξ1 −α p1 )1 ···(ξl1 −α p1 )l1 (ξl1 +1 −α p2 )1 ···(ξl1 +l2 −α p2 )l2 ···(ξm2 −κnk +1 −αk )1 ···(ξm1 −αk )κnk −1
1 ϕ0 (ξ0 ) ϕ0 (ξ1 ) · · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κnk +1 ) · · · ϕ0 (ξm1 )
' ϕ1 (ξ0 ) ' ϕ0 (ξ0 ) ' ϕ1 (ξ1 ) ' ϕ0 (ξ1 ) ' . . ' . . ' . . ' ϕ0 (ξl1 ) ϕ1 (ξl1 ) ' ' ϕ (ξ ) ϕ (ξ ) 0 1 l +1 l ' 1 1 +1 ×'' . . . . ' . . ' ϕ0 (ξl1 +l2 ) ϕ1 (ξl1 +l2 ) ' ' ' . . ' . . ' . . ' ϕ0 (ξm1 −1 ) ϕ1 (ξm1 −1 ) ' ϕ0 (ξm1 ) ϕ1 (ξm1 )
···
ϕl1 (ξ0 )
ϕl1 +1 (ξ0 )
···
ϕl1 +l2 (ξ0 )
···
ϕm2 −κnk +1 (ξ0 )
···
···
ϕl1 (ξ1 )
ϕl1 +1 (ξ1 )
···
ϕl1 +l2 (ξ1 )
···
···
..
.
. . .
. . .
..
.
. . .
ϕm2 −κnk +1 (ξ1 )
..
.
. . .
. . .
..
.
. . .
..
.
. . .
. . .
..
.
. . .
..
.
. . .
..
.
. . .
..
.
. . .
..
.
··· ϕl1 (ξl1 ) ϕl1 +1 (ξl1 ) ··· ϕl1 +l2 (ξl1 ) ··· ϕm2 −κnk +1 (ξl1 ) ··· ··· ϕl1 (ξl1 +1 ) ϕl1 +1 (ξl1 +1 ) ··· ϕl1 +l2 (ξl1 +1 ) ··· ϕm2 −κnk +1 (ξl1 +1 ) ···
..
.
··· ϕl1 (ξl1 +l2 ) ϕl1 +1 (ξl1 +l2 ) ··· ϕl1 +l2 (ξl1 +l2 ) ··· ϕm2 −κnk +1 (ξl1 +l2 ) ···
..
.
··· ϕl1 (ξm1 −1 ) ϕl1 +1 (ξm1 −1 ) ··· ϕl1 +l2 (ξm1 −1 ) ··· ϕm2 −κnk +1 (ξm1 −1 ) ··· ··· ϕl1 (ξm1 ) ϕl1 +1 (ξm1 ) ··· ϕl1 +l2 (ξm1 ) ··· ϕm2 −κnk +1 (ξm1 ) ···
' ' ' ' ' . ' . ' . ' ϕ(n−1)K+k−1 (ξl1 ) ' ϕ(n−1)K+k−1 (ξl1 +1 ) '' ' . ' . ' . ' ϕ(n−1)K+k−1 (ξl1 +l2 ) ' ' ' . ' . ' . ϕ(n−1)K+k−1 (ξm1 −1 ) ' ' ϕ(n−1)K+k−1 (ξm1 ) ϕ(n−1)K+k−1 (ξ0 ) ϕ(n−1)K+k−1 (ξ1 )
(ξ0 ,ξ1 ,...,ξl ,ξl +1 ,...,ξl +l ,...,ξm −κ +1 ,...,ξ(n−1)K+k−1 ) =: G 2 nk 1 1 1 2
=
1 · · · dμ(ξσ (0) ) dμ(ξσ (1) ) · · · dμ(ξσ (l1 ) )dμ(ξσ (l1 +1) ) · · · m2 ! R σ ∈S m2 R R (n−1)K+k
× ×
··· dμ(ξσ (l1 +l2 ) )··· dμ(ξσ (m2 −κnk +1) ) ··· dμ(ξσ (m1 ) )
(ξσ (0) −αk )0 (ξσ (1) −α p1 )1 ···(ξσ (l1 ) −α p1 )l1 (ξσ (l1 +1) −α p2 )1 ···(ξσ (l1 +l2 ) −α p2 )l2 ···(ξσ (m2 −κnk +1) −αk )1 ···(ξσ (m1 ) −αk )κnk −1
1 ϕ0 (ξσ (0) ) ϕ0 (ξσ (1) ) · · · ϕ0 (ξσ (l1 ) ) ϕ0 (ξσ (l1 +1) ) · · · ϕ0 (ξσ (l1 +l2 ) ) · · · ϕ0 (ξσ (m2 −κnk +1) ) · · · ϕ0 (ξσ (m1 ) )
(ξσ (0) , ξσ (1) , . . . , ξσ (l ) , ξσ (l +1) , . . . , ξσ (l +l ) , . . . , ξσ (m −κ +1) , . . . , ξσ ((n−1)K+k−1) ) ×G 2 1 1 1 2 nk
1 = m2 !
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R R (n−1)K+k
,ξ ,...,ξ ,ξ × · · · dμ(ξm −κ +1 ) · · · dμ(ξm1 ) G(ξ 0 1 l1 l1 +1 , . . . , ξl1 +l2 , . . . , ξm2 −κ nk +1 , . . . , ξ(n−1)K+k−1 ) 2 nk
×
σ) sgn(σ
σ ∈Sm2
×
1 (ξσ (0) −αk )0 (ξσ (1) −α p1 )1 ···(ξσ (l1 ) −α p1 )l1 (ξσ (l1 +1) −α p2 )1 ···(ξσ (l1 +l2 ) −α p2 )l2 ···(ξσ (m2 −κnk +1) −αk )1 ···(ξσ (m1 ) −αk )κnk −1
1 ϕ0 (ξσ (0) ) ϕ0 (ξσ (1) ) · · · ϕ0 (ξσ (l1 ) ) ϕ0 (ξσ (l1 +1) ) · · · ϕ0 (ξσ (l1 +l2 ) ) · · · ϕ0 (ξσ (m2 −κnk +1) ) · · · ϕ0 (ξσ (m1 ) )
1 = m2 !
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R R (n−1)K+k
,ξ ,...,ξ ,ξ × · · · dμ(ξm −κ +1 ) · · · dμ(ξm1 ) G(ξ 0 1 l1 l1 +1 , . . . , ξl1 +l2 , . . . , ξm2 −κ nk +1 , . . . , ξ(n−1)K+k−1 ) 2 nk
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal... ' (ϕ (ξ ))−1 ' 0 0 0 ' (ξ0 −αk ) ' ' (ϕ (ξ ))−1 ' 0 1 ' (ξ1 −αk )0 ' ' ' . ' . . ' ' (ϕ (ξ ))−1 ' 0 l1 ' (ξl −αk )0 ' 1 ' ' (ϕ0 (ξl1 +1 ))−1 ' ' (ξl1 +1 −αk )0 × '' . ' . ' . ' ' (ϕ0 (ξl +l ))−1 ' 1 2 ' (ξl1 +l2 −αk )0 ' ' ' . ' . . ' ' '(ϕ0 (ξm1 −1 ))−1 ' (ξm −1 −α )0 k ' 1 ' ' (ϕ0 (ξm1 ))−1 ' (ξm −α )0 =
1 m2 !
···
( ϕ0 (ξ0 ))−1 (ξ0 −α p1 )l1
( ϕ0 (ξ0 ))−1 (ξ0 −α p2 )
···
( ϕ0 (ξ0 ))−1 (ξ0 −α p2 )l2
( ϕ0 (ξ1 ))−1 (ξ1 −α p1 )
···
( ϕ0 (ξ1 ))−1 (ξ1 −α p1 )l1
( ϕ0 (ξ1 ))−1 (ξ1 −α p2 )
···
( ϕ0 (ξ1 ))−1 (ξ1 −α p2 )l2
. . .
..
. . .
. . .
..
. . .
( ϕ0 (ξl ))−1 1 (ξl −α p2 ) 1
···
( ϕ0 (ξl ))−1 1 (ξl −α p2 )l2 1
.
.
( ϕ0 (ξl ))−1 1 (ξl −α p1 ) 1
···
( ϕ0 (ξl ))−1 1 (ξl −α p1 )l1 1
( ϕ0 (ξl +1 ))−1 1 (ξl +1 −α p1 ) 1
···
( ϕ0 (ξl +1 ))−1 1 (ξl +1 −α p1 )l1 1
( ϕ0 (ξl +1 ))−1 1 (ξl +1 −α p2 ) 1
···
( ϕ0 (ξl +1 ))−1 1 (ξl +1 −α p2 )l2 1
. . .
..
. . .
. . .
..
. . .
( ϕ0 (ξl +l ))−1 1 2 (ξl +l −α p1 ) 1 2
···
. . .
..
( ϕ0 (ξm −1 ))−1 1 (ξm −1 −α p1 ) 1
···
( ϕ0 (ξm1 ))−1 (ξm1 −α p1 )
···
k
1
( ϕ0 (ξ0 ))−1 (ξ0 −α p1 )
.
.
( ϕ0 (ξl +l ))−1 ( ϕ0 (ξl +l ))−1 1 2 1 2 (ξl +l −α p1 )l1 (ξl1 +l2 −α p2 ) 1 2
. . .
. . .
( ϕ0 (ξm −1 ))−1 ( ϕ0 (ξm −1 ))−1 1 1 (ξm −1 −α p1 )l1 (ξm1 −1 −α p2 ) 1 ( ϕ0 (ξm1 ))−1 (ξm1 −α p1 )l1
( ϕ0 (ξm1 ))−1 (ξm1 −α p2 )
.
···
( ϕ0 (ξl +l ))−1 1 2 (ξl +l −α p2 )l2 1 2
..
. . .
.
···
( ϕ0 (ξm −1 ))−1 1 (ξm −1 −α p2 )l2 1
···
( ϕ0 (ξm1 ))−1 (ξm1 −α p2 )l2
239
' ' ' ' ' ( ϕ0 (ξ1 ))−1 ( ϕ0 (ξ1 ))−1 ' ··· ··· (ξ1 −αk ) (ξ1 −αk )κnk −1 ' ' ' ' . . .. . ' . .. . . . . ' ' −1 −1 ( ϕ0 (ξl )) ( ϕ0 (ξl )) ' 1 1 ··· ··· ' (ξl −αk ) (ξl −αk )κnk −1 ' 1 1 ' ' ( ϕ0 (ξl +1 ))−1 ( ϕ0 (ξl +1 ))−1 1 ' ··· (ξ 1 −α ) ··· l1 +1 k (ξl +1 −αk )κnk −1 ' 1 ' ' . . ' .. . . .. . ' . . . ' ' ( ϕ0 (ξl +l ))−1 ( ϕ0 (ξl +l ))−1 ' 2 1 2 ··· (ξ 1 −α ··· l1 +l2 k) (ξl +l −αk )κnk −1 ' 1 2 ' ' ' . . .. . ' . . . . . . . ' ' ( ϕ0 (ξm −1 ))−1 ( ϕ0 (ξm −1 ))−1 ' 1 ··· (ξ 1 −α ) ··· ' m1 −1 k (ξm −1 −αk )κnk −1 ' 1 ' ' ( ϕ0 (ξm1 ))−1 ( ϕ0 (ξm1 ))−1 ··· ··· κnk −1 ' (ξm −αk ) ···
( ϕ0 (ξ0 ))−1 (ξ0 −αk )
1
···
( ϕ0 (ξ0 ))−1 (ξ0 −αk )κnk −1
(ξm1 −αk )
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k ,ξ ,...,ξ ,ξ · · · dμ(ξm −κ +1 ) · · · dμ(ξm1 ) G(ξ 0 1 l1 l1 +1 , . . . , ξl1 +l2 , . . . , ξm2 −κ nk +1 , . . . , ξ(n−1)K+k−1 ) 2 nk × ϕ02 (ξ0 ) ϕ02 (ξ1 ) · · · ϕ02 (ξl ) ϕ02 (ξl +1 ) · · · ϕ02 (ξl +l ) · · · ϕ02 (ξm −κ +1 ) · · · ϕ02 (ξm1 ) 2 1 1 1 2 nk
' ϕ1 (ξ0 ) ··· ϕl1 (ξ0 ) ϕl1 +1 (ξ0 ) ··· ϕl1 +l2 (ξ0 ) ··· ϕm2 −κnk +1 (ξ0 ) ··· ϕ(n−1)K+k−1 (ξ0 ) ' ' ϕ0 (ξ0 ) ' ' ' ϕ1 (ξ1 ) ··· ϕl1 (ξ1 ) ϕl1 +1 (ξ1 ) ··· ϕl1 +l2 (ξ1 ) ··· ϕm2 −κnk +1 (ξ1 ) ··· ϕ(n−1)K+k−1 (ξ1 ) ' ' ϕ0 (ξ1 ) ' ' ' ' . . . . . . . ' ' .. .. .. .. . . . . . . . ' ' . . . . . . . . . . . ' ' ' ' ϕ0 (ξl1 ) ϕ1 (ξl1 ) ··· ϕl1 (ξl1 ) ϕl1 +1 (ξl1 ) ··· ϕl1 +l2 (ξl1 ) ··· ϕm2 −κnk +1 (ξl1 ) ··· ϕ(n−1)K+k−1 (ξl1 ) ' ' ' ' ' ϕ1 (ξl1 +1 ) ··· ϕl1 (ξl1 +1 ) ϕl1 +1 (ξl1 +1 ) ··· ϕl1 +l2 (ξl1 +1 ) ··· ϕm2 −κnk +1 (ξl1 +1 ) ··· ϕ(n−1)K+k−1 (ξl1 +1 ) ' ' ϕ0 (ξl1 +1 ) ' ' ' ×' ' . . . . . . . . . . . ' ' . . .. . . .. . .. . .. . ' ' . . . . . . . ' ' ' ' ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ··· ϕ (ξ ) ··· ϕ (ξ ) +1 0 1 m − κ l +l l +l l l +l l +1 l +l l +l l +l l +l (n−1)K+k−1 l +l 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2 1 2 ' nk ' ' ' ' ' . . . . . . . .. .. .. .. ' ' . . . . . . . . . . . ' ' . . . . . . . ' ' ' ϕ1 (ξm1 −1 ) ··· ϕl1 (ξm1 −1 ) ϕl1 +1 (ξm1 −1 ) ··· ϕl1 +l2 (ξm1 −1 ) ··· ϕm2 −κnk +1 (ξm1 −1 ) ··· ϕ(n−1)K+k−1 (ξm1 −1 ) '' ' ϕ0 (ξm1 −1 ) ' ' ϕ0 (ξm1 ) ϕ1 (ξm1 ) ··· ϕl1 (ξm1 ) ϕl1 +1 (ξm1 ) ··· ϕl1 +l2 (ξm1 ) ··· ϕm2 −κnk +1 (ξm1 ) ··· ϕ(n−1)K+k−1 (ξm1 )
1 = m2 !
R
(ξ0 ,ξ1 ,...,ξl ,ξl +1 ,...,ξl +l ,...,ξm −κ +1 ,...,ξ(n−1)K+k−1 ) =G 2 nk 1 1 1 2
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R
R
(n−1)K+k
× · · · dμ(ξm2 −κnk +1 ) · · · dμ(ξm1 ) !2 (ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξm2 −κnk +1 , . . . , ξ(n−1)K+k−1 ) G × ; ϕ0 (ξ0 ) ϕ0 (ξ1 )· · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 )· · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κnk +1 ) · · · ϕ0 (ξm1 )
240
K. T.-R. McLaughlin et al.
but, noting the determinantal factorisation (ξ0 , ξ1 , . . . , ξl , ξl +1 , . . . , ξl +l , . . . , ξm −κ +1 , . . . , ξ(n−1)K+k−1 ) G 2 nk 1 1 1 2 = D V2 (ξ0 , ξ1 , . . . , ξ(n−1)K+k−1 ),
where D :=
' ' a0,0 ' ' a0,1 ' ' ' .. ' . ' ' a0,l 1 ' ' a ' 0,l1 +1 ' ' . ' .. ' ' a0,l +l ' 1 2 ' ' .. ' . ' ' a0,m −1 1 ' ' a 0,m1
a1,0
···
···
a1,1
.. .
..
a1,l
a1,l
1
1 +1
.. .
a1,l
.. .
1 ,l1
1 ,l1 +1
. .
··· ···
.. .
··· al
..
1 +1,l1 +l2
.
.. .
..
.
··· ···
am
···
am
.
···
1 +l2 ,l1 +1
··· am
.. .
1 +l2 ,l1 +l2
.. .
.. .
. .
··· ···
..
···
2 −κnk +1,l1 +1
.. .
..
.
···
2 −κnk +1,l1 +l2
.. .
.
···
2 −κnk +1,l1
··· am
..
al +l ,m −1 1 2 1 al +l ,m 1 2 1
···
2 −κnk +1,1
am
..
···
2 −κnk +1,0
al
··· al
al +1,m −1 1 1 al +1,m 1 1
···
..
1 +l2 ,l1
1 +1,l1 +1
al
al ,m −1 1 1 al ,m 1 1
.
···
.. .
1 ,l1 +l2
··· al
..
.. .
al al
.. .
al
···
1 +1,l1
al
1 +l2 ,0 al +l ,1 1 2
···
.. .
··· al
..
a1,m −1 1 a1,m 1
1 +1,0 al +1,1 1
al
.. .
.
···
..
1 +l2
1 ,0 al ,1 1
al
..
am −κ +1,m −1 2 1 nk am −κ +1,m 2 1 nk
.
··· ···
' ' ' a(n−1)K+k−1,1 '' ' .. ' ' . ' ' a(n−1)K+k−1,l 1 ' a(n−1)K+k−1,l +1 '' 1 ' ( = 0), ' .. ' . ' a(n−1)K+k−1,l +l ' 1 2 ' ' .. ' ' . ' a(n−1)K+k−1,m −1 ' 1 ' ' a
a(n−1)K+k−1,0
(n−1)K+k−1,m1
(24)
and V2 (ξ0 , ξ1 , . . . , ξ(n−1)K+k−1 ) := ' 1 ' ' ξ0 ' ' . ' .. ' l ' ξ1 ' 0 ' l1 +1 ' ξ0 ' ' . ' . ' . ' l1 +l2 ' ξ0 ' ' . ' . ' . ' m1 −1 ' ξ0 ' m1 ' ξ0
1 ξ1
··· ···
.. .
..
l
···
l +1 ξ11
···
.. .
..
l +l2
ξ11
.. .
m −1 ξ1 1 m ξ1 1
.. .
.
ξ11
.
'
1 ··· 1 ··· 1 ··· 1 ' ξl1 +1 ··· ξl1 +l2 ··· ξm2 −κnk +1 ··· ξ(n−1)K+k−1 '
1 ξl1
l
l
ξl 1
1 l +1 ξl 1 1
.. .
l +l2 1
··· ξl 1
..
.
··· ···
.. .
m −1 ξl 1 1 m ξl 1 1
.. .
..
.
l
.. .
..
.
l
ξl 1+1 ··· ξl 1+l
··· ξm1
···
··· ξm1
1 l +1 ξl 1+1 1
.. .
..
l +l
.
1 2 l +1 ξl 1+l 1 2
.. .
2 −κnk +1 l +1 2 −κnk +1
..
l +l
.
.. .
m1 −1 ξl +1 1 m1 ξl +1 1
1
..
.
··· ···
.. .
2
m1 −1 ξl +l 1 2 m1 ξl +l 1 2
..
.
··· ···
..
···
..
.. .
..
m −1 ξm 1−κ +1 2 nk m ξm 1−κ +1 2 nk
.
···
.. .
l +l2 2 −κnk +1
ξl 1+12 ··· ξl 1+l 2 ··· ξm1 1
.. .
.
···
.
··· ···
' ' ' ' l1 ξ(n−1)K+k−1 '' ' l1 +1 ' ξ(n−1)K+k−1 ' ' (n−1)K+k−1 .. (ξi −ξ j ), ' = i, j=0 ' . j
it follows that, for n ∈ N and k = 1, 2, . . . , K, cˆ D =
1 ((n−1)K+k)!
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k
· · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξm1 )(D )2 ×
=
(n−1)K+k−1 i, j=0
(ξi −ξ j )2
j
( ϕ0 (ξ0 ) ϕ0 (ξ1 ) · · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξ(n−1)K+k−1 ))2 (D )2 ((n−1)K+k)!
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · R R
(n−1)K+k
× · · · dμ(ξl1 +l2 ) · · · dμ(ξ(n−1)K+k−κ nk +1 ) · · · dμ(ξ(n−1)K+k−1 )
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
×
(n−1)K+k−1 $
(n−1)K+k−1 $
(ξi −ξ j)
2
241
!−2 ϕ0 (ξm )
m=0
i, j=0 j
(D )2 = ((n−1)K+k)!
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · R R R (n−1)K+k
× · · · dμ(ξl1 +l2 ) · · · dμ(ξ(n−1)K+k−κnk +1 ) · · · dμ(ξ(n−1)K+k−1 ) ×
⎛
(n−1)K+k−1 $
(ξi −ξ j)2 ⎝
(n−1)K+k−1 s −1 $ $
i, j=0
l=0
⎞−2 (ξl −α pq )κnkkq (ξl −αk )κnk −1 ⎠
⇒
q=1
j
cˆ D =
(D )2 ((n−1)K+k)!
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξ(n−1)K+k−1 ) R R R (n−1)K+k
×
(n−1)K+k−1 $
⎛
(ξi −ξ j)2 ⎝
i, j=0
(n−1)K+k−1 s −1 $ $ m=0
⎞−2 (ξm −α pq )κnkkq (ξm −αk )κnk −1⎠ (> 0).
q=1
j
(25) Hence, for n ∈ N and k = 1, 2, . . . , K, (23) and (25) establish the existence and the uniqueness of the monic FPC ORF, π nk : N × {1, 2, . . . , K} × C \ {α1 , α2 , . . . , α K } → C, (n, k, z) → π nk (z) = Y11 (z)/(z−αk ). There remains, still, the question of the existence and the uniqueness of Y21 : N × {1, 2, . . . , K} × C \ {α1 , α2 , . . . , α K } → C, which necessitates a detailed analysis of the elements of the second row of Y (z), that is, Y2m (z), m = (ξ ) dξ < 1, 2; thus the gist of the subsequent calculations. Recalling that R ξ m w ∞ and R (ξ −α j)−(m+1) w (ξ ) dξ < ∞, m ∈ Z+ , j= 1, 2, . . . , K, it follows via 0 l+1 i z z l + 1 2 the expansion z1 −z = i=0 zi+1 + zl+1 (z2 −z ) , l ∈ Z0 , the integral representation 2 1 2 1 (z−αk )Y21 (ξ1 ) w (ξ ) dξ (cf. (14)) Y22 (z) = R (ξ −αk )(ξ −z) , z ∈ C \ R, and the second of each of the 2π i asymptotic conditions (11), (12), and (13) that, for n ∈ N and k = 1, 2, . . . , K, w (ξ ) Y21 (ξ ) dξ = 0, p= 0, 1, . . . , κnk −2, (26) ξ −α (ξ −αk ) p k R w (ξ ) Y21 (ξ ) dξ = 2π i, (27) κ −1 ξ −α (ξ −α k k ) nk R w (ξ ) Y21 (ξ ) dξ = 0, q = 1, 2, . . . , s −1, r = 1, 2, . . . , κnkkq . (28) r ξ −α (ξ −α k pq ) R
242
K. T.-R. McLaughlin et al.
Note: for n = 1 and k = 1, 2, . . . , K, depending on the distribution of the real poles α1 , α2 , . . . , α K , it can happen that the corresponding κ1k < 2, in which case, (26) is vacuous (of course, for n 2, κnk 2, k = 1, 2, . . . , K); moreover, if, for n ∈ N and k = 1, 2, . . . , K, the set {αk ; k ∈ {1, 2, . . . , K}, αk = αk } = ∅, then the corresponding (28) are vacuous (actually, this can only occur if n = 1). Via the above ordered disjoint partition for the repeated real pole sequence {α1 , α2 , . . . , α K } ∪ · · · ∪ {α1 , α2 , . . . , α K } ∪ {α1 , α2 , . . . , αk } and (26)–(28), one writes, in the indicated order, lq ls s−1
/ νq,r (n, k) Y21 (z) / νs,m (n, k) = + r m−1 z−αk (z−α ) (z−α i(q)kq i(s)ks ) q=1 r=1 m=1 l =κ
nkkq l =κ s−1 q
/ νq,r (n, k) s nk / νs,m (n, k) = + r (z−α pq ) (z−αk )m−1 q=1 r=1 m=1
l1 =κnkk
1/ ν1,r (n, k) +· · ·+ (z−α p1 )r r=1
=
+
ls =κnk m=1
/ νs,m (n, k) , (z−αk )m−1
ls−1 =κnkk
s−1
r=1
/ νs−1,r (n, k) (z−α ps−1 )r
/ νs,ls (n, k) = 0.
Substituting the latter partial fraction expansion for Y21 (z)/(z−αk ) into (26)–(28), one arrives at, for n ∈ N and k = 1, 2, . . . , K, the orthogonality conditions ⎛
lm =κnk km
m=1
q=1
⎝
s−1
R
⎞ l =κ / νm,q (n, k) s nk / νs,q (n, k) (ξ ) ⎠ w + dξ = 0, (ξ −α pm )q (ξ −αk )q−1 (ξ −αk )r q=1
r = 0, 1, . . . , κnk −2, ⎛
lm =κnk km
m=1
q=1
⎝
s−1
R
⎛
(30)
q=1
s−1
lm =κnk km
m=1
q=1
⎝ R
⎞ l =κ / νm,q (n, k) s nk / νs,q (n, k) w (ξ ) ⎠ + dξ = 2π i, (ξ −α pm )q (ξ −αk )q−1 (ξ −αk )κnk −1
(29)
⎞ l =κ / νm,q (n, k) s nk / νs,q (n, k) (ξ ) ⎠ w + dξ = 0, (ξ − α pm )q (ξ −αk )q−1 (ξ −α pi ) j q=1
i = 1, 2, . . . , s −1, j= 1, 2, . . . , li .
(31)
Incidentally, for n ∈ N and k = 1, 2, . . . , K, via the orthogonality conditions (29) and (31), condition (30) can be manipulated thus: ⎛ ⎞ κnkkm l =κ s−1 lm =
/ νm,q (n, k) s nk / νs,q (n, k) νs,ls (n, k) ⎠ / 2π i = ⎝ + q q−1 (ξ −α ) (ξ −α ) (ξ −αk )κnk −1 p k R m=1 m q=1 q=1
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
w (ξ ) × dξ = / νs,ls (n, k)
⎞ κnkkm l =κ s−1 lm =
/ νm,q (n, k) s nk / νs,q (n, k) ⎝ ⎠ + (ξ −α pm )q (ξ −αk )q−1 R m=1 q=1 q=1
⎛
⎛
κnkkm s−1 lm =
/ νm,q (n, k) ×⎝ + (ξ −α pm )q m=1 q=1
w (ξ ) dξ = × / νs,ls (n, k)
243
ls −1= κnk −1
q=1
⎞
/ νs,q (n, k) / νs,ls (n, k) ⎠ + (ξ −αk )q−1 (ξ −αk )κnk −1
⎞ κnkkm l =κ s−1 lm =
/ νm,q (n, k) s nk / νs,q (n, k) ⎝ ⎠ + q q−1 (ξ −α ) (ξ −α ) p k R m=1 m q=1 q=1
⎛
= Y21 (ξ )/(ξ −αk )
⎛
⎞ κnkkm ls =κnk s−1 lm =
/ νm,q (n, k) / νs,q (n, k) (ξ ) ⎠ w dξ ×⎝ + q q−1 (ξ −α ) (ξ −α ) / ν pm k s,ls (n, k) m=1 q=1 q=1 = Y21 (ξ )/(ξ −αk )
=
R
Y21 (ξ ) ξ −αk
2
w (ξ ) dξ ; / νs,ls (n, k)
hence, for n ∈ N and k = 1, 2, . . . , K, one arrives at the interesting ‘normalisation formula’: Y21 (ξ ) 2 w (ξ ) dξ = 2π i/ νs,ls (n, k). ξ −αk R conditions (29)–(31) For n ∈ N and k = 1, 2, . . . , K, the orthogonality s−1 s give rise to a total of (cf. Subsection 1.2; recall that r=1 lr = r=1 lr + l s = s−1 kr + κnk = (n−1)K+k)(n−1)K+k linear inhomogeneous algebraic r=1 κnk ν1,l1 (n, k), equations for the (n−1)K+k real unknowns / ν1,1 (n, k), . . . ,/ ν2,l2 (n, k), . . . ,/ νs,1 (n, k), . . . ,/ νs,ls (n, k), namely: / ν2,1 (n, k), . . . ,/ ⎛
R
(ξ0 −αk )0 dμ(ξ0 ) (ξ0 −α p1 )
(ξ0 −αk )0 dμ(ξ0 ) (ξ0 −α p1 )l1 (ξ1 −α p1 )−1 dμ(ξ1 ) R (ξ1 −α p1 )l1
···
R
(ξ0 −αk )0 dμ(ξ0 ) (ξ0 −α p2 )
···
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −α p2 )
···
R
(ξ0 −αk )0 dμ(ξ0 ) (ξ0 −α p2 )l2 (ξ1 −α p1 )−1 dμ(ξ1 ) R (ξ1 −α p2 )l2 R
⎜ (ξ −α p )−1 dμ(ξ ) (ξ1 −α p1 )−1 dμ(ξ1 ) 1 1 ⎜ 1 ··· ··· R (ξ1 −α p2 ) ⎜ R (ξ1 −α p1 ) ⎜ .. .. .. .. ⎜ .. .. ⎜ . . . . . ⎜ (ξl −α p .)−l1 dμ(ξl ) −l −l (ξl1 −α p1 ) 1 dμ(ξl1 ) (ξl1 −α p1 )−l1 dμ(ξl1 ) (ξl1 −α p1 ) 1 dμ(ξl1 ) 1 ⎜ 1 1 ··· ··· R R R (ξl −α p1 ) (ξl −α p2 ) ⎜ R (ξl −α p1 )l1 (ξl −α p2 )l2 1 1 1 1 ⎜ ⎜ (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) ··· (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) ··· (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) ⎜ R R R R (ξl +1 −α p1 ) (ξl +1 −α p2 ) (ξl +1 −α p1 )l1 (ξl +1 −α p2 )l2 1 1 ⎜ 1 1 ⎜ . . . .. . . ⎜ .. .. .. .. .. . ⎜ ⎜ (ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) (ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) (ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) (ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) ··· ··· ⎜R R R R (ξl +l −α p1 ) (ξl +l −α p2 ) (ξl +l −α p1 )l1 (ξl +l −α p2 )l2 1 2 1 2 ⎜ 1 2 1 2 ⎜ . . . .. .. .. ⎜ . . . . . . . . . ⎝
R
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −α p1 )
···
R
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −α p1 )l1
R
R
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −α p2 )l2
244
K. T.-R. McLaughlin et al. ···
···
···
···
···
···
.. .. .. . . . ···
···
···
···
···
···
.. .. .. . . . ···
···
···
.. .. .. . . . ···
···
···
(ξ0 −αk )0 dμ(ξ0 ) (ξ0 −αk )0 (ξ1 −α p1 )−1 dμ(ξ1 ) R (ξ1 −αk )0 R
.. .
(ξl −α p1 )−l1 dμ(ξl ) 1 1 R (ξl −αk )0 1 (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) R (ξl +1 −αk )0 1
.. .
(ξl +l −α p2 )−l2 dμ(ξl +l ) 1 2 1 2 R (ξl +l −αk )0 1 2
.. .
(ξm1 −αk )−κnk +1 dμ(ξm1 ) R (ξm1 −αk )0
⎛
⎞ / ν1,1 (n, k) ⎛ .. ⎜ ⎟ ⎜ ⎟ ⎜ . ⎜ ⎟ ⎜ ⎜/ ⎟ ⎜ ν (n, k) 1,l ⎜ 1 ⎟ ⎜ ⎜/ ⎟ ⎜ (n, k) ν 2,1 ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ .. ⎜ ⎟ ⎜ . ⎟=⎜ ×⎜ ⎜/ ⎟ ⎜ ν2,l2 (n, k) ⎟ ⎜ ⎜ ⎟ ⎜ .. ⎜ ⎟ ⎜ . ⎜ ⎟ ⎜ ⎜/ ⎟ ⎜ ν (n, k) ⎜ s,1 ⎟ ⎜ ⎜ ⎟ ⎝ . .. ⎝ ⎠ / νs,ls (n, k)
··· ···
..
.
··· ···
..
.
···
..
.
···
(ξ0 −αk )0 dμ(ξ0 ) R (ξ −α )κnk −1 0 k (ξ1 −α p1 )−1 dμ(ξ1 ) R (ξ −α )κnk −1 1 k
⎞
⎟ ⎟ ⎟ ⎟ .. ⎟ ⎟ . ⎟ (ξl1 −α p1 )−l1 dμ(ξl1 ) ⎟ R ⎟ (ξl −αk )κnk −1 1 ⎟ (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) ⎟ ⎟ κ −1 R (ξl +1 −αk ) nk 1 ⎟ ⎟ .. ⎟ . ⎟ (ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) ⎟ ⎟ κ −1 R (ξl +l −αk ) nk ⎟ 1 2 ⎟ .. ⎟ . ⎠ −κ +1
R
(ξm1 −αk )
nk dμ(ξm1 ) (ξm1 −αk )κnk −1
⎞ 0 .. ⎟ . ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ .. ⎟ , . ⎟ ⎟ 0 ⎟ ⎟ .. ⎟ . ⎟ ⎟ 0 ⎠ 2π i
(32)
where m1 := (n−1)K+k−1. For n ∈ N and k = 1, 2, . . . , K, the linear system (32) of (n−1)K+k inhomogeneous algebraic equations for the (n− 1)K+k real unknowns / ν1,1 (n, k), . . . ,/ ν1,l1 (n, k),/ ν2,1 (n, k), . . . ,/ ν2,l2 (n, k), . . . , / νs,1 (n, k), . . . ,/ νs,ls (n, k) admits a unique solution if, and only if, the determinant of the coefficient matrix, denoted N (n, k), is non-zero: this fact will now be established. Via the multi-linearity property of the determinant, one shows that, for n ∈ N and k = 1, 2, . . . , K, N (n, k) is given by
N (n, k) =
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k
×
···dμ(ξm2 −κnk +1 )···dμ(ξm1 )
(ξ0 −αk )0 (ξ1 −α p1 )1 ···(ξl1 −α p1 )l1 (ξl1 +1 −α p2 )1 ···(ξl1 +l2 −α p2 )l2 ···(ξm2 −κnk +1 −αk )1 ···(ξm1 −αk )κnk −1
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
' ' (ξ0 −α1 p1 ) ' ' (ξ1 −α1 p1 ) ' . ' . ' ' (ξ −α1. p ) ' l1 1 ' 1 × '' (ξl1 +1 −α p1 ) . ' . ' ' (ξ 1.−α ) ' l1 +l2 p1 ' . ' . ' . 1 ' (ξm −α p ) 1
···
1 (ξ0 −α p1 )l1
1 (ξ0 −α p2 )
···
1 (ξ0 −α p2 )l2
···
1 (ξ0 −αk )0
···
···
1 (ξ1 −α p1 )l1
1 (ξ1 −α p2 )
···
1 (ξ1 −α p2 )l2
···
1 (ξ1 −αk )0
···
. . .
..
. . .
..
..
···
..
1
1 (ξl −α p2 ) 1
···
1 (ξl +1 −α p2 ) 1
···
. . .
. . .
..
1 (ξl +l −α p2 ) 1 2
···
. . .
..
. . .
.
···
..
. . .
..
1 (ξl −α p1 )l1 1 1 (ξl +1 −α p1 )l1 1
.
···
1 (ξl +l −α p1 )l1 1 2
. . .
.
···
1 (ξm1 −α p1 )l1
1 (ξm1 −α p2 )
.
1 (ξl −α p2 )l2 1 1 (ξl +1 −α p2 )l2 1
.
··· ···
. . .
..
1 (ξl +l −α p2 )l2 1 2
···
. . .
..
.
.
1 (ξm1 −α p2 )l2
···
···
. . .
..
1 (ξl +l −αk )0 1 2
···
. . .
..
.
.
···
.
1 (ξl −αk )0 1 1 (ξl +1 −αk )0 1
1 (ξm1 −αk )0
···
.
.
···
245
1 (ξ0 −αk )κnk −1 1 (ξ1 −αk )κnk −1
. . .
1 (ξl −αk )κnk −1 1 1 (ξl +1 −αk )κnk −1 1
. . .
1 (ξl +l −αk )κnk −1 1 2
. . .
1 (ξm1 −αk )κnk −1
' ' ' ' ' ' ' ' ' ' ', ' ' ' ' ' ' ' ' '
where m2 := (n−1)K+k. Recalling, for n ∈ N and k = 1, 2, . . . , K, the (n−1)K+ k linearly independent (on R) functions ϕ0 (z), ϕ1 (z), . . . , ϕl1 (z), ϕl1 +1 (z), . . . , ϕl1 +l2 (z), . . . , ϕ(n−1)K+k−κnk +1 (z), . . . , ϕ(n−1)K+k−1 (z) introduced above for the analysis of cˆ D , one shows that N (n, k) can be written as N (n, k) =
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k
× · · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξm1 ) ×
(−1)(n−1)K+k−κnk ( ϕ0 (ξ0 ) ϕ0 (ξ1 )··· ϕ0 (ξl1 ) ϕ0 (ξl1 +1 )··· ϕ0 (ξl1 +l2 )··· ϕ0 (ξm2 −κnk +1 )··· ϕ0 (ξm1 ))−1
(ξ0 −αk )0 (ξ1 −α p1 )1 ···(ξl1 −α p1 )l1 (ξl1 +1 −α p2 )1 ···(ξl1 +l2 −α p2 )l2 ···(ξm2 −κnk +1 −αk )1 ···(ξm1 −αk )κnk −1
' ϕ0 (ξ0 ) ϕ1 (ξ0 ) ··· ϕl (ξ0 ) ϕl +1 (ξ0 ) ··· ϕl +l (ξ0 ) ··· ϕm −κ +1 (ξ0 ) ··· ϕ(n−1)K+k−1 (ξ0 ) ' 2 nk 1 1 1 2 ' ' ' ϕ0 (ξ1 ) ϕ1 (ξ1 ) ··· ϕl1 (ξ1 ) ϕl1 +1 (ξ1 ) ··· ϕl1 +l2 (ξ1 ) ··· ϕm2 −κnk +1 (ξ1 ) ··· ϕ(n−1)K+k−1 (ξ1 ) ' ' ' ' ' . . . . . . . .. .. .. .. . . . . . . . ' ' . . . . . . . . . . . ' ' ' ϕ0 (ξl1 ) ϕ1 (ξl1 ) ··· ϕl1 (ξl1 ) ϕl1 +1 (ξl1 ) ··· ϕl1 +l2 (ξl1 ) ··· ϕm2 −κnk +1 (ξl1 ) ··· ϕ(n−1)K+k−1 (ξl1 ) ' ' ϕ0 (ξl +1 ) ϕ1 (ξl +1 ) ··· ϕl (ξl +1 ) ϕl +1 (ξl +1 ) ··· ϕl +l (ξl +1 ) ··· ϕm −κ +1 (ξl +1 ) ··· ϕ(n−1)K+k−1 (ξl +1 ) ' 2 nk 1 1 1 1 1 1 1 2 1 1 1 ' ' '. × '' . . . . . . . .. .. .. .. ' . . . . . . . ' ' . . . . . . . . . . . ' ϕ0 (ξl +l ) ϕ1 (ξl +l ) ··· ϕl (ξl +l ) ϕl +1 (ξl +l ) ··· ϕl +l (ξl +l ) ··· ϕm2 −κ +1 (ξl +l ) ··· ϕ(n−1)K+k−1 (ξl +l ) ' nk 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2 1 2 ' ' ' ' . . . . . . . .. .. .. .. ' ' . . . . . . . . . . . ' ' . . . . . . . ' ϕ0 (ξm1 −1 ) ϕ1 (ξm1 −1 ) ··· ϕl1 (ξm1 −1 ) ϕl1 +1 (ξm1 −1 ) ··· ϕl1 +l2 (ξm1 −1 ) ··· ϕm2 −κnk +1 (ξm1 −1 ) ··· ϕ(n−1)K+k−1 (ξm1 −1 ) ' ' ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ··· ϕ ' (ξ ) ··· ϕ (ξ ) 0
m1
1
m1
l1
m1
l1 +1
l1 +l2
m1
m1
m2 −κnk +1
m1
(n−1)K+k−1
m1
Modulo the factor (−1)(n−1)K+k−κnk , this is the same determinantal expression encountered while studying cˆ D ; therefore, for n ∈ N and k = 1, 2, . . . , K, N (n, k) =
(−1)(n−1)K+k−κ nk (D )2 ((n−1)K+k)!
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξ(n−1)K+k−1 ) R R
(n−1)K+k
⎞−2
⎛
(n−1)K+k−1 s$ −1 $
×⎝
m=0
×
(ξm −α pq )
(ξi −ξ j )2
j
(ξm −αk )
κ nk −1⎠
q=1
(n−1)K+k−1 $ i, j=0
κ nk kq
( = 0),
(33)
246
K. T.-R. McLaughlin et al.
whence follows, for n ∈ N and k = 1, 2, . . . , K, the existence and the uniqueness of Y21 (z)/(z−αk ). Corollary 2.1 Let V: R \{α1 , α2 , . . . , α K } → R satisfy conditions (3)–(5). Let Y : N × {1, 2, . . . , K} × C \ R → SL2 (C) be the unique solution of RHPFPC with integral representation given in Lemma 2.1, where, in particular, π nk : N× {1, 2, . . . , K} × C \ {α1 , α2 , . . . , α K } → C, (n, k, z) → π nk (z) := (μn,κnk (n, k))−1 ·φkn (z) = Y11 (z)/(z−αk ) is the monic FPC ORF defined in Subsection 1.2, with μn,κnk (n, k) the associated norming constant, and φkn (z) the corresponding FPC orthonormal rational function. Then, for n ∈ N and k = 1, 2, . . . , K, 1
0
μn,κnk (n, k) = (n−1)K+k+1
ς(n, k) , λ(n, k)
where ς (n, k) := (D )2 · · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξ(n−1)K+k−1 ) R R R (n−1)K+k
⎞−2
⎛
(n−1)K+k−1 s$ −1 $
× ⎝
m=0
×
(ξm −α pq )
κ nk kq
(ξm −αk )κ nk −1 ⎠
q=1
(n−1)K+k−1 $
(ξi −ξ j )2 ,
i, j=0 j
· · · dμ(τ0 ) dμ(τ1 ) · · · dμ(τ(n−1)K+k ) R R
λ(n, k) := D2
R
(n−1)K+k+1
⎛ × ⎝
(n−1)K+k −1 $ s$ m=0
×
⎞−2 (τm −α pq )
κ nk kq
(τm −αk )κ nk ⎠
q=1
(n−1)K+k $
(τi −τ j )2 ,
i, j=0 j
with dμ(z) = exp(−N V(z)) dz, N ∈ N, ⎛
A (1)
⎜ ⎜ A (2) ⎜ ⎜ . ⎜ (n−1)K+k . D = (−1) det ⎜ . ⎜ ⎜ ⎜ A (s−1) ⎝ A (s)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
where (with the convention i(r) = 1+
r−1
q(i(r), r) = i(r)−
(A (r))i(r) j(r) =
⎧ ⎪ ⎪ ⎨
lm , 2+
m=1
r−1 m=1
0
m=1 ∗ := 0),
r−1 m=1
l m , . . . , lr +
for r = 1, . . . , s −1, s,
r−1
lm ,
m=1
lm ,
q(i(r),r) (−1)q(i(r),r) ∂ a j(r)−1 (α), j(r) = 1, 2, . . . , (n−1)K+k−q(i(r), r)+1, q(i(r),r)−1 (lr −m) ∂α pr m=0
⎪ ⎪ ⎩ 0,
am˜ (α) := 1
×
247
j(r) = (n−1)K+k−q(i(r), r)+2, . . . , (n−1)K+k,
(−1)(n−1)K+k−m˜ 1
i p =0,1,...,l p p∈{1,2,...,s−1} is =0,1,...,κ nk s ˜1 m=1 im =(n−1)K+k−m s−1 $
s−1 $
l j! κnk ! i !(l −i j )! is !(κnk −is )! j=1 j j
(α pm )im (αk )is ,
m=1
and ⎛
A (1)
⎜ ⎜ A (2) ⎜ ⎜ .. (n−1)K+k+1 D = (−1) det ⎜ . ⎜ ⎜ ⎜ A (s −1) ⎝ A (s)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠
where, for r = 1, . . . , s −1, s, i(r) = 1+
r−1 m=1
q(i(r), r) = i(r)−
(A (r))i(r) j(r) =
am˜ (α) := 1
⎧ ⎪ ⎪ ⎨
lm , 2+
r−1 m=1
r−1 m=1
lm , . . . , lr −δrs +
m=1
lm ,
lm ,
(−1)q(i(r),r)
q(i(r),r)−1
m=0 ⎪ ⎪ ⎩ 0,
(lr −δrs −m)
q(i(r),r) ∂ a j(r)−1 (α), ∂α pr
j(r) = 1, 2, . . . , (n−1)K+k−q(i(r), r),
j(r) = (n−1)K+k−q(i(r), r)+1, . . . , (n−1)K+k−1,
(−1)(n−1)K+k−m˜ 1 −1
i p =0,1,...,l p p∈{1,2,...,s−1} is =0,1,...,κ nk −1 s ˜ 1 −1 m=1 im =(n−1)K+k−m
×
r−1
s−1 $ (κnk −1)! (α pm )im (αk )is . is !(κnk −1−is )! m=1
s−1 $
l j!
j=1
i j !(l j −i j )!
248
K. T.-R. McLaughlin et al.
Proof Recall from the proof of Lemma 2.1 that, for n ∈ N and k = 1, 2, . . . , K, (μn,κnk (n, k))−2 = cˆ N /ˆc D , where cˆ N is given in (23) (with the determinant D given in (22)), and cˆ D is given in (25) (with the determinant D given in (24)); hence, substituting (23) and (25) into the formula for (μn,κnk (n, k))−2 and taking the positive square root of both sides of the resulting quotient, one arrives at the formula for μn,κnk (n, k) stated in the Corollary. Lemma 2.2 Let V: R \ {α1 , α2 , . . . , α K } → R satisfy conditions (3)–(5). Let Y : N × {1, 2, . . . , K} × C \ R → SL2 (C) be the unique solution of RHPFPC with integral representation given in Lemma 2.1, where, in particular, π nk : N× {1, 2, . . . , K} × C \ {α1 , α2 , . . . , α K } → C, (n, k, z) → π nk (z) := (μn,κnk (n, k))−1 ·φkn (z) = Y11 (z)/(z−αk ) is the monic FPC ORF defined in Subsection 1.2, with μn,κnk (n, k) given in Corollary 2.1, and φkn (z) is the corresponding FPC orthonormal rational function. Then, for n ∈ N and k = 1, 2, . . . , K, π nk (z) and Y21 (z)/(z−αk ) have, respectively, the following integral representations: ⎛ ⎞−1 s −1 $ Y (z) D ϒ N (n, k; z) 11 = ⎝ (z−α pq )κnkkq (z−αk )κnk ⎠ π nk (z) = , z−αk D q=1 ϒ D (n, k) where D and D are given in Corollary 2.1, ϒ N (n, k; z) :=
R
· · · dμ(ξ0 )dμ(ξ1 ) · · · dμ(ξ(n−1)K+k−1 ) R R
(n−1)K+k
⎛
(n−1)K+k−1 s −1 $ $
× ⎝
m=0
×
⎞−2 (ξm −α pq )κnkkq (ξm −αk )κnk⎠
q=1
(n−1)K+k−1 $
(n−1)K+k−1 $
i, j=0
l=0
(ξi −ξ j)2
(ξl −αk )(z−ξl ),
j
ϒ D (n, k) :=
· · · dμ(τ0 ) dμ(τ1 ) · · · dμ(τ(n−1)K+k−1 ) R R R (n−1)K+k
⎛
(n−1)K+k−1 s −1 $ $
× ⎝
m=0
×
⎞−2 (τm −α pq )κnkkq (τm −αk )κnk −1 ⎠
q=1
(n−1)K+k−1 $
(τi −τ j)2 ,
i, j=0 j
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
249
with dμ(z) = exp(−N V(z)) dz, N ∈ N; and ⎛ ⎞−1 s−1 Y21 (z) D ⎝$
N (n, k; z) = 2π i((n−1)K+k) (z−α pq )κnkkq (z−αk )κnk −1 ⎠ , z−αk D q=1
D (n, k) where ⎛
A (1)
⎜ ⎜ A (2) ⎜ ⎜ ..
(n−1)K+k D = (−1) det ⎜ . ⎜ ⎜ ⎜ A (s −1) ⎝ A (s) where (with the convention
0
m=1 ∗ := 0),
⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠
for r = 1, . . . , s −1, s,
r−1 l , 2+ r−1 l , . . . , l −2δ + r−1 l , i(r) = 1+m=1 m r rs m=1 m m=1 m r−1 lm , q(i(r), r) = i(r)−m=1
⎧ (−1)q(i(r),r) ∂ q(i(r),r) ⎪ ⎨ a j(r)−1(α), j(r) = 1,2, . . . , (n−1)K+k−q(i(r),r)−1, (A (r))i(r) j(r) = q(i(r),r)−1(lr −2δrs−m) ∂α pr ⎪ ⎩ m=0 0, j(r) = (n−1)K+k−q(i(r), r), . . . , (n−1)K+k−2,
am˜ (α) := 1
(−1)(n−1)K+k−m˜ 1
i p =0,1,...,l p p∈{1,2,...,s−1} is =0,1,...,κ nk −2 s ˜ 1 −2 m=1 im =(n−1)K+k−m
×
s−1 $
l j!
j=1
i j !(l j −i j )!
s−1 $ (κnk −2)! i (α pm )im (αks ), is !(κnk −2−is )! m=1
and Λ N (n, k; z) :=
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξ(n−1)K+k−2 ) R R R (n−1)K+k−1
⎛ × ⎝
(n−1)K+k−2 s −1 $ $ m=0
×
⎞−2 (ξm −α pq )κnkkq (ξm −αk )κnk −1 ⎠
q=1
(n−1)K+k−2 $
(n−1)K+k−2 $
i, j=0
l=0
(ξi −ξ j)2
j
(ξl −αk )(z−ξl ),
250
K. T.-R. McLaughlin et al.
Λ D (n, k) :=
R
· · · dμ(τ0 ) dμ(τ1 ) · · · dμ(τ(n−1)K+k−1 ) R R
(n−1)K+k
⎛ × ⎝
(n−1)K+k−1 s −1 $ $ m=0
×
⎞−2 (τm −α pq )κnkkq (τm −αk )κnk −1 ⎠
q=1
(n−1)K+k−1 $
(τi −τ j)2 .
i, j=0 j
Proof For n ∈ N and k = 1, 2, . . . , K, recall, from the proof of Lemma 2.1, the ordered disjoint partition for {α1 , α2 , . . . , α K } ∪ · · · ∪ {α1 , α2 , . . . , α K } ∪ {α1 , α2 , . . . , αk } and the corresponding formula for the monic FPC ORF,
π nk (z) =
κnkkm s−1 lm =
νm,q (n, k) Y11 (z) = φ0 (n, k)+ z−αk (z−α pm )q m=1 q=1
+
ls −1= κnk −1
q=1
νs,q (n, k) 1 + , (z−αk )q (z−αk )κnk
0 (n, k), ν1,1 (n, k), . . . , ν1,l1 (n, k), where the (n−1)K+k+1 real coefficients φ ν2,1 (n, k), . . . , ν2, l2 (n, k), . . . , νs, 1 (n, k), . . . , νs,ls −1 (n, k), μn,κnk (n, k), with μn,κnk (n, k) the ‘norming constant’ (cf. Corollary 2.1), satisfy the linear inhomogeneous algebraic system (21) given in the proof of Lemma 2.1. Via the multi-linearity property of the determinant and an application of Cramer’s Rule to system (21), one arrives at, for n ∈ N and k = 1, 2, . . . , K, the following (ordered) determinantal representation for the monic FPC ORF:
π nk (z) =
Y11 (z) Ξkn (z) = , z−αk cˆ D
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
251
where cˆ D is given in the proof of Lemma 2.1, (25), and, with m1 := (n−1)K+k−1, ' dμ(ξ0 ) ' ' R (ξ −α )0 0 k ' ' dμ(ξ1 ) ' ' R (ξ1−α p1 ) ' ' ' .. ' . ' ' dμ(ξ ) l ' ' R (ξ −α p1 )l1 l1 ' 1 ' ' dμ(ξl1 +1 ) ' R (ξ −α p ) l1 +1 2 Ξkn (z) := '' ' . ' .. ' ' ' dμ(ξl1 +l2 ) ' R (ξ −α )l2 p2 ' l1 +l2 ' ' .. ' . ' ' dμ(ξm1 ) ' ' R (ξm −αk )κnk−1 1 ' ' 1 '
···
..
···
R
R
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −α p1 ) 1
..
···
.. .
..
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −α p1 )
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −α p2 )l2 1 (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) R (ξl +1 −α p2 )l2 1
R
.. .
···
(ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) R (ξl +l −α p2 )l2 1 2
..
.. .
. R
···
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −α p2 )l2 1 (z−α p2 )l2
··· ···
R
.. .
.. .. . .
(ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) R (ξl +l −α p2 ) 1 2
··· ···
.. .
.. .. . .
R
R
(ξ0 −αk )0 dμ(ξ0 ) (ξ0 −αk )
···
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −αk )
···
..
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −α p2 )
···
..
R
R
..
R
R
···
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −αk )κnk
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −αk )κnk 1
(ξl +1 −α p2 )−1 dμ(ξl +1 ) 1 1 (ξl +1 −αk )κnk 1
.. .
R
(ξl +l −α p2 )−l2 dμ(ξl +l ) 1 2 1 2 (ξl +l −αk )κnk 1 2
.. .
.
···
(ξ0 −αk )0 dμ(ξ0 ) (ξ0 −αk )κnk
.. .
.
.. .
1 z−αk
.
(ξl +1 −α p2 )−1 dμ(ξl +1 ) 1 1 (ξl +1 −αk ) 1
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −αk )
··· ···
1 z−α p2
.. .. . .
··· ···
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −α p2 ) 1
1 (z−α p1 )l1
···
R
R
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −α p1 )l1
··· ···
.. .
(ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) R (ξl +l −αk ) 1 2
··· ···
.. .. . .
(ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) R (ξl +l −α p1 )l1 1 2
.. .
.. .. . .
··· ···
.. .
.. .
···
R
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p2 )
··· ···
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −αk ) 1
··· ···
(ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) R (ξl +1−α p2 ) 1
.. .
··· ···
(ξ0−αk )0 dμ(ξ0 ) (ξ0−α p2 )
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −α p1 )l1 1 (ξl1 +1−α p2 )−1 dμ(ξl1 +1 ) R (ξl +1−α p1 )l1 1
.
R
.. .. . .
R
R
R
.. .
···
R
···
1 z−α p1
(ξ0−αk )0 dμ(ξ0 ) (ξ0−α p1 )l1
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p1 )l1
.
(ξl1 +l2 −α p2 )−l2 dμ(ξl1 +l2 ) R (ξl +l −α p1 ) 1 2
··· ···
R
···
.. .
R
.
···
.. .
..
(ξl1 +1−α p2 )−1 dμ(ξl1 +1 ) R (ξl +1 −α p1 ) 1
R
···
.. .
···
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p1 )
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p2 )l2
.
···
=
(ξ0−αk )0 dμ(ξ0 ) (ξ0−α p1 )
··· ···
.
···
..
R
R
(ξ0 −αk )0 dμ(ξ0 ) (ξ0 −α p2 )l2
R
···
R
(ξm1 −αk )−κnk +1 dμ(ξm1 ) (ξm1 −αk )κnk 1 (z−αk )κnk
··· ··· ··· ···
' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R R (n−1)K+k
×
··· dμ(ξm2 −κnk +1 ) ··· dμ(ξm1 ) (ξ0 −αk )0 (ξ1 −α p1 )1 ···(ξl1 −α p1 )l1 (ξl1 +1 −α p2 )1 ···(ξl1 +l2 −α p2 )l2 ···(ξm2 −κnk +1 −αk )1 ···(ξm1−αk )κnk −1
252
K. T.-R. McLaughlin et al.
' '1 ' '1 ' ' ' .. '. '1 ' ' '1 ' × '. '. '. '1 ' ' ' .. '. '1 ' ' '1
1 (ξ0 −α p1 )
···
1 (ξ1 −α p1 )
···
.. .1
(ξl −α p1 ) 1 1 (ξl +1 −α p1 ) 1
.. .1
(ξl +l −α p1 ) 1 2
.. .1
(ξm1 −α p1 ) 1 z−α p1
..
.
··· ···
..
.
···
..
.
··· ···
1 (ξ0 −α p1 )l1 1 (ξ1 −α p1 )l1
.. .1
(ξl −α p1 )l1 1 1 (ξl +1 −α p1 )l1 1
.. .1
(ξl +l −α p1 )l1 1 2
.. .1
(ξm1 −α p1 )l1 1 (z−α p1 )l1
1 (ξ0 −α p2 )
···
1 (ξ1 −α p2 )
···
.. .1
(ξl −α p2 ) 1 1 (ξl +1 −α p2 ) 1
.. .1
(ξl +l −α p2 ) 1 2
.. .1
(ξm1 −α p2 ) 1 z−α p2
..
.
··· ···
..
.
···
..
.
··· ···
1 (ξ0 −α p2 )l2 1 (ξ1 −α p2 )l2
.. .1
(ξl −α p2 )l2 1 1 (ξl +1 −α p2 )l2 1
.. .1
(ξl +l −α p2 )l2 1 2
.. .1
(ξm1 −α p2 )l2 1 (z−α p2 )l2
···
1 (ξ0 −αk )
···
1 (ξ0 −αk )κnk
···
1 (ξ1 −αk )
···
..
.. .1
1 (ξ1 −αk )κnk
..
1 (ξl +1 −αk ) 1
···
.
··· ···
..
.
···
..
.
··· ···
(ξl −αk ) 1
···
.. .1
..
.. .1
..
1 z−αk
···
(ξl +l −αk ) 1 2
(ξm1 −αk )
.. .1
.
(ξl −αk )κnk 1 1 (ξl +1 −αk )κnk 1
.. .1
.
···
(ξl +l −αk )κnk 1 2
.. .1
.
···
(ξm1 −αk )κnk 1 (z−αk )κnk
' ' ' ' ' ' ' ' ' ' ' ' ' ', ' ' ' ' ' ' ' ' ' ' '
where dμ(z) is given in the Lemma. For n ∈ N and k = 1, 2, . . . , K, recalling, from the proof of Lemma 2.1, the (n−1)K+k+1 linearly independent (on −1 R) functions ϕ0 (z) := sm=1 (z−α pm )lm (z−αk )κnk =: (n−1)K+k a j,0 z j, ϕq(r1 ) (z) := j=0 1 −1 {ϕ0 (z)(z−α pr1 )−m(r1 ) =: (n−1)K+k a j,q(r1 ) z j}, r1 = 1, 2, . . . , s, q(r1 ) = ri=1 li +1, j=0 r1 −1 r1 −1 , and the (n−1)K+k linearly i=1 li +2, . . . , i=1 li +lr1 , m(r1 ) = 1, 2, . . . , lr1 −1 ϕ0 (z) := sm=1 (z−α pm )lm (z−αk )κnk −1 =: independent (on R) functions (n−1)K+k−1 j a j,0 z , ϕq(r2 ) (z):={ ϕ0 (z)(z−α pr2 )−m(r2 )=: (n−1)K+k−1 a j,q(r2 ) z j}, r2 = j=0 r2−1 r2 −1 r2 −1 j=0 1, 2, . . . , s, q(r2 ) = i=1 li +1, i=1 li +2, . . . , i=1 li +lr2 −δr2 s , m(r2 ) = 1, 2, . . . , lr2 −δr2 s , one proceeds, via the latter determinantal expression, with the analysis of Ξkn (z) (with m2 := (n−1)K+k):
Ξkn (z) =
R
· · · dμ(ξ0 )dμ(ξ1 ) · · · dμ(ξl1 )dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k
× ×
··· dμ(ξm2 −κnk +1 ) ··· dμ(ξm1 )
(ξ0−αk )0 (ξ1 −α p1 )1 ···(ξl1−α p1 )l1 (ξl1 +1−α p2 )1 ···(ξl1 +l2−α p2 )l2 ···(ξm2 −κnk +1−αk )1 ···(ξm1−αk )κnk−1
(ϕ0 (z))−1 ϕ0 (ξ0 )ϕ0 (ξ1 ) · · · ϕ0 (ξl1 )ϕ0 (ξl1+1 ) · · · ϕ0 (ξl1+l2 ) · · · ϕ0 (ξm2−κnk+1 ) · · · ϕ0 (ξm1 )
' ϕ (ξ ) ϕ1 (ξ0 ) ··· ϕl (ξ0 ) ϕl +1 (ξ0 ) ··· ϕl +l (ξ0 ) ··· ϕm −κ +1 (ξ0 ) ··· ϕ(n−1)K+k (ξ0 ) '' ' 0 0 2 1 1 1 2 nk ' ' ' ϕ (ξ ) ' ϕ (ξ ) ··· ϕ (ξ ) ϕ 0 1 1 1 l1 1 l1 +1 (ξ1 ) ··· ϕl1 +l2 (ξ1 ) ··· ϕm2 −κ nk +1 (ξ1 ) ··· ϕ(n−1)K+k (ξ1 ) ' ' ' ' ' ' . . . . . . . . . . . ' ' . . .. . . .. . .. . .. . ' ' . . . . . . . ' ' ' ϕ0 (ξ ) ' (ξ ) ··· ϕ (ξ ) ··· ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ϕ (ξ ) ··· ϕ m2 −κ nk +1 l1 1 l1 (n−1)K+k l1 l1 l1 l1 l1 +1 l1 l1 +l2 l1 ' ' ' ϕ (ξ ' ' 0 l +1 ) ϕ1 (ξl +1 ) ··· ϕl (ξl +1 ) ϕl +1 (ξl +1 ) ··· ϕl +l (ξl +1 ) ··· ϕm −κ +1 (ξl +1 ) ··· ϕ(n−1)K+k (ξl +1 ) ' 2 1 1 1 1 1 1 1 1 2 1 nk 1 ' ' ' × '' ' . . . . . . . .. .. .. .. ' ' . . . . . . . ' ' . . . . . . . . . . ' ϕ (ξ . ' ' 0 l +l ) ϕ1 (ξl +l ) ··· ϕl (ξl +l ) ϕl +1 (ξl +l ) ··· ϕl +l (ξl +l ) ··· ϕm2 −κ +1 (ξl +l ) ··· ϕ(n−1)K+k (ξl +l ) ' 1 2 1 2 ' 1 2 1 2 1 1 2 1 1 2 1 2 1 2 nk ' ' ' ' ' . . . . . . . .. .. .. .. ' ' . . . . . . . . . . . ' ' . . . . ' ϕ (ξ. ) ϕ (ξ. ) ··· ϕ (ξ. ' ' 0 m1 1 m1 l1 m1 ) ϕl1 +1 (ξm1 ) ··· ϕl1 +l2 (ξm1 ) ··· ϕm2 −κ nk +1 (ξm1 ) ··· ϕ(n−1)K+k (ξm1 ) '' ' ' ϕ0 (z) ϕ1 (z) ··· ϕl (z) ϕl +1 (z) ··· ϕl +l (z) ··· ϕm −κ +1 (z) ··· ϕ(n−1)K+k (z) ' 2 1 1 1 2 nk =: G (ξ0 ,ξ1 ,...,ξl ,ξl +1 ,...,ξl +l ,...,ξ(n−1)K+k−1 ;z) 1 1 1 2
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
=
(ϕ0 (z))−1 m2 !
253
· · · dμ(ξσ (0) ) dμ(ξσ (1) ) · · · dμ(ξσ (l1 ) ) dμ(ξσ (l1 +1) ) · · · R R
σ ∈S m2
R
(n−1)K+k
× ×
··· dμ(ξσ (l1 +l2 ) )··· dμ(ξσ (m2 −κnk +1) ) ··· dμ(ξσ (m1 ) )
(ξσ (0) −αk )0 (ξσ (1) −α p1 )1 ···(ξσ (l1 ) −α p1 )l1 (ξσ (l1 +1) −α p2 )1 ···(ξσ (l1 +l2 ) −α p2 )l2 ···(ξσ (m2 −κnk +1) −αk )1 ···(ξσ (m1 ) −αk )κnk −1
1 ϕ0 (ξσ (0) )ϕ0 (ξσ (1) ) · · · ϕ0 (ξσ (l1 ) )ϕ0 (ξσ (l1 +1) ) · · · ϕ0 (ξσ (l1 +l2 ) ) · · · ϕ0 (ξσ (m2 −κ nk +1) ) · · · ϕ0 (ξσ (m1 ) )
× G (ξσ (0) , ξσ (1) , . . . , ξσ (l1 ) , ξσ (l1 +1) , . . . , ξσ (l1 +l2 ) , . . . , ξσ ((n−1)K+k−1) ; z) =
(ϕ0 (z))−1 m2 !
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) . . . R R
(n−1)K+k
× · · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξm1 )G (ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξ(n−1)K+k−1 ; z) ×
σ) sgn(σ
1 (ξσ (0)−αk )0(ξσ (1)−α p1)1···(ξσ (l1 )−α p1)l1(ξσ (l1+1)−α p2)1···(ξσ (l1+l2)−α p2)l2···(ξσ (m2−κnk+1)−αk )1···(ξσ (m1 )−αk )κnk−1
σ∈S m2
×
1 ϕ0 (ξσ (0) )ϕ0 (ξσ (1) ) · · · ϕ0 (ξσ (l1 ) )ϕ0 (ξσ (l1 +1) ) · · · ϕ0 (ξσ (l1 +l2 ) ) · · · ϕ0 (ξσ (m2−κ nk +1) ) · · · ϕ0 (ξσ (m1 ) )
=
(ϕ0 (z))−1 m2 !
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
R
(n−1)K+k
×
· · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξm1 )G (ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξ(n−1)K+k−1 ; z) ϕ0 (ξ0 )ϕ0 (ξ1 ) · · · ϕ0 (ξl1 )ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξm1 )
' ' ' ' ' ' ' ' ' ' ' ' ' ' ×' ' ' ' ' ' ' ' ' ' ' ' ' =
1 (ξ0 −αk )0 1 (ξ1 −αk )0
1 (ξ0 −α p1 )
···
1 (ξ1 −α p1 )
···
.. .
.. .
1 (ξl −αk )0 1 1 (ξl +1 −αk )0 1
.. .
1 (ξl −α p1 ) 1
··· ···
.. .
..
1 (ξl +l −α p1 ) 1 2
.. .
.. .
(ϕ0 (z))−1 m2 !
.
1 (ξl +1 −α p1 ) 1
1 (ξl +l −αk )0 1 2
1 (ξm1 −αk )0
..
1 (ξm1 −α p1 )
.
···
..
.
···
1 (ξ0 −α p1 )l1 1 (ξ1 −α p1 )l1
.. .
1 (ξ0 −α p2 )
···
1 (ξ1 −α p2 )
···
.. .
..
.
1 (ξl −α p1 )l1 1 1 (ξl +1 −α p1 )l1 1
1 (ξl +1 −α p2 ) 1
···
.. .
..
1 (ξl +l −α p1 )l1 1 2
1 (ξl +l −α p2 ) 1 2
.. . .. .
1 (ξm1 −α p1 )l1
1 (ξl −α p2 ) 1
.. .
1 (ξm1 −α p2 )
···
.
···
..
.
···
1 (ξ0 −α p2 )l2 1 (ξ1 −α p2 )l2
.. .
1 (ξl −α p2 )l2 1 1 (ξl +1 −α p2 )l2 1
.. .
1 (ξl +l −α p2 )l2 1 2
.. .
1 (ξm1 −α p2 )l2
···
1 (ξ0 −αk )
···
···
1 (ξ1 −αk )
···
..
..
···
.. .
1 (ξl −αk ) 1
···
···
1 (ξl +1 −αk ) 1
···
.
.
···
.. .
..
1 (ξl +l −αk ) 1 2
···
..
.. .
..
..
.
.
···
1 (ξm1 −αk )
.
.
···
1 (ξ0 −αk )κnk −1 1 (ξ1 −αk )κnk −1
.. .
1 (ξl −αk )κnk −1 1 1 (ξl +1 −αk )κnk −1 1
.. .
1 (ξl +l −αk )κnk −1 1 2
1 (ξm1 −αk )κnk −1
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k
×
.. .
· · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξm1 )G (ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξ(n−1)K+k−1 ; z) ϕ0 (ξ0 )ϕ0 (ξ1 ) · · · ϕ0 (ξl1 )ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξm1 )
' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '
254
K. T.-R. McLaughlin et al.
1 ϕ0 (ξ0 ) ϕ0 (ξ1 ) · · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξm1 ) ' ϕ1 (ξ0 ) ··· ϕl1 (ξ0 ) ϕl1 +1 (ξ0 ) ··· ϕl1 +l2 (ξ0 ) ··· ϕm2 −κnk +1 (ξ0 ) ' ϕ0 (ξ0 ) ' ϕ1 (ξ1 ) ··· ϕl1 (ξ1 ) ϕl1 +1 (ξ1 ) ··· ϕl1 +l2 (ξ1 ) ··· ϕm2 −κnk +1 (ξ1 ) ' ϕ0 (ξ1 ) ' .. .. .. .. .. .. .. .. .. ' ' . . . . . . . . . ' ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ··· ϕ (ξ ) 0 1 m − κ l l l l l +1 l l +l l ' 2 nk +1 l1 1 1 1 1 1 1 1 2 1 ' ϕ0 (ξl1 +1 ) ϕ1 (ξl1 +1 ) ··· ϕl1 (ξl1 +1 ) ϕl1 +1 (ξl1 +1 ) ··· ϕl1 +l2 (ξl1 +1 ) ··· ϕm2 −κnk +1 (ξl1 +1 ) × '' .. .. .. .. .. .. .. .. .. ' ' . . . . . . . . . ' ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ··· ϕ (ξ ) 0 1 m − κ l1 +l2 l1 +l2 l1 l1 +l2 l1 +1 l1 +l2 l1 +l2 l1 +l2 2 nk +1 l1 +l2 ' ' .. .. .. .. .. .. .. .. .. ' ' . . . . . . . . . '
×
ϕ0 (ξm1 )
ϕ1 (ξm1 )
···
ϕl1 (ξm1 )
= D
=
D (ϕ0 (z))−1 m2 !
ϕl1 +1 (ξm1 )
(n−1)K+k−1 i, j=0 j
···
ϕl1 +l2 (ξm1 )
···
(ξi −ξ j )
ϕm2 −κnk +1 (ξm1 )
··· ···
..
.
··· ···
..
.
···
..
.
···
' ' ' ' ' .. ' ' . ϕ(n−1)K+k−1 (ξl1 ) '' ϕ(n−1)K+k−1 (ξl1 +1 ) ' ' ' .. ' ' . ϕ(n−1)K+k−1 (ξl1 +l2 ) '' ' .. ' ' . ϕ(n−1)K+k−1 (ξm1 ) ' ϕ(n−1)K+k−1 (ξ0 ) ϕ(n−1)K+k−1 (ξ1 )
(cf. proof of Lemma 2.1)
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
R
(n−1)K+k
· · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξm1 ) × ×
i, j=0
(ξi − ξ j )
j
ϕ0 (ξ0 )ϕ0 (ξ1 ) · · · ϕ0 (ξl1 )ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξm1 ) 1 ϕ0 (ξ0 ) ϕ0 (ξ1 ) · · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξm1 )
' ϕ0 (ξ0 ) ϕ1 (ξ0 ) ' ' ϕ0 (ξ1 ) ϕ1 (ξ1 ) ' ' .. .. ' ' . . ' ϕ0 (ξl1 ) ϕ1 (ξl1 ) ' ' ϕ0 (ξl1 +1 ) ϕ1 (ξl1 +1 ) ' ×' .. .. ' . . ' ' ϕ0 (ξl1 +l2 ) ϕ1 (ξl1 +l2 ) ' ' .. .. ' . . ' ' ϕ0 (ξm1 ) ϕ1 (ξm1 ) ' ϕ0 (z) ϕ1 (z)
··· ···
..
DD (ϕ0 (z))−1 ((n−1)K+k)!
ϕl1 (ξ0 ) ϕl1 (ξ1 )
.
.. .
.
.. .
ϕl1 +1 (ξ0 ) ϕl1 +1 (ξ1 )
··· ···
.. .
..
.. .
..
.. .
..
ϕl1 +l2 (ξ0 ) ϕl1 +l2 (ξ1 )
··· ···
.
.. .
..
.
.. .
..
.. .
..
ϕm2 −κnk +1 (ξ0 ) ϕm2 −κnk +1 (ξ1 )
··· ···
.
.. .
..
.
.. .
..
.. .
..
.
··· ϕl1 (ξl1 ) ϕl1 +1 (ξl1 ) ··· ϕl1 +l2 (ξl1 ) ··· ϕm2 −κnk +1 (ξl1 ) ··· ··· ϕl1 (ξl1 +1 ) ϕl1 +1 (ξl1 +1 ) ··· ϕl1 +l2 (ξl1 +1 ) ··· ϕm2 −κnk +1 (ξl1 +1 ) ···
..
.
··· ϕl1 (ξl1 +l2 ) ϕl1 +1 (ξl1 +l2 ) ··· ϕl1 +l2 (ξl1 +l2 ) ··· ϕm2 −κnk +1 (ξl1 +l2 ) ···
..
.
··· ···
=D
=
(n−1)K+k−1
R
.. .
ϕl1 (ξm1 ) ϕl1 (z)
ϕl1 +1 (ξm1 ) ϕl1 +1 (z)
(n−1)K+k−1 i, j=0 j
(ξi −ξ j )
.
··· ···
ϕl1 +l2 (ξm1 ) ϕl1 +l2 (z)
(n−1)K+k−1 m=0
(z−ξm )
.
··· ···
ϕm2 −κnk +1 (ξm1 ) ϕm2 −κnk +1 (z)
.
··· ···
' ' ' ' ' .. ' ' . ϕ(n−1)K+k (ξl1 ) ' ' ϕ(n−1)K+k (ξl1 +1 ) ' ' ' .. ' . ' ϕ(n−1)K+k (ξl1 +l2 ) ' ' ' .. ' . ' ϕ(n−1)K+k (ξm1 ) ' ' ϕ(n−1)K+k (z) ϕ(n−1)K+k (ξ0 ) ϕ(n−1)K+k (ξ1 )
(cf. Proof of Lemma 2.1)
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · R R
(n−1)K+k
× · · · dμ(ξl1 +l2 ) · · · dμ(ξ(n−1)K+k−1 ) ×
×
( ϕ0 (ξ0 ) ϕ0 (ξ1 ) · · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξ(n−1)K+k−1 ))−1 ϕ0 (ξ0 )ϕ0 (ξ1 ) · · · ϕ0 (ξl1 )ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξ(n−1)K+k−1 ) (n−1)K+k−1 $
(n−1)K+k−1 $
i, j=0
m=0
(ξi −ξ j )2
j
(z−ξm );
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
255
but, recalling that, for n ∈ N and k = 1, 2, . . . , K, ϕ0 (z) = ϕ0 (z)/(z−αk ), one arrives at DD (ϕ0 (z))−1 n Ξk (z) = · · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · ((n−1)K+k)! R R R (n−1)K+k
× · · · dμ(ξl1 +l2 ) · · · dμ(ξ(n−1)K+k−1 )
(n−1)K+k−1 $
!−2 ϕ0 (ξl )
l=0
×
(n−1)K+k−1 $
(n−1)K+k−1 $
i, j=0
m=0
(ξi −ξ j)2
(ξm −αk )(z−ξm ) ⇒
j
⎛ ⎞−1 s −1 $ DD κ ⎝ (z−α pq ) nkkq (z−αk )κnk ⎠ Ξkn (z) = ((n−1)K+k)! q=1 ×
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξ(n−1)K+k−1 ) R R R (n−1)K+k
⎛ ×⎝
(n−1)K+k−1 s −1 $ $ m=0
×
⎞−2 (ξm −α pq )κnkkq (ξm −αk )κnk ⎠
q=1
(n−1)K+k−1 $
(ξi −ξ j)
i, j=0
2
(n−1)K+k−1 $
(ξl −αk )(z−ξl ).
l=0
j
Recalling that, for n ∈ N and k = 1, 2, . . . , K, π nk (z) = Y11 (z)/(z−αk ) = Ξkn (z)/ cˆ D , where Ξkn (z) is given directly above, and cˆ D is given in the proof of Lemma 2.1, (25), one arrives at the integral representation for π nk (z) stated in the Lemma. The determinantal representation for Y21 (z)/(z−αk ) is now studied. For n ∈ N and k = 1, 2, . . . , K, recall the ordered disjoint partition for {α1 , α2 , . . . , α K } ∪ · · · ∪ {α1 , α2 , . . . , α K } ∪ {α1 , α2 , . . . , αk } introduced in the proof of Lemma 2.1. For this ordered disjoint partition, introduce, for n ∈ N and k = 1, 2, . . . , K, the following notation (cf. Subsection 1.2; recall that lm := κnkkm , m = 1, 2, . . . , s −1, and ls := κnk ): / ϕ0 (z) :=
s −1 $
m=1
(z−α pm )lm (z−αk )κnk −2 =:
(n−1)K+k−2
j=0
/ a j,0 z j,
256
K. T.-R. McLaughlin et al.
and, for r = 1, 2, . . . , s, q(r) = m(r) = 1, 2, . . . , lr −2δrs ,
r−1
i=1 li +1,
r−1
i=1 li +2, . . . ,
r−1
i=1 li +lr −2δrs ,
and
⎫ ⎧ (n−1)K+k−2 ⎬ ⎨
/ a j,q(r) z j ; / ϕq(r) (z) := / ϕ0 (z)(z−α pr )−m(r) =: ⎭ ⎩ j=0
e.g., for r = 1, the notation / ϕq(1) (z) := {/ ϕ0 (z)(z−α p1 )−m(1) =:
(n−1)K+k−2 j=0
/ a j,q(1) z j}, q(1) = 1, 2, . . . , l1 , m(1) = 1, 2, . . . , l1 , encapsulates the l1 = κnkk1 functions
/ ϕ1 (z) =
/ ϕ2 (z) =
...,/ ϕl1 (z) =
/ ϕ0 (z) =: z−α p1
(n−1)K+k−2
/ a j,1 z j,
j=0
/ ϕ0 (z) =: (z−α p1 )2
(n−1)K+k−2
/ ϕ0 (z) =: (z−α p1 )l1
(n−1)K+k−2
/ a j,2 z j, . . .
j=0
/ a j,l1 z j,
j=0
/ for r = 2, the notation / ϕq(2) (z) := {/ ϕ0 (z)(z−α p2 )−m(2) =: (n−1)K+k−2 a j,q(2) z j}, j=0 q(2) =l1 +1, l1 +2, . . . , l1 +l2 , m(2) = 1, 2, . . . , l2 , encapsulates the l2 = κnkk2 functions
/ ϕl1 +1 (z) =
/ ϕ0 (z) =: z−α p2
(n−1)K+k−2
/ a j,l1 +1 z j,
j=0
/ ϕ0 (z) / ϕl1 +2 (z) = =: (z−α p2 )2
(n−1)K+k−2
/ ϕ0 (z) =: (z−α p2 )l2
(n−1)K+k−2
...,/ ϕl1 +l2 (z) =
/ a j,l1 +2 z j, . . .
j=0
/ a j,l1 +l2 z j,
j=0
s−1 etc., and, for r = s (cf. Subsection 1.2; recall that α ps := αk and i=1 li = (n−1) (n−1)K+k−2 −m(s) / ·K+k− κnk ), the notation/ a j,q(s) z j}, ϕq(s) (z) := {/ ϕ0 (z)(z−αk ) =: j=0
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
257
q(s) = (n−1)K+k− κnk +1, (n−1)K+k− κnk +2, . . . , (n−1)K+k−2, m(s) = 1, 2, . . . , ls −2, encapsulates the ls −2 = κnk −2 functions
/ ϕ0 (z) / ϕ(n−1)K+k−κnk +1 (z) = =: z − αk
(n−1)K+k−2
/ ϕ0 (z) =: / ϕ(n−1)K+k−κnk +2 (z) = (z − αk )2
(n−1)K+k−2
/ ϕ0 (z) ...,/ ϕ(n−1)K+k−2 (z) = =: (z − αk )κnk −2
(n−1)K+k−2
/ a j,(n−1)K+k−κnk +1 z j,
j=0
/ a j,(n−1)K+k−κnk +2 z j, . . .
j=0
/ a j,(n−1)K+k−2 z j.
j=0
s (z)(z−α pr )−m(r) } =lr −2δrs , r = 1, 2, . . . , s, and # ∪r=1 {/ ϕ0 (z)(z− (Note: #{/ ϕ0 s −m(r) α pr ) } = r=1 lr − 2 = (n−1)K+k−2.) One notes that, for n ∈ N and k = 1, 2, ϕ0 (z), / ϕ1 (z), . . . , . . . , K, the 1+l1 +l2 +· · ·+(ls −2) = (n−1)K+k−1 functions / ϕl1 +1 (z), . . . , / ϕl1 +l2 (z), . . . , / ϕ(n−1)K+k−κnk +1 (z), . . . , / ϕ(n−1)K+k−2 (z) are lin/ ϕl1 (z), / (n−1)K+k−2 / cj / early independent on R, that is, for z ∈ R, ϕ j(z) = 0 ⇒ (via j=0 a Vandermonde-type argument) / c j = 0, j= 0, 1, . . . , (n−1)K+k−2 (see the ((n−1)K+k−1) × ((n−1)K+k−1) non-zero determinant D in (34) below). For n ∈ N and k = 1, 2, . . . , K, corresponding to the ordered disjoint partion above, recall the formula for Y21 (z)/(z−αk ) given in the proof of Lemma 2.1:
l =κ s−1 lm =κnkk νm,q (n, k) s nk / νs,q (n, k) Y21 (z) m / = + , q−1 z−αk m=1 q=1 (z−α pm )q (z−α k) q=1
/ νs,ls (n, k) = 0,
where the (n − 1)K+k real coefficients / ν1,1 (n, k), . . . ,/ ν1,l1 (n, k),/ ν2,1 (n, k), . . . , νs,1 (n, k), . . . ,/ νs,ls (n, k), with / νs,ls (n, k) = 0, satisfy the linear / ν2,l2 (n, k), . . . ,/ inhomogeneous algebraic system of equations (32) given in the proof of Lemma 2.1. Via the multi-linearity property of the determinant and an application of Cramer’s Rule to system (32), one arrives at, for n ∈ N and k = 1, 2, . . . , K, the following (ordered) determinantal representation for Y21 (z)/(z−αk ): /n (z) Ξ 1 Y21 (z) = k , 2π i z−αk N (n, k)
258
K. T.-R. McLaughlin et al.
where N (n, k) is given in the proof of Lemma 2.1, and, with n1 := (n−1)K+ k−2, ' (ξ0 −αk )0 dμ(ξ0 ) (ξ0 −αk )0 dμ(ξ0 ) (ξ0 −αk )0 dμ(ξ0 ) ' ··· ··· ' R R R (ξ0 −α p1 ) (ξ0 −α p2 ) (ξ0 −α p1 )l1 ' ' (ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p1 )−1 dμ(ξ1 ) ' ··· ··· R R R ' (ξ1 −α p1 ) (ξ1 −α p2 ) l 1 (ξ1 −α p1 ) ' ' . . . .. .. ' . . . . . ' . . . ' −l (ξl1 −α p1 ) 1 dμ(ξl1 ) ' (ξl1 −α p1 )−l1 dμ(ξl1 ) ··· (ξl1 −α p1 )−l1 dμ(ξl1 ) ··· ' R R R (ξl −α p1 ) (ξl −α p2 ) (ξl −α p1 )l1 1 1 ' 1 ' −1 dμ(ξ −1 dμ(ξ ' (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) (ξl1 +1 −α p2 ) (ξl1 +1 −α p2 ) l1 +1 ) l1 +1 ) ··· R ··· ' R R (ξl +1 −α p1 ) (ξ −α p2 ) l n 1 l +1 (ξl +1 −α p1 ) /k (z) := ' 1 1 Ξ 1 ' ' . . . . . . .. . . .. ' . . . ' ' (ξl +l −α p2 )−l2 dμ(ξl +l ) (ξl +l −α p2 )−l2 dμ(ξl +l ) (ξl +l −α p2 )−l2 dμ(ξl +l ) 1 2 1 2 1 2 1 2 ··· 1 2 ··· ' R 1 2 (ξ R R (ξl +l −α p2 ) ' l1 +l2 −α p1 ) (ξl +l −α p1 )l1 1 2 1 2 ' ' . . . .. .. ' . . . . . . . . ' ' (ξn −α )−κ nk +2 dμ(ξn ) (ξn1 −αk )−κ nk +2 dμ(ξn1 ) (ξn1 −αk )−κ nk +2 dμ(ξn1 ) ' R 1 k 1 ··· R ··· R (ξn1 −α p1 ) (ξn1 −α p2 ) ' (ξn1 −α p1 )l1 ' ' 1 1 1 ··· ··· z−α p1 z−α p2 ' (z−α p )l1 1
(ξ0 −αk )0 dμ(ξ0 ) (ξ0 −α p2 )l2
··· ··· ···
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −α p2 )l2
··· ··· ···
R
R
.. .
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −α p2 )l2 1 (ξl1 +1 −α p2 )−1 dμ(ξl1 +1 ) R (ξl +1 −α p2 )l2 1
R
.. .
R
R
(ξl +l −α p2 )−l2 dμ(ξl +l ) 1 2 1 2 (ξl +l −α p2 )l2 1 2
(ξn1 −αk )−κnk +2 dμ(ξn1 ) (ξn1 −α p1 )l2 1 (z−α p2 )l2
=
R
.. .
R
R
··· ··· ···
R
R
.. . . . . . . . ··· ··· ··· ··· ··· ···
(ξl −α p1 )−l1 dμ(ξl ) 1 1 (ξl −αk )0 1
(ξl +1 −α p2 )−1 dμ(ξl +1 ) 1 1 (ξl +1 −αk )0 1
.. .
.. .. .. . . . ··· ··· ···
(ξ1 −α p1 )−1 dμ(ξ1 ) (ξ1 −αk )0
.. .
.. .. .. . . .
··· ··· ···
(ξ0 −αk )0 dμ(ξ0 ) (ξ0 −αk )0
R
R
(ξl +l −α p2 )−l2 dμ(ξl +l ) 1 2 1 2 (ξl +l −αk )0 1 2
.. .
(ξn1 −αk )−κnk +2 dμ(ξn1 ) (ξn1 −αk )0 1 (z−αk )0
' ' ' ' ' ··· ' ' ' .. .. ' . . ' ' (ξl1 −α p1 )−l1 dμ(ξl1 ) ··· ' R −1 κ (ξl −αk ) nk ' 1 ' −1 (ξl1 +1 −α p2 ) dμ(ξl1 +1 ) ' ··· R ' (ξl +1 −αk )κnk −1 1 ' ' . .. ' . . . ' −l (ξl +l −α p2 ) 2 dμ(ξl1 +l2 ) ' ' ··· R 1 2 (ξl +l −αk )κnk −1 ' 1 2 ' ' . .. .. ' . ' (ξn1 −αk )−κnk +2 dμ(ξn1 ) ' ··· R −1 κ ' (ξn1 −αk ) nk ' 1 ' ··· κ −1 ···
(ξ0 −αk )0 dμ(ξ0 ) R (ξ −α )κnk −1 0 k (ξ1 −α p1 )−1 dμ(ξ1 ) R (ξ −α )κnk −1 1 k
(z−αk ) nk
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k−1
×
··· dμ(ξm2 −κnk +1 ) ··· dμ(ξn1 )(−1)(n−1)K+k−κnk 0 1 (ξ0 −αk ) (ξ1 −α p1 ) ···(ξl1 −α p1 )l1 (ξl1 +1 −α p2 )1 ···(ξl1 +l2 −α p2 )l2 ···(ξm2 −κnk +1 −αk )1 ···(ξn1 −αk )κnk −2
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
'1 ' ' '1 ' '. ' .. ' '1 ' '1 ' × '' . ' .. '1 ' ' '. ' .. ' '1 ' '1
1 (ξ0 −α p1 )
···
1 (ξ1 −α p1 )
···
.. .
..
.
1 (ξl −α p1 ) 1 1 (ξl +1 −α p1 ) 1
··· ···
.. .
..
1 (ξl +l −α p1 ) 1 2
···
.
.. .
..
1 (ξn1 −α p1 )
···
1 z−α p1
···
1 (ξ0 −α p1 )l1 1 (ξ1 −α p1 )l1
1 (ξ0 −α p2 )
···
1 (ξ1 −α p2 )
···
.. .
..
1 (ξl −α p1 )l1 1 1 (ξl +1 −α p1 )l1 1
1 (ξl −α p2 ) 1 1 (ξl +1 −α p2 ) 1
···
.. .
..
1 (ξl +l −α p1 )l1 1 2
1 (ξl +l −α p2 ) 1 2
···
.. .
..
1 (ξn1 −α p1 )l1 1 (z−α p1 )l1
1 (ξn1 −α p2 )
···
1 z−α p2
···
.. .
.. . .. .
.
.
···
.
.
1 (ξ0 −α p2 )l2 1 (ξ1 −α p2 )l2
.. .
1 (ξl −α p2 )l2 1 1 (ξl +1 −α p2 )l2 1
···
1 (ξ0 −αk )
···
···
1 (ξ1 −αk )
···
.. .
..
..
···
.. .
..
.. .
..
1 (ξl +l −α p2 )l2 1 2
1 (ξn1 −α p2 )l2 1 (z−α p2 )l2
.
···
.
···
.
1 (ξl −αk ) 1 1 (ξl +1 −αk ) 1
··· ···
.. .
..
1 (ξl +l −αk ) 1 2
···
.
···
.. .
..
1 (ξn1 −αk )
···
···
1 z−αk
···
.
.
1 (ξ0 −αk )κnk −1 1 (ξ1 −αk )κnk −1
.. .
1 (ξl −αk )κnk −1 1 1 (ξl +1 −αk )κnk −1 1
.. .
1 (ξl +l −αk )κnk −1 1 2
.. .
1 (ξn1 −αk )κnk −1 1 (z−αk )κnk −1
259
' ' ' ' ' ' ' ' ' ' ' ' '. ' ' ' ' ' ' ' ' ' ' '
For n ∈ N and k = 1, 2, . . . , K, recalling the (n−1)K+k linearly independent −1 (on R) functions ϕ0 (z) := sm=1 (z−α pm )lm (z−αk )κnk −1 =: (n−1)K+k−1 a j,0 z j, j=0 ϕq(r1 ) (z) := { ϕ0 (z)(z−α pr1 )−m(r1 ) =: (n−1)K+k−1 a j,q(r1 ) z j}, r1 = 1, 2, . . . , s,q(r1 ) = r1 −1 r1 −1 j=0 r1 −1 , and i=1 li +1, i=1 li +2, . . . , i=1 li +lr1 −δr1 s , m(r1 ) = 1, 2, . . . , lr1 −δ r1 s−1 ϕ0 (z) := sm=1 (z− the (n−1)K+k−1 linearly independent (on R) functions / (n−1)K+k−2 j −m(r2 ) / α pm )lm (z−αk )κnk −2 =: a z , / ϕ (z) := {/ ϕ (z)(z − α ) =: j,0 q(r2 ) 0 pr j=0 r2 −1 r2 −1 2 (n−1)K+k−2 j / a j, q(r2 ) z }, r2 = 1, 2, . . . , s, q(r2 ) = i=1 li + 1, i=1 li + 2, . . . , rj=0 2 −1 i=1 li + lr2 − 2δr2 s , m(r2 ) = 1, 2, . . . , lr2 −2δr2 s , one proceeds, via the latter /n (z) := (−1)(n−1)K+k−κnk Ξˇ n (z): determinantal expression, with the analysis of Ξ k k Ξˇ kn (z) =
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k−1
× ×
··· dμ(ξm2 −κnk +1 ) ··· dμ(ξn1 )
(ξ0 −αk )0 (ξ1 −α p1 )1 ···(ξl1 −α p1 )l1 (ξl1 +1 −α p2 )1 ···(ξl1 +l2 −α p2 )l2 ···(ξm2 −κnk +1 −αk )1 ···(ξn1 −αk )κnk −2
( ϕ0 (z))−1 ϕ0 (ξ0 ) ϕ0 (ξ1 ) · · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξn1 )
' ' ϕ1 (ξ0 ) ··· ϕl1 (ξ0 ) ϕl1 +1 (ξ0 ) ··· ϕl1 +l2 (ξ0 ) ··· ϕm2 −κnk +1 (ξ0 ) ··· ϕ(n−1)K+k−1 (ξ0 ) ' ' ϕ0 (ξ0 ) ' ϕ0 (ξ1) ϕ1 (ξ1) ··· ϕl1 (ξ1) ϕl1 +1 (ξ1) ··· ϕl1 +l2 (ξ1) ··· ϕm2 −κnk +1 (ξ1) ··· ϕ(n−1)K+k−1 (ξ1) ' ' ' .. .. .. .. .. .. .. ' ' .. .. .. .. ' ' . . . . . . . . . . . ' ϕ0 (ξl1) ϕ1 (ξl1) ··· ϕl1 (ξl1) ϕl1 +1 (ξl1) ··· ϕl1 +l2 (ξl1) ··· ϕm2 −κnk +1 (ξl1) ··· ϕ(n−1)K+k−1 (ξl1) '' ' ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ϕ (ξ ) ··· ϕ (ξ ) ··· ϕ (ξ ) ··· ϕ (ξ ) +1 0 1 m − κ l +1 l +1 l l +1 l +1 l +1 l +l l +1 l +1 (n−1)K+k−1 l +1 ' ' 2 nk 1 1 1 1 1 1 1 2 1 1 1 ' × '' .. .. .. .. .. .. .. .. .. .. .. ' . . . . ' ' . . . . . . . ϕ0 (ξl1 +l2) ϕ1 (ξl1 +l2) ··· ϕl1 (ξl1 +l2) ϕl1 +1 (ξl1 +l2) ··· ϕl1 +l2 (ξl1 +l2) ··· ϕm2 −κnk +1 (ξl1 +l2) ··· ϕ(n−1)K+k−1 (ξl1 +l2) ' ' ' ' . . . . . . . .. .. .. .. ' ' .. .. .. .. .. .. .. ' ' . . . . ' ϕ0 (ξn1) ϕ1 (ξn1) ··· ϕl1 (ξn1) ϕl1 +1 (ξn1) ··· ϕl1 +l2 (ξn1) ··· ϕm2 −κnk +1 (ξn1) ··· ϕ(n−1)K+k−1 (ξn1) ' ' ϕ1 (z) ··· ϕl1 (z) ϕl1 +1 (z) ··· ϕl1 +l2 (z) ··· ϕm2 −κnk +1 (z) ··· ϕ(n−1)K+k−1 (z) ' ϕ0 (z) =: G∨ (ξ0 ,ξ1 ,...,ξl1 ,ξl1 +1 ,...,ξl1 +l2 ,...,ξ(n−1)K+k−2 ;z)
=
( ϕ0 (z))−1 m1 !
σ ∈S m1
R
· · · dμ(ξσ (0) ) dμ(ξσ (1) ) · · · dμ(ξσ (l1 ) ) dμ(ξσ (l1 +1) ) · · · R R
(n−1)K+k−1
260 × ×
K. T.-R. McLaughlin et al. ··· dμ(ξσ (l1 +l2 ) ) ··· dμ(ξσ (m2 −κnk +1) ) ··· dμ(ξσ (n1 ) )
(ξσ (0) −αk )0 (ξσ (1) −α p1 )1 ···(ξσ (l1 ) −α p1 )l1 (ξσ (l1 +1) −α p2 )1 ···(ξσ (l1 +l2 ) −α p2 )l2 ···(ξσ (m2 −κnk +1) −αk )1 ···(ξσ (n1 ) −αk )κnk −2
1 ϕ0 (ξσ (0) ) ϕ0 (ξσ (1) ) · · · ϕ0 (ξσ (l1 ) ) ϕ0 (ξσ (l1 +1) ) · · · ϕ0 (ξσ (l1 +l2 ) ) · · · ϕ0 (ξσ (m2 −κ nk +1) ) · · · ϕ0 (ξσ (n1 ) )
× G∨ (ξσ (0) , ξσ (1) , . . . , ξσ (l1 ) , ξσ (l1 +1) , . . . , ξσ (l1 +l2 ) , . . . , ξσ ((n−1)K+k−2) ; z) =
( ϕ0 (z))−1 m1 !
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
R
(n−1)K+k−1
× · · · dμ(ξm2−κ nk +1 ) · · · dμ(ξn1 )G∨ (ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξ(n−1)K+k−2 ; z)
×
σ ) (ξσ (0) −αk )0 (ξσ (1) −α p1 )1 ···(ξσ (l ) −α p1 )l1 (ξσ (l +1) −α p2 )1 ···(ξ1σ (l +l ) −α p2 )l2 ···(ξσ (m2 −κ +1) −αk )1 ···(ξσ (n1 ) −αk )κnk −2 sgn(σ 1 1 1 2 nk
σ ∈S m1
×
1 ϕ0 (ξσ (0) ) ϕ0 (ξσ (1) ) · · · ϕ0 (ξσ (l1 ) ) ϕ0 (ξσ (l1 +1) ) · · · ϕ0 (ξσ (l1 +l2 ) ) · · · ϕ0 (ξσ (m2 −κ nk +1) ) · · · ϕ0 (ξσ (n1 ) )
=
( ϕ0 (z))−1 m1 !
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
R
(n−1)K+k−1
×
· · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξn1 )G∨ (ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξ(n−1)K+k−2 ; z) ϕ0 (ξ0 ) ϕ0 (ξ1 ) · · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξn1 )
' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ×' ' ' ' ' ' ' ' ' ' ' ' ' '
=
1 (ξ0 −αk )0 1 (ξ1 −αk )0
.. .
1 (ξl −αk )0 1 1 (ξl +1 −αk )0 1
.. .
1 (ξl +l −αk )0 1 2
.. .
1 (ξn1 −αk )0
( ϕ0 (z))−1 m1 !
1 (ξ0 −α p1 )
···
1 (ξ1 −α p1 )
···
.. .
..
1 (ξl −α p1 ) 1
···
1 (ξl +1 −α p1 ) 1
···
.
.. .
..
1 (ξl +l −α p1 ) 1 2
···
.. .
..
1 (ξn1 −α p1 )
.
.
···
1 (ξ0 −α p1 )l1 1 (ξ1 −α p1 )l1
.. .
1 (ξl −α p1 )l1 1 1 (ξl +1 −α p1 )l1 1
.. .
1 (ξl +l −α p1 )l1 1 2
.. .
1 (ξn1 −α p1 )l1
1 (ξ0 −α p2 )
···
1 (ξ1 −α p2 )
···
.. .
..
1 (ξl −α p2 ) 1
···
1 (ξl +1 −α p2 ) 1
···
.
.. .
..
1 (ξl +l −α p2 ) 1 2
···
.. .
..
1 (ξn1 −α p2 )
.
.
···
1 (ξ0 −α p2 )l2 1 (ξ1 −α p2 )l2
···
1 (ξ0 −αk )
···
···
1 (ξ1 −αk )
···
.. .
..
1 (ξl −α p2 )l2 1 1 (ξl +1 −α p2 )l2 1
···
.. .
..
1 (ξl −αk ) 1
···
···
1 (ξl +1 −αk ) 1
···
.
.. .
..
1 (ξl +l −α p2 )l2 1 2
···
.. .
..
1 (ξn1 −α p2 )l2
.
.
···
.. .
.
.. .
..
1 (ξl +l −αk ) 1 2
···
.. .
..
1 (ξn1 −αk )
1 (ξ0 −αk )κnk −2 1 (ξ1 −αk )κnk −2
1 (ξl −αk )κnk −2 1 1 (ξl +1 −αk )κnk −2 1
.. .
.
1 (ξl +l −αk )κnk −2 1 2
.
···
R
.. .
1 (ξn1 −αk )κnk −2
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · R R
(n−1)K+k−1
×
· · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξn1 )G∨ (ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξ(n−1)K+k−2 ; z) ϕ0 (ξ0 ) ϕ0 (ξ1 ) · · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξn1 )
×
1 / ϕ0 (ξ0 )/ ϕ0 (ξ1 ) · · · / ϕ0 (ξl1 )/ ϕ0 (ξl1 +1 ) · · · / ϕ0 (ξl1 +l2 ) · · · / ϕ0 (ξm2 −κ nk +1 ) · · · / ϕ0 (ξn1 )
' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal... ' / / ϕ1 (ξ0 ) ' ϕ0 (ξ0 ) ' / / ϕ1 (ξ1 ) ' ϕ0 (ξ1 ) ' .. .. ' ' . . ' ϕ0 (ξl1 ) / ϕ1 (ξl1 ) ' / '/ ϕ1 (ξl1 +1 ) ' ϕ0 (ξl1 +1 ) / ×' ' . .. ' .. . ' '/ ϕ1 (ξl1 +l2 ) ' ϕ0 (ξl1 +l2 ) / ' .. .. ' ' . . '
/ ϕ0 (ξn1 )
/ ϕ1 (ξn1 )
··· ···
..
/ ϕl1 (ξ0 )
/ ϕl1 +1 (ξ0 )
···
/ ϕl1 +1 (ξ1 )
···
.. .
.. .
..
/ ϕl1 (ξ1 )
.
···
.
/ ϕl1 (ξl1 )
/ ϕl1 +1 (ξl1 )
···
.. .
.. .
..
.. .
..
/ ϕl1 +l2 (ξ0 )
···
/ ϕl1 +l2 (ξ1 )
···
.. .
..
.
/ ϕl1 +l2 (ξl1 )
···
.. .
..
.. .
..
/ ϕm2 −κnk +1 (ξ0 )
···
/ ϕm2 −κnk +1 (ξ1 )
···
.. .
..
.
/ ϕm2 −κnk +1 (ξl1 )
···
.. .
..
.. .
..
··· / ϕl1 (ξl1 +1 ) / ϕl1 +1 (ξl1 +1 ) ··· / ϕl1 +l2 (ξl1 +1 ) ··· / ϕm2 −κnk +1 (ξl1 +1 ) ···
..
.
.
.
.
··· / ϕl1 (ξl1 +l2 ) / ϕl1 +1 (ξl1 +l2 ) ··· / ϕl1 +l2 (ξl1 +l2 ) ··· / ϕm2 −κnk +1 (ξl1 +l2 ) ···
..
.. .
.
···
/ ϕl1 (ξn1 )
/ ϕl1 +1 (ξn1 )
.
···
/ ϕl1 +l2 (ξn1 )
.
···
/ ϕm2 −κnk +1 (ξn1 )
/(ξ0 ,ξ1 ,...,ξl ,ξl +1 ,...,ξl +l ,...,ξm −κ +1 ,...,ξ(n−1)K+k−2 ) =: G 2 nk 1 1 1 2
.
···
261
' ' ' ' ' .. ' ' . ' / ϕ(n−1)K+k−2 (ξl1 ) ' ' / ϕ(n−1)K+k−2 (ξl1 +1 ) ' '; ' .. ' . ' / ϕ(n−1)K+k−2 (ξl1 +l2 ) ' ' ' .. ' ' . ' / ϕ(n−1)K+k−2 (ξn1 ) / ϕ(n−1)K+k−2 (ξ0 ) / ϕ(n−1)K+k−2 (ξ1 )
but, noting the determinantal factorisation /(ξ0 , ξ1 , . . . , ξl1 , ξl1 +1 , . . . , ξl1 +l2 , . . . , ξm2 −κnk +1 , . . . , ξ(n−1)K+k−2 ) G = D V3 (ξ0 , ξ1 , . . . , ξ(n−1)K+k−2 ), where D :=
' /a0,0 /a1,0 ' ' /a0,1 /a1,1 ' ' .. ' .. ' . . ' ' /a0,l1 /a1,l1 ' /a ' 0,l1 +1 /a1,l1 +1 ' ' . .. ' . . ' . ' /a0,l +l /a1,l +l ' 1 2 1 2 ' . .. ' . ' . . ' /a ' 0,n1 −1 /a1,n1 −1 ' /a / a 0,n1
1,n1
··· ···
/ al1 ,0 / al1 ,1
/ al1 +1,0 / al1 +1,1
··· ···
/ al1 +l2 ,0 / al1 +l2 ,1
··· ···
/ am2 −κnk +1,0 / am2 −κnk +1,1
..
.. .
.. .
..
.. .
..
.. .
.
.
/ al1 ,l1
/ al1 +1,l1
···
.
.. .
.. .
..
.
.. .
.. .
..
···
.
/ al1 +l2 ,l1
···
.
.. .
..
.
.. .
..
/ am2 −κnk +1,l1
··· / al1 ,l1 +1 / al1 +l2 ,l1 +1 ··· / am2 −κnk +1,l1 +1 al1 +1,l1 +1 ··· /
..
.
.. .
.
.. .
··· / al1 ,l1 +l2 / al1 +l2 ,l1 +l2 ··· / am2 −κnk +1,l1 +l2 al1 +1,l1 +l2 ··· /
..
··· / al1 ,n1 −1 / al1 +l2 ,n1 −1 ··· / am2 −κnk +1,n1 −1 al1 +1,n1 −1 ··· / / ··· / al1 ,n1 al1 +l2 ,n1 ··· / am2 −κnk +1,n1 al1 +1,n1 ··· /
' ' ' ' ' . ' .. . ' . . ' ··· / a(n−1)K+k−2,l1 ' ' ··· / a(n−1)K+k−2,l1 +1 ' ' ' .. .. ' . . ' ··· / a(n−1)K+k−2,l1 +l2 ' ' ' .. ' .. ' . . ' ··· / a(n−1)K+k−2,n1 −1 ' ' ··· / a ··· ···
/ a(n−1)K+k−2,0 / a(n−1)K+k−2,1
( = 0),
(n−1)K+k−2,n1
(34)
and
' 1 ' ξ ' 0 ' . ' . ' .l ' ξ01 ' ' ξ l1 +1 ' 0 ' . ' . ' . ' ξ l1 +l2 ' 0 ' . ' . ' n .−1 'ξ 1 ' 0n ' ξ1 0
V3 (ξ0 , ξ1 , . . . , ξ(n−1)K+k−2 ) := 1 ξ1
.. .
l ξ11 l1 +1 ξ1
··· ···
..
··· ···
.. .
..
.. .
..
l +l ξ11 2
n −1 ξ1 1 n ξ1 1
.
.
···
.
··· ···
1 ξl1
.. .
.. .
l ξl 1 1 l1 +1 ξl 1
l ξl 1+1 1 l1 +1 ξl +1 1
l +l ξl 1 2 1
l +l ξl 1+12 1
n −1 ξl 1 1 n ξl 1 1
n −1 ξl 1+1 1 n ξl 1+1 1
.. . .. .
' ' ' ' .. ' . ' l1 ξ(n−1)K+k−2 '' l1 +1 ' (n−1)K+k−2 ξ(n−1)K+k−2 ' $ ' .. (ξi −ξ j), '= ' . l1 +l2 i, j=0 ' ξ(n−1)K+k−2 ' j
1 ··· 1 ··· 1 ··· 1 ξl1 +1 ··· ξl1 +l2 ··· ξm2 −κnk +1 ··· ξ(n−1)K+k−2
..
.
··· ···
.. .
..
.. .
..
.
···
.
··· ···
.. .
l ξl 1+l 1 2 l +1 ξl 1+l 1 2
..
··· ···
.. .
..
.. .
..
l +l ξl 1+l 2 1 2
n −1 ξl 1+l 1 2 n ξl 1+l 1 2
.
.
···
.
··· ···
.. .
l ξm1 −κ +1 2 nk l +1 ξm1 −κ +1 2 nk
..
···
.. .
..
.. .
..
l +l ξm1 −2κ +1 2 nk
n −1 ξm1 −κ +1 2 nk n ξm1 −κ +1 2 nk
.
···
.
···
.
··· ···
(n−1)K+k−2
262
K. T.-R. McLaughlin et al.
it follows that, for n ∈ N and k = 1, 2, . . . , K, Ξˇ kn (z) =
D ( ϕ0 (z))−1 m1 !
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · R R
R
(n−1)K+k−1
· · · dμ(ξl1 +l2 ) · · · dμ(ξm2 −κ nk +1 ) · · · dμ(ξn1 ) × ×
i, j=0
(ξi − ξ j )
j
ϕ0 (ξ0 ) ϕ0 (ξ1 ) · · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 )· · · ϕ0 (ξl1 +l2 )· · · ϕ0 (ξm2 −κ nk +1 )· · · ϕ0 (ξn1 ) 1 / ϕ0 (ξ0 )/ ϕ0 (ξ1 )· · ·/ ϕ0 (ξl1 )/ ϕ0 (ξl1 +1 )· · ·/ ϕ0 (ξl1 +l2 )· · ·/ ϕ0 (ξm2 −κ nk +1 )· · ·/ ϕ0 (ξn1 )
' ϕ1 (ξ0 ) ' ϕ0 (ξ0 ) ' ϕ1 (ξ1 ) ' ϕ0 (ξ1 ) ' . .. ' .. ' . ' ϕ1 (ξl1 ) ' ϕ0 (ξl1 ) ϕ (ξ ) ϕ (ξ ) ' ' 0 l1 +1 1 l1 +1 ×' .. .. ' . . ' ' ϕ (ξ ) ϕ (ξ ) ' 0 l1 +l2 1 l1 +l2 ' .. .. ' . . ' ' ϕ (ξ ) ϕ1 (ξn1 ) ' 0 n1 ϕ1 (z) ϕ0 (z)
··· ···
..
ϕl1 (ξ0 ) ϕl1 (ξ1 )
.
.. .
. .
ϕl1 +1 (ξ0 ) ϕl1 +1 (ξ1 )
··· ···
.. .
..
.. .
.. .
..
.. .
.. .
..
ϕl1 +l2 (ξ0 ) ϕl1 +l2 (ξ1 )
··· ···
.
.. .
..
.
.. .
..
.
.. .
..
ϕm2 −κnk +1 (ξ0 ) ϕm2 −κnk +1 (ξ1 )
··· ···
.
.. .
..
..
.
.. .
..
.
.. .
..
.
.
··· ϕl1 (ξl1 +l2 ) ϕl1 +1 (ξl1 +l2 ) ··· ϕl1 +l2 (ξl1 +l2 ) ··· ϕm2 −κnk +1 (ξl1 +l2 ) ···
..
··· ···
ϕl1 (ξn1 ) ϕl1 (z)
D D ( ϕ0 (z))−1 ((n−1)K+k−1)!
ϕl1 +1 (ξn1 ) ϕl1 +1 (z)
(n−1)K+k−2 i, j=0 j
(ξi −ξ j )
··· ···
ϕl1 +l2 (ξn1 ) ϕl1 +l2 (z)
(n−1)K+k−2 m=0
(z−ξm )
··· ···
ϕm2 −κnk +1 (ξn1 ) ϕm2 −κnk +1 (z)
.
··· ···
' ' ' ' ' .. ' ' . ϕ(n−1)K+k−1 (ξl1 ) ' ' ϕ(n−1)K+k−1 (ξl1 +1 ) ' ' ' .. ' . ' ϕ(n−1)K+k−1 (ξl1 +l2 ) ' ' ' .. ' . ' ϕ(n−1)K+k−1 (ξn1 ) ' ' ϕ(n−1)K+k−1 (z) ϕ(n−1)K+k−1 (ξ0 ) ϕ(n−1)K+k−1 (ξ1 )
··· ϕl1 (ξl1 ) ϕl1 +1 (ξl1 ) ··· ϕl1 +l2 (ξl1 ) ··· ϕm2 −κnk +1 (ξl1 ) ··· ··· ϕl1 (ξl1 +1 ) ϕl1 +1 (ξl1 +1 ) ··· ϕl1 +l2 (ξl1 +1 ) ··· ϕm2 −κnk +1 (ξl1 +1 ) ···
= D
=
(n−1)K+k−2
(cf. Proof of Lemma 2.1)
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) dμ(ξl1 +1 ) · · · R R
R
(n−1)K+k−1
× · · · dμ(ξl1 +l2 ) · · · dμ(ξ(n−1)K+k−2 ) ×
×
ϕ0 (ξ1 ) · · · / ϕ0 (ξl1 )/ ϕ0 (ξl1 +1 ) · · · / ϕ0 (ξl1 +l2 ) · · · / ϕ0 (ξm2 −κ nk +1 ) · · · / ϕ0 (ξ(n−1)K+k−2 ))−1 (/ ϕ0 (ξ0 )/ ϕ0 (ξ0 ) ϕ0 (ξ1 ) · · · ϕ0 (ξl1 ) ϕ0 (ξl1 +1 ) · · · ϕ0 (ξl1 +l2 ) · · · ϕ0 (ξm2 −κ nk +1 ) · · · ϕ0 (ξ(n−1)K+k−2 ) (n−1)K+k−2 $
(n−1)K+k−2 $
i, j=0
m=0
(ξi −ξ j )2
(z−ξm );
j
but, recalling that, for n ∈ N and k = 1, 2, . . . , K, / ϕ0 (z) = ϕ0 (z)/(z−αk ) and /n (z) := (−1)(n−1)K+k−κnk Ξˇ n (z), one arrives at Ξ k k /kn (z) = Ξ
(−1)(n−1)K+k−κnk D D ( ϕ0 (z))−1 ((n−1)K+k−1)!
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξl1 ) R R R (n−1)K+k−1
× dμ(ξl1 +1 ) · · · dμ(ξl1 +l2 ) · · · dμ(ξ(n−1)K+k−2 )
(n−1)K+k−2 $ l=0
!−2 ϕ0 (ξl )
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
×
(n−1)K+k−2 $
(ξi −ξ j)
2
263
(n−1)K+k−2 $
(ξm −αk )(z−ξm ) ⇒
m=0
i, j=0 j
/kn (z) = Ξ
(n−1)K+k−κnk
⎛
⎞−1
s −1 $
DD ⎝ (−1) (z−α pq )κnkkq (z−αk )κnk −1 ⎠ ((n−1)K+k−1)! q=1
×
R
· · · dμ(ξ0 ) dμ(ξ1 ) · · · dμ(ξ(n−1)K+k−2 ) R R
(n−1)K+k−1
⎛
(n−1)K+k−2 s −1 $ $
×⎝
m=0
×
⎞−2 (ξm −α pq )κnkkq (ξm −αk )κnk −1 ⎠
q=1
(n−1)K+k−2 $
(n−1)K+k−2 $
i, j=0
l=0
(ξi −ξ j)2
(ξl −αk )(z−ξl ).
j
/n (z)/ Recalling that, for n ∈ N and k = 1, 2, . . . , K, Y21 (z)/(z − αk ) = 2π i Ξ k n / N (n, k), where Ξk (z) is given directly above, and N (n, k) is given in the proof of Lemma 2.1, (33), one arrives at the integral representation for Y21 (z)/(z−αk ) stated in the Lemma. Remark 2.3 For n ∈ N and k = 1, 2, . . . , K, the integral representation for the FPC orthonormal rational function φkn : N × {1, 2, . . . , K} × C \ {α1 , π nk (z), α2 , . . . , α K } → C is obtained via (cf. Subsection 1.2) φkn (z) = μn,κnk (n, k)π where the ‘norming constant’, μn,κnk : N × {1, 2, . . . , K} → R+ , is given in Corollary 2.1, and the integral representation for the monic FPC ORF, π nk : N × {1, 2, . . . , K} × C \ {α1 , α2 , . . . , α K } → C, is given in Lemma 2.2.
3 The Monic FPC ORF Family of Variational Problems and Transformed RHPs In this section, the analysis of the family of K (different) variational problems associated with the monic FPC ORFs is undertaken; more precisely, the corresponding family of equilibrium measures and generalised weighted Fekete sets (see Lemmata 3.1–3.3 and Lemmata 3.5–3.8), as well as the corresponding family of complex potentials, or ‘g-functions’ (see Lemma 3.4), are studied. Furthermore, Lemma RHPFPC is reformulated as a family of K equivalent matrix RHPs on R (see Lemma 3.4). One begins by establishing the existence of the family of equilibrium measures.
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R \ {α1 , α2 , . . . , α K } → R satisfy condiLemma 3.1 Let the external field V: tions (6)–(8), and set (z) := exp(−V(z)). For n ∈ N and k = 1, 2, . . . , K, with 6 EQ EQ associated measure μ (= μ (n, k)) ∈ M1 (R), define the weighted (n- and k-dependent) logarithmic energy functional 2 3 2 3 IV : N × {1, 2, . . . , K} × M1 (R) → R, (n, k, μEQ ) → IV n, k; μEQ := IV μEQ ⎛
=
R2
ln ⎝|s−t|
1 n
|s−t| |s−αk ||t−αk |
κnkn −1 s$ −1 q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
⎞−1 × (s) (t)⎠
dμEQ (s) dμEQ (t),
and consider the family of minimisation problems 2 3 EV (n, k) := EV = inf IV μEQ ; μEQ ∈ M1 (R) . Then, for n ∈ N and k = 1, 2, . . . , K: (1) EV is finite; (2) the infimum is attained, that is, there exists μV (= μV (n, k)) ∈ M1 (R) such that IV [μV ] = EV , and μV has finite weighted logarithmic energy, that is, −∞ < IV [μV ] < +∞; and (3) ({x ∈ R; (x) > 0} ⊃) J := supp(μV ) is a proper compact subset of R \ {α1 , α2 , . . . , α K }. Sketch of Proof For n ∈ N and k = 1, 2, . . . , K, let μEQ (= μEQ (n, k)) ∈ M1 (R), and set, as in the Lemma, (z) := exp(−V(z)), where (cf. Subsection 1.3) V: R \ {α1 , α2 , . . . , α K } → R satisfies conditions (6)–(8). For n ∈ N and k = 1, 2, . . . , K, from the definition of IV : N × {1, 2, . . . , K} × M1 (R) → R given in the Lemma, one shows that IV [μEQ ] = KV (s, t) dμEQ (s) dμEQ (t), R2
where the symmetric kernel KV : N × {1, 2, . . . , K} × R2 → R, (n, k, s, t) → KV (n, k; s, t) := KV (s, t) is given by ' '−1 ! ' ' ) κ −1 1 ( 1 1 nk ' KV (s, t) = KV (t, s) := ln |s−t|−1 + ln '' − n n t−αk s−αk ' +
s−1
κnkkq
q=1
n
' ' 1 ln '' t−α
' ! 1 ''−1 − + V(s)+ V(t). s−α pq ' pq
6 The measure, etc., vary according to the parameters n (∈ N) and k (= 1, 2, . . . , K), and thus, strictly speaking, all the introduced variables in the proof, too, depend on n and k, which would necessitate the introduction of additional, superfluous notation(s) to encode these n- and k-dependencies; however, for simplicity of notation, unless where absolutely necessary, such cumbersome n- and kdependencies will not be introduced, and the reader should be cognizant of this fact: this comment applies, mutatis mutandis, throughout the remainder of the paper.
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(Of course, the definition of IV [μEQ ] only makes sense provided both integrals exist and are finite.) Recall the following inequalities (see, e.g., Chapter 6 of [56]): for s, t ∈ R, |s−t| (1+s2 )1/2 (1+t2 )1/2 , ( )1/2 ( )1/2 , | (t−αk )−1 −(s−αk )−1 | 1+(t−αk )−2 1+(s−αk )−2 ( )−1 ( )−1 ( )1/2 ( )1/2 | t−α pq − s−α pq | 1+(t−α pq )−2 1+(s−α pq )−2 , whence ( ( ) ) ln(|s−t|−1 ) − 12 ln 1+s2 − 12 ln 1+t2 , ( ( ) ) ) ( ln |(t−αk )−1 −(s−αk )−1 |−1 − 21 ln 1+(t−αk )−2 − 12 ln 1+(s−αk )−2 , - ( - ( )−1 ( )−1 −1 . )−2 . − − s−α pq | − 12 ln 1+ t−α pq ln | t−α pq
1 2
- ( )−2 . ; ln 1+ s−α pq
thus, for n ∈ N and k = 1, 2, . . . , K, ⎛ 1⎝ κnk −1 1 (2 ) KV (s, t) 2V(s)− ln s +1 − ln((s−αk )−2 +1) 2 n n
−
s−1
κnkkq
q=1
−
n
ln
-(
s−α pq
)−2
⎛ ⎞ . 1 1 ln(t2 +1) +1 ⎠ + ⎝2V(t)− 2 n
⎞ s−1 -( .
κ ) κnk −1 nk k −2 q ln((t−αk )−2 +1) − ln t−α pq +1 ⎠ . n n q=1
R \ {α1 , α2 , . . . , α K } → R, Recalling conditions (6)–(8) for the external field V: in particular, there exists some arbitrarily fixed, sufficiently small positive real −1 (1+ number δ∞ (= δ∞ (n, k)) such that, for x ∈ O∞ := {x ∈ R; |x| > δ∞ }, V(x) 2 c∞ ) ln(x +1), where c∞ (= c∞ (n, k)) is some bounded, positive real number, and, for q = 1, 2, . . . , s, there exist arbitrarily fixed, sufficiently small positive real numbers δ˜q (= δ˜q (n, k)) such that, for x ∈ Oδ˜q (α pq ) := {x ∈ R; |x−α pq | < (1+ cq ) ln((x−α pq )−2 +1), where cq (= cq (n, k)), q = 1, 2, . . . , s, are δ˜q }, V(x) bounded, positive real numbers, it follows that, for n ∈ N and k = 1, 2, . . . , K, ( ) 1 ( 2 ) κnk −1 2V(x)− ln x +1 − ln (x−αk )−2 +1 n n −
s−1
κnkkq
q=1
n
( ) ln (x−α pq )−2 +1 Cˆ V (n, k) := Cˆ V > −∞,
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whence KV (s, t) Cˆ V (> −∞), which shows that, for n ∈ N and k = 1, 2, . . . , K, KV (s, t) is bounded from below (on R2 ); hence, for n ∈ N and k = 1, 2, . . . , K, IV [μ ] EQ
R2
Cˆ V dμEQ (s) dμEQ (t) = Cˆ V
dμ (s) EQ
R
R
dμEQ (t) Cˆ V
(> −∞).
It follows from the above inequality and the definition of EV stated in the Lemma that, for n ∈ N and k = 1, 2, . . . , K, for all μEQ ∈ M1 (R), EV Cˆ V > −∞, which shows that, for n ∈ N and k = 1, 2, . . . , K, EV is bounded from below. Let ε (= ε(n, k)) be an arbitrarily fixed, sufficiently small positive real number, and ∞ set ε := {x ∈ R; (x) ε}; then ε is compact, and 0 := ∪∞ j=1 1/j = ∪ j=1 {x ∈ R \ {α1 , α2 , . . . , α K } → R satisR; (x) > 1/j} = {x ∈ R; (x) > 0}. Since, for V: fying conditions (6)–(8), is an admissible weight [60], it follows that, for n ∈ N and k = 1, 2, . . . , K, there exits j ∗ (= j ∗ (n, k)) ∈ N such that cap(1/j ∗ ) = exp(− inf{IV [μEQ ]; μEQ ∈ M1 (1/j ∗ )}) > 0, which, in turn, means that, for n ∈ N EQ and k = 1, 2, . . . , K, there exists a probability measure μEQ j ∗ (= μ j ∗ (n, k)), say, with supp(μ j ∗ ) ⊆ 1/j ∗ such that ⎛
1/j∗ × 1/j ∗
1 ln ⎝|s−t| n
×
s −1 $
q=1
|s−t| |s−αk ||t−αk |
|s−t| |s−α pq ||t−α pq |
κnkn −1
κnknkq
⎞−1 ⎠
EQ dμEQ j∗ (s) dμ j∗ (t) < +∞,
∗ with 1/j∗ × 1/j∗ ⊆ R2 . For x ∈ supp(μEQ j∗ ) ⊆ 1/j∗ , it follows that (x) 1/j (bounded from below), whence
1/j∗ × 1/j∗
EQ ∗ ln( (s) (t))−1 dμEQ j∗ (s) dμ j∗ (t) 2 ln( j ) < +∞,
which implies that, for n ∈ N and k = 1, 2, . . . , K, ⎛
IV [μ ] = EQ j∗
×
1/j ∗ × 1/j ∗ s −1 $
q=1
ln ⎝|s−t|
|s−t| |s−α pq ||t−α pq |
1 n
|s−t| |s−αk ||t−αk |
κnknkq
κnkn −1
⎞−1 (s) (t)⎠
EQ dμEQ j∗ (s) dμ j∗ (t) < +∞;
thus, for n ∈ N and k = 1, 2, . . . , K, −∞ < EV := inf{IV [μEQ ]; μEQ ∈ M1 (R)} < +∞. This establishes the first claim of the Lemma.
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For n ∈ N and k = 1, 2, . . . , K, choose a sequence of probability measures 1 ∞ EQ EQ {μEQ [μm ] E V + m . From the analysis m = μm (n, k)}m=1 in M1 (R) such that IV above, it follows that, for n ∈ N and k = 1, 2, . . . , K, 1 1 EQ EQ EQ IV [μm ] = KV (s, t) dμm (s) dμm (t) 2V(s)− ln(s2 +1) 2 2 2 n R R ⎞ s−1
κ κnk −1 nk k q − ln((s−αk )−2 +1) − ln((s−α pq )−2 +1)⎠ n n q=1 1 κnk −1 1 2V(t)− ln(t2 +1)− ln((t−αk )−2 +1) 2 n n ⎞⎞ s−1
κnkkq EQ − ln((t−α pq )−2 +1)⎠⎠ dμEQ m (s) dμm (t). n q=1 +
For n ∈ N and k = 1, 2, . . . , K, set
1 κnk −1 2 ˆ ψV (z) := 2V(z)− ln(z +1)− ln((z−αk )−2 +1) n n −
s−1
κnkkq
n
q=1
ln((z−α pq )−2 +1).
(35)
1 ˆ EQ EQ Then, for n ∈ N and k = 1, 2, . . . , K, IV [μEQ ] R2 ( 21 ψˆ V (s)+ 2 ψ (t))dμm (s)dμm (t) V m 1 EQ EQ EQ ˆ ˆ (ξ ) dμm (ξ ) ⇒ E V + m IV [μm ] R ψV (ξ )dμm (ξ ). For n ∈ N and k = R ψV 1, 2, . . . , K, recalling that, for x ∈ O∞ , ∃ c∞ (= c∞ (n, k)) > 0 such (1+ c∞ ) ln(x2 +1), and, for q = 1, 2, . . . , s, ∃ cq (= cq (n, k)) > 0 such that V(x) (1+ cq ) ln((x−α pq )−2 + 1), it follows that, that, for x ∈ Oδ˜q (α pq ), V(x) for any b (= b (n, k)) > 0, there exists (some) M (= M(n, k)) > 1, for which O 1 (α pi ) ∩ O 1 (α p j ) = ∅, i = j∈ {1, 2, . . . , s}, such that ψˆ V (z) > b ∀ M M s z ∈ D M := {|x| M} ∪ (∪q=1 O 1 (α pq )), which implies that, upon writing M R = (R \ D M ) ∪ D M , with (R \ D M ) ∩ D M = ∅, 1 EQ ˆ EV + ψV (ξ )dμm (ξ ) = ψˆ V (ξ )dμEQ m (ξ ) m (R\D M )∪D M D M
>b
+
ψˆ V (ξ ) dμEQ (ξ ) b m
R\D M
− |Cˆ V |
−|Cˆ V |
dμEQ m (ξ ) DM
dμm (ξ ) b
ˆ |; dμEQ m (ξ )−|C V
EQ
R\D M
∈ [0,1]
DM
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ˆ V |+ 1 , whence thus, for n ∈ N and k = 1, 2, . . . , K, b D M dμEQ +|C m (ξ ) E V m . ( ) −1 −1 ˆ E . dμEQ (ξ ) lim sup +| C |+m lim sup b V V m m→∞
m→∞
DM
Recall that, for n ∈ N and k = 1, 2, . . . , K, EV is bounded above. By the Archimedean property, it follows that, for n ∈ N and k = 1, 2, . . . , K, ∀ ε˜ (= ˜ m−1 < ε˜ ; thus, ˜ ε˜ (n, k)) > 0, ∃ N˜ (= N(n, k)) ∈ N such that, ∀ m (= m(n, k)) N, −1 choosing b (= b (n, k)) = [ε (EV +|Cˆ V |+ ε˜ )], where [∗∗] denotes the largest integer less than or equal to ∗ , with ε (= ε(n, k)) some arbitrarily fixed, sufficiently small positive real number, it follows that, for n ∈ N and k = 1, 2, . . . , K, with m > b (∈ N), lim sup dμEQ m (ξ ) ε, m→∞
DM
∞ which implies that the sequence of probability measures {μEQ m }m=1 in M1 (R) is tight [61] (that is, for n ∈ N and k = 1, 2, . . . , K, given ε (= ε(n, k)) > 0, ∃ M (= M(n, k)) > 1, for which O 1 (α pi ) ∩ O 1 (α p j ) = ∅, i = j∈ {1, 2, . . . , s}, M M EQ EQ such that lim supm→∞ μEQ m (D M ) ε, where μm (D M ) := D M dμm (ξ )). Since, for ∞ n ∈ N and k = 1, 2, . . . , K, the sequence of probability measures {μEQ m }m=1 in M1 (R) is tight, by a Helly selection theorem [60], there exists, for n ∈ N and k = 1, 2, . . . , K, a (weak-∗ convergent) subsequence of probability mea∞ EQ sures {μEQ m j = μm j (n, k)} j=1 (with j= j(n, k)) in M1 (R) converging (weakly) to ∗ EQ a probability measure μEQ (= μEQ (n, k)) ∈ M1 (R), symbolically μEQ as mj → μ ∗ 7 EQ EQ EQ j→ ∞. One now shows that, if μm → μ , then lim infm→∞ IV [μm ] IV [μEQ ]. is upper semi-continuous (resp., lower semi-continuous), Since (resp., V) there exists, for n ∈ N and k = 1, 2, . . . , K, a decreasing (resp., an increasing) ∞ sequence { m = m (n, k)}∞ m=1 (resp., { Vm = Vm (n, k)}m=1 ) of continuous m+1 (·) V m (·))8 , and, pointfunctions on R such that m+1 (·) m (·) (resp., V m (x) # V(x)) wise, m (x) " (x) (resp., V as m . Not→ ∞ for every x ∈ REQ ] = R2 KV (s, t) dμEQ (s) dμm (t) ing that, for n ∈ N and k = 1, 2, . . . , K, IV [μEQ m m EQ EQ ˆ ˆ dμEQ [μm ] m (s)dμm (t), it follows that, for any (= (n, k)) ∈ R, IV R2 KVm (s, t) EQ EQ 2 R2 g(s, t) dμm (s) dμm (t), where g: N ×{1, 2, . . . , K}× R → R, (n, k, s, t) → ˆ g(n, k, s, t) := g(s, t) = g(t, s) = min{, KVm (s, t)} is, by virtue of conditions (6)– R \ {α1 , α2 , . . . , α K } → R, bounded and con(8) on the external field V: tinuous on R2 . Let ε (= ε(n, k)) > 0 be given, and choose M0 (= M0 (n, k)) > 1, for which O 1 (α pi ) ∩ O 1 (α p j ) = ∅, i = j∈ {1, 2, . . . , s}, such M0 M0 that lim supm→∞ D M dμEQ m (ξ ) ε. For n ∈ N and k = 1, 2, . . . , K, with M0 > 1, 0
sequence of probability measures {μEQ m }m∈N in M1 (D) is said to converge weakly as m → ∞ ∗ EQ EQ EQ → μ , if μEQ to μEQ ∈ M1 (D), symbolically μEQ m m ( f ) := D f (λ) dμm (λ) → D f (λ) dμ (λ) =: 0 0 EQ Cb (D), where C b (D) denotes the set of all bounded, continuous μ ( f ) as m → ∞ for all f ∈C functions on D with compact support 8 Adding a suitable constant, if necessary, which does not change μEQ , or the regularity of V: R\ m 0 and V m 0, m ∈ N. {α1 , α2 , . . . , α K } → R, one may assume that V 7A
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269
let (the test function) C 0b (R) h M : N × {1, 2, . . . , K} × R → [0, 1], (n, k, x) → h M (n, k; x) := h M (x) be such that: (i) 0 h M (x) 1, x ∈ R; (ii) h M (x) = 1, x ∈ R \ s D M0 , where D M0 := {|x| M0 } ∪ (∪q=1 O 1 (α pq )); and (iii) h M (x) = 0, x ∈ D M0 +1 . M0
Note the splitting (decomposition) [56] R2
EQ g(t, s) dμEQ m (t) dμm (s) = Ia + Ib + Ic ,
where Ia :=
R2
EQ g(t, s)(1−h M (s)) dμEQ m (t) dμm (s),
Ib :=
R2
EQ g(t, s)h M (s)(1−h M (t)) dμEQ m (t) dμm (s),
Ic :=
R2
EQ g(t, s)h M (t)h M (s) dμEQ m (t) dμm (s).
One shows that
|Ia |
R2
⎛ ×⎝
EQ |g(t, s)|(1−h M (s)) dμEQ m (t) dμm (s) sup |g(t, s)|
R
(t,s)∈R2
⎞
(1−h M (s)) dμEQ m (s)+ D M +1 0
R\D M0
=0
dμEQ m (t)
⎠ (1−h M (s)) dμEQ m (s) , =1
whence lim sup|Ia | sup |g(t, s)| lim sup m→∞
(t,s)∈R2
m→∞
R\D M0 +1
dμEQ m (ξ ) ε sup |g(t, s)|; (t,s)∈R2
similarly, lim sup |Ib | ε sup |g(t, s)|. m→∞
(t,s)∈R2
Since, for n ∈ N and k = 1, 2, . . . , K, g(t, s) is bounded and continuous on R2 , Cb (R2 ), there exists, by an appropriate generalisation of the Stone– that is, g ∈C Weierstrass theorem for the single-variable case, a sequence of polynomials, with n- and k-dependent coefficients (suppressed for notational simplicity), { pm (t, s) = pm (n, k; t, s)}∞ m=1 in R[t, s] (the algebra of polynomials in the indeterminates t and s with coefficients in R) such that sup(t,s)∈R2 | pm (t, s)−
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g(t, s)| → 0 as m → ∞, that is, there exists a (symmetric) p(s, t) = polynomial, p(t, s), say, in R[s, t], with representation p(t, s) = ii0 j j0 γijt i s j, where γij = γij(n, k), with |γij(n, k)| < +∞, such that |g(t, s)− p(t, s)| ε (= ε(n, k)); thus, |h M (t)h M (s)g(t, s)−h M (t)h M (s) p(t, s)| ε for t, s ∈ R. Split Ic as follows: β Ic = Icα + Ic , where
Icα
:=
Icβ :=
R2
R2
EQ h M (s)h M (t) p(t, s) dμEQ m (t) dμm (s),
EQ h M (s)h M (t)(g(t, s)− p(t, s)) dμEQ m (t) dμm (s).
One now shows that |Icβ |
EQ h M (s)h M (t) |g(t, s)− p(t, s)| dμEQ m (t) dμm (s)
R2
ε
ε
R
h M (s) dμEQ m (s)
⎛ ⎝ ε
h M (t) dμEQ m (t) ⎞2
D M0 +1
h (s) dμEQ m (s)+ M =0
!2
ε
R
dμm (s)
R\D M0
ε
EQ
R\D M0
R
⎠ h (s) dμEQ m (s) M =1
2 dμEQ (s) ε, m
β
whence lim supm→∞ |Ic | ε, and Icα =
=
R2
h M (s)h M (t)
i i0 j j0
=
Icα =
R2
EQ γij ti s j dμEQ m (t) dμm (s)
i i0 j j0
⎛ ⎝
γij
i i0 j j0
→
R
γij
R
h M (t)ti dμEQ m (t)
R
h M (s)s j dμEQ m (s)
∗ EQ h M (t)ti dμEQ (t) h M (s)s j dμEQ (s) (since μEQ as m → ∞) m →μ R
⎞ i j⎠
γij t s
h M (t)h M (s) dμEQ (t) dμEQ (s)
i i0 j j0
R2
p(t, s)h M (t)h M (s) dμEQ (t) dμEQ (s).
⇒
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
Furthermore, recalling that |h M (t)h M (s)g(t, s)−h M (t)h M (s) p(t, s)| ε, Icα
R2
g(t, s)h M (t)h M (s) dμEQ (t) dμEQ (s)
⎛ ⎜ + ε⎝
⎞2
D M0 +1
h (s) dμEQ (s)+ M
R\D M0
=0
⎟ h (s) dμEQ (s)⎠ M =1
R2
g(t, s)h M (t)h M (s) dμ
EQ
(t) dμ
EQ
(s)+ε
dμ
R\D M0
!2
R2
g(t, s)h M (t)h M (s) dμEQ (t) dμEQ (s)+ε
EQ
(s)
2
R
dμEQ (s)
= Icα
R2
g(t, s)h M (t)h M (s) dμEQ (t) dμEQ (s)+ε
⇒
R2
g(t, s)|1+(h M (t)−1)||1+(h M (s)−1)| dμEQ (t) dμEQ (s)+ε
R2
g(t, s) dμEQ (t) dμEQ (s)+2
R2
g(t, s)|h M (s)−1| dμEQ (t) dμEQ (s)+ε
+
R2
g(t, s)|h M (t)−1||h M (s)−1| dμEQ (t) dμEQ (s)
R2
g(t, s) dμEQ (t) dμEQ (s)+ε+2 sup |g(t, s)| (t,s)∈R2
dμEQ (t) R
=1
⎛ ⎜ ×⎝
+
⎞
D M0 +1
| h M (s)−1| dμ
EQ
(s) +
=0
⎛ ⎜ sup |g(t, s)| ⎝
(t,s)∈R2
R\D M0
| h M (s)−1| dμ
⎞2
D M0 +1
⎟ (s)⎠
=1
| h M (s)−1| dμEQ (s)+ =0
EQ
R\D M0
⎟ | h M (s)−1| dμEQ (s)⎠ =1
R2
g(t, s) dμEQ (t) dμEQ (s)+ 2 sup |g(t, s)| (t,s)∈R2
dμEQ (s)
D M0 +1
ε
⎛ +
⎞2
⎜ ⎟ ⎜ ⎟ EQ ⎟ +ε sup |g(t, s)| ⎜ dμ (s) ⎜ ⎟ 2 ⎝ D M0 +1 ⎠ (t,s)∈R ε
!
=
R2
g(t, s) dμEQ (t) dμEQ (s)+ε 1+2 sup |g(t, s)| + O (ε 2 ), (t,s)∈R2
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K. T.-R. McLaughlin et al.
whereupon, neglecting the O (ε2 ) term, and setting κ (= κ (n, k)) := 1+ 2 sup(t,s)∈R2 |g(t, s)|, one obtains Icα g(t, s) dμEQ (t) dμEQ (s)+κ ε. R2
β
Hence, assembling the above-derived bounds for Ia , Ib , Ic , and Icα , one arrives at, for n ∈ N and k = 1, 2, . . . , K, upon setting ε (= ε (n, k)) := 2κ ε, EQ g(t, s) dμEQ (t) dμ (s)− g(t, s) dμEQ (t) dμEQ (s) ε ; m m R2
R2
thus,
R2
EQ g(t, s) dμEQ m (t) dμm (s) →
R2
g(t, s) dμEQ (t) dμEQ (s)
as m → ∞.
ˆ V (t, s)}, (, ˆ m) ∈ R × N, it follows that, for n ∈ N Recalling that g(t, s) := min{,K m and k = 1, 2, . . . , K, ˆ lim inf IV [μEQ ] min , K (t, s) dμEQ (t) dμEQ (s) : Vm m m→∞
R2
ˆ KV (t, s)} → KV (t, s), after letting ˆ ↑ +∞ and m → ∞, and noting that min{, m an application of the monotone convergence theorem, one arrives at, for n ∈ N and k = 1, 2, . . . , K, ] KV (t, s) dμEQ (t) dμEQ (s) = IV [μEQ ]. lim inf IV [μEQ m m→∞
R2
Since, from the analysis above, it was shown that, for n ∈ N and k = 1, 2, . . . , K, there exists a weakly (weak-∗) convergent subsequence of probability mea∞ EQ ∞ EQ sures {μEQ m j } j=1 (⊂ M1 (R)) of {μm }m=1 (⊂ M1 (R)) with a weak-∗ limit μ ∈ ∗
1 EQ M1 (R), that is, μEQ as j→ ∞, upon recalling that IV [μEQ + m, mj → μ m ] EV m ∈ N, it follows that, for n ∈ N and k = 1, 2, . . . , K, in the limit as m → ∞, IV [μEQ ] EV = inf{IV [μEQ ]; μEQ ∈ M1 (R)}; from the latter two inequalities, it follows, thus, that for n ∈ N and k = 1, 2, . . . , K, there exists μEQ (= μEQ (n, k)) := μV ∈ M1 (R), the equilibrium measure, such that IV [μV ] = inf{IV [μEQ ]; μEQ ∈ M1 (R)}, that is, for n ∈ N and k = 1, 2, . . . , K, the infimum is attained. This establishes the second claim of the Lemma (the unicity of the equilibrium measure is proven in Lemma 3.3 below). It remains, finally, to establish the third claim of the Lemma, that is, to show that, for n ∈ N and k = 1, 2, . . . , K, ({x ∈ R; (x) > 0} ⊃) J := supp(μV ) ⊂ R \ {α1 , α2 , . . . , α K } is compact. The following argument is valid for any μEQ (= μEQ (n, k)) ∈ M1 (R) achieving the above minimum; in particular, for μEQ = μV . Without loss of generality, therefore, for n ∈ N and k = 1, 2, . . . , K, EQ EQ let M1 (R) μEQ [μ ] = E V , andlet D (= D(n, k)) (= μ (n, k)) be such that IV EQ be any proper, measurable subset of R for which μEQ (D) = D dμ (ξ ) > 0. As in [61], set, for n ∈ N and k = 1, 2, . . . , K, ( EQ ) −1 με (n, k; z) := με (z) = (1+εμEQ μ (z)+ε(μEQ (D)) D )(z) , ε ∈ (−1, 1),
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
273
EQ ε where μEQ D denotes the restriction of μ to D. (Note that R dμ (ξ ) = 1; in ε particular, μ ∈ M1 (R), ε ∈ (−1, 1).) Using the fact that KV (s, t) = KV (t, s), one shows that, for n ∈ N and k = 1, 2, . . . , K,
IV [με ] =
−2 KV (s, t) dμε (s) dμε (t) = (1+εμEQ (D))
R2
EQ × IV [μ ]+2ε + ε
2
R2
EQ KV (s, t)dμEQ (s) d(μ D )(t)
KV (s, t) d(μ D )(s) d(μ D )(t) EQ
R2
EQ
(note that all the above integrals are finite due to the argument at the beginning of the proof). By the minimal property of μEQ ∈ M1 (R), it follows that [61], for n ∈ N and k = 1, 2, . . . , K, ∂ε IV [με ] = 0, which implies that R2
EQ EQ (KV (s, t)−IV [μEQ ]) dμ (s) d(μ D )(t) = 0;
but, recalling that, for n ∈ N and k = 1, 2, . . . , K, KV (t, s) 12 (ψˆ V (s)+ ψˆ V (t)), where ψˆ V (z) is defined in (35), it follows, from the latter minimisation condition, that
IV [μ ] dμ (s) d(μ D )(t) EQ
R2
EQ
EQ
R2
1 1 ˆ ˆ ψ (s)+ ψV (t) 2 V 2
EQ × dμEQ (s) d(μ D )(t) ⇒ 2 EQ 3 1 1 EQ ˆ ˆ 0 ψ (s)+ ψV (t)−IV μ dμEQ (s) d(μ D )(t), 2 V 2 R2
whence, for n ∈ N and k = 1, 2, . . . , K, 0
R
ψˆ V (t)+
R
2 EQ 3 EQ ˆ ψV (s) dμ (s) −2IV μ d(μEQ D )(t).
R\ But, for n ∈ N and k = 1, 2, . . . , K, from the growth conditions on V: {α1 , α2 , . . . , α K } → R, that is, ∃ c∞ (= c∞ (n, k)) > 0 such that, for x ∈ O∞ , (1 + c∞ ) ln(x2 + 1), and, for q = 1, 2, . . . , s, ∃ cq (= cq (n, k)) > 0 such V(x) that, for x ∈ Oδ˜q (α pq ), V(x) (1 + cq ) ln((x − α pq )−2 + 1), it follows that, for n ∈ N and k = 1, 2, . . . , K, there exists (some) T M (= T M (n, k)) > 1 (take, say, T M = max{δ∞ , maxq=1,2,...,s {δ˜q−1 }}), with O 1 (α pi ) ∩ O 1 (α p j ) = ∅, TM
TM
s O 1 (α pq )), ψˆ V (τ ) − i = j ∈ {1, 2, . . . , s}, such that, for τ ∈ {|x| T M } ∪ (∪q=1 TM EQ ˆ 2IV [μEQ ] + ψ (ξ ) dμ (ξ ) 1; hence, if D is such that D ⊂ {|x| T M } ∪ R V
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1 TM
(α pq )), with T M > 1, it follows from the above calculations that, for
n ∈ N and k = 1, 2, . . . , K, 2 EQ 3 ψˆ V (t)+ μ d(μEQ 0 (s) −2I ψˆ V (s) dμEQ V D )(t) 1, R
R
which is a contradiction; hence, for n ∈ N and k = 1, 2, . . . , K, supp(μEQ )⊆ s R \ ({|x| T M } ∪ (∪q=1 O 1 (α pq ))), for T M > 1; in particular, J := supp(μV ) ⊆ TM s R \ ({|x| T M } ∪ (∪q=1 O 1 (α pq ))), which establishes the compactness of the TM support of the equilibrium measure μV ∈ M1 (R): it is a straightforward conse R \ {α1 , α2 , . . . , α K } → quence that supp(μV ) ∩ {α1 , α2 , . . . , α K } = ∅. Since V: R is real analytic on R \ {α1 , α2 , . . . , α K } and supp(μV ) ∩ {α1 , α2 , . . . , α K } = / (= m /(n, k)) V(x) =: m ∅, in which case, for x ∈ supp(μV ), infx∈J V(x) / / M (= M(n, k)) := supx∈J V(x), whence, for n ∈ N and k = 1, 2, . . . , K, −∞ < (ξ ) < +∞, it follows, too, that for n ∈ N and k = 1, 2, . . . , K, the J V(ξ ) dμV weighted logarithmic energy of μV ∈ M1 (R) is finite, that is (recalling that (x) := exp(−V(x))), ⎛ κnkn −1 |s−t| 1 ln ⎝|s−t| n −∞ < IV [μV ] = |s−αk ||t−αk | J×J
×
s −1 $
q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
⎞−1 (s) (t)⎠
dμV (s) dμV (t) < +∞,
which gives rise to the corollary that, for n ∈ N and k = 1, 2, . . . , K, the logarithmic energy of μV ∈ M1 (R) is finite, that is, ⎛ κnkn −1 |s−t| 1 −∞ < IV [μV ] = ln ⎝|s−t| n |s−αk ||t−αk | J×J
×
s −1 $
q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
⎞−1 ⎠
dμV (s) dμV (t) < +∞.
Furthermore, it is a straightforward consequence of the facts established that, for n ∈ N and k = 1, 2, . . . , K, J := supp(μV ) (a proper compact subset of R \ {α1 , α2 , . . . , α K }) has positive logarithmic capacity, that is, for n ∈ N and k = 1, 2, . . . , K, cap(J) = exp(−EV ) > 0. This establishes the third claim of the Lemma; and thus concludes the proof. Prior to establishing, for n ∈ N and k = 1, 2, . . . , K, the unicity of the equilibrium measure μV (= μV (n, k)) ∈ M1 (R), whose support, supp(μV ), as proven in Lemma 3.1, is a proper compact subset of R \ {α1 , α2 , . . . , α K }, the variational inequality of Lemma 3.2 is requisite.
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
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R \ {α1 , α2 , . . . , α K } → R satisfy condiLemma 3.2 Let the external field V: tions (6)–(8), and set (z) := exp(−V(z)). For n ∈ N and k = 1, 2, . . . , K, let EQ EQ μ j (= μ j (n, k)) ∈ M1 (R), j= 1, 2, be non-negative, finite-moment measures on R supported on distinct compact sets, that is, supp(μEQ ) ξ m dμEQ j (ξ ) < ∞ and j EQ −m EQ dμ j (ξ ) < ∞, m ∈ N, i = 1, 2, . . . , K, j= 1, 2, and supp(μEQ 1 )∩ supp(μ ) (ξ −αi ) j
EQ EQ supp(μEQ (= μEQ (n, k)) := μEQ 2 ) = ∅. For n ∈ N and k = 1, 2, . . . , K, let μ 1 −μ2 be the (unique) Jordan decomposition of the finite-moment signed measure on R with compact support and mean zero, that is, supp(μEQ ) dμEQ (ξ ) = 0. Suppose, furthermore, that for n ∈ N and k = 1, 2, . . . , K, the measures μEQ j , j= 1, 2, have finite logarithmic energy:
⎛
−∞ <
ln ⎝|s−t| n
|s−t| |s−αk ||t−αk |
1
R2
κnkn −1 s$ −1
EQ × dμEQ j (s) dμ j (t) < +∞,
q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
⎞−1 ⎠
j= 1, 2.
Then, for n ∈ N and k = 1, 2, . . . , K, ⎛
ln ⎝|s−t| n 1
R2
|s−t| |s−αk ||t−αk |
κnkn −1 s$ −1 q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
⎞−1 ⎠
× dμEQ (s) dμEQ (t) ⎛ κnkn −1 s$ κnknkq −1 |s−t| |s−t| 1 ln ⎝|s−t| n = |s−αk ||t−αk | |s−α pq ||t−α pq | R2 q=1 !−1 × (s) (t) = where / μ(λ) :=
R
dμEQ (s) dμEQ (t)
(n−1)K+k n
+∞
ξ −1 |/ μ(ξ )|2 dξ 0,
0
eiλξ dμEQ (ξ ), and equality holds if, and only if, μEQ = 0.
Remark 3.1 Lemma 3.2 says that, for n ∈ N and k = 1, 2, . . . , K, upon EQ EQ EQ EQ EQ writing dμEQ (s)dμEQ (t) = d(μEQ 1 − μ2 )(s)d(μ1 − μ2 )(t) = dμ1 (s) dμ1 (t) + EQ EQ EQ EQ EQ EQ dμ2 (s) dμ2 (t) − dμ1 (s)dμ2 (t) − dμ2 (s) dμ1 (t), and taking note of the symmetry of the (n- and k-dependent) integrand under the interchange s ↔ t, R2
⎛ ln⎝|s−t|
1 n
κnkn −1 s$ κnknkq −1 |s−t| |s−t| |s−αk ||t−αk | |s−α pq ||t−α pq | q=1
276
K. T.-R. McLaughlin et al.
!−1 × (s) (t) ⎛
R2
ln ⎝|s−t|
1 n
× (s) (t)⎠ ⎛
=2
R2
ln ⎝|s−t|
) EQ EQ EQ dμEQ 1 (s) dμ1 (t) + dμ2 (s) dμ2 (t)
|s−t| |s−αk ||t−αk |
⎞−1
(
1 n
κnkn −1 s$ −1 q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
( EQ ) EQ EQ dμ1 (s) dμEQ 2 (t)+dμ2 (s) dμ1 (t)
|s−t| |s−αk ||t−αk |
κnkn −1 s$ −1 q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
!−1 × (s) (t) ⎛
=2
R2
ln ⎝|s−t|
1 n
EQ dμEQ 1 (s) dμ2 (t)
|s−t| |s−αk ||t−αk |
κnkn −1 s$ −1 q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
!−1 × (s) (t)
EQ dμEQ 2 (s) dμ1 (t);
( ) κnkn −1 s−1 1( hence, if, for n ∈ N and k = 1, 2, . . . , K, ln |s − t| n |s−α|s−t| q=1 k ||t−αk | κ nkkq ) ( ) |s−t| n is integrable with respect to the product measures dμEQ 1 (λ) |s−α p ||t−α p | q
q
EQ EQ ·dμEQ 1 (τ ) and dμ2 (λ)dμ2 (τ ), which is the supposition of the Lemma, then it is integrable with respect to the ‘mixed’ product measures dμEQ 1 (λ) EQ EQ ·dμEQ (τ ) and dμ (λ)dμ (τ ). 2 2 1
Proof of Lemma 3.2 Recall the following identity [56] (see pg. 147, (6.44)): for λ ∈ R and (any) ε > 0, +∞ (iν)−1 (eiλν −1)e−εν dν ; ln(λ2 +ε2 ) = ln(ε2 )+2 Im 0
thus, for n ∈ N and k = 1, 2, . . . , K, with r ∈ {s, t}, 1 1 2 2 EQ EQ 2 ln((s−t) + ε ) dμ (s) dμ (t) = dμEQ (s) dμEQ (t) ln(ε ) 2n R2 2n R2 +
1 n
Im R2
0
+∞
(iν)−1 (ei(s−t)ν −1)e−εν dν dμEQ (s) dμEQ (t),
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
277
1 κnk −1 κnk −1 2 2 EQ EQ ln((s−t) + ε ) dμ (s) dμ (t) = ln(ε2 ) n 2 n R2 +∞ κnk −1 × dμEQ (s) dμEQ (t) + Im (iν)−1 (ei(s−t)ν −1)e−εν dν n R2 R2 0 1 2
×dμEQ (s) dμEQ (t), κnkkq
2n +
ln((s−t) + ε ) dμ (s) dμ (t) = 2
R2
κnkkq
n
+∞
Im R2
2
EQ
EQ
ln(ε )
κnkkq
2
2n
R2
dμEQ (s) dμEQ (t)
(iν)−1 (ei(s−t)ν −1)e−εν dν dμEQ (s) dμEQ (t),
0
1 κnk −1 κnk −1 2 2 EQ EQ ln(ε2 ) ln((r−αk ) + ε ) dμ (s) dμ (t) = n 2 n R2 +∞ κnk −1 EQ EQ −1 i(r−αk )ν −εν dμ (s) dμ (t) + Im (iν) (e −1)e dν × n R2 R2 0 1 2
×dμEQ (s) dμEQ (t), κnkkq
2n +
R2
ln((r−α pq )2 +ε2 )dμEQ (s)dμEQ (t) =
κnkkq
n
Im R2
+∞
−1
(iν) (e
i(r−α pq )ν
−1)e
κnkkq
−εν
2n
ln(ε2 )
R2
dμEQ (s)dμEQ (t)
dν dμEQ (s) dμEQ (t).
0
( )2 Noting that, for n ∈ N and k = 1, 2, . . . , K, R2 dμEQ (s)dμEQ (t) = R dμEQ(ξ ) = 0, after some re-arrangement, the above simplifies to, with r ∈ {s, t}, 1 2n
R2
ln((s−t)2 +ε2 ) dμEQ (s) dμEQ (t)
+∞ 1 Im e−εν (iν)−1 (ei(s−t)ν −1) dμEQ (s) dμEQ (t) dν , n 0 R2 1 κnk −1 ln((s−t)2 +ε2 ) dμEQ (s) dμEQ (t) 2 n R2 +∞ κnk −1 −εν −1 i(s−t)ν EQ EQ Im = e (iν) (e −1) dμ (s)dμ (t) dν , n R2 0 κnkkq ln((s−t)2 +ε2 ) dμEQ (s) dμEQ (t) 2 2n R =
278
=
K. T.-R. McLaughlin et al.
κnkkq
n
+∞
e
Im
−εν
0
−1
R2
(iν) (e
i(s−t)ν
−1) dμ (s) dμ (t) dν , EQ
EQ
κnk −1 ln((r−αk )2 +ε2 ) dμEQ (s) dμEQ (t) 2 n R +∞ κnk −1 Im = e−εν (iν)−1 (ei(r−αk )ν −1)dμEQ (s) dμEQ (t) dν , n R2 0 κnkkq ln((r−α pq )2 +ε2 ) dμEQ (s) dμEQ (t) 2n R2 +∞ κnkkq −εν −1 i(r−α pq )ν EQ EQ Im e (iν) (e −1) dμ (s)dμ (t) dν . = n 0 R2 1 2
Noting that, for n ∈ N and k = 1, 2, . . . , K, 1 −1 i(s−t)ν EQ EQ (iν) (e −1) dμ (s) dμ (t) = ei(s−t)ν dμEQ (s) dμEQ (t) iν R2 R2 1 dμEQ (s) dμEQ (t) − iν R2 =0
=
1 iν
R
eisν dμEQ (s)
R
e−itν dμEQ (t),
and setting / μ(λ) (= / μ(n, k; λ)) := R eiλξ dμEQ (ξ ), one gets that (iν)−1 (ei(s−t)ν −1) dμEQ (s) dμEQ (t) = (iν)−1 |/ μ(ν)|2 ; R2
similarly, one shows that, for n ∈ N and k = 1, 2, . . . , K, with r ∈ {s, t}, −1 i(r−αk )ν EQ EQ (iν) (e −1) dμ (s) dμ (t) = (iν)−1 (ei(r−α pq )ν −1) R2
R2
× dμEQ (s) dμEQ (t) = 0. Hence, for n ∈ N and k = 1, 2, . . . , K, with r ∈ {s, t}, 1 ln((s−t)2 +ε2 ) dμEQ (s) dμEQ (t) 2n R2 +∞ 1 = Im (iν)−1 |/ μ(ν)|2 e−εν dν , n 0 1 2
κnk −1 n
R2
ln((s−t)2 +ε2 ) dμEQ (s) dμEQ (t)
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
=
+∞ κnk −1 −1 2 −εν Im (iν) |/ μ(ν)| e dν , n 0
κnkkq
2n = 1 2 =
279
κnkkq
n
R2
ln((s−t)2 +ε2 ) dμEQ (s) dμEQ (t)
+∞
Im
κnk −1 n
−1
2 −εν
(iν) |/ μ(ν)| e
dν ,
0
R2
ln((r−αk )2 +ε2 ) dμEQ (s) dμEQ (t)
κnkkq
ln((r−α pq )2 +ε2 ) dμEQ (s) dμEQ (t) = 0. 2n R2 Noting that / μ(0) = R dμEQ (ξ ) = 0, a Taylor expansion about λ = 0 shows that / μ(λ) =λ→0 / μ (0)λ+ O (λ2 ), where / μ (0) := ∂λ / μ(λ)|λ=0 ; thus, λ−1 |/ μ(λ)|2 =λ→0 2 2 |/ μ (0)| λ+ O (λ ), which means that there is no singularity in the integrand at λ = 0 (in fact, λ−1 |/ μ(λ)|2 is real analytic in an open neighbourhood of the origin), whence, for n ∈ N and k = 1, 2, . . . , K, 1 +∞ −1 1 ln((s−t)2 +ε2 )− 2n dμEQ (s) dμEQ (t) = ξ |/ μ(ξ )|2 e−εξ dξ, 2 n R 0 +∞ (κnk −1) κnk −1 ln((s−t)2 +ε2 )− 2n dμEQ (s) dμEQ (t) = ξ −1 |/ μ(ξ )|2 e−εξ dξ, n 0 R2 κ κnkkq +∞ −1 nkkq ln((s−t)2 +ε2 )− 2n dμEQ (s) dμEQ (t) = ξ |/ μ(ξ )|2 e−εξ dξ. n 0 R2 Adding the above, and recalling that R2 ln((r− rˆ)2 +ε2 ) dμEQ (s) dμEQ (t) = 0, s−1 r ∈ {s, t}, rˆ ∈ {αk , α pq }, and κnk + q=1 κnkkq = (n−1)K+k, for n ∈ N and k = 1, 2, . . . , K, ⎛ κnkn −1 ( )1 (s−t)2 +ε2 2 2 2n ⎝ ln (s−t) +ε ((s−αk )2 +ε2 )((t−αk )2 +ε2 ) R2
×
s −1 $
q=1
(s−t)2 +ε2 ((s−α pq )2 +ε2 )((t−α pq )2 +ε2 )
(n−1)K+k = n
+∞
κ
nk kq n
⎞−1 ⎠
dμEQ (s) dμEQ (t)
ξ −1 |/ μ(ξ )|2 e−εξ dξ.
0
Now, using the fact that ln((s−t)2 +ε2 )−1 (resp., ln((r− rˆ)2 +ε2 ), r ∈ {s, t}, rˆ ∈ {αk , α pq }) is (resp., are) bounded from below (resp., above) uniformly with
280
K. T.-R. McLaughlin et al.
respect to ε and that the measures have compact support, letting ε ↓ 0 and invoking the monotone and dominated convergence theorems, one arrives at, for n ∈ N and k = 1, 2, . . . , K, ⎛
.1 2n ln ⎝ (s−t)2 +ε2
R2
×
s$ −1
(s−t)2 +ε2 ((s−α pq )2 +ε2 )((t−α pq )2 +ε2 )
q=1
⎛
=
ε↓0
R2
(s−t)2 +ε2 2 ((s−αk ) +ε2 )((t−αk )2 +ε2 )
1 ⎜ ln ⎝|s−t| n
|s−t| |s−αk ||t−αk |
× dμEQ (s)dμEQ (t) =
!
κnk −1 s$ −1 n
q
2n
⎞−1 ⎟ ⎠
dμEQ (s) dμEQ (t)
|s−t| |s−α pq ||t−α pq |
q=1
(n−1)K+k n
κnk k
−1 κnk 2n
+∞
!
κnk k
q
n
⎞−1 ⎟ ⎠
ξ −1 |/ μ(ξ )|2 dξ 0,
0
where, trivially, equalityholds if, and only if, μEQ = 0. Furthermore, recalling that, by assumption, R dμEQ (ξ ) = 0, it follows that ln( (s) (t))−1 2 R EQ EQ ·dμ (s) dμ (t) = 0; hence, letting ε ↓ 0 and using, again, monotone and dominated convergence, one arrives at, for n ∈ N and k = 1, 2, . . . , K, R2
⎛ .1 2n ln ⎝ (s−t)2 +ε2
×
s$ −1 q=1
=
R2
(s−t)2 +ε2 ((s−α pq )2 +ε2 )((t−α pq )2 +ε2 )
⎛ .1 2n ln ⎝ (s−t)2 +ε2
×
s$ −1 q=1
(s−t)2 +ε2 2 ((s−αk ) +ε2 )((t−αk )2 +ε2 ) !
κnk k
q
2n
⎞−1 ⎟ ⎠
(s−t)2 +ε2 2 ((s−αk ) +ε2 )((t−αk )2 +ε2 )
(s−t)2 +ε2 ((s−α pq )2 +ε2 )((t−α pq )2 +ε2 )
!
−1 κnk 2n
dμEQ (s) dμEQ (t)
−1 κnk 2n
⎞−1
κnk k
q
2n
⎟ (s) (t)⎠
× dμEQ (s) dμEQ (t) =
ε↓0
R2
⎛ 1 ⎜ ln ⎝|s−t| n
|s−t| |s−αk ||t−αk |
κnk −1 s$ −1 n q=1
|s−t| |s−α pq ||t−α pq |
!
κnk k
q
n
× (s) (t))−1 dμEQ (s) dμEQ (t) =
(n−1)K+k n
+∞
ξ −1 |/ μ(ξ )|2 dξ 0,
(36)
0
where, again, and trivially, equality holds if, and only if, μEQ = 0.
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With the variational inequality of Lemma 3.2 at hand, the unicity of the equilibrium measure μV (∈ M1 (R)) will now be established. R \ {α1 , α2 , . . . , α K } → R satisfy condiLemma 3.3 Let the external field V: tions (6)–(8), and set (z) := exp(−V(z)). For n ∈ N and k = 1, 2, . . . , K, define IV : N × {1, 2, . . . , K} × M1 (R) → R as in Lemma 3.1, and consider the associated minimisation problem EV = inf{IV [μEQ ]; μEQ ∈ M1 (R)}. Then, for n ∈ N and k = 1, 2, . . . , K, there exists unique μV ∈ M1 (R) such that IV [μV ] = EV . Proof It was shown in Lemma 3.1 that, for n ∈ N and k = 1, 2, . . . , K, there exists μV (= μV (n, k)) ∈ M1 (R), the equilibrium measure, such that IV [μV ] = EV (= EV (n, k)), with ({x ∈ R; (x) > 0} ⊃) supp(μV ) =: J a proper compact subset of R \ {α1 , α2 , . . . , α K }, in particular, for (some) T M (= T M (n, k)) > 1, s supp(μV ) ⊂ R \ ({|x| T M } ∪ (∪q=1 O 1 (α pq ))); therefore, it remains to estabTM lish the uniqueness of the equilibrium measure. For n ∈ N and k = 1, 2, . . . , K, let μV (= μV (n, k)) ∈ M1 (R) be a second probability measure for which IV [ μV ] = EV = IV [μV ]: the argument of Lemma 3.1 shows that, for n ∈ N and k = 1, 2, . . . , K, ({x ∈ R; (x) > 0} ⊃) supp( μV ) =: J is a proper compact subset of R \ {α1 , α2 , . . . , α K }, and −∞ < IV [ μV ] < +∞. Define, for n ∈ N and k = 1, 2, . . . , K, the finite-moment signed measure (on R) μ (= μ (n, k)) := μV − μV , where μV , μV ∈ M1 (R), and J ∩ J = ∅ (the measures are supported on distinct proper compact subsets of R \ {α1 , α2 , . . . , α K }); thus, from Lemma 3.1 (with the change μ → μ ), that is, R2
=
R2
⎛ 1 ln ⎝|s−t| n
|s−t| |s−αk ||t−αk |
κnkn −1 s$ −1 q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
⎞−1 ⎠
× dμ (s) dμ (t) ⎛ κnkn −1 s$ κnknkq −1 |s−t| |s−t| 1 ln ⎝|s−t| n |s−αk ||t−αk | |s−α pq ||t−α pq | q=1 × (s) (t))−1 dμ (s) dμ (t) 0,
it follows that, for n ∈ N and k = 1, 2, . . . , K, via the symmetry of the associated integrand under the interchange s ↔ t, ⎛
R2
ln ⎝|s−t|
κnkn −1 s$ κnknkq −1 |s−t| |s−t| |s−αk ||t−αk | |s−α pq ||t−α pq | q=1
1 n
⎞−1 × (s) (t)⎠
. μV (t) + dμV (s) dμV (t) d μV (s) d
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K. T.-R. McLaughlin et al.
⎛
R2
ln ⎝|s−t|
1 n
|s−t| |s−αk ||t−αk |
⎞−1 × (s) (t)⎠ ⎛
=2
R2
ln ⎝|s−t|
1 n
κnkn −1 s$ −1 q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
( ) d μV (s) dμV (t)+dμV (s) d μV (t)
|s−t| |s−αk ||t−αk |
κnkn −1 s$ −1 q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
⎞−1 × (s) (t)⎠ ⎛
=2
R2
ln ⎝|s−t|
1 n
d μV (s) dμV (t)
|s−t| |s−αk ||t−αk |
κnkn −1 s$ −1 q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
⎞−1 × (s) (t)⎠
dμV (s) d μV (t).
The above shows that (cf. Lemma 3.1), for n ∈ N and k = 1, 2, . . . , K, since both ( ) κnkn −1 s−1 ( |s−t| ) κnknkq ) 1( IV [μV ] and IV [ μV ] are finite, ln |s−t| n |s−α|s−t| q=1 |s−α pq ||t−α pq | k ||t−αk | is integrable with respect to both ‘product’ measures d μV (s) dμV (t) and dμV (s) d μV (t). From an argument on pg. 149 of [56], it follows that, for ) κnkn −1 s−1 ( ) κnknkq ) ( 1( |s−t| is n ∈ N and k = 1, 2, . . . , K, ln |s−t| n |s−α|s−t| q=1 |s−α pq ||t−α pq | k ||t−αk | integrable with respect to the one-parameter family of ‘product’ measures EQ EQ dμEQ μV −μV )(z), (z, λ) ∈ R × [0, 1]. For (z)+λ( λ (s) dμλ (t), where μλ (z) := μV n ∈ N and k = 1, 2, . . . , K, set Fμ (λ) := IV [μEQ λ ], that is, EQ Fμ : N ×{1, 2, . . . , K}× M1 (R)×[0, 1] → R, (n, k, μEQ λ ) → Fμ (n, k; μλ ) := Fμ (λ) ⎛ κnkn −1 s$ κnknkq −1 |s−t| |s−t| 1 = ln ⎝|s−t| n |s−αk ||t−αk | |s−α pq ||t−α pq | R2 q=1
⎞−1 × (s) (t)⎠
EQ dμEQ λ (s) dμλ (t).
(37)
EQ Noting that dμEQ μV − μV )(t) + (s)dμV (t) + λdμV (s)d( λ (s)dμλ (t) = dμV λdμV (t)d( μV − μV )(s) + λ2 d( μV − μV )(s)d( μV − μV )(t), it follows from the
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above definition of Fμ (λ), Lemma 3.1, and the symmetry of the integrand under the interchange s ↔ t that, for n ∈ N and k = 1, 2, . . . , K, ⎛ κnkn −1 |s−t| 1 Fμ (λ) = IV [μV ]+2λ ln ⎝|s−t| n |s−αk ||t−αk | R2
×
s −1 $
|s−t| |s−α pq ||t−α pq |
q=1
⎛
+λ
2 R2
ln ⎝|s−t|
1 n
κnknkq
⎞−1 (s) (t)⎠
|s−t| |s−αk ||t−αk |
dμV (s) d( μV −μV )(t)
κnkn −1 s$ −1 q=1
κnknkq |s−t| |s−α pq ||t−α pq |
⎞−1 × (s) (t)⎠
d( μV −μV )(s) d( μV −μV )(t).
Since μ ∈ M1 (R) is a finite-moment signed measure on R with compact support and mean zero, that is, R dμ (ξ ) = R d( μV −μV )(ξ ) = 0, it follows from the expression above and the variational inequality of Lemma 3.2 that, for n ∈ N and k = 1, 2, . . . , K, Fμ (λ) is a twice-differentiable, real-valued function of λ on [0, 1], and, with d2λ := d2 /dλ2 , ⎛ κnkn −1 1 2 |s−t| 1 ln ⎝|s−t| n d Fμ (λ) = 2 λ |s−αk ||t−αk | R2
×
s −1 $
q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
⎞−1 (s) (t)⎠
dμ (s) dμ (t) 0,
that is, Fμ (λ) is a real-valued convex9 function of λ on [0, 1]; thus, for n ∈ N and k = 1, 2, . . . , K, with [0, 1] λ, IV [μV ] Fμ (λ) = IV [μEQ λ ] = Fμ (λ+(1−λ)0) λFμ (1)+(1−λ)Fμ (0) = λIV [ μV ]+(1−λ)IV [μV ] = λIV [μV ]+(1−λ)IV [μV ] ⇒ IV [μV ] IV [μEQ [μV ], λ ] IV
9 If f is twice differentiable on (a, b ), then f (x) 0 on (a, b ) is both a necessary and sufficient condition that f be convex on (a, b ).
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K. T.-R. McLaughlin et al.
EQ whence IV [μEQ [μV ] = EV (= E V (n, k)). Since IV [μλ ] = Fμ (λ) = E V , it λ ] = IV follows, in particular, that for n ∈ N and k = 1, 2, . . . , K, d2λ Fμ (λ)|λ=0 = 0, in which case ⎛ κnkn −1 s$ κnknkq −1 |s−t| |s−t| 1 0= ln ⎝|s−t| n |s−αk ||t−αk | |s−α pq ||t−α pq | R2 q=1
⎞−1 × (s) (t)⎠ ⎛
=
R2
ln ⎝|s−t|
1 n
d( μV −μV )(s) d( μV −μV )(t)
|s−t| |s−αk ||t−αk |
κnkn −1 s$ −1 q=1
× d( μV −μV )(s) d( μV −μV )(t) + 2 ×
d( μV −μV )(t)
R2
κnknkq
⎞−1 ⎠
d( V(s) μV −μV )(s)
⇒
=0
⎛
0=
R
R
|s−t| |s−α pq ||t−α pq |
1 ln ⎝|s−t| n
|s−t| |s−αk ||t−αk |
κnkn −1 s$ −1 q=1
|s−t| |s−α pq ||t−α pq |
κnknkq
⎞−1 ⎠
× d( μV −μV )(s) d( μV −dμV )(t); but, in Lemma 3.2, it was shown that, for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎞ κnkn −1 s$ κnknkq −1 −1 |s−t| |s−t| 1 ⎠ ln ⎝|s−t| n |s−αk ||t−αk | |s−α pq ||t−α pq | R2 q=1 × d( μV −μV )(s)d( μV −dμV )(t) +∞ (n−1)K+k 4V − μ = τ −1 |( μ 4V )(τ )|2 dτ 0, n 0 whence, for n ∈ N and k = 1, 2, . . . , K, +∞ 4V − μ τ −1 |( μ 4V )(τ )|2 dτ = 0 0
⇒
4V (τ ) = μ 4V (−τ ) = e−iτ ξ d 4V (τ ) and μ 4V (τ ), τ 0. Noting that μ μV (ξ ) = μ R −iτ ξ 4 μ 4V (−τ ) = R e dμV (ξ ) = μ 4V (τ ), it follows from μV (τ ) = μ 4V (τ ), τ 0, via 4V (−τ ) = μ 4V (−τ ), τ 0; hence, for a complex-conjugation argument, that μ 4V (τ ) = μ n ∈ N and k = 1, 2, . . . , K, μ 4V (τ ), τ ∈ R. The latter relation shows that,
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285
for n ∈ N and k = 1, 2, . . . , K, R eiτ ξ d( μV −μV )(ξ ) = 0 ⇒ μV = μV ; thus the uniqueness of the equilibrium measure. For n ∈ N and k = 1, 2, . . . , K, RHPFPC , that is, (Y (n, k; z) := Y (z), I+ e−nV(z) σ+ , R), is now reformulated as a family of K equivalent, auxiliary RHPs. Notational Remark 3.1 For completeness, the integrand appearing in the definition of g(z) in Lemma 3.4 below is defined as follows: for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎞ κnkn −1 s$ κnknkq −1 (z−s) (z−s) 1 ⎠ ln ⎝(z−s) n (z−αk )(s−αk ) (z−α pq )(s−α pq ) q=1 1 := ln(z−s) + n +
s−1
κnkkq (
q=1
n
κnk −1 (ln(z−s)−ln(z−αk )−ln(s−αk )) n
) ln(z−s)−ln(z−α pq )−ln(s−α pq ) ,
where, for x < 0, ln x := ln|x|+iπ . R \ {α1 , α2 , . . . , α K } → R satisfy condiLemma 3.4 Let the external field V: tions (6)–(8). For the associated equilibrium measure μV ∈ M1 (R), set J := supp(μV ), where J is a proper compact subset of R \ {α1 , α2 , . . . , α K }, and, for n ∈ N and k = 1, 2, . . . , K, let Y : N × {1, 2, . . . , K} × C \ R → SL2 (C) be the unique solution of RHPFPC . For n ∈ N and k = 1, 2, . . . , K, set Y(n, k; z) := Y(z) = e− 2
n
ad(σ3 )
Y (z)e−n(g(z)−P0 (n,k))σ3 ,
where : N × {1, 2, . . . , K} → R, the variational constant, is given in Lemma 3.8 below, g : N × {1, 2, . . . , K} × C \ −∞, max max {α pq }, max{J} → C, q=1,2,...,s
⎛ κnkn −1 (z−s) 1 ⎝ n (n, k, z) → g(n, k; z) := g(z) = ln (z−s) (z−αk )(s−αk ) J
×
⎞ κnknkq (z−s) ⎠dμV (s), (z−α pq )(s−α pq )
s −1 $
q=1
and
P0 (n, k) =
P+ 0, P− 0,
z ∈ C+ , z ∈ C− ,
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with P± 0 :=
⎛
ln ⎝|s−αk | n 1
J
q=1
− iπ ± iπ
s −1 $
κnk −1 n
|s−αk | |s−α pq ||α pq −αk |
J∩R< α
dμV (s) − iπ
J∩R> α
⎞ ⎠ dμV (s)
s−1
κnkkq
n
q=1
k
(n−1)K+k n
κnknkq
dμV (s)∓iπ
κnkkq ˆ 2 (k) q∈
k
J∩R< α pq
n
dμV (s)
,
ˆ 2 (k) := { j∈ {1, 2, . . . , s −1}; α p j > αk }. Then, for n ∈ N and k = 1, 2, where . . . , K, Y : N × {1, 2, . . . , K} × C \ R → SL2 (C) solves the following RHP: (i) Y(z) is holomorphic for z ∈ C \ R; (ii) the boundary values Y± (z) := limε↓0 Y(z± iε) satisfy the jump condition Y+ (z) = Y− (z)V(z),
z ∈ R,
where, with g± (z) := limε↓0 g(z±iε), V: N × {1, 2, . . . , K} × R → GL2 (C), −n(g (z)−g (z)+P− −P+ ) n(g (z)+g (z)−P− −P+ −V(z)−) + − − 0 0 0 0 e e + z → V(z) := ; − + 0 en(g+ (z)−g− (z)+P0 −P0 )
(iii) Y(z)
=
C± z→αk
I+O (z−αk );
(iv) Y(z)
=
O (1);
=
O (1).
C± z→∞
(v) for q ∈ {1, 2, . . . , s −1}, Y(z)
C± z→α pq
Proof For (arbitrary) z1 , z2 ∈ C± , and for n ∈ N and k = 1, 2, . . . , K, note from z the definition of g(z) stated in the Lemma that g(z2 )−g(z1 ) = iπ z12 F(ξ ) dξ , where, with D := C \ (supp(μV ) ∪ {α1 , α2 , . . . , α K }), F: N × {1, 2, . . . , K} × D → C, (n, k, z) → F(n, k; z) := F(z)
⎛ ⎞ s−1 (n−1)K+k dμV (ξ ) ⎠ 1 ⎝ (κnk −1) κnkkq + + , =− iπ n(z−αk ) q=1 n(z−α pq ) n J ξ −z
(38)
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and, since−msupp(μV ) ∩ {α1 , α2 , . . . , α K } = ∅ and μV ∈ M1 (R), in particular, dμV (ξ ) < ∞, m ∈ N, J (ξ − αk ) F(z)+
1 (κnk −1) iπ n(z−αk )
=
D z→αk
O (1)
(e.g., 0 < |z−αk | % min{infξ ∈J {||ξ |−αk |}, min j=1,2,...,s−1 {|α p j −αk |}}), and, for q ∈ {1, 2, . . . , s −1}, F(z)+
(e.g., 0 < |z−α pq | % min{inf
1 κnkkq iπ n(z−α pq ) ξ ∈J
j=1,2,...,s−1
=
D z→α pq
O (1)
{||ξ |−α p j |}, min j=1,2,...,s−1 {|α p j −αk |}}); thus,
|g(z2 )−g(z1 )| π |z2 −z1 | supz∈C± |F(z)|. Thus, for n ∈ N and k = 1, 2, . . . , K, → C, with D := (−∞, max{maxq=1,2,...,s {α pq }, from the definition of g : C \ D max{supp(μV )}}), given in the Lemma: (1) for ξ ∈ J (J ∩ {α1 , α2 , . . . , α K } = , with 0 < |z−αk | % min{infξ ∈J {||ξ |−αk |}, min j=1,2,...,s−1 {|α p j −αk |}}, ∅), z ∈ C \ D and μV ∈ M1 (R), in particular, J (ξ −αk )−m dμV (ξ ) < ∞, m ∈ N, it follows j (z−αk )l+1 + 1 k) from the expansions (z−αk )−(ξ = − lj=0 (ξ(z−α + (ξ −α l+1 (z−ξ ) , l ∈ Z0 , and −αk ) −αk ) j+1 k) ∗j ln(1−∗∗) =|∗∗|→0 − ∞ j=1 j , and a careful analysis of the branch cuts, that κnk −1 1 ln(z−αk ) + ln |ξ −αk | n − n J ⎞ κnknkq s −1 $ |ξ −αk | ⎠ dμV (ξ ) × |ξ −α pq ||α pq −αk | q=1
g(z)
=
C± z→αk
− iπ ± iπ
κnk −1 n
J∩R< α
(n−1)K+k n
dμV (τ )−iπ
s−1
κnkkq
q=1
k
J∩R> α
k
dμV (τ )∓iπ
n
J∩R< α pq
dμV (τ )
κnkkq ˆ 2 (k) q∈
n
⎛ ∞ s−1
κnkkq 1 ⎝ (n−1)K+k + − (ξ −αk )−m m m n(α −α ) n pq k J m=1 q=1 ⎞ × dμV (ξ )⎠ (z−αk )m ,
(39)
, with ˆ 2 (k) is defined in the Lemma; (2) for ξ ∈ J, z ∈ D where |z| & max{supξ ∈J {|ξ |}, maxq=1,2,...,s {|α pq |}}, and μV ∈ M1 (R), in particular, m (ξ ) = 1 and J ξ dμV (ξ ) < ∞, m ∈ N, it follows from the expansions J dμV
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∞ 1 ∗ j j l+1 = − lj=0 zξj+1 + zl+1ξ(ξ −z) , l ∈ Z+ j=1 j ( z ) , 0 , and ln(z − ∗ ) =|z|→∞ ln(z) − and a careful analysis of the branch cuts, that 1 ξ −z
1 ln(z)− g(z) = C± z→∞ n −iπ
⎛
ln ⎝|ξ −αk |
κnk −1 n
J
κnk −1 n
s −1 $
⎞ κ
|ξ −α pq |
nk kq n
⎠ dμV (ξ )
q=1
dμV (τ ) − iπ
J∩R< α
s−1
κnkkq
n
q=1
k
J∩R< α pq
dμV (τ )
⎛ ∞ s−1
κnkkq 1 ⎝ κnk −1 + (αk )m + (α pq )m m n n m=1 q=1 −
(n−1)K+k n
⎞
J
ξ m dμV (ξ )⎠ z−m ;
(40)
, with 0 < |z−α pq | % and (3) for q ∈ {1, 2, . . . , s −1}, ξ ∈ J, z ∈ D min{inf ξ ∈J {||ξ |−α p j |}, min j=1,2,...,s−1 {|α p j −αk |}}, and μV ∈ M1 (R), in parj=1,2,...,s−1 ticular, J (ξ − α pq )−mdμV (ξ ) < ∞, m ∈ N, it follows from the expansions (z−α p ) j (z−α pq )l+1 1 ∗) =|∗∗|→0 = − lj=0 (ξ −α p q) j+1 + (ξ −α p )l+1 , l ∈ Z+ 0 , and ln(1−∗ (z−α pq )−(ξ −α pq ) (z−ξ ) q q ∞ ∗ j − j=1 j , and a careful analysis of the branch cuts, that
g(z)
=
C± z→α pq
−
×
κnkkq
n
s −1 $
j=1
⎛
⎜ ln(z−α pq ) + ln ⎜ ⎝ J
|ξ −α pq | |ξ −α p j ||α pq −α p j |
|ξ −α pq |
κnk n
(|ξ −αk ||α pq −αk |)
κ
nk kj n
κnk −1 n
⎞ ⎟ ⎟ dμ (ξ ) ⎠ V
j =q
− iπ
−iπ
κnk −1 n
J∩R< α
n
J∩R< αp
j =q
∓iπ
˜ 2 (k,q) j∈
n
k
s−1
κnkk j
j=1
dμV (τ )−iπ
κnkkq
κnkk j
n
dμV (τ ) ∓ iπ
J∩R< α pq
dμV (τ )
κnk −1 ε(k, q) n
j
±iπ
(n−1)K+k n
J∩R> α pq
dμV (τ )
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⎛ ∞
s−1
κnkk j (κnk −1) 1 ⎜ ⎜ + + m ⎝ n(αk −α pq )m j=1 n(α p j −α pq )m m=1 j =q
−
(n−1)K+k n
⎞
J
⎟ m (ξ −α pq )−m dμV (ξ )⎟ ⎠ (z−α pq ) ,
(41)
where ε(k, q) =
1, α pq < αk , 0, α pq > αk ,
˜ 2 (k, q) := { j∈ {1, 2, . . . , s −1}; α p j > α pq }. Using the definition of Y(z) (in and terms of Y (z)) stated in the Lemma, and recalling that Y : N × {1, 2, . . . , K} × C \ R → SL2 (C) solves, uniquely, RHPFPC , via the above asymptotic expansions for g(z), one arrives at the RHP (Y(n, k; z) := Y(z), V(z), R) stated in the Lemma. In order to prove the following Lemma 3.5, which is requisite for the proof of Lemma 3.6 (see below), one follows closely the idea of the proof of Theorem 6.6.2 of Deift [56]. R \{α1 , α2 , . . . , α K } → R satisfy conditions (6)–(8). For n ∈ N Lemma 3.5 Let V: and k = 1, 2, . . . , K, let
V d(n−1)K+k :=
×
1 ((n−1)K+k)((n−1)K+k−1) ⎛ ⎛ (n−1)K+k
1 ⎝ ⎝ n inf ln |xi −x j|
{x1 ,x2 ,...,x(n−1)K+k } ⊂ R
i, j=1
|xi −x j| |xi −αk ||x j −αk |
κnkn −1
i = j
×
s −1 $
q=1
|xi −x j| |xi −α pq ||x j −α pq |
κnknkq
⎞
⎞−1 ⎠ +2((n−1)K+k−1)
(n−1)K+k
j=1
V Then, for k = 1, 2, . . . , K, limn→∞ d(n−1)K+k exists,
= EV = inf IV [μEQ ]; lim dV n→∞ (n−1)K+k
μEQ ∈ M1 (R) = IV [μV ],
V and limn→∞ exp(−d(n−1)K+k ) = exp(−EV ) > 0 and finite.
j)⎟ V(x ⎠.
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For n ∈ N and k = 1, 2, . . . , K, let x∗1 , x∗2 , . . . , x∗(n−1)K+k denote the generalised weighted Fekete points, that is,
V d(n−1)K+k =
1 ((n−1)K+k)((n−1)K+k−1)
(n−1)K+k
-
1 ln |xi∗ −x∗j | n
i, j=1 i = j
|xi∗ −x∗j | |xi∗ −αk ||x∗j −αk |
×
+ 2((n−1)K+k−1)
! κnkn −1
s −1 $
q=1 (n−1)K+k
|xi∗ −x∗j | |xi∗ −α pq ||x∗j −α pq |
! κnknkq
⎞−1 ⎟ ⎠
⎞ ∗j )⎠ . V(x
j=1
For n ∈ N and k = 1, 2, . . . , K, with {x∗1 , x∗2 , . . . , x∗(n−1)K+k } a generalised weighted ((n−1)K+k)-Fekete set, denote by λ(n−1)K+k :=
1 ((n−1)K+k)
(n−1)K+k
δx∗j ,
j=1
where δx∗j , j= 1, 2, . . . , (n−1)K+k, are the Dirac delta (atomic) masses concentrated at x∗j , the normalised counting measure. Then, in the weak-∗ topology of measures, λ(n−1)K+k converges (weakly) to μV as n → ∞. R \ {α1 , α2 , . . . , α K } → R Sketch of Proof For n ∈ N and k = 1, 2, . . . , K, with V: satisfying conditions (6)–(8), set, with N := (n−1)K+k, ⎛
V δN =
⎝
sup
{x1 ,x2 ,...,xN } ⊂ R
where
sup
⎝
{x1 ,x2 ,...,xN } ⊂ R
∗) := i< j (∗
N −1 N i=1
(f(xi , x j))2 e−2V(xi ) e−2V(x j ) ⎠
i< j
⎛ =
⎞ N (N1 −1)
$
$
(f(xi , x j))2 e−2(N −1)
N
⎞ N (N1 −1)
j=1 V(x j ) ⎠
,
i< j
∗), j=i+1 (∗
and
f : N × {1, 2, . . . , K} × R2 → R+ , (n, k, x, y) → f(n, k; x, y) := f(x, y) = f(y, x)
= |x− y|
1 n
|x− y| |x−αk ||y−αk |
κnkn −1 s$ −1 q=1
|x− y| |x−α pq ||y−α pq |
κnknkq
.
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
291
(Note: n → ∞ ⇒ N → ∞). One begins by showing that, for n ∈ N and k = V is finite, and (the supremum) is attained at some (or many) 1, 2, . . . , K, δN finite point set(s) {x∗1 , x∗2 , . . . , x∗N } ⊂ R, with x∗j = x∗j (n, k), j= 1, 2, . . . , N (actually, as will be shown below, {x∗1 , x∗2 , . . . , x∗N } ⊂ R \ {α1 , α2 , . . . , α K }). Recall, from the (sketch of the) proof of Lemma 3.1, the following inequality: for s, t ∈ R, |s−t| (1+s2 )1/2 (1+t2 )1/2 . Using this inequality, one shows that, for n ∈ N and k = 1, 2, . . . , K, .1/2 $( $ )1/2 1+xi2 |xi −x j| 1+x2j i= j
i= j
⎛ ⎞N −1 N . $ =⎝ 1+x2j ⎠ , j=1
$ i= j
|xi −x j| |xi −αk ||x j −αk |
κnk −1
$
1 1+ (xi −αk )2
i= j
1 1+ (x j −αk )2
×
⎛ N $ = ⎝ 1+
i= j q=1
|xi −x j| |xi −α pq ||x j −α pq |
κnkkq
−1 $ s$
1+
i= j q=1
×
1/2 !κnk −1
1 (x j −αk )2
j=1 −1 $ s$
⎛ N s −1 $ $ = ⎝ 1+
κnk −1
1 (xi −α pq )2
1 1+ (x j −α pq )2
j=1 q=1
1/2
⎞N −1 ⎠
,
1/2
1/2 !κnkkq
1 (x j −α pq )2
κnkkq
⎞N −1 ⎠
;
hence, using the fact that f(x, y) = f(y, x), via the above inequalities, one arrives at, for n ∈ N and k = 1, 2, . . . , K: ⎛ κnkn −1 N .1 $ $ 1 i ) −2V(x j) 2 −2V(x 2 n ⎝ (f(xi , x j)) e e 1+x j 1+ (x j −αk )2 i< j j=1
×
s −1 $
1+
q=1
1 (x j −α pq )2
κnknkq
⎞N −1
e−2V(x j )⎠
.
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K. T.-R. McLaughlin et al.
R \ {αi , α2 , . . . , α K } → R satisfies conditions (6)–(8), one Recalling that V: shows that, for n ∈ N and k = 1, 2, . . . , K: (i) for x j ∈ R \ {α1 , α2 , . . . , α K }, j= 1, 2, . . . , N , . n1 0 1+x2j 1+
1 (x j −αk )2
κnkn −1 s$ −1 1+ q=1
1 (x j −α pq )2
κnknkq e−2V(x j )
c(n, k) < +∞; −1 (ii) for x j ∈ O∞ := {x ∈ R; |x| > δ∞ }, j= 1, 2, . . . , N , where δ∞ (= δ∞ (n, k)) is an arbitrarily fixed, sufficiently small positive real number, there exists c∞ j) (1+ c∞ ) ln(x2 +1), it follows (= c∞ (n, k)) > 0 and bounded such that V(x j
that (1+x2j )1/n e−2V(x j ) → 0 as |x j| → +∞, j= 1, 2, . . . , N ; hence, for x j ∈ O∞ , j= 1, 2, . . . , N , .1 2 n 0 1+x j 1+
1 (x j −αk )2
κnkn −1 s$ −1 1+ q=1
1 (x j −α pq )2
κnknkq
e−2V(x j )
c(n, k) < +∞;
and (iii) for x j ∈ Oδ˜k := {x ∈ R; |x−αk | < δ˜k }, j= 1, 2, . . . , N , k = 1, 2, . . . , K, where δ˜k (= δ˜k (n, k)) are arbitrarily fixed, sufficiently small positive real j) (1+ numbers, there exists ci (= ci (n, k)) > 0 and bounded such that V(x κ −1 −2 −2 nkn j)) → 0 as ck ) ln((x j −αk ) +1), it follows that (1+(x j −αk ) ) exp(−2V(x x j → αk , j= 1, 2, . . . , N , k = 1, 2, . . . , K; hence, for x j ∈ Oδ˜k , j= 1, 2, . . . , N , k = 1, 2, . . . , K, .1 2 n 0 1+x j 1+
1 (x j −αk )2
κnkn −1 s$ −1 1+ q=1
1 (x j −α pq )2
κnknkq
e−2V(x j )
c(n, k) < +∞.
Hence, for n ∈ N and k = 1, 2, . . . , K, with xm ∈ R, m ∈ {1, 2, . . . , N }, $ i< j
(f(xi , x j))2 e−2V(xi ) e−2V(x j ) =
$
f(xi , x j)e−2(N −1)
N j=1
j) V(x
i= j
⎛ (N −1)2
(c(n, k))
⎝(1+x2m ) n1
1 1+ (xm −αk )2
κnkn −1
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal... s −1 $
×
q=1
1 1+ (xm −α pq )2
293
⎞N −1
κnknkq e
m) −2V(x ⎠
→ 0 as |xm | → +∞ and as xm → αk .
V is finite, and (the supremum) is Hence, for n ∈ N and k = 1, 2, . . . , K, δN attained at some (or many) finite point set(s) {x∗1 , x∗2 , . . . , x∗N } ⊂ R, where x∗j = x∗j (n, k), j= 1, 2, . . . , N (actually, {x∗j }N j=1 ⊂ R \ {α1 , α2 , . . . , α K }). For n ∈ N and k = 1, 2, . . . , K, one agrees to call, with some abuse of nomenclature, a maximising set, that is, a set {x∗1 , x∗2 , . . . , x∗N } for which ⎛ ⎞ N (N1 −1) $ ∗ ∗ V δN = ⎝ (f(xi∗ , x∗j ))2 e−2V(xi ) e−2V(x j ) ⎠ i< j
⎞ N (N1 −1) ⎛ N $ ∗ = ⎝ f(xi∗ , x∗j )e−2(N −1) j=1 V(x j ) ⎠ , i= j
a generalised weighted N -Fekete set, and the N points x∗1 , x∗2 , . . . , x∗N will be called general weighted Fekete points. For n ∈ N and k = 1, 2, . . . , K, define
KNV (x1 , x2 , . . . , xN ) :=
N
KV (xi , x j),
i, j=1 i = j
where, from the (sketch of the) proof of Lemma 3.1, 1 κnk −1 ln KV (s, t) = KV (t, s) := ln(|s−t|−1 )+ n n +
s−1
κnkkq
n
q=1
' ' 1 ln '' t−α
' '−1 ! ' 1 ' 1 ' ' ' t−α − s−α ' k k
' ! 1 ''−1 + V(s)+ V(t). − s−α pq ' pq
An algebraic calculation shows that, for n ∈ N and k = 1, 2, . . . , K, ⎛ N
|xi −αk ||x j −αk | 1 1 κnk −1 V ⎝ KN (x1 , x2 , . . . , xN ) = ln + ln n |xi −x j| n |xi −x j| i, j=1 i = j
+
s−1
κnkkq
q=1
n
⎞ N
|xi −α pq ||x j −α pq | j). ⎠ + 2(N −1) ln V(x |xi − x j| j=1
For n ∈ N and k = 1, 2, . . . , K, set
V dN :=
1 N (N −1)
inf
{x1 ,x2 ,...,xN } ⊂ R
KNV (x1 , x2 , . . . , xN ).
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K. T.-R. McLaughlin et al.
One shows, via the definition of KNV (x1 , x2 , . . . , xN ), that, for n ∈ N and k = 1, 2, . . . , K, ⎛ κnkn −1 $ |xi −x j| 1 V −KN (x1 ,x2 ,...,xN ) ⎝ n e = |xi −x j| |xi −αk ||x j −αk | i< j
×
s −1 $
q=1
=
|xi −x j| |xi −α pq ||x j −α pq |
$
(f(xi , x j))2 e−2(N −1)
κnknkq
N j=1
⎞2 ⎠ e−2(N −1)
j) V(x
N j=1
j) V(x
,
i< j
whence ⎛ e
1 V − N (N −1) KN
(x1 ,x2 ,...,xN )
=⎝
$
(f(xi , x j))2 e−2(N −1)
N
⎞ N (N1 −1)
j=1 V(x j ) ⎠
.
i< j
Using the fact that − sup(−∗∗) = inf(∗∗), one arrives at, for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎞ N (N1 −1) N $ V V ⎝ (f(xi , x j))2 e−2(N −1) j=1 V(x j ) ⎠ e−dN = sup = δN . {x1 ,x2 ,...,xN } ⊂ R
i< j
The above calculations also show that, for n ∈ N and k = 1, 2, . . . , K, ⎛ N . n1 ( $ ) κnk −1 V e−KN (x1 ,x2 ,...,xN ) ⎝ 1+(x j −αk )−2 n 1+x2j j=1
×
s −1 $
(
⎞N −1 κ nkkq ) 1+(x j −α pq )−2 n e−2V(x j ) ⎠ ,
q=1
whence e
⎛ V −dN
×
sup
{x1 ,x2 ,...,xN } ⊂ R
s −1 $
(
⎛ N .1 ( $ ) κnk −1 ⎝ ⎝ 1+x2j n 1+(x j −αk )−2 n j=1
1+(x j −α pq )−2
)
κ
nk kq n
⎞N −1 ⎞N (N1 −1) ⎟ e−2V(x j )⎠ < +∞; ⎠
q=1
V hence, for n ∈ N and k = 1, 2, . . . , K, dN > −∞, that is, (N (N −1))−1 V · inf{x1 ,x2 ,...,xN }⊂R KN (x1 , x2 , . . . , xN ) > −∞.
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295
At the beginning of the proof it was shown that, for n ∈ N and k = 1, 2, . . . , K, a generalised weighted N -Fekete set for KNV (x1 , x2 , . . . , xN ) exists, that is, V a set {x∗1 , x∗2 , . . . , x∗N } (⊂ R \ {α1 , α2 , . . . , α K }) such that dN = (N (N −1))−1 ∗ V ∗ ∗ ∗ ·KN (x1 , x2 , . . . , xN ) (the points x j , j= 1, 2, . . . , N , are distinct). For n ∈ N and k = 1, 2, . . . , K, define the normalised counting measure for the generalised weighted N -Fekete set as follows:
λN :=
N 1 δ x∗ , N j=1 j
where δx∗j , j= 1, 2, . . . , N , are the Dirac delta (atomic) masses concentrated at x∗j (note that, for n ∈ N and k = 1, 2, . . . , K, R dλN (ξ ) = N1 R N j=1 δ(ξ − x∗j ) dξ = 1). One now shows that, in the weak-∗ topology of measures, for k = 1, 2, . . . , K, as n → ∞ (⇒ N → ∞) λN converges (weakly) to μV (∈ M1 (R)), that is, for k = 1, 2, . . . , K, λN μV as n → ∞. One proceeds via a contradiction argument, namely, one assumes that, for k = 1, 2, . . . , K and C0b (R) (the space of real-valued, bounded, (some) g ∈C continuous functions on R with compact support), R g(ξ ) dλN (ξ ) R g(ξ ) dμV (ξ ) as n → ∞, that is, ˆ ∃ ε > 0 and a subsequence Nkˆ → ∞ such that, for all k ∈ N, | R g(ξ ) dλNkˆ (ξ )− (ξ )| ε. One first shows, though, that for n ∈ N and k = 1, 2, . . . , K, R g(ξ ) dμV the sequence of probability measures {λN }N ∈N is tight (cf. (sketch of the) proof of Lemma 3.1). Since, for n ∈ N and k = 1, 2, . . . , K, 1 1 K V (x1 , x2 , . . . , xN ) inf K V (x1 , x2 , . . . , xN ) N (N −1) {x1 ,x2 ,...,xN }⊂R N N (N −1) N
1 sup K V (x1 , x2 , . . . , xN ), N (N −1) {x1 ,x2 ,...,xN } ⊂ R N
V upon recalling the definition of dN , one notes that (N (N −1))−1 KNV (x1 , V x2 , . . . , xN ) dN : integrating both sides of this latter inequality with respect to the ‘product measure’ (cf. Remark 3.1) dμV (x1 ) dμV (x2 ) · · · dμV (xN ), one arrives at, for n ∈ N and k = 1, 2, . . . , K, after a straightforward calculation:
⎛
ln ⎝ R2
1 1
|s−τ | n
|s−αk ||τ −αk | |s−τ |
κnkn −1 s$ κnknkq −1 |s−α pq ||τ −α pq |
. × eV(s) eV(τ ) dμV (s) dμV (τ ) =
V = IV [μV ] = EV dN
(> −∞).
q=1
R2
|s−τ |
KV (s, τ ) dμV (s) dμV (τ )
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Via this inequality and the definition of ψˆ V (z) given in (35) (cf. (sketch of the) proof of Lemma 3.1), one proceeds thus; for n ∈ N and k = 1, 2, . . . , K:
V EV dN =
=
1 N (N −1)
inf
{x1 ,x2 ,...,xN } ⊂ R
KNV (x1 , x2 , . . . , xN )
N
1 1 KNV (x∗1 , x∗2 , . . . , x∗N ) = K (x∗ , x∗ ) N (N −1) N (N −1) i, j=1 V i j i = j
N N . 1
1 1 ψˆ (x∗ ) ψˆ V (xi∗ )+ ψˆ V (x∗j ) = 2 N (N −1) i, j=1 N j=1 V j i = j
= 2(N −1)
=
R
N j=1
ψˆ V (x∗j )
ψˆ V (ξ ) dλN (ξ ) ⇒
V EV = inf IV [μEQ ]; μEQ ∈ M1 (R) = IV [μV ] dN
R
ψˆ V (ξ ) dλN (ξ ).
s For n ∈ N and k = 1, 2, . . . , K, writing D M := {|x| M} ∪ (∪q=1 O 1 (α pq )), for M some suitably chosen M (= M(n, k)) > 1, one proceeds as in the (sketch of the) proof of Lemma 3.1 to show that
R
ψˆ V (ξ ) dλN (ξ ) =
ψˆ V (ξ ) dλN (ξ )+ R\D M −|Cˆ V |
DM
ψˆ V (ξ ) dλN (ξ ), >b
whence follows the tightness of the sequence of probability measures {λN }N ∈N inM1 (R), that is, for (some) sufficiently small ε (= ε(n, k)) > 0, lim supN →∞ D M dλN (ξ ) ε. For n ∈ N and k = 1, 2, . . . , K, with L ∈ R,
V dN =
1 N (N −1)
inf
{x1 ,x2 ,...,xN } ⊂ R
KNV (x1 , x2 , . . . , xN )
N
1 1 ∗ ∗ ∗ V = K (x , x , . . . , xN ) = K (x∗ , x∗ ) N (N −1) N 1 2 N (N −1) i, j=1 V i j i = j
N
1 N2 min L, KV (xi∗ , x∗j ) = N (N −1) i, j=1 N (N −1) i = j
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
×
N N 1 1 ∗ min L, KV (s, τ ) δ(s−xi ) ds δ(τ −x∗j ) dτ N i=1 N j=1 R2
= dλN (s)
V dN
N2 N (N −1)
R2
297
⇒
= dλN (τ )
min L, KV (s, τ ) dλN (s) dλN (τ ).
(42)
Now, and as a—direct—consequence of tightness, for n ∈ N and k = 1, 2, . . . , K, as the (sub) sequence of probability measures {λNkˆ }∞ in M1 (R) is also tight, ˆ k=1 there exists, by a Helly selection theorem, a further weak-∗ convergent subsequence of probability measures {λNkˆ }∞ j=1 in M1 (R) converging weakly to j λ as j→ ∞. Proceeding, verbaa probability measure λ ∈ M1 (R), that is, λNkˆ j
tim, as in the (sketch of the) proof of Lemma 3.1, one shows, via inequality (42), that, as j→ ∞ (using N ˆ2 (Nkˆ j (Nkˆ j −1))−1 → 1) N ˆ2 (Nkˆ j (Nkˆ j −1))−1 kj kj · R2 min{L, KV (s, τ )} dλNkˆ (s)dλNkˆ (τ ) → R2 min{L, KV (s, τ )} dλ(s) dλ(τ ) ⇒ j
j
V lim inf dN ˆ j→∞
kj
R2
min L, KV (s, τ ) dλ(s) dλ(τ ) :
letting L ↑ +∞ and using monotone convergence (cf. (sketch of the) proof of Lemma 3.1), one arrives at V KV (s, τ ) dλ(s) dλ(τ ) = IV [λ] (> −∞); lim inf dN ˆ j→∞
kj
R2
but,
V V EV lim sup dN lim inf dN IV [λ] ˆ ˆ j→∞
j→∞
kj
kj
EV = inf IV [μEQ ]; μEQ ∈ M1 (R) = IV [μV ]
⇒
IV [λ] = IV [μV ], which, by the unicity of the equilibrium measure (cf. Lemma 3.3), implies C0b (R), R g(ξ ) dλNkˆ (ξ ) → R g(ξ ) dμV (ξ ) as that λ = μV , in particular, for g ∈C j μV as n → ∞ (⇒ N → ∞). Since, j→ ∞, which is a contradiction; hence, λN V V from the above calculation, lim sup j→∞ dN and lim inf j→∞ dN exist, it follows ˆ ˆ kj
kj
V exists, and equals EV = inf{IV [μEQ ]; μEQ ∈ M1 (R)} = IV [μV ]; in that lim j→∞ dN ˆ kj
V fact, limN →∞ dN = EV = IV [μV ]. Observe from the calculations above that V EV R ψˆ V (ξ ) dλN (ξ ) > −∞. In order to prove that, indeed, limN →∞ dN = ∞ V EV , it is sufficient to show that if {dN }N =1 converges for some subse V quence, the limit is always equal to EV . So, suppose that E♣ := liml→∞ dN l
V ∞ for some such subsequence {dN } . Using that λN μV as N → ∞, one l l=1 must have that, in particular, λNl μV as l → ∞; but, then, as shown above,
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V V EV lim supl→∞ dN lim infl→∞ dN IV [μV ] = EV ; thus, by the unicity of the l l
V = EV . equilibrium measure, one arrives at E♣ = EV , that is, limN →∞ dN
With the result of Lemma 3.5 at hand, one now establishes, for n ∈ N and k = 1, 2, . . . , K, the regularity of the family of K equilibrium measures M1 (R) μV . R \ {α1 , α2 , . . . , α K } → R satisfy conditions (6)–(8). Then, Lemma 3.6 Let V: for n ∈ N and k = 1, 2, . . . , K, the equilibrium measure M1 (R) μV is absolutely continuous with respect to Lebesgue measure. R \ {α1 , α2 , . . . , α K } → R Sketch of Proof For n ∈ N and k = 1, 2, . . . , K, with V: satisfying conditions (6)–(8), set f : N × {1, 2, . . . , K} × R \ {α1 , α2 , . . . , α K } → R, N
∗ j=1 (x−x j ) lq q=1 (x−α pq )
(n, k, x) → f (n, k; x) := f (x) = s
,
∗ ∗ where N := (n−1)K+k, {x∗j }N j=1 (⊂ R \ {α1 , α2 , . . . , α K }), with xi < x j for i < j, are the generalised weighted Fekete points described in Lemma 3.5, (cf. Subsection 1.2) lq := κnkkq , q = 1, 2, . . . , s −1, ls := κnk , α ps := αk , and the poles are enumerated as per the ordered partition introduced in Subsection 1.2. One shows, for n ∈ N and k = 1, 2, . . . , K, proceeding from the definition of f given above, and using the identity ⎛ ⎞2 ! N N N N
1 1 1 1 1 ⎝ ⎠ − ∗ ∗ 2 = ∗− ∗ ∗ ∗ x−x (x−x ) x−x x−x x −x j j j j k k j=1 j=1 j=1 k =1 k = j
=
N N
1
(x−x∗j )(x−x∗k ) j=1 k =1 k = j
=2
N N
1
, (x−x∗j )(x∗j −x∗k ) j=1 k =1 k = j
that f (x) f N (x) = , f (x) f D (x) where f N (x) := 2
N $ N N
(x−xl∗ )
j=1 l=1 l = j
k =1 k = j
N $ N s
lq 1 ∗ −2 (x−x ) l ∗ ∗ x j −xk x−α pq q=1 j=1
l=1 l = j
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
⎛ +
f D (x) :=
N $
s ⎜ (x−x∗j ) ⎝
j=1
q=1
N $ N
lq (x−α pq )2
(x−x∗k )−
j=1 k =1
N $
⎛ s
+⎝ q=1 s
(x−x∗j )
q=1
j=1
k = j
lq x−α pq
299
⎞2 ⎞ ⎠⎟ ⎠,
lq , x−α pq
whence, using the fact that, for m = 1, 2, . . . , N , N $ N
(x−xl∗ )
N
k =1
j=1 l=1
k = j
l = j
N $ N
s
j=1 l=1
q=1
(x−xl∗ )
l = j
N $ N
(x−x∗k )−
j=1 k =1 k = j
' ' 1 '' ' x∗j −x∗k ' ' ' ' lq '' ' x−α pq ' '
N
l=1
x∗ −x∗k k =1 m
=
N $
N $
s
j=1
q=1
(x∗m −xl∗ )
s
l=1 l =m
x=x∗m
' ' lq '' ' x−α pq ' '
1
,
k =m
l =m
x=x∗m
(x−x∗j )
N $
(x∗m −xl∗ )
=
lq ∗ x −α pq q=1 m
= x=x∗m
N $
,
(x∗m −x∗k ),
k =1 k =m
one arrives at, for n ∈ N and k = 1, 2, . . . , K, N s
lq 1 1 f (x∗m ) = − , ∗ −x∗ ∗ −α 2 f (x∗m ) x x pq m k q=1 m
m = 1, 2, . . . , N
(43)
k =1 k =m
(recall that {x∗j }N j=1 ∩ {α p1 , α p2 , . . . , α ps } = ∅). Recall from the (sketch of the) proof of Lemma 3.5 that, for n ∈ N and k = 1, 2, . . . , K,
V δN =
sup N {xl }l=1 ⊂ R\{α1 ,α2 ,...,α K }
⎛ ⎞ N (N1 −1) N $ j) ⎝ (f(xi , x j))2 e−2(N −1) j=1 V(x ⎠ , i< j
where f(x, y) = f(y, x) = |x− y|
1 n
|x− y| |x−α ps ||y−α ps |
lsn−1 s$ −1 q=1
|x− y| |x−α pq ||y−α pq |
lnq
:
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V one shows from this formula for δN that, for n ∈ N and k = 1, 2, . . . , K,
(x∗m ) + − (N −1)V
N s−1
lq /n N 1 −( N −1) ∗ ∗ ∗ n xm −xk x −α pq q=1 m
k =1 k =m
− (N −1)
(ls −1)/n = 0, x∗m −α ps
m = 1, 2, . . . , N .
(44)
Combining (43) and (44), one shows that, for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎞ s
l /n n ⎝ ( N −1)/n q ∗ ∗ (xm ) + ⎠ f (x∗m ) = 0, − (N −1)V + ∗ f (xm )+2 ∗ −α N x x −α pq ps m q=1 m m = 1, 2, . . . , N , whence, since {x ∈ R; f (x) = 0} = {x∗1 , x∗2 , . . . , x∗N }, via an ODE argument à la Polya (see Chapter VI of [8]), one arrives at, for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎞ s
lq /n n ⎝ (N −1)/n ⎠ (x)+ + f (x)+2 −(N −1)V f (x) = Q(x) f (x), N x−α x−α ps pq q=1 (45) with Q(x∗m ) = 0, m = 1, 2, . . . , N , to be determined below. Writing, for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎞ s
l /n f (x) N −1)/n n ⎝ ( q (x)+ ⎠ f (x) , Q(x) = +2 −(N −1)V + f (x) N x−α pq x−α ps f (x) q=1 where, for x ∈ / {x∗1 , x∗2 , . . . , x∗N }, N
lq f (x) 1 = − , f (x) x−x∗j q=1 x−α pq j=1 s
⎛ ⎞2 N s N s
lq lq 1 f (x) ⎝ 1 ⎠ − = − + , ∗ ∗ 2 f (x) x−x j q=1 x−α pq (x−x j ) q=1 (x−α pq )2 j=1 j=1 one arrives at, for n ∈ N and k = 1, 2, . . . , K, after a straightforward calculation, N
Q(x) = 2
+2
s−1
lq (N −1) ∗ N (x −α pq )(x−α pq ) j j=1 q=1 N s
lq (N −1) (ls −1) + ∗ 2 N (x −α )(x−α ) (x−α ps ps pq ) j q=1 j=1
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⎛ ⎞2 s s (N −2) ⎝ lq ⎠ (N −1) 1 lq + −2 N x−α pq N x−α ps q=1 x−α pq q=1 +2
s N (x∗ )
V (x)− V (N −1) lq n j nV (x) − 2 (N −1) . ∗ N x−α pq N x−x j q=1 j=1
(46)
Since, from the (sketch of the) proof of Lemma 3.5, for n ∈ N and k = 1, 2, . . . , K, {x∗j }N j=1 ⊂ R \ {α1 , α2 , . . . , α K }, choose 2(N1 +1) real points N1 +1 enumerated so that −∞ < B0 < A1 < · · · < B j−1 < (N1 ∈ Z+ 0 ) {B j−1 , A j } j=1
1 +1 [B j−1 , A j] ⊇ A j < · · · < B N1 < A N1 +1 < +∞, with R \ {α1 , α2 , . . . , α K } ⊃ ∪ Nj=1 ∗ ∗ ∗ {x1 , x2 , . . . , xN } and [B j−1 , A j] ∩ {α1 , α2 , . . . , α K } = ∅, j∈ {1, 2, . . . , N1 +1}. 1 +1 [B j−1 , A j], which is the disjoint union of N1 +1 compact real intervals, (∪ Nj=1 is the ‘pre-confinement for the generalised weighted Fekete points Ndomain’ 1 +1 {x∗j }N .) Let Z (x) := (z− B j−1 )(z− A j ). A calculation shows that Z (x) j=1 j=1 0 can be presented as Z (x) = p =2(N1 +1) c2(N1 +1)− p x p , where ! N +1 ! N$ 1 +1 1 $
1 1 (−1)q cq = (Bi−1 )ki (Am )lm , k l i m m=1 i=1
ki ,lm =0,1 i,m∈{1,2,...,N1 +1} N1 +1 N1 +1 i=1 ki + m=1 lm =q
q = 0, 1, . . . , 2(N1 +1),
with c0 = 1 and c2(N1 +1) =
N1 +1
Br−1 Ar . For n ∈ N and k = 1, 2, . . . , K, one (x∗ ) (x)−V V j proceeds to manipulate the term 2 Nn (N −1) N , which appears j=1 x−x∗ r=1
j
in the expression for Q(x) (cf. (46)): N N (x∗ )
V (x)− V n (N −1) n j = 2 2 (N −1) N x−x∗j N Z (x) j=1 j=1
(x∗ )Z (x) (x)Z (x)− V V j
!
x−x∗j
! N (x)Z (x)− V (x∗ )Z(x∗ ) n (N −1) V j j =2 N Z (x) j=1 x−x∗j N
n (N −1) − 2 N Z (x) j=1
(x∗ )(Z (x)− Z (x∗ )) V j j x−x∗j
! ;
recalling the expression above for Z (x), and using the identity an − b n = (a−b )(an−1 +an−2 b +· · ·+ab n−2 +b n−1 ), one shows that ⎛ ⎞ p −1 3
Z (x)− Z (x∗j ) = c2(N1 +1)− p ⎝ x p −r−1 (x∗j )r ⎠ x−x∗j r=0 p =2(N1 +1)
+ c2(N1 +1)−2 (x+x∗j )+c2(N1 +1)−1 ;
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thus, via (44),
2
(x)Z (x)− V (x∗ )Z (x∗ ) ! V j j
N V N (x)− V (x∗ )
n n (N −1) j =2 (N −1) ∗ N x−x j N Z (x) j=1
x−x∗j
j=1
⎛ −2
N N s −1 lq /n 1 n 1 ⎜ ⎜N −(N −1) ⎜ ∗ ∗ ⎝ n x j −xk N Z (x) x∗j −α pq k =1
j=1
q=1
k = j
(ls −1)/n − (N −1) ∗ x j −α ps
!
N n (N −1) =2 N Z (x)
Z (x)− Z (x∗j ) x−x∗j
(x∗ )Z (x∗ ) ! (x)Z (x)− V V j j x−x∗j
j=1
N N 1 2
− ∗ −x∗ Z (x) x k j=1 k =1 j
Z (x)− Z (x∗j )
Z (x)− Z (x∗j )
N s−1 lq 2 (N −1) Z (x) N x∗j −α pq N 2 (N −1) (ls −1) Z (x) N x∗j −α ps
Z (x)− Z (x∗j )
N n (N −1) N Z (x)
(x)Z (x)− V (x∗ )Z (x∗ ) ! V j j x−x∗j
j=1
⎛ N N 1 2
⎝ − Z (x) x∗j −x∗k j=1 k =1 k = j
⎛ ×⎝
−1 p
!
x−x∗j
j=1
=2
3
p =2(N1 +1)
c2(N1 +1)− p
⎞ x
p −r−1
(x∗j )r ⎠ + c2(N1 +1)−2 (x+x∗j )
r=0
⎞ + c2(N1 +1)−1⎠ +
N s−1
lq 2 (N −1) Z (x) N x∗j −α pq j=1 q=1
⎛ ×⎝
3
p =2(N1 +1)
!
x−x∗j
j=1 q=1
+
!
x−x∗j
k = j
+
!
⎛ c2(N1 +1)− p ⎝
−1 p
r=0
⎞ x p −r−1 (x∗j )r ⎠
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal... ⎞ 2 (N −1) + c2(N1 +1)−2 (x+x∗j )+c2(N1 +1)−1 ⎠ + Z (x) N ⎛ N
(ls −1) ⎝ × x∗j −αps j=1
⎛ ×⎝
−1 p
3
p =2(N1 +1)
c2(N1 +1)− p
⎞ x p −r−1 (x∗j )r ⎠ + c2(N1 +1)−2 (x+x∗j )
r=0
⎞ + c2(N1 +1)−1⎠ .
For n ∈ N and k = 1, 2, . . . , K, let ⎛ ⎛ ⎞ p −1 N N 3
1 ⎝ A := c2(N1 +1)− p ⎝ x p −r−1 (x∗j )r ⎠ ∗ ∗ x −x j k r=0 j=1 p =2(N +1) k =1
1
k = j
⎞ + c2(N1 +1)−2 (x+x∗j )+c2(N1 +1)−1⎠ . For n ∈ N and k = 1, 2, . . . , K, one manipulates the expression for A thus: ⎛ ⎛ N N 3
1 ⎝ A= c2(N1 +1)− p ⎝x p −1 +x p −2 x∗j ∗ ∗ x j −xk p =2(N +1) j=1 k =1
1
k = j
⎞
+
p −1
⎞
x p −r−1 (x∗j )r ⎠ +c2(N1 +1)−2 (x+x∗j ) + c2(N1 +1)−1 ⎠
r=2
⎛ =
3
c2(N1
+1)− p
p =2(N1 +1)
N N N N p −1
x p −2 x∗j ⎜ x ⎝ ∗ ∗ + x j −xk j=1 x∗j −x∗k j=1
k =1 k = j
k =1 k = j
⎞ p −1 N N N N
x p −r−1 (x∗j )r ⎟
x+x∗j + + c ⎠ 2(N1 +1)−2 x∗j −x∗k x∗j −x∗k r=2 j=1 j=1 k =1
k =1
k = j
+ c2(N1 +1)−1
k = j
N N
1
x∗ −x∗k j=1 k =1 j k = j
;
303
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K. T.-R. McLaughlin et al.
recall from the (sketch of the) proof of Lemma 3.5 that, for n ∈ N and k = 1, 2, . . . , K, given an arbitrary function h, say,
h(x∗j ) :=
j
h(x∗k ) :=
N −1
N
j=1
k = j+1
N −1
j
N
h(x∗j ) = (N −1)h(x∗1 )+
N −1
m=2
h(x∗k ) = (N −1)h(x∗N )+
N −1
j=1 k = j+1
h(x∗j )+
j
h(x∗m )(N −m),
h(x∗m )(m−1),
m=2 N
h(x∗k ) = (N −1)
j
h(x∗m ),
m=1
whence one derives, via the identity an −b n = (a−b )(an−1 +an−2 b +· · ·+ab n−2 + b n−1 ), the following relations:
N N
j=1 k =1 k = j
1 = 0, x∗j −x∗k
N N
x+x∗j
j=1 k =1 k = j
x∗j −x∗k
N N
(x∗j )r
j=1
k =1 k = j
x∗j −x∗k
=
N N
x∗j
j=1 k =1
x∗j −x∗k
k = j
=
N −1
j=1
=
=
N −1
N (N −1) , 2 N r−1
k = j+1
(x∗j )r−m−1 (x∗k )m
m=0
(x∗k )r−1
j=1 k = j+1 r−1
k = j+1
1=
j=1
N −1 N
(x∗j )r−1+
N −1 N
= (N −1)
N
j=1 k = j+1
x∗j −x∗k
k = j+1
N −1 N
j=1
N −1
N
(x∗j )r −(x∗k )r
j=1 k = j+1
+
=
(x∗j )r−m (x∗k )m−1
m=2
N N −1 N r−1
(x∗j )r−1 + (x∗j )r−m (x∗k )m−1 ,
j=1
j=1 k = j+1 m=2
r 2;
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
305
hence, via the above relations, one shows that, for n ∈ N and k = 1, 2, . . . , K,
A =
⎛
3
p =2(N1 +1)
+
N −1
⎛ p −1 N
N ( N −1) p −2 ⎝ ⎝ x c2(N1 +1)− p + (N −1) (x∗j )r−1 2 r=2 j=1
N r−1
⎞
⎞
(x∗j )r−m (x∗k )m−1 ⎠x p −r−1 ⎠ +
j=1 k = j+1 m=2
N (N −1) c2(N1 +1)−2 , 2
whence, for n ∈ N and k = 1, 2, . . . , K, one arrives at 2
(x∗ )Z (x∗ ) ! (x)Z (x)− V V j j
N V N (x∗ ) (x)− V
n n (N −1) j (N −1) =2 ∗ N x−x j N Z (x) j=1
x−x∗j
j=1
⎧ 2 ⎨ − Z (x) ⎩
3
p =2(N1 +1)
+
−1 p
⎛ ⎝(N −1)
r=2
·
r−1
m=2
N
⎛ N (N −1) p −2 x c2(N1 +1)− p ⎝ 2
(x∗j )r−1 +
N −1
j=1
N
j=1 k = j+1
⎞
⎞
N (N −1) (x∗j )r−m (x∗k )m−1 ⎠ x p −r−1 ⎠ + 2
⎫ ⎧ ⎬ 2 (N −1)⎨ × c2(N1 +1)−2 + ⎭ Z (x) N ⎩
3
c2(N1 +1)− p
p =2(N1 +1)
⎛
⎞
N s −1 lq (x+x∗ )
j
N s −1
−1 p N s −1 lq (x∗ )r
j ×⎝ x p −r−1 ⎠ + c2(N1 +1)−2 x∗j −α pq r=0 j=1 q=1
×
j=1 q=1
x∗j −α pq
+c2(N1 +1)−1
⎧ 2(ls −1) (N −1) ⎨ + ⎩ Z (x) N ⎛
−1 p N
×⎝
r=0 j=1
(x∗j )r
x∗j −α ps
j=1 q=1 3
p =2(N1 +1)
⎫ lq ⎬ x∗j −α pq ⎭
c2(N1 +1)− p
⎞ x p −r−1⎠ + c2(N1 +1)−2
⎫ N N ⎬ x+x∗j
1 × +c2(N1 +1)−1 . x∗j −α ps x∗j −α ps ⎭ j=1 j=1
(47)
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K. T.-R. McLaughlin et al.
Substituting (47) into (46), one arrives at, for n ∈ N and k = 1, 2, . . . , K: ⎧⎛ ⎞⎛ 0 N s−1 ⎨
lq 1 (N −1) ⎝ c2(N1 +1)− p x p ⎠⎝2 Q(x) = ∗ Z (x) ⎩ p =2(N +1) N (x −α pq )(x−α pq ) j j=1 q=1 1
+ 2
N s
lq (N −1) (ls −1) (N −2) + + ∗ 2 N (x j −α ps )(x−α ps ) q=1 (x−α pq ) N j=1
⎛ s
×⎝ q=1
×
s
q=1
⎛ + 2⎝
⎞2 s
lq lq (N −1) ⎠ − 2 (N −1) 1 +2 nV (x) x−α pq N x−α ps q=1 x−α pq N
⎞ ! N (x∗ )Z (x∗ ) (x)Z (x)− V
V lq j j ⎠ − 2n (N −1) x−α pq N x−x∗j j=1 ⎛
⎛ p −1 N
N ( N −1) ⎝(N −1) (x∗j )r−1 x p −2 + c2(N1 +1)− p ⎝ 2 r=2 j=1
3
p =2(N1 +1)
+
N −1 N
r−1
⎞
⎞
(x∗j )r−m (x∗k )m−1⎠x p −r−1⎠ +
j=1 k = j+1 m=2
⎛ (N −1) ⎝ −2 N
+ c2(N1 +1)−2
⎛
3
c2(N1 +1)− p ⎝
p =2(N1 +1)
N (N −1) c2(N1 +1)−2 ⎠ 2
x∗j −α pq
⎞
p −1 N s−1
lq (x∗j )r r=0 j=1 q=1
N s−1
lq (x+x∗j )
j=1 q=1
⎞
+c2(N1 +1)−1
x∗j −α pq
N s−1
j=1 q=1
x p −r−1 ⎠
⎞ lq ⎠ x∗j −α pq
⎛ ⎛ ⎞ p −1 N 3
(x∗j )r (N −1) ⎝ − 2(ls −1) c2(N1 +1)− p ⎝ x p −r−1 ⎠ ∗ N x −α p s j r=0 j=1 p =2(N +1) 1
+ c2(N1 +1)−2
N
x + x∗j
j=1
x∗j −α ps
+c2(N1 +1)−1
N
j=1
⎞⎫ ⎬ 1 ⎠ . ∗ x j − α ps ⎭
(48)
For n ∈ N and k = 1, 2, . . . , K, recall the second-order, non-constant coefficient ODE for f (x) given in (45), with Q(x) given in (48). As per the proof of Lemma 2.15 in [62], set, for n ∈ N and k = 1, 2, . . . , K, 1 x ˆ f (x) = F(x) exp − P (ξ ) dξ , (49) 2
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
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where ⎛ ⎞ s
l /n n ( N −1) 1 q (x)+ ⎝−(N −1)V ⎠. Pˆ (x) := 2 + N x−α n x−α p p q s q=1 Integrating, and substituting (49) into (45), one arrives at, for n ∈ N and k = 1, 2, . . . , K, the following second-order, non-constant coefficient ODE for F(x):
1 ˆ 1 ˆ 2 F (x) = Q(x)+ P (x)+ (P (x)) F(x), 2 4
(50)
where, after simplification, 1 (N −1) 1 (N −1) 1 1 Q(x) + Pˆ (x)+ (Pˆ (x))2 = − nV (x)− 2 4 N N N (x−α ps )2 lq (N −1) + + 2 N (x−α pq ) q=1 s
(x) + 2nV
N −1 N
2 s q=1
N −1 N
2
(x))2 (nV
lq N −1 2 −2 x−α pq N
s 1 lq N −1 2 nV (x) × −2 x−α ps q=1 x−α pq N x−α ps +
×
N −1 N
N s−1
j=1 q=1
×
N
j=1
×
2
⎛ ⎝
s
q=1
⎞2 lq ⎠ + 2 (N −1) x−α pq N
lq (N −1) +2 (x∗j −α pq )(x−α pq ) N
(ls −1) 1 (N −1) + −2n ∗ (x j −α ps )(x−α ps ) Z (x) N
N
(x)Z (x)− V (x∗ )Z (x∗ ) V j j
j=1
x−x∗j
⎛
!
⎛ N (N −1) p −2 + 2⎝ c2(N1 +1)− p ⎝ x 2 p =2(N +1) 3
1
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+
p −1
⎛
N N −1 N r−1
⎝(N −1) (x∗j )r−1 + (x∗j )r−m
r=2
j=1 k = j+1 m=2
j=1
⎞
⎞
⎞ N ( N −1) × (x∗k )m−1⎠ x p −r−1 ⎠ + c2(N1 +1)−2 ⎠ 2
⎛ 3 (N −1) ⎝ c2(N1 +1)− p −2 N p =2(N +1) 1
⎛ ×⎝
r=0 j=1 q=1
×
x∗j −α pq
N s−1
lq (x+x∗j )
j=1 q=1
x∗j −α pq
− 2(ls −1)
×⎝
×
3
j=1
x∗j −α ps
x∗j −α ps
N s−1
⎞
lq ⎠ ∗ x − α pq q=1 j
c2(N1 +1)− p
p =2(N1 +1)
⎞
p −1 N
(x∗j )r
N
x+x∗j
j=1
(N −1) ⎝ N
r=0 j=1
x p −r−1 ⎠ + c2(N1 +1)−2
+c2(N1 +1)−1 ⎛
⎛
⎞
p −1 N s−1
lq (x∗j )r
x p −r−1 ⎠ + c2(N1 +1)−2
+ c2(N1 +1)−1
N
j=1
⎞⎫ ⎬ 1 ⎠ . (51) x∗j −α ps ⎭
1 +1 Since, for n ∈ N and k = 1, 2, . . . , K, {x∗1 , x∗2 , . . . , x∗N } ⊆ ∪ Nj=1 [B j−1 , A j] (⊂ ∗ ∗ R \ {α1 , α2 , . . . , α K }) and xi < x j for i < j, with i, j∈ {1, 2, . . . , N }, it follows from the (sketch of the) proof of Lemma 3.5 that, for any two consecutive generalised weighted Fekete points x∗j and x∗j+1 , there are, for r ∈ {1, 2, . . . , N1 +1}, the following cases to consider: (1) Br−1 x∗j < x∗j+1 1 (Br−1 + Ar ); (2) 12 (Br−1 + Ar ) x∗j < x∗j+1 Ar ; and (3) Br−1 x∗j < 12 (Br−1 + 2 Ar ) < x∗j+1 Ar . This is the (up to a linear scaling) ‘one-interval-case’ result given in Lemma 2.15 of [62], wherein a lower bound for the distance between two consecutive Fekete points is estimated; in order to use the result of Lemma 2.15 of [62], one needs to: (i) map the compact real intervals [Br−1 , Ar ], r = 1, 2, . . . , N1 +1, onto the compact real interval [−1, 1]; and (ii) for n ∈ N and k = 1, 2, . . . , K, estimate an upper bound for Q(x)+ 12 Pˆ (x)+ 14 (Pˆ (x))2 , which is given by (51). For the former problem (i), one makes the linear change of variables λr : C → C, x → λr (x) := (2x−(Ar + Br−1 ))/(Ar − Br−1 ), r = 1, 2, . . . , N1 + 1, which maps, one-to-one, the compact real intervals [Br−1 , Ar ] onto the compact real interval [−1, 1]; and, for the latter problem (ii), noting that,
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
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for n ∈ N and k = 1, 2, . . . , K, inf{|α pq −x∗j |; q = 1, 2, . . . , s, j= 1, 2, . . . , N } > 0 and inf{|α pq −x|; q = 1, 2, . . . , s, x ∈ [Br−1 , Ar ], r = 1, 2, . . . , N1 +1} > 0, writ 1 +1 ing Z (x) = (x− Br−1 )(x− Ar ) Nj=1 (x− B j−1 )(x− A j), r ∈ {1, 2, . . . , N1 +1}, and j =r
R \ {α1 , α2 , . . . , α K } → R, one shows from (51) using the real analyticity of V: that, for n ∈ N and k = 1, 2, . . . , K, ' ' 2 ' ' ' Q(x)+ 1 Pˆ (x)+ 1 (Pˆ (x))2 ' Cr (n, k)N , r ∈ {1, 2, . . . , N1 +1}, ' ' 2 4 (x− Br−1 )(Ar −x) where Cr (n, k) > 0. One now uses this inequality in conjunction with the result of Lemma 2.15 of [62] to show that, for n ∈ N and k = 1, 2, . . . , K, two consecutive generalised weighted Fekete points x∗j and x∗j+1 satisfy the following ‘nearest-neighbour-distance’ inequality: x∗j+1 − x∗j
)( ).1/2 min{(3Cr (n, k))−1/2 , 1/4} -( Ar −x∗j Ar −x∗j+1 , ((n−1)K+k)
r ∈ {1, 2, . . . , N1 +1},
j = 1, 2, . . . , N −1
(52)
(note that ((Ar −x∗j )(Ar −x∗j+1 ))1/2 > 0)10 . One now uses the estimate (52) and the fact (cf. Lemma 3.5) that the associated normalised counting measure converges (as n → ∞) weakly (in the weak-∗ topology of measures) to the associated equilibrium measure μV in order to proceed, mutatis mutandis, as in the proof of Lemma 2.26 of [62] to conclude that, for n ∈ N and k = 1, 2, . . . , K, the family of equilibrium measures μV is absolutely continuous with respect to Lebesgue measure. R \ {α1 , α2 , . . . , α K } → R satisfy condiLemma 3.7 Let the external field V: is regular. For n ∈ N and k = tions (6)–(8). Suppose, furthermore, that V 1, 2, . . . , K, set, for the associated equilibrium measure μV ∈ M1 (R), J := supp(μV ), with J a proper compact subset of R \ {α1 , α2 , . . . , α K }. Then, + for n ∈ N and k = 1, 2, . . . , K : (1) J = ∪ N+1 j=1 [b j−1 , a j ], with N ∈ Z0 and finite, [b j−1 , a j] ∩ {α1 , α2 , . . . , α K } = ∅, j= 1, 2, . . . , N+1, [bi−1 , ai ] ∩ [b j−1 , a j] = ∅, i = j∈ {1, 2, . . . , N+1}, and −∞ < b0 < a1 < b1 < a2 < · · · < b N < a N+1 < +∞, and {b j−1 , a j} N+1 j=1 satisfy the locally solvable system of 2(N+1) real moment equations ⎞ ⎞ ⎛ ⎛ s−1
(ξ ) κnkkq 2 ( V ξj κ −1) nk ⎠+ ⎠ dξ = 0, ⎝ ⎝ + 1/2 iπ n(ξ −αk ) q=1 n(ξ −α pq ) iπ J (R(ξ ))+ j= 0, 1, . . . , N,
for n ∈ N and k = 1, 2, . . . , K, two consecutive generalised weighted Fekete points x∗j and x∗j+1 lie, respectively, in the disjoint compact real intervals [Br1 −1 , Ar1 ] and [Br2 −1 , Ar2 ], r1 0. 10 If,
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⎛
ξ N+1 J
1/2
(R(ξ ))+
⎛ ⎞ ⎞ s−1
κ 2 ( κ −1) V (ξ ) nkkq ⎝ ⎝ nk ⎠+ ⎠ dξ + iπ n(ξ −αk ) q=1 n(ξ −α pq ) iπ
(n−1)K+k = −2 , n ⎛ ⎛ ⎛ ⎞ bj s−1
κ 2 ( κ −1) nkkq ⎝(R(ς))1/2 (R(ξ ))−1/2 ⎝ ⎝ nk ⎠ + + iπ n(ξ −αk ) q=1 n(ξ −α pq ) aj J ⎞ ⎞ ' ' ' ' s−1
(ξ ) κnkkq ' b j −α pq ' ' b j −αk ' dξ κ −1 V nk ' ' ' ' ⎠ ⎠ dς = 2 ln ' ln ' + +2 ' iπ ξ −ς n a j −αk ' n a −α j p q q=1 j)− V(a j)), j= 1, 2, . . . , N, +(V(b where
⎛ (R(z))1/2 := ⎝
N+1 $
⎞1/2 (z−b j−1 )(z−a j)⎠
,
j=1 1/2
with (R(z))± := limε↓0 (R(z±iε))1/2 , and the branch of the square root is chosen so that z−(N+1) (R(z))1/2 ∼C± z→∞ ±1; and (2) the associated equilibrium measure, which is absolutely continuous with respect to Lebesgue measure, is given by dμV (x) := ψV (x) dx = where 1 hV (z) = 2
(n−1)K+k n
1 1/2 (R(x))+ hV (x)1 J (x) dx, 2π i ⎛
⎛ ⎞ s−1
κnkkq 2i κ −1) ( nk ⎝ ⎝ ⎠ + π n(ξ −αk ) q=1 n(ξ −α pq ) CV
−1 *
⎞ (ξ ) (R(ξ ))−1/2 iV ⎠ + dξ π ξ −z (real analytic for z ∈ R \ {α1 , α2 , . . . , α K }), with CV (⊂ C \ {α1 , α2 , . . . , α K }) the simple boundary of any open (s +1)-connected punctured region of the type s CV = ∂ U R ∪ (∪q=1 ∂ Uδq (α pq )), where U R := {z ∈ C; |z| < R}, with ∂ U R = {z ∈ C; |z| = R} oriented clockwise, and, for q = 1, 2, . . . , s, U δq (α pq ) := {z ∈ C; |z− α pq | < δq }, with ∂ Uδq (α pq ) = {z ∈ C; |z−α pq | = δq } oriented counter-clockwise, and where the numbers 0 < δq < R < +∞, q = 1, 2, . . . , s, are chosen so that, for (that is, C \ {α1 , α2 , . . . , α K }), (any) non-real z in the domain of analyticity of V • •
∂ U δi (α pi ) ∩ ∂ Uδ j (α p j ) = ∅, i = j∈ {1, 2, . . . , s}, ∂ U δ j (α p j ) ∩ ∂ U R = ∅, j∈ {1, 2, . . . , s},
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
• • •
311
U ( δ j (α p j ) ∪ ∂ Uδ j (α p j )) ∩ J = ∅, j∈ {1, 2, . . . , s}, ∂ U R ∩ J =-∅, . s ( U int(CV ) := ext(∪q=1 δq (α pq ) ∪ ∂ Uδq (α pq ))) ∩ U R ⊃ {z} ∪ J,
1 J (x) is the characteristic function of the set J, and ψV (x) 0 (resp., ψV (x) > 0) for x ∈ J (resp., x ∈ int(J)). Proof One begins by showing that, for n ∈ N and k = 1, 2, . . . , K, the support of each member of the family of equilibrium measures, that is, supp(μV ) =: J, consists of the union of a finite number of disjoint and bounded (real) intervals. Recalling from Lemma 3.1 that, for n ∈ N and k = 1, 2, . . . , K, J ⊂ s R \ ({|x| T M } ∪ (∪q=1 O 1 (α pq ))), for some T M (= T M (n, k)) > 1, and that TM R \ {α1 , α2 , . . . , α K } → R is real analytic, thus real analytic on J, and with V: an analytic extension to, say, the following open neighbourhood of J, U := {z ∈ C; infq∈J |z−q|
+
s−1
q=1
⎞
2 ⎠ ψV (z) + 4i (n−1)K+k ψV (z)(H ψV )(z), (53) n(z−α pq ) n κnkkq
f (ξ ) dξ denotes the where H : LM2 2 (C) (∗∗) → LM2 2 (C) (∗∗), f → (H f )(z) := z−ξ π R Hilbert transform, with the principle value integral. For n ∈ N and k = 1, 2, . . . , K, set H : N × {1, 2, . . . , K} × C \ (J ∪ {α p1 , . . . , α ps−1 , αk }) → C, S(ξ ) dξ (n, k, z) → H(n, k; z) := H(z) = (F(z))2 − , J ξ −z 2π i
(54)
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where, from the proof of Lemma 3.4, F : N × {1, 2, . . . , K} × C \ (J ∪ {α p1 , . . . , α ps−1 , αk }) → C,
⎛ s−1 1 ⎝(κnk −1) κnkkq (n, k, z) → F(n, k; z) := F(z) = − + iπ n(z−αk ) q=1 n(z−α pq ) +
(n−1)K+k n
J
⎞ dμV (ξ ) ⎠ , ξ −z
(55)
dμV (ξ ) with J ξ −z the Stieltjes transform of the equilibrium measure. Via the distri−1 butional identities (x−(x0 ±i0)) = (x−x0 )−1 ±iπ δ(x−x0 ), where δ(·) is the λ2 Dirac delta function, and λ1 f (ξ )δ(ξ −λ)dξ = f (λ) if λ ∈ (λ1 , λ2 ) or 0 if λ ∈ R \ (λ1 , λ2 ), it follows that ⎧ ⎪ ⎨(F± (z))2 − S(ξ ) dξ ∓ 12 S(z), z ∈ J, J ξ − z 2π i H± (z) = S(ξ ) dξ ⎪ 2 ⎩(F(z)) − J z∈ / J, ξ − z 2π i , where !± (z) := limε↓0 !(z±i0), ! ∈ {H, F}. Using the representation dμV (x) = ψV (x) dx, x ∈ J, recall the definition of g given in Lemma 3.4: g : N × {1, 2, . . . , K} × C \ −∞, max max {α pq }, max{J} → C, q=1,2,...,s
⎛
ln⎝(z−ξ )
(n, k, z) → g(n, k; z) := g(z) =
1 n
J
×
s −1 $
q=1
(z−ξ ) (z−α pq )(ξ −α pq )
κnknkq
κnkn −1 (z−ξ ) (z−αk )(ξ −αk )
⎞ ⎠ ψV (ξ ) dξ ;
taking note of the above distributional identities and the fact that (ξ )dξ = 1, one shows that (assuming differentiation commutes with J ψV taking boundary values) (g± (z)) =
⎧ . . s −1 κ ⎪ ψV (ξ ) nkkq (κnk −1) ⎪ ⎪ − (n−1)K+k dξ ∓iπ (n−1)K+k ψV (z), ⎨− n(z−αk ) − n(z−α pq ) n ξ −z n J q=1 . −1 κ ⎪ (κnk −1) s ψV (ξ ) nkkq (n−1)K+k ⎪ ⎪ − − − dξ, ⎩ n(z−αk ) n(z−α pq ) n ξ −z q=1
J
z ∈ J, z∈ / J,
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
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where (g± (z)) := limε↓0 (g) (z±iε), whence one concludes that (g+ +g− ) (z) = −
s −1 κnk
ψV (n−1)K+k 2(κnk −1) kq (ξ ) −2 −2 dξ J ξ −z n(z−αk ) n(z−α pq ) n q=1
=−
s −1 κnk
2(κnk −1) (n−1)K+k kq −2 + 2π (H ψ V )(z), n(z−αk ) n(z−α pq ) n
z ∈ J,
q=1
(g+ −g− ) (z) =
. + k ψ (z), −2πi (n − 1)K V n 0,
z ∈ J, z∈ / J.
Demanding that, in a piecewise-continuous sense (see Lemma 3.8 below), (z) = 0, z ∈ J, one shows from the above formula that, for z ∈ J, (g+ +g− ) (z)− V ⎛
⎞ ⎞ ⎛ s −1 s −1 κnk κnk
( ( κ −1) κ −1) k k q q nk nk ⎝g (z)+ ⎠ + ⎝g (z)+ ⎠ + + n(z−αk ) n(z−α pq ) n(z−αk ) n(z−α pq ) q=1
= 2π
q=1
+
−
s −1 κnk
(n−1)K+k 2(κnk −1) kq (z) ⇒ (H ψ V +2 +V )(z) = n n(z−αk ) n(z−α pq ) q=1
⎛ ⎞ s −1 κ
1 (n−1)K+k −1⎝2(κnk −1) nk kq ⎠ (H ψ V +2 + V (z) , )(z) = 2π n n(z−αk ) n(z−α pq )
z ∈ J.
(56)
q=1
From (54) and the above distributional identities, one shows that F± (z)=
⎧ ⎪ ⎪ ⎨ − iπ1
(κ nk −1) n(z−αk ) +
⎪ ⎪ ⎩ − iπ1
(κ nk −1) n(z−αk ) +
s −1
q=1 s −1 q=1
κ nk kq
n(z−α pq κ nk kq
) −π
n(z−α pq ) +
-
-
(n−1)K+k n
(n−1)K+k n
.
.
! (H ψ V )(z) ∓ !
ψV (ξ ) J ξ −z
-
(n−1)K+k n
.
dξ ,
ψV (z), z ∈ J, z∈ / J;
(57) / thus, for z ∈ R \ (J ∪ {α p1 , . . . , α ps−1 , αk }), F+ (z) = F− (z) = F(z). Hence, for z ∈ J ∪ {α p1 , . . . , α ps−1 , αk }, one shows that, upon recalling (53), H+ (z) = H− (z). For z ∈ J, one notes that H+ (z)− H− (z) = (F+ (z))2 −(F− (z))2 − S(z),
and
⎛ ⎞2 s−1
κ κ −1) 1 ( nk k nk q ⎠ (F± (z))2 = − 2 ⎝ + π n(z−αk ) q=1 n(z−α pq ) +
2 π
⎛
s−1
κnkkq
⎞
(n−1)K+k ⎝ (κnk −1) ⎠ (H ψV )(z) + n n(z−αk ) q=1 n(z−α pq )
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2i ∓ π
⎛ ⎞ s−1 (n−1)K+k ⎝ (κnk −1) κnkkq ⎠ + ψV (z) n n(z−αk ) q=1 n(z−α pq )
(n−1)K+k ± 2i n −
(n−1)K+k n
2 (H ψV )(z)ψV (z)
2
((H ψV )(z))2 +
whence, for z ∈ J, (F+ (z))2 −(F− (z))2 = −
4i π
(n−1)K+k n
⎛
s−1
2 (ψV (z))2 , ⎞
κnkkq
(n−1)K+k ⎝ (κnk −1) ⎠ ψV (z) + n n(z−αk ) q=1 n(z−α pq )
(n−1)K+k + 4i n
2 (H ψV )(z)ψV (z),
which implies that H+ (z)− H− (z) = 0; thus, for z ∈ J, H+ (z) = H− (z). The above argument shows, therefore, that H(z) is analytic across R \ {α p1 , . . . , α ps−1 , αk }; in fact, H(z) is entire (resp., meromorphic) for z ∈ C \ {α p1 , . . . , α ps−1 , αk } (resp., z ∈ C). Recalling that, for n ∈ N and k = 1, 2, . . . , K, μV ∈ M1 (R), in particular, J (ξ −αk )−m dμV (ξ ) = J (ξ −αk )−m ψV (ξ ) dξ < ∞, m ∈ N, for ξ ∈ J and z ∈ / J such that |(z−αk )/(ξ −αk )| % 1 (e.g., 0 < |z−αk | % min{minq=1,2,...,s−1 {|αk − j 1 k) = − lj=0 (ξ(z−α + α pq |}, infξ ∈J {||ξ |−αk |}}), via the expansion (z−αk )−(ξ −αk ) −αk ) j+1 (z−αk )l+1 , (ξ −αk )l+1 (z−ξ )
l ∈ Z+ 0,
1 1 1 κnk −1 2 2 κnk −1 (F(z)) = − + z→αk (z−αk )2 π n (z−αk ) π 2 n ⎛ ⎞ s−1
κnkkq (n−1)K+k × ⎝ − (ξ −αk )−1 ψV (ξ ) dξ ⎠ n(α −α ) n p k J q q=1 2
+ O (1), whence, recalling the definition of H(z) (cf. (54)), in particular, for ξ ∈ J and z∈ / J such that |(z−αk )/(ξ −αk )| % 1 (e.g., 0 < |z−αk | % min{minq=1,2,...,s−1 {|αk − j 1 k) α pq |}, infξ ∈J {||ξ |−αk |}}), via the expansion (z−αk )−(ξ = − lj=0 (ξ(z−α + −αk ) −αk ) j+1 (z−αk )l+1 , (ξ −αk )l+1 (z−ξ )
l ∈ Z+ 0,
J
it follows that 1 H(z) = − z→αk (z−αk )2
1 π
S(ξ ) dξ = O (1), ξ −z 2π i z→αk
κnk −1 n
2
1 + (z−αk )
2 π2
κnk −1 n
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
⎛ × ⎝
s−1
q=1
κnkkq
(n−1)K+k − n(α pq −αk ) n
315
J
⎞
(ξ −αk )−1 ψV (ξ ) dξ ⎠
+ O (1), which shows that H(z) has a pole of order 2 at z = αk , with ⎛ s−1 (n−1)K+k 2 κnk −1 ⎝ κnkkq − Res(H(z); αk ) = 2 π n n(α pq −αk ) n q=1 × J
⎞ ψV (ξ ) ⎠ dξ , ξ −αk
k = 1, 2, . . . , K.
One from the analysis above that, for n ∈ N and k = 1, 2, . . . , K, s−1learns, then, 2 2 q=1 (z−α pq ) (z−αk ) H(z) is entire; look, in particular, at the behaviour of s−1 2 2 q=1 (z−α pq ) (z−αk ) H(z) as z → ∞. Towards this end, though, some notation is requisite (cf. Subsection 1.2). Proceeding as per the detailed discussion of Subsection 1.2, write, for n ∈ N and k = 1, 2, . . . , K, the ordered disjoint partition for the repeated real pole sequence (with the convention ∪0m=1 {∗∗, ∗ , . . . , ∗ } := ∅): 1
n−1
n
K
K
k
{α1 , α2 , . . . , α K } ∪ · · · ∪ {α1 , α2 , . . . , α K } ∪ {α1 , α2 , . . . , αk }
:=
s −1
q=1
:=
s −1
q=1
{αi(q)kq , αi(q)kq , . . . , αi(q)kq } ∪ {αi(s)ks , αi(s)ks , . . . , αi(s)ks } ls =κni(s)k
lq =κnkkq
s
{α pq , α pq , . . . , α pq } ∪ {αk , αk , . . . , αk }, ls =κnk
lq =κnkkq
s−1 s−1 s where kq + κnk = (n−1)K+k. With this orq=1 lq = q=1 lq +ls = q=1 κnk dered disjoint partition, one writes, for n ∈ N and k = 1, 2, . . . , K, via (53), upon s−1 setting S(z) := i/ S(z) and noting that deg( q=1 (z−α pq )2 (z−αk )2 ) = 2s, ⎛ ⎞2 s−1 s−1 (κnk −1)2 2(κnk −1) κnkkq 1 ⎝ κnkkq ⎠ H(z) = − 2 2 − − π n (z−αk )2 π 2 n(z−αk ) q=1 n(z−α pq ) π 2 q=1 n(z−α pq ) 2(κnk −1) − 2 π n(z−αk )
(n−1)K+k n
J
dμV (ξ ) 2 − 2 ξ −z π
(n−1)K+k n
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×
−
s−1
κnkkq
q=1
n(z−α pq )
/ S(ξ ) dξ , J ξ −z 2π
J
1 dμV (ξ ) − 2 ξ −z π
(n−1)K+k n
2 J
dμV (ξ ) ξ −z
2
z ∈ C \ (J ∪ {α p1 , . . . , α ps−1 , αk }).
(58)
Recalling that μV ∈ M1 (R), in particular, J ξ m dμV (ξ ) = J ξ m ψV (ξ ) dξ < ∞, / J such that |ξ/z| % 1 (e.g., |z| & max{max j=1,2,...,N+1 m ∈ Z+ 0 , for ξ ∈ J and z ∈ j l+1 1 {|bj−1 −a j|}, maxq=1,2,...,s {|α pq |}}), via the expansion ξ −z = − lj=0 zξj+1 + zl+1ξ(ξ −z) , 2s l ∈ Z+ h (n, m=1 0 , one shows that, for n ∈ N and k = 1, 2, . . . , K, H(z) =z→∞ m (ξ )ψV k)z−m + O (z−(2s+1) ), where, in particular, h1 (n, k) = π12 ( (n−1)K+k V ) J n s−1 2s−1 2 2 m −1 (ξ ) dξ ; hence, q=1 (z−αpq ) (z−αk ) H(z)− m=0 ρm (n, k)z =z→∞ O (z ), 1 (n−1)K+k (ξ )ψV (ξ ) dξ , which implies where, in particular, ρ2s−1 (n, k) = π 2 ( ) JV n s−1 that, for n ∈ N and k = 1, 2, . . . , K, since q=1 (z−α pq )2 (z−αk )2 H(z) is entire, via a generalisation of Liouville’s theorem, 2s−1
ρm (n, k)zm H(z) = s−1m=0 , 2 2 q=1 (z−α pq ) (z−αk )
z ∈ C \ (J ∪ {α p1 , . . . , α ps−1 , αk }).
(59)
It remains, therefore, to determine ρm (n, k), m = 0, 1, . . . , 2s −1. Recalling (58) for H(z), one shows that, for n ∈ N and k= 1, 2, . . . , K, upon recalling that μV ∈ M1 (R), in particular, J ξ m dμV (ξ ) = J ξ m ψV (ξ ) dξ < ∞, m ∈ Z+ 0, for ξ ∈ J and z ∈ / J such that |ξ/z| % 1 (e.g., |z| & max{max j=1,2,...,N+1 {|bj−1 − j l+1 1 = − lj=0 zξj+1 + zl+1ξ(ξ −z) , l ∈ Z+ a j|}, maxq=1,2,...,s {|α pq |}}), via the expansion ξ −z 0,
−
s −1 $
! / S(ξ ) dξ J ξ −z 2π !
(z−α pq )2 (z−αk )2
q=1
=
z→∞
1 − 2 π
=
z→∞
−
s −1 $
q=1
2 s−1
2 s−1
m=0
q=m
ωq−m (n, k)cq+1 (n, k) zm + O (z−1 ),
(n−1)K+k n
2 s$ −1
2( s−1)
2( s−1)
m=0
q=m
(z−α pq ) (z−αk ) 2
2 J
q=1
dμV (ξ ) ξ −z
2
!
μˇ q−m (n, k)cq+2 (n, k) zm + O (z−1 ), ⎛
(z−α pq )2 (z−αk )2 ⎝
2(κnk −1) π 2 n(z−αk )
(n−1)K+k n
J
dμV (ξ ) ξ −z
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
⎞ s−1 (n−1)K+k κnkkq dμV (ξ ) ⎠ n n(z−α pq ) J ξ −z q=1 ! s−1) 2( s−1) 2( = νˇ q−m (n, k)cq+1 (n, k) zm + O (z−1 ), 2 + 2 π
z→∞
q=m
m=0
s −1 $
⎛
s−1 (κnk −1)2 2(κnk −1) κnkkq ⎝ − (z−α pq ) (z−αk ) + π 2 n2 (z−αk )2 π 2 n(z−αk ) q=1 n(z−α pq ) q=1 ⎛ ⎞2 ⎞ 2( s−1) s−1
κ 1 nkkq ⎠⎟ c m (n, k)zm + O (z−1 ), = + 2 ⎝ ⎠ z→∞ π n(z−α ) p q q=1 m=0
2
2
where ωr (n, k) :=
1 π2
μˇ m (n, k) =
(n−1)K+k n
m
J
(ξ )ψV (ξ ) dξ, ξrV
νˇ q (n, k)ˇνm−q (n, k),
r ∈ Z+ 0,
m = 0, 1, . . . , 2s −2,
q=0
νˇr (n, k) := J
cl (n, k) :=
r ∈ Z+ 0
ξ r ψV (ξ ) dξ,
2s−l
(−1)
(2!)
s
ir =0,1,2
s} r∈{1,2,..., s r=1 ir =2s−l
(ν0 (n, k) = 1),
! s−1 $ 1 (α p j )i j (αk )is , i !(2−i )! m m m=1 j=1 s $
l = 0, 1, . . . , 2s, cl (n, k)
1 := − 2 π
×
(n−1)K+k n
s −1 $
(α p j )i j (αk )is ,
2
(−1)2s−l (2!)s
ir =0,1,2 s} r∈{1,2,..., s r=1 ir =2s−l
s $
1 i !(2−im )! m=1 m
l = 0, 1, . . . , 2s,
j=1
cl (n, k)
2 := 2 π
κnk −1 n
(n−1)K+k n
ir =0,1,2 is =0,1 s−1} r∈{1,2,..., s i =2s−l−1 r=1 r
(−1)2s−l−1 (2!)s−1 is !(1−is )!
317
318
K. T.-R. McLaughlin et al. s −1 $
s−1
$ 1 2 × (α p j )i j (αk )is + 2 i !(2−i )! π m m=1 m j=1 ×
s−1
κnkkq
n
q=1
×
iq =0,1 ir =0,1,2 s}\{q} r∈{1,2,..., s m=1 im =2s−l−1
s−1 $ (α p j )i j (αk )is ,
(n−1)K+k n
s (−1)2s−l−1 (2!)s−1 $ 1 iq !(1−iq )! i !(2−i m )! m=1 m m =q
l = 0, 1, . . . , 2s −1,
j=1
cl (n, k) := −
1 π2
κnk −1 n
s −1 $
2
(−1)2(s−1)−l (2!)s−1
iq =0,1,2 q∈{1,2,...,s−1} s−1 r=1 ir =2(s−1)−l s−1
$ 1 2 × (α p j )i j − 2 i !(2−i )! π m m=1 m j=1
×
s−1 κnk −1 κnkkq n n q=1
s−1 s −1 $ (−1)2(s−1)−l (2!)s−2 $ 1 (α p j )i j (αk )is iq !(1−iq )!is !(1−is )! m=1 im !(2−im )! j=1
iq ,is =0,1 ir =0,1,2 s−1}\{q} r∈{1,2,..., s m=1 im =2(s−1)−l
−
m =q
s−1 1 κnkkq 2 π 2 q=1 n
(−1)2(s−1)−l (2!)s−1
ir ,is =0,1,2 s−1}\{q} r∈{1,2,..., s m=1 im =2(s−1)−l m =q
s −1 s−2 s−1 $ 1 2 κnkk p κnkkq ij is × (α p j ) (αk ) − 2 i !(2−im )! j=1 π p =1 q= p +1 n n m=1 m s $
m =q
j =q
×
(−1)2(s−1)−l (2!)s−2 i p !(1−i p )!iq !(1−iq )!is !(2−is )!
i p ,iq =0,1 ir ,is =0,1,2 r∈{1,2,...,s−1}\{q, p } −1 ir =2(s−1)−l i p +iq +is + sr=1 r =q, p
×
s −1 $
m=1
m =q, p
s −1 $ 1 i p iq (αp ) (α pq ) (α p j )i j (αk )is , l = 0, 1, . . . , 2s −2. im !(2−im )! p j=1 j =q, p
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
319
Hence, via (54) and (59), one arrives at, for n ∈ N and k = 1, 2, . . . , K, 2s−1 / ρm (n, k)zm S(ξ ) dξ (F(z))2 = , + s−1m=0 2 2 J ξ −z 2π q=1 (z−α pq ) (z−αk ) z ∈ C \(J ∪{α p1 , . . . , α ps−1 , αk }), where ρ2s−1 (n, k) = ω0 (n, k) =
1 π2
(n−1)K+k n
J
(ξ )ψV (ξ ) dξ, V
and ρm (n, k) = ω2s−m−1 (n, k) + c m (n, k) +
2( s−1)
(ωq−m (n, k)cq+1 (n, k)
q=m + νˇ q−m (n, k)cq+1 (n, k) + μˇ q−m (n, k)cq+2 (n, k)), m = 0, 1, . . . , 2s − 2,
with ωr (n, k), μˇ m (n, k), νˇr (n, k), cl (n, k), cl (n, k), cl (n, k), and cl (n, k) defined above. Recalling that S(z) := i/ S(z), where S(z) is defined in (53), one shows (z)ψV (z), S(z) = π2 ( (n−1)K+k )V that, via (56), for n ∈ N and k = 1, 2, . . . , K, / n whence 1 (n−1)K+k V (ξ )ψV (ξ ) 2 (F(z)) = 2 dξ π n ξ −z J 2s−1 ρm (n, k)zm + s−1m=0 , z ∈ C \ (J ∪ {α p1 , . . . , α ps−1 , αk }). (60) 2 2 q=1 (z−α pq ) (z−αk ) But, via (55), 1 π2
1 = 2 π
(n−1)K+k n
(z) V + 2 π
V (ξ )ψV (ξ ) dξ ξ −z J
(n−1)K+k n
J
(ξ )− V (z))ψV (ξ ) (V dξ ξ −z
(n−1)K+k ψV (ξ ) dξ n ξ −z J (κ
s −1 κ nkkq −1) − n(z−α pq ) k) q=1
nk =−iπ F(z)− n(z−α
=
1 π2
(n−1)K+k n
J
⎛
(ξ )− V (z))ψV (ξ ) (V dξ ξ −z
⎞ s−1
κ i V (z) (κnk −1) nkkq ⎠: − V (z)F(z)− 2 ⎝ + π π n(z−αk ) q=1 n(z−α pq )
320
K. T.-R. McLaughlin et al.
substituting the above into (60), one arrives at, for n ∈ N and k = 1, 2, . . . , K, upon completing the square and re-arranging terms, !2 (z) iV q (z) F(z)+ + V 2 = 0, z ∈ C \ (J ∪ {α p1 , . . . , α ps−1 , αk }), (61) 2π π where (z) V 2
qV (z) := −
⎞ ⎛ s−1
κ nkkq (z)⎝(κnk −1) + ⎠ +V n(z−αk ) q=1 n(z−α pq )
!2
(n−1)K+k n
J
(ξ )− V (z))ψV (ξ ) (V dξ ξ −z
2s−1 ρ /m (n, k)zm − s−1m=0 2 2 q=1 (z−α pq ) (z−αk ) (z) V 2
= −
⎞ ⎛ s−1
κ nkkq (z)⎝(κnk −1) + ⎠ +V n(z−αk ) q=1 n(z−α pq )
!2
(n−1)K+k n
J
1 0
(λξ +(1−λ)z) dλ dμV (ξ ) V
2s−1
ρ /m (n, k)zm − s−1m=0 , 2 2 q=1 (z−α pq ) (z−αk ) where ρ /m (n, k) := π 2 ρm (n, k), m = 0, 1, . . . , 2s −1. Perusing the expression for ρm (n, k), m = 0, 1, . . . , 2s −1, derived above (see, in particular, the formu+ lae notes that integrals of the type rfor ωr (n, k) and νˇr (n,rk), r ∈ Z0 ), one (ξ ) dμV (ξ ) and ξ dμV (ξ ), r ∈ Z+ , are encountered; in fact, one may V ξ 0 J J use such integrals in order to evaluate the Stieltjes transform of the equilibrium measure on the real pole set {α1 , α2 , . . . , α K }, that is, J (ξ −α pq )−1 dμV (ξ ), q = 1, 2, . . . , s, as follows. Recalling that, for n ∈ N and k = 1, 2, . . . , K, μV ∈ M1 (R), in particular, J (ξ −αk )−m ψV (ξ ) dξ < ∞ and J ξ m ψV (ξ ) dξ < ∞, m ∈ N, with J ψV (ξ ) dξ = 1, for ξ ∈ J and z ∈ / J such that |(z−αk )/(ξ −αk )| % 1 (e.g., 0 < |z−αk | % min{min j=1,2,...,N+1 {||bj−1 −a j|−αk |}, minq=1,2,...,s−1 {|α pq − j (z−αk )l+1 + 1 k) αk |}}), via the expansion (z−αk )−(ξ = − lj=0 (ξ(z−α + (ξ −α l+1 (z−ξ ) , l ∈ Z0 , −αk ) −αk ) j+1 k) one shows, via (58) and (59), that H(z) = − z→αk
s−1 (κnk −1)2 2(κnk −1) κnkkq + (z−αk )2 π 2 n2 (z−αk )π 2 n q=1 n(α pq −αk )
2(κnk −1) − (z−αk )π 2 n
(n−1)K+k n
J
dμV (ξ ) + O (1), ξ −αk
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
1 1 P(αk ) H(z) = + z→αk (z−αk )2 F(αk ) (z−αk )
321
P (αk ) P(αk ) F (αk ) − F(αk ) F(αk ) F(αk )
+ O (1), s−1 s−1 ρm (n, k)(αk )m , P (αk ) = 2m=1 mρm (n, k)(αk )m−1 , F(αk ) = where P(αk ) = 2m=0 s−1 s−1 2 −1 = 2 q=1 (αk −α pq )−1 , and, for q = 1, 2, q=1 (αk −α pq ) , and F (αk )(F(αk )) . . . , s −1, H(z)
=
z→α pq
×
−
2 (κnk kq )
π 2 n2 (z−α
pq
)2
s−1
κnk kr
r=1
n(α pq −α pr )
+
2(κnk −1) π 2 n(αk −α pq )
−
2 π2
κnk kq
n(z−α pq )
(n−1)K+k n
−
2 κnk kq π 2 n(z−α pq )
κnk kq
n(z−α pq )
J
dμV (ξ ) + O (1), ξ −α pq
r =q
H(z)
5(α pq ) P 1 1 + 2 (z−α pq ) (z−α pq ) G(α pq )
=
z→α pq
5 (α p ) P q G(α pq )
−
5(α p ) P q
!
G(α pq )
G (α pq )
!
G(α pq )
+O (1),
5(α pq ) = 2s−1 ρm (n, k)(α pq )m , P 5 (α pq ) = 2s−1 mρm (n, k)(α pq )m−1 , where P m=0 m=1 s−1 G(α pq ) = (α pq −αk )2 r=1 (α pq −α pr )2 , and G (α pq )(G(α pq ))−1 = 2(α pq −αk )−1 + r =q s−1 (α pq −α pr )−1 ; whence, equating coefficients of like powers of (z−αk )−1 2 r=1 r =q
and (z−α pq )−1 , q = 1, 2, . . . , s −1, respectively, in the above pair of asymptotic expansions for H(z), one obtains, for n ∈ N and k = 1, 2, . . . , K, the following set of 2(s +1) real linear equations: 2 s−1 m=0
1 ρm (n, k)(αk ) = − 2 π m
2 s−1 2 s−1
κnk −1 n
2 (κnk kq )
m=0
(αk −α pq )2 ,
q=1
mρm (n, k)(αk )m−1 = / " (n, k),
m=0
ρm (n, k)(α pq )m = −
2 s$ −1
π 2 n2
(α pq −αk )2
s$ −1
(αpq−αpr )2 , q = 1, 2, . . . , s −1,
r=1 r=q
2 s−1
/(n, k; q), q = 1, 2, . . . , s −1, mρm (n, k)(α pq )m−1 = Ξ
m=0
where / " (n, k) := − +
2 π2
κnk −1 n
(n−1)K+k n
s$ −1
(αk −α pq )2
q=1
J
dμV (ξ ) , ξ −αk
s−1
(κnk + κnk kr −1) r=1
n(αk −α pr )
322
K. T.-R. McLaughlin et al.
⎛ /(n, k; q) := − Ξ
2 κnk kq (α pq −αk )2 2 n π
s$ −1 r=1
s−1 (κ ⎜ nk kq + κnk kr ) (α pq −α pr )2 ⎜ ⎝ n(α pq −α pr ) r=1
r=q
+
(κnk + κnk kq −1) n(α pq −αk )
+
(n−1)K+k n
r=q
⎞
J
dμV (ξ )⎟ ⎟ , q = 1, 2, . . . , s −1, ξ −α pq ⎠
which can be presented in the form ⎛ 1 αk (αk )2 ... (αk )2s−1 2 2s−1 ⎜1 α p1 (α ) . . . (α p1 p1 ) ⎜ ⎜1 α p2 (α p2 )2 . . . (α p2 )2s−1 ⎜ ⎜ .. .. .. .. . .. ⎜. . . . ⎜ ⎜1 α ps−1 (α ps−1 )2 . . . (α )2s−1 p s−1 ⎜ ⎜0 1 2αk . . . (2s −1)(αk )2s−2 ⎜ ⎜0 1 2α p1 . . . (2s −1)(α p1 )2s−2 ⎜ ⎜0 1 2α p2 . . . (2s −1)(α p2 )2s−2 ⎜ ⎜ .. .. .. .. .. ⎝. . . . .
0
1
2α ps−1
⎞⎛
ρ0 (n, k) ρ1 (n, k) ρ2 (n, k) .. .
⎞
⎟⎜ ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ ⎟ ⎜ ρs−1 (n, k) ⎟ ⎟⎜ ⎟ ⎟ ⎜ ρs (n, k) ⎟ ⎟⎜ ⎟ ⎟ ⎜ ρs+1 (n, k) ⎟ ⎟⎜ ⎟ ⎟ ⎜ ρs+2 (n, k) ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ .. ⎠⎝ ⎠ . 2s−2 ρ2s−1 (n, k) . . . (2s −1)(α ps−1 )
:= D
⎛
(κnk − 1)2 s−1 2 − q=1 (αk −α pq ) π 2 n2 (κnkk )2 s−1 − 2 12 (α p1 −αk )2 q=1 (α p1 −α pq )2 π n q =1 (κnkk2 )2 2 s−1 2 − 2 2 (α p2 −αk ) q=1 (α p2 −α pq ) π n q =2 .. .
⎞
⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟. ⎜ =⎜ ⎟ 2 ⎜ (κnkks−1 ) s −1 2 2⎟ ⎟ ⎜− (α −α ) (α −α ) p k p p s −1 s −1 q q=1 ⎜ ⎟ π 2 n2 q =s−1 ⎟ ⎜ /(n, k) ⎟ ⎜ Γ ⎟ ⎜ /(n, k; 1) ⎟ ⎜ Ξ ⎟ ⎜ / ⎟ ⎜ Ξ (n, k; 2) ⎟ ⎜ ⎟ ⎜ .. ⎠ ⎝ . /(n, k; s −1) Ξ
(62)
For n ∈ N and k = 1, 2, . . . , K, via the formula for ρm (n, k), m = 0, 1, . . . , 2s −1, derived above, the first s equations of system (62) give rise to remarkable identities, whereas the latter s equations of system (62) allow one to express the Stieltjes transforms, J (ξ −α pq )−1 dμV (ξ ), q = 1, 2, . . . , s (with α ps := (ξ ) dμV (ξ ), r1 = 0, 1, . . . , 2s −1, αk ), in of the moment integrals J ξ r1 V terms r2 and J ξ dμV (ξ ), r2 = 0, 1, . . . , 2s −2, which proves, incidentally, and as a
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
323
by-product of the above calculations, that the Stieltjes transforms, J (ξ − α pq )−1 dμV (ξ ), q = 1, 2, . . . , s, are real valued and bounded. Alternatively, given the Stieltjes transforms, J (ξ −α pq )−1 dμV (ξ ), q = 1, 2, . . . , s, one may regard system (62) as a means by which real linear combinations r of the 2 (ξ ) dμV (ξ ), r1 = 0, 1, . . . , 2s −1, and moment integrals J ξ r1 V (ξ ), J ξ dμV r2 = 0, 1, . . . , 2s −2, may be expressed in terms of them (also as real linear combinations); thus, one must verify that the determinant of the 2s × 2s coefficient matrix, D, is not equal to zero. Towards this end, one uses Theorem 20 of [71] to show that det(D) = (−1)
s(s−1) 2
s −1 $
q=1
(α pq −αk )4
s −1 $
(α p j −α pi )4 = 0.
i, j=1 i< j
R \ {α1 , α2 , . . . , α K } → R satisfies conditions (6)–(8), it Note that, since V: l−2 l−2 l−1 follows from the identity zl1 −zl2 = (z1 −z2 )(zl−1 1 +z1 z2 +· · ·+z1 z2 +z2 ), l ∈ N, that, for n ∈ N and k = 1, 2, . . . , K, qV (z) is real analytic on R \ {α1 , α2 , . . . , α K } ⊃ J. For x ∈ J, set z := x±iε, and consider the ε ↓ 0 limit of is real analytic on J); (61): limε↓0 (F(x±iε)+ iV (x±iε) )2 = (F± (x)+ iV2π(x) )2 (as V 2π κnk −1) + recalling that (cf. (57)), for n ∈ N and k = 1, 2, . . . , K, F± (x) = − iπ1 ( (n(x−α k) s−1 κnkkq (n−1)K+k (n−1)K+k )(H ψV )(x))∓( )ψV (x), x ∈ J, via (56), it q=1 n(x−α p ) −π( n n q
follows that F± (x) = − iV2π(x) ∓( (n−1)K+k )ψV (x), which implies that (F± (x)+ n (x) 2 iV ) = ( (n−1)K+k )2 (ψV (x))2 , 2π n
whence (ψV (x))2 = −(π( (n−1)K+k ))−2 qV (x), x ∈ J, n whereupon, using the fact that ψV (x) 0 ∀ x ∈ J, it follows that qV (x) 0, x ∈ J; moreover, as a by-product, decomposing qV (x), for x ∈ J, into its positive and negative parts, that is, qV (x) = (qV (x))+ −(qV (x))− , x ∈ J, where (qV (x))± := max{±qV (x), 0} ( 0), one learns from the above analysis that, for x ∈ J, (qV (x))+ ≡ 0 and ψV (x) = (π( (n−1)K+k ))−1 ((qV (x))− )1/2 ; and, since n (n−1)K+k −1 − 1/2 )) dξ = 1, which (ξ ) dξ = 1, it follows that (π( (ξ )) ) J ψV J ((q V n gives rise to the interesting fact that the function (qV (x))− ≡ 0 on J. For x∈ / J, set z := x±iε, and, again, study the ε ↓ 0 limit of (61): in this case, )2 = (F± (x)+ iV2π(x) )2 = (F(x)+ iV2π(x) )2 ; recalling that limε↓0 (F(x±iε)+ iV (x±iε) 2π κnk −1) s−1 κnkkq + q=1 n(x−α p ) + (cf. (57)), for n ∈ N and k = 1, 2, . . . , K, F(x) = − iπ1 ( (n(x−α k) q κ ψ (ξ ) nk k s −1 (n−1)K+k q i (κnk −1) V ( (n−1)K+k ) dξ ) = ( + )−i( )( H ψ )(x), x ∈ / J, V q=1 J n ξ −x π n(x−αk ) n(x−α pq ) n (x) 2 iV 1 (κnk −1) s−1 κnkkq which implies that, via (61), (F(x)+ 2π ) = ( π ( n(x−αk ) + q=1 n(x−α p ) )− )(H ψV )(x)+ V2π(x) )2 = qV (x)/π 2 , x ∈ / J (since V ( (n−1)K+k n on (R \ {α1 , α2 , . . . , α K }) \ J, it follows that qV (x), too, is
q
is real analytic real analytic on (R \ {α1 , α2 , . . . , α K }) \ J, in which case, this latter relation merely states that, for x = αk , k = 1, 2, . . . , K, +∞ = +∞), whence qV (x) > 0, x ∈ / J. Now, recalling that, on a compact subset of R, in particular, one whose intersection with {α1 , α2 , . . . , α K } equals ∅, a rational, in fact, mermorphic, function changes sign an at most countable number of times, it follows from the argument above, the R \ {α1 , α2 , . . . , α K } → R satisfies conditions (6)–(8), in particular, fact that V:
324
K. T.-R. McLaughlin et al.
is real analytic in the open neighbourhood U := {z ∈ C; infq∈J |z−q|
κnkkq 1 ⎝ κnk −1 1 m F(z) = − (αk ) + (α pq )m z→∞ iπ nz iπ z m=1 n n q=1 −
(n−1)K+k n
⎞
J
ξ m ψV (ξ ) dξ ⎠ z−m ;
(63)
and (ii) for ξ ∈ J and z ∈ / J such that |(z−αk )/(ξ −αk )| % 1 (e.g., 0 < |z− αk | % min{minq=1,2,...,s−1 {|αk −α pq |}, min j=1,2,...,N+1 {||bj−1 −a j|−αk |}}), via the j (z−αk )l+1 + 1 k) expansion (z−αk )−(ξ = − lj=0 (ξ(z−α + (ξ −α l+1 (z−ξ ) , l ∈ Z0 , one gets that −αk ) −αk ) j+1 k) κnkk
nk −1) F(z) =z→αk − iπ(κn(z−α + O (1), and F(z) =z→α pq − iπ n(z−αq p ) + O (1), q = 1, 2, . . . , k) q
s −1. Recalling, too, the formulae for F± (z) given in (57), one deduces (z)/iπ , z ∈ J, and F+ (z)− F− (z) = 0, z ∈ that, via (56), F+ (z)+ F− (z) = V / J. Thus, one learns that, for n ∈ N and k = 1, 2, . . . , K, F : N × {1, 2, . . . , K} × C \ (J ∪ {α p1 , . . . , α ps−1 , αk }) → C solves the following (scalar and homogeneous) RHP: (1) F(z) is holomorphic (resp., meromorphic) for z ∈ C \ (J ∪ {α p1 , . . . , α ps−1 , αk }) (resp., z ∈ C \ J); (2) F± (z) := limε↓0 F(z±iε) satisfy (z)/iπ , z ∈ J, and F+ (z) = F− (z) = the boundary condition F+ (z)+ F− (z) = V 1 −2 F(z), z ∈ / J; (3) F(z) =z→∞ iπ nz + O (z ); and (4) Res(F(z); αk ) = −(κnk −1)/iπ n, and, for q ∈ {1, 2, . . . , s − 1}, Res(F(z); α pq ) = −κnkkq /iπ n. For n ∈ N and
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
325
k = 1, 2, . . . , K, an explicit representation for the solution of this RHP can be presented as (see, for example, Chapter VI of [72]) ⎛ ⎞ ⎛ ⎛ s−1 1 ⎝ (κnk −1) κnkkq ⎠ ⎝ 2 ⎝ (κnk −1) + + (R(z))1/2 F(z) = − iπ n(z−αk ) q=1 n(z−α pq ) iπ n(ξ −αk ) J +
s−1
q=1
⎞ (ξ ) (R(ξ ))−1/2 V dξ + ⎠+ ⎠ , n(ξ −α pq ) iπ ξ −z 2π i κnkkq
⎞
z ∈ C \ (J ∪ {α p1 , . . . , α ps−1 , αk }),
(64)
1/2 where (R(z))1/2 is defined in item (1) of the Lemma, with (R(z))± := limε↓0 (R(z±iε))1/2 , and the branch of the square root is chosen so that (for |z| & max{max j=1,2,...,N+1 {|bj−1 −a j|}, maxq=1,2,...,s {|α pq |}}) z−(N+1) (R(z))1/2 ∼C± z→∞ ±1. It follows from the above integral representation (64) that, for n ∈ N and k = 1, 2, . . . , K, for ξ ∈ J and z ∈ / J such that |ξ/z| % 1 (e.g., |z| & 1 max{max j=1,2,...,N+1 {|bj−1 −a j|}, maxq=1,2,...,s {|α pq |}}), via the expansion ξ −z = l ξ l+1 ξj − j=0 z j+1 + zl+1 (ξ −z) , l ∈ Z+ 0,
(z N+1 +· · · ) F(z) = − z→∞ 2π iz
⎛
⎛ ⎞ ⎞ s−1
κ 2 ( V κ −1) (ξ ) nk k q ⎝ ⎝ nk ⎠+ ⎠ + 1/2 iπ (R(ξ ))+ iπ n(ξ −αk ) q=1 n(ξ −α pq ) 1
J
s−1 1 κnkkq 1 (1+· · · ) ξ ξ2 ξN 1 1 κnk −1 − − 1+ + 2 +· · ·+ N dξ − z z z z iπ n z q=1 iπ n z 2π i ⎛ ⎛ ⎞ ⎞ s−1 (ξ ) ξ N+1 ⎝ 2 ⎝ (κnk −1) κnkkq ⎠ V ⎠ dξ − 1 × + + 1/2 iπ n(ξ −α ) n(ξ −α ) iπ π iz k pq J (R(ξ ))+ q=1
×
⎛ ⎞ ∞ s−1
κ (1+· · · ) ξ N+2 1 nkkq ⎝ κnk −1 (αk )m + × (α pq )m ⎠ m − 1/2 2 n n z 2π iz J (R(ξ ))+ m=1 q=1 ⎞ ⎞ ⎛ ⎛ s−1
κ ( V 2 κ −1) (ξ ) nkkq nk ⎠+ ⎠ 1+ ξ +· · · dξ : + × ⎝ ⎝ iπ n(ξ −αk ) q=1 n(ξ −α pq ) iπ z
now, recalling from above that F(z) =z→∞ iπ1nz + O (z−2 ), it follows, upon removing secular terms, that ⎞ ⎞ ⎛ ⎛ s−1
(ξ ) κnkkq ( V ξj 2 κ −1) nk ⎠+ ⎠ dξ = 0, ⎝ ⎝ + 1/2 iπ n(ξ −α ) n(ξ −α ) iπ k p J (R(ξ ))+ q q=1 j= 0, 1, . . . , N,
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which gives N+1 real moment equations, and, upon equating z−1 terms, ⎛ ⎛ ⎞ ⎞ s−1 N+1
κ ξ nkkq ⎝ 2 ⎝(κnk −1) + ⎠ + V (ξ )⎠ dξ 1/2 iπ n(ξ −α ) n(ξ −α ) iπ k pq J (R(ξ ))+ q=1 (n−1)K+k = −2 ; n
it remains, therefore, to determine an additional 2(N+1)−(N+1)−1 = N real moment equations. From the integral representation (64), a residue calculation shows that, for n ∈ N and k = 1, 2, . . . , K, for non-real z in the domain of (that is, C \ {α1 , α2 , . . . , α K }), analyticity of V ⎞ ⎛ ⎛ s−1
(z) (R(z))1/2 * κ V ( 2 κ −1) nk k q ⎠ ⎝ ⎝ nk F(z) = + + 2π i 2 iπ n(ξ −αk ) q=1 n(ξ −α pq ) CV ⎞ V (ξ )⎠ (R(ξ ))−1/2 dξ + , (65) iπ ξ −z 2π i where CV (⊂ C \ {α1 , α2 , . . . , α K }) is the simple boundary of an open (s +1)s connected punctured region of the type CV = ∂ U R ∪ (∪q=1 ∂ Uδq (α pq )), with U R = {z ∈ C; |z| < R}, and ∂ U R = {z ∈ C; |z| = R} oriented clockwise, and, for q = 1, 2, . . . , s, U δq (α pq ) := {z ∈ C; |z−α pq | < δq }, with ∂ Uδq (α pq ) = {z ∈ C; |z− α pq | = δq } oriented counter-clockwise, and where the numbers 0 < δq (sufficiently small) < R (sufficiently large) < +∞, q = 1, 2, . . . , s, are chosen so that, for (any) non-real z in the domain of analyticity of V, • • • • •
U ∂ δi (α pi ) ∩ ∂ Uδ j (α p j ) = ∅, i = j∈ {1, 2, . . . , s}, ∂ U δ j (α p j ) ∩ ∂ U R = ∅, j∈ {1, 2, . . . , s}, U ( δ j (α p j ) ∪ ∂ Uδ j (α p j )) ∩ J = ∅, j∈ {1, 2, . . . , s}, ∂ U R ∩ J = ∅, s int(CV ) := (ext(∪q=1 ( U δq (α pq ) ∪ ∂ Uδq (α pq ))) ∩ U R ) ⊃ {z} ∪ J.
Recall from (57) that, for z ∈ / J,
⎛ s−1 1 ⎝ (κnk −1) κnkkq + F+ (z) = F− (z) = F(z) = − iπ n(z−αk ) q=1 n(z−α pq ) ⎞ (n−1)K+k ψV (ξ ) ⎠ + dξ , n J ξ −z
whence, via (55), ⎛ ⎞ s−1
κ 1 (κnk −1) nkkq ⎠ = −i (n−1)K+k (H ψV )(z); F(z)+ ⎝ + iπ n(z−αk ) q=1 n(z−α pq ) n
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
thus, using (64), one arrives at, for z ∈ R \ J, −1
⎛
327
⎛
(n−1)K+k ⎝ 2 ⎝ (κnk −1) (R(z))1/2 n iπ n(ξ −αk ) J ⎞ ⎞ s−1
(ξ ) (R(ξ ))−1/2 κnkkq V dξ + ⎠+ ⎠ . + n(ξ −α pq ) iπ ξ −z 2π i q=1
(H ψV )(z) = i
A contour integration argument shows that, for j= 1, 2, . . . , N, ⎛ ⎛ −1 bj 1 (n−1)K+k ⎝(H ψV )(ξ )− ⎝2(κnk −1) 2π n n(ξ −αk ) aj + 2
s−1
κnkkq
q=1
n(ξ −α pq )
⎞⎞ (ξ )⎠⎠ dξ = 0, +V
(66)
whence, using the above expression for (H ψV )(z), z ∈ ∪ Nj=1 (a j, bj), for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎛ ⎛ ⎞ ⎞ bj s−1
(ξ ) κ 2 ( V κ −1) nk k nk q ⎝(R(ς))1/2 ⎝ ⎝ ⎠+ ⎠ + n(ξ −α ) iπ p aj J iπ n(ξ −αk ) q q=1 −1/2
(R(ξ ))+ × ξ −ς
⎞ dξ⎠ dς =
⎛ bj
aj
⎞ s−1
κnkkq 2( κ −1) nk (ξ )⎠ dξ, ⎝ +2 +V n(ξ −αk ) n(ξ −α ) p q q=1
j= 1, 2, . . . , N. If no αk , k = 1, 2, . . . , K, belongs to any gap (a j, bj), j= 1, 2, . . . , N, then, via : R \ {α1 , α2 , . . . , α K } → R and an application of the the real analyticity of V fundamental theorem of calculus, the N integrals on the right-hand side of the expression above are easily evaluated; if, however, an αk , k = 1, 2, . . . , K, belongs to any gap (a j, bj), j= 1, 2, . . . , N, then the integrals appearing on the b right-hand side of the expression above, a j j !(ξ ) dξ , j= 1, 2, . . . , N, need to be α −ε b understood in the Cauchy principal value sense, limε↓0 (( a jk + αkj+ε )!(ξ ) dξ ), j= 1, 2, . . . , N; in either case, one arrives at, for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎛ ⎛ ⎞ ⎞ bj s−1
(ξ ) κ 2 ( κ −1) V nk k nk q ⎝(R(ς))1/2 ⎝ ⎝ ⎠+ ⎠ + n(ξ −α ) iπ p aj J iπ n(ξ −αk ) q q=1 −1/2
(R(ξ ))+ × ξ −ς
⎞
' ' ' ' s−1
κnkkq ' bj −α pq ' ' bj −αk ' κnk −1 ' ' ' ' ⎠ +2 dξ dς = 2 ln ' ln ' ' n a j −αk ' n a −α j p q q=1
j)− V(a j)), +(V(b
j = 1, 2, . . . , N,
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which yields the remaining N real moment equations determining the 2(N+1) end-points of the support of the equilibrium measure. Since R \ {α1 , α2 , . . . , α K } → R is real analytic J ∩ {α1 , α2 , . . . , α K } = ∅ and V: 1/2 on J, (R(x)) =x↓bj−1 O ((x−bj−1 )1/2 ) and (R(x))1/2 =x↑a j O ((a j −x)1/2 ), j= 1, 2, . . . , N+1, which shows that all the integrals above constituting the system of 2(N+1) real moment equations for the end-points of the support of μV have integrable singularities at bj−1 , a j, j= 1, 2, . . . , N+1. Recall from (57) that, for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎞ s−1 (n−1)K+k 1 ⎝ (κnk −1) κnkkq + −π (H ψV )(z)⎠ F± (z) = − iπ n(z−αk ) q=1 n(z−α pq ) n ∓
(n−1)K+k ψV (z), n
z∈ J :
R \ {α1 , using, for z ∈ J, (56), it follows from the real analyticity of V: α2 , . . . , α K } → R that, for n ∈ N and k = 1, 2, . . . , K, (z) (n−1)K+k V ∓ ψV (z), F± (z) = z ∈ J. (67) 2π i n R \ {α1 , α2 , . . . , α K } → R, it follows From (64) and the real analyticity of V: that, for n ∈ N and k = 1, 2, . . . , K, with z ∈ J, ⎛ ⎛ ⎞ * s−1
(z) (R(z))1/2 κ 2 ( κ −1) V nkkq ± ⎝ ⎝ nk ⎠ + + F± (z) = 2π i 2 iπ n(ξ −αk ) q=1 n(ξ −α pq ) CV ⎞ (ξ ) (R(ξ ))−1/2 dξ V ⎠ + ; iπ ξ −z 2π i thus, equating the two expressions above for F± (z), z ∈ J, one arrives at, 1/2 for n ∈ N and k = 1, 2, . . . , K, ψV (x) = 2π1 i (R(x))+ hV (x)1 J (x), where hV (z) is defined in item (2) of the Lemma, and 1 J (x) is the characteristic function of the set J, which gives rise to the formula for the equilibrium measure, dμV (x) = ψV (x) dx (the integral representation for hV (z) shows that it is analytic in some open subset of C \ {α p1 , . . . , α ps−1 , αk } containing J). Now, recalling R \ {α1 , α2 , . . . , α K } → R satisfies conditions (6)–(8), and that, for ξ ∈ J that V: 1/2 (resp., ξ ∈ int(J)), ψV (ξ ) 0 (resp., ψV (ξ ) > 0) and (R(ξ ))+ = i(|R(ξ )|)1/2 ∈ iR 1/2 (resp., (R(ξ ))+ = i(|R(ξ )|)1/2 ∈ iR± ), it follows from the formula ψV (ξ ) = 1/2 1 (R(ξ ))+ hV (ξ )1 J (ξ ) and the regularity assumption, that is, hV (ξ ) = 0 for 2π i ξ ∈ J, that (|R(ξ )|)1/2 hV (ξ ) 0, ξ ∈ J (resp., (|R(ξ )|)1/2 hV (ξ ) > 0, ξ ∈ int(J)). Finally, it will be shown that, if J = ∪ N+1 j=1 [b j−1 , a j ], the end-points of the support of the equilibrium measure, which satisfy the system of 2(N+1) real moment equations stated in item (1) of the Lemma, are real-analytic functions of zo , thus establishing the (local) solvability of the system of 2(N+1)
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329
real moment equations. Towards this end, one follows closely the idea of the proof of Theorem 1.3 (iii) in [63] (see, also, Section 8 of [62]). Recall := zo V(z), where zo : N × N → R+ , (N, n) → zo := from Subsection 1.3 that V(z) N/n, and, in the double-scaling limit N, n → ∞, zo = 1+o(1). Furthermore, from the analysis above, it was shown that the end-points of the support of the equilibrium measure were the simple roots of the (meromorphic) function qV (z), that is, with the enumeration −∞ < b 0 < a1 < b1 < a2 < · · · < b N < a N+1 < +∞, {b 0 , a1 , b1 , a2 , . . . , b N , a N+1 } = {x ∈ R \ {α1 , α2 , . . . , α K }; qV (x) = 0} (these are the only roots for the regular case studied in this work). The (meromorphic) function qV (x) ∈ R(x) (the algebra of rational functions in the indeterminate x with coefficients in R) is real rational (resp., real analytic) on R (resp., R \ {α1 , α2 , . . . , α K }), it has analytic extension (indepen/ / dent of zo ) to the open neighbourhood / U = ∪ N+1 j=1 U j , where U j := {z ∈ C \
Ui ∩ {α1 , α2 , . . . , α K }; inf p∈(bj−1 ,a j ) |z− p|
/ U j = ∅, i = j∈ {1, 2, . . . , N+1}, and depends continuously on zo ; thus, its simple roots, that is, bj−1 = bj−1 (zo ) and aj = aj(zo ), j= 1, 2, . . . , N+1, are continuous functions of zo . Write the large-z (e.g., |z| & max{max j=1,2,...,N+1 {|b j−1 −a j|}, maxq=1,2,...,s {|α pq |}}) asymptotic expansion for F(z) (cf. (64)) as follows: for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎛ ⎞ s−1 ∞
κnkkq 1 ⎝ κnk −1 1 κnk −1 ⎝ ⎠ F(z) = − + − (αk )m z→∞ iπ z n n iπ z n q=1 m=1 +
s−1
κnkkq
n
q=1
where T j :=
⎛ ξ
j
1/2 J (R(ξ ))+
⎞ (α pq )m ⎠ z−m −
⎛
∞ (R(z))1/2 T j z− j , 2π iz j=0
s−1
⎝ 2 ⎝ (κnk −1) + iπ n(ξ −αk ) q=1
⎞ ⎠ + V (ξ ) ⎠ dξ, n(ξ −α pq ) iπ κnkkq
⎞
j∈ Z+ 0. (68)
Set, for n ∈ N and k = 1, 2, . . . , K, ⎛ ⎛ −1 bj 1 (n−1)K+k ⎝(H ψV )(ξ )− ⎝2(κnk −1) N j := 2π n n(ξ −αk ) aj
+ 2
s−1
κnkkq
q=1
n(ξ −α pq )
⎞⎞ (ξ )⎠⎠ dξ, j = 1, 2, . . . , N. +V
With the above definitions, the system of 2(N+1) real moment equations are, thus, equivalent to T j = 0, j= 0, 1, . . . , N, T N+1 = −2((n−1)K+k)/n, and R \ {α1 , α2 , N j = 0, j= 1, 2, . . . , N. It will first be shown that, for regular V:
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. . . , α K } → R satisfying conditions (6)–(8), the Jacobian of the transformation{b0 (zo ), b1(zo ), . . . , b N (zo ), a1 (zo ), a2 (zo ), . . . , a N+1 (zo )} → {T0 , T1 , . . . , T N+1 , N1 , N2 , . . . , N N } is non-zero whenever bj−1 = bj−1 (zo ) and a j = a j(zo ), j= 1, 2, . . . , N+1, are chosen so that J = ∪ N+1 j=1 [b j−1 , a j ]. Using the equation (cf. (55)) ⎛ ⎞ s−1
κnkkq 1 (n−1)K+k −1 ⎝ 2(κnk −1) ⎠, 2π iF(z)+ (H ψV )(z) = +2 2π n n(z−αk ) n(z−α pq ) q=1 and (64), one follows the analysis on pp. 778–779 of [63] to show that, for n ∈ N and k = 1, 2, . . . , K, with i = 1, 2, . . . , N+1: ∂T j ∂ T j−1 1 = bi−1 + T j−1 , ∂bi−1 ∂bi−1 2
j∈ N,
∂ T j−1 1 ∂T j = ai + T j−1 , ∂ai ∂ai 2
1 ∂ F(z) =− ∂bi−1 2π i
∂ T0 ∂bi−1
j∈ N,
(70)
(R(z))1/2 , z−bi−1
z ∈ C \ (J ∪ {α p1 , . . . , αps−1 , αk }), ∂ F(z) 1 =− ∂ai 2π i
∂ T0 ∂ai
(71)
(R(z))1/2 , z−ai
z ∈ C \ (J ∪ {α p1 , . . . , α ps−1 , αk }), ∂N j 1 =− ∂bi−1 2π
(n−1)K+k n
−1
∂ T0 ∂bi−1
bj
aj
(72)
(R(ξ ))1/2 dξ, ξ −bi−1
j= 1, 2, . . . , N, ∂N j 1 =− ∂ai 2π
(n−1)K+k n
−1
(69)
∂ T0 ∂ai
(73)
bj
aj
j= 1, 2, . . . , N;
(R(ξ ))1/2 dξ, ξ −ai (74)
furthermore, if one evaluates (69) and (70) on the solution of the system of 2(N+1) real moment equations, that is, T j = 0, j= 0, 1, . . . , N, T N+1 = −2((n− 1)K+k)/n, and Nl = 0, l = 1, 2, . . . , N, one arrives at, for i = 1, 2, . . . , N+1, ∂T j ∂ T0 = (bi−1 ) j ∂bi−1 ∂bi−1
and
∂T j ∂ T0 = (ai ) j , ∂ai ∂ai
j= 0, 1, . . . , N+1. (75)
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331
Via (73), (74), and (75), one computes, for n ∈ N and k = 1, 2, . . . , K, the Jacobian of the transformation {b0 (zo ), b1 (zo ), . . . , b N (zo ), a1 (zo ), a2 (zo ), . . . , a N+1 (zo )} → {T0 , T1 , . . . , T N+1 , N1 , N2 , . . . , N N } on the solution of the system of 2(N+1) real moment equations:
Jac(T0 , T1 , . . . , T N+1 , N1 , N2 , . . . , N N ) :=
∂(T0 , T1 , . . . , T N+1 , N1 , N2 , . . . , N N ) ∂(b0 , b1 , . . . , b N , a1 , a2 , . . . , a N+1 )
' ∂T 0 ' ' ∂b0 ' ∂ T1 ' ∂b ' 0 ' . ' . ' . ' ' ∂ T N+1 ' = ' ∂b0 ' ∂ N1 ' ∂b0 ' ∂N 2 ' ' ∂b0 ' . ' . ' . ' ∂ NN '
∂ T0 ∂b1 ∂ T1 ∂b1
×⎝
.. .
··· ··· ···
∂ T0 ∂a1 ∂ T1 ∂a1
.. .
.
∂ T0 ∂a2 ∂ T1 ∂a2
.. .
··· ···
.. .
..
∂ T N+1 ∂ T N+1 ∂ T N+1 ∂b N ∂a1 ∂a2 ∂ N1 ∂ N1 ∂ N1 ∂b N ∂a1 ∂a2 ∂ N2 ∂ N2 ∂ N2 ∂b N ∂a1 ∂a2
.. .
..
. ···
∂ NN ∂b1
N $ j=1
..
∂ T N+1 ∂b1 ∂ N1 ∂b1 ∂ N2 ∂b1
(−1) N = (2π ) N ⎛
···
.. .
∂b0
∂ T0 ∂b N ∂ T1 ∂b N
···
∂ NN ∂b N
(n−1)K+k n
.. .
.. .
∂ NN ∂a1
−N
m=1
··· ··· ··· ..
. ···
∂ NN ∂a2 N+1 $
.
' ' ' ' ' ' .. '' . ' ' ∂ T N+1 ' ∂a N+1 ' ' ∂ N1 ' ∂a N+1 ' ' ∂ N2 ' ∂a N+1 ' .. '' . ' ∂ N N '' ∂ T0 ∂a N+1 ∂ T1 ∂a N+1
∂a N+1
∂ T0 ∂ T0 ∂bm−1 ∂am
!
⎞ bj
aj
(R(ξ j))1/2 dξ j⎠ d (ξ ),
(76)
where ' ' 1 1 ' ' ' b0 b1 ' ' . .. ' . . ' . ' '(b0 ) N+1 (b1 ) N+1 ' ' 1 d (ξ ) := ' 1 ' ξ1 −b0 ξ1 −b1 ' 1 ' 1 ' ξ2 −b0 ξ2 −b1 ' ' . .. ' . . ' . ' ' 1 1 ' ξ −b ξ −b N
0
N
1
···
1
1
1
···
bN .. .
a1 .. .
a2 .. .
..
.
· · · (bN ) N+1 (a1 ) N+1 (a2 ) N+1 ···
1 ξ1 −bN
1 ξ1 −a1
1 ξ1 −a2
···
1 ξ2 −bN
1 ξ2 −a1
1 ξ2 −a2
..
.. .
.. .
.. .
1 ξ N −bN
1 ξ N −a1
1 ξ N −a2
.
···
' ' ' ' · · · a N+1 ' ' ' .. .. ' . . ' ' N+1 ' · · · (a N+1 ) ' ' · · · ξ1 −a1 N+1 '' . (77) ' · · · ξ2 −a1 N+1 '' ' ' .. .. ' . . ' ' ' 1 ' · · · ξ −a ···
1
N
N+1
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The above determinant d (ξ ) has been calculated on pg. 780 of [63]: ! ! - . N N+1 N+1 N+1 j=1 i=1 (bi−1 −a j ) i, j=1 (ai −a j )(bi−1 −b j−1 ) i, j=1 (ξi −ξ j )
j
d (ξ ) =
(−1) N
N N+1 j=1
i=1
j
(ξ j −ai )(ξ j −bi−1 )
;
(78) but, for −∞ < b 0 < a1 < ξ1 < b1 < a2 < ξ2 < b2 < · · · < bN−1 < a N < ξ N <
bN < a N+1 < +∞, d (ξ ) = 0 (which means that it is of fixed sign), and bj 1/2 dξ j = 0, j= 1, 2, . . . , N; thus, for n ∈ N and k = 1, 2, . . . , K, a j (R(ξ j )) ⎛ ⎝
N $ j=1
⎞ bj
aj
(R(ξ j))1/2 dξ j⎠ d (ξ ) = 0.
(79)
It remains to show that ∂ T0 /∂b j−1 and ∂ T0 /∂a j, j= 1, 2, . . . , N+1, too, are non-zero: for this purpose, one exploits the fact that, for n ∈ N and k = 1, 2, . . . , K, ⎞ ⎞ ⎛ ⎛ s−1
(ξ ) κnkkq ( V 1 2 κ −1) nk ⎠+ ⎠ dξ ⎝ ⎝ + (80) T0 = 1/2 iπ n(ξ −αk ) q=1 n(ξ −α pq ) iπ J (R(ξ ))+ is independent of z. It follows from (64), the integral representation for hV (z) given in item (2) of the Lemma, and (71) and (72) that, for n ∈ N and k = 1, 2, . . . , K, ) ( ( ) ∂hV (z) z−b j−1 ∂ F(z) (n−1)K+k 1 z−b j−1 =− √ ∂b 2π i n ∂b j−1 R(z) j−1 1 − hV (z) , 2 (z−a j) ∂ F(z) 1 =− √ 2π i R(z) ∂a j −
j = 1, 2, . . . , N+1,
(n−1)K+k n
1 hV (z) , 2
(z−a j)
∂hV (z) ∂a j
j = 1, 2, . . . , N+1 :
using, now, the z-independence of T0 , and the fact that, for the case of regular R \ {α1 , α2 , . . . , α K } → R satisfying conditions (6)–(8), hV (b j−1 ), hV (aj) = 0, V: j= 1, 2, . . . , N+1, one shows that, for n ∈ N and k = 1, 2, . . . , K, ' (z−b j−1 ) ∂ F(z) '' 1 (n−1)K+k = hV (b j−1 ) = 0, j= 1, 2, . . . , N+1, √ ' n R(z) ∂b j−1 z=b j−1 4π i
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
' (z−a j) ∂ F(z) '' 1 (n−1)K+k = hV (a j) = 0, √ ' n R(z) ∂a j z=a j 4π i
333
j= 1, 2, . . . , N+1;
thus, via (71) and (72), one arrives at, for n ∈ N and k = 1, 2, . . . , K, ∂ T0 1 =− ∂b j−1 2
(n−1)K+k hV (b j−1 ), n
j= 1, 2, . . . , N+1,
and 1 ∂ T0 =− ∂a j 2
(n−1)K+k hV (a j), n
j= 1, 2, . . . , N+1,
whence, for n ∈ N and k = 1, 2, . . . , K, N+1 $ m=1
N+1 1 (n−1)K+k 2(N+1) $ ∂ T0 ∂ T0 = hV (b j−1 )hV (a j) = 0. ∂bm−1 ∂am 2 n j=1
Hence, Jac(T0 , T1 , . . . , T N+1 , N1 , N2 , . . . , N N ) = 0. In order to invoke the implicit function theorem, it remains, still, to show that T j, j= 0, 1, . . . , N+1, and Ni , i = 1, 2, . . . , N, are (real) analytic functions + of {b j−1 (zo ), a j(zo )} N+1 j=1 . From the definition of T j , j∈ Z0 , above, using the fact that they are independent of z, thus giving rise to zero residue contributions, a Residue Calculus calculation shows that, equivalently, for n ∈ N and k = 1, 2, . . . , K, 1 Tj= 2
* CV
⎞ ⎞ ⎛ ⎛ s−1
(ξ ) κnkkq ξj ( κ −1) V 2 nk ⎠+ ⎠ dξ, ⎝ ⎝ + (R(ξ ))1/2 iπ n(ξ −αk ) q=1 n(ξ −α pq ) iπ
j∈ Z+ 0,
where the (simple closed) contour CV has been defined heretofore: the 1/2 . As (R(z))1/2 is analytic only factor depending on {bj−1 , aj} N+1 j=1 is (R(z)) N+1 for z ∈ C \ ∪ N+1 ⊂ C \ ∪ j=1 [bj−1 , aj ], with int(C V ) ⊃ j=1 [bj−1 , aj ], and since C V N+1 1/2 {z} ∪ (∪ j=1 [bj−1 , aj]), it follows that, in particular, (R(z)) CV is an analytic function of {bj−1 , aj} N+1 j=1 , which implies, via the above (equivalent) contour integral representation for T j, j∈ Z+ 0 , that, for n ∈ N and k = 1, 2, . . . , K, Ti , i = 0, 1, . . . , N+1, are (real) analytic functions of {bj−1 , aj} N+1 j=1 . Recalling from the analysis above that, for n ∈ N and k = 1, 2, . . . , K, with z ∈ ∪ Nj=1 (aj, bj), 2π
s−1
κnkkq (n−1)K+k (z)− 2(κnk −1) −2 ( ) (H ψV )(z)− V n n(z−αk ) n z−α pq q=1 =−
)1/2 (n−1)K+k ( R(z) hV (z), n
334
K. T.-R. McLaughlin et al.
it follows from the definition of N j, j= 1, 2, . . . , N, that 1 Nj=− 2π
bj
aj
(R(ξ ))1/2 hV (ξ ) dξ,
j= 1, 2, . . . , N :
making the linear change of variables uj : C → C, ξ → uj(ξ ) := (bj −aj)−1 (ξ −aj), j= 1, 2, . . . , N, which take each of the compact intervals [aj, bj], j= 1, 2, . . . , N, onto [0, 1], and setting ⎛ 1/2 ˆ ( R(z)) := ⎝
j $
(z−bi1 −1 )
i1 =1
j−1 $
(z−ai2 )
i2 =1
N+1 $
(ai3 −z)
i3 = j+1
N+1 $
⎞1/2 (b i4 −1 −z)⎠
,
i4 = j+2
one arrives at, for n ∈ N and k = 1, 2, . . . , K, 1 )1/2 - ( ( ).1/2 1 2 N j = − (bj −aj) Rˆ (bj −aj)uj +aj uj(1−uj) 2π 0 ( ) × hV (bj −aj)uj +aj duj, j= 1, 2, . . . , N. Recalling that hV (z) is real analytic on R \ {α1 , α2 , . . . , α K }, in particular, h (bj−1 ), hV (aj) = 0, j= 1, 2, . . . , N+1, and that it is an analytic function of V N+1 1/2 ˆ bi−1 (zo ), ai (zo ) i=1 , and noting from the definition of ( R(z)) above that, it, too, is an analytic function of (bj −aj)uj +aj, and thus an analytic function N+1 of b i−1 (zo ), ai (zo ) i=1 , it follows that N j, j= 1, 2, . . . , N, are (real) analytic N+1 functions of b i−1 (zo ), ai (zo ) i=1 . Thus, as the Jacobian of the transformation b0 (zo ), b1 (zo ), . . . , bN (zo ), a1 (zo ), a2 (zo ), . . . , a N+1 (zo ) → T0 , T1 , . . . , T N+1 , N1 , N2 , . . . , N N is non N+1 R\ zero whenever bj−1 (zo ), aj(zo ) j=1 are chosen so that, for regular V: 3 2 N+1 {α1 , α2 , . . . , α K } → R satisfying conditions (6)–(8), J = ∪ j=1 bj−1 (zo ), aj(zo ) , and T j, j= 0, 1, . . . , N+1, and Ni , i = 1, 2, . . . , N, are (real) analytic functions N+1 of b m−1 (zo ), am (zo ) m=1 , it follows, via the implicit function theorem, that, for n ∈ N and k = 1, 2, . . . , K, bj−1 (zo ), aj(zo ), j= 1, 2, . . . , N+1, are real analytic functions of zo . R \ {α p1 , . . . , α ps−1 , α ps } → R (with Remark 3.2 It turns out that, for a V: α ps := αk ) satisfying conditions (6)–(8) of the form V(z) =
s −1
q=1 j=−2mq
∞
( ) j 2m ρ j,q z − α pq + ρ j,∞ z j,
j=0
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
335
where, for q = 1, 2, . . . , s, mq ∈ N, m∞ ∈ N, and ρ−2mq ,q , ρ2m∞ ,∞ > 0, a closedform expression for
1 hV (z) = 2
⎞ ⎛ s−1 κ
κ −1) 2i ( nk k q ⎝ ⎝ nk )⎠ ( + π n(ξ −αk ) n ξ −α pq CV
(ξ ) iV + π
!
⎛
−1 *
(n−1)K+k n
q=1
(R(ξ ))−1/2 dξ ξ −z
can be derived, where the definition of the contour CV (⊂ C \ {α1 , α2 , . . . , α K }) ( N+1 )1/2 , is given in item (2) of Lemma 3.7, (R(z))1/2 := j=1 (z−bj−1 )(z−aj ) N+1 and {bj−1 , aj} j=1 satisfy the locally solvable system of 2(N+1) real moment equations stated in item (1) of Lemma 3.7. A Residue Calculus calculation shows that
hV (z) =
z2m∞ −N−2 ℵ ⎛
· ⎝
−N−2 2m∞
2
p =0 j p =0
- ×
⎞⎛
l =0
⎞ q −1 N l$ $ 1 ⎠⎝ q ⎠ +m 2 =0 q =0 m
+ j p
N k p p =0 (b p )
- N
k0 ,k1 ,...,k N l0 ,l1 ,...,l N ˆ l| ˆ 2m∞ − j−N−2 0|k|+| ki 0, li 0, i∈{0,1,...,N}
j=0
p −1 N k$ $ 1
(2m∞ − j )ρ2m∞ − j,∞
q
. -
kl !
N lq q =0 (aq +1 )
. -
N ! =0 lm m
.
.
ˆ
ˆ
z− j−|k|−|l|
N+1 s 1 (−1)N (α pq ) ( i=1 |bi−1 −α pq ||ai −α pq |)−1/2 − ℵ q=1 (z−α pq )2mq +1 ·
0
(2mq + j)$−2mq − j,q
j=−2mq +1
k0 ,k1 ,...,k N l0 ,l1 ,...,l N ˆ l| ˆ 2mq + j 0|k|+| ki 0, li 0, i∈{0,1,...,N}
- N
). - N lq −1 (1 ). + j + m p q q =0 q =0 2 m j p =0 2 . - . - . - . · - N N N N k l p q (b −α ) (a −α ) k ! l ! pq pq =0 m p =0 p q =0 q +1 l =0 l m p =0
k p −1 (1
336
K. T.-R. McLaughlin et al. N (α pq ) s−1 2 κnkkq (−1) ˆ l|− ˆ j |k|+| × (z−α pq ) + ℵ q=1 n
+
2 ℵ
κnk −1 n
- N+1 i=1
|bi−1 −α pq ||ai −α pq |
.−1/2
(z−α pq )
.−1/2 (−1)N (αk ) N+1 |bi−1 −αk ||ai −αk | i=1
(z−αk )
,
z ∈ C \ {α p1 , . . . , α ps−1 , αk }, where ℵ := ((n−1)K+k)/n, ˆ := k0 +k1 +· · ·+k N , |k| ⎧ 0, α pq ∈ (a N+1 , +∞), ⎪ ⎪ ⎪ ⎨ N (αpq ) = N+1, α pq ∈ (−∞, b0 ), ⎪ ⎪ ⎪ ⎩ N− j+1, α pq ∈ (aj, bj),
ˆ :=l0 +l1 +· · ·+l N , |l|
q = 1, 2, . . . , s,
j= 1, 2, . . . , N,
and the primes (resp., double primes) on the summations mean that all N N possible sums over {kl }l=0 and {lk }k=0 must be taken for which 0 k0 +k1 +· · ·+ k N +l0 +l1 +· · ·+l N 2m∞ − j− N−2, j= 0, 1, . . . , 2m∞ − N−2, ki 0, li 0, i ∈ {0, 1, . . . , N} (resp., 0 k0 +k1 +· · ·+k N +l0 +l1 +· · ·+l N 2mq + j, j= −2mq + 1, −2mq +2, . . . , 0, ki 0, li 0, i ∈ {0, 1, . . . , N}). It is important to note that all of the above sums are finite sums: any sums for which the upper limit is less than the lower limit are defined to be equal to zero, and any products in which the upper limit is less than the lower limit are defined to be equal to one; for ∗) := 0 and −1 ∗) := 1. example, −1 j=0 (∗ j=0 (∗ Remark 3.3 It is interesting to note that, for n ∈ N and k = 1, 2, . . . , K, one may derive explicit formulae equilibrium mfor the various mmoments of theassociated−m measure, that is, ξ dμ (ξ ) = ξ ψ (ξ ) dξ and (ξ −α ) dμV (ξ ) = pq V V R J R −m (ξ −α ) ψ (ξ ) dξ , m ∈ N , q = 1, 2, . . . , s , in terms of the external field pq V J R \ {α1 , α2 , . . . , α K } → R satisfying conditions (6)–(8), and the multi-valued V: N+1 function (R(z))1/2 := ( i=1 (z−bi−1 )(z−ai ))1/2 , where {bj−1 , aj} N+1 j=1 satisfy the locally solvable system of 2(N+1) moment equations stated in item (1) of Lemma 3.7. Without loss of generality, and for demonstrative purposes only,consider, say, the following non-trivial moments: J (ξ −α pq )−r dμV (ξ ) and J ξ r dμV (ξ ), q = 1, 2, . . . , s, r= 1, 2, 3 (the scheme of the calculations below generalises to the moments J (ξ −α pq )−(r+3) dμV (ξ ) and J ξ r+3 dμV (ξ ), q = 1, 2, . . . , s, r ∈ N). Recall the representations for F(z) given in (the proof of) Lemma 3.7 (cf. (55) and (64)): ⎛ F(z) = −
s−1
κnkkq
1 ⎝ (κnk −1) (n−1)K+k + + iπ n(z−αk ) q=1 n(z−α pq ) n
⎞
J
ψV (ξ ) ⎠ dξ , ξ −z
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
337
⎛ ⎞ s−1 1 ⎝ (κnk −1) κnkkq ⎠ 2 (κnk −1) F(z) = − + + (R(z))1/2 iπ n(z−αk ) q=1 n(z−α pq ) iπ n(ξ −αk ) J +
⎞ (ξ ) (R(ξ ))−1/2 V dξ + ⎠+ ⎠ . n(ξ −α pq ) iπ ξ −z 2π i
s−1
⎞
κnkkq
q=1
that, for n ∈ N and k = 1, 2, . . . , K, μV ∈ M1 (R), in particular, Recalling −m (ξ −α ) dμV (ξ ) < ∞ and J ξ m dμV (ξ ) < ∞, m ∈ N, q = 1, 2, . . . , s, with p q J (ξ ) = 1, one derives, for n ∈ N and k = 1, 2, . . . , K, the following asJ dμV ymptotic expansions: (1) for ξ ∈ J and z ∈ / J such that |(z−αk )/(ξ −αk )| % 1 (e.g., 0 < |z − αk | % min{min j=1,2,...,N+1 {|bj−1 − αk |}, min j = 1,2,...,N+1 {|aj − αk |}, min j=1,2,...,N+1 {||bj−1 −aj|−αk |}, minq=1,2,...,s−1 {|α pq −αk |}}), via the expansions j (z−αk )l+1 + 1 ∗m k) ∗) = − ∞ = − lj=0 (ξ(z−α + (ξ −α l+1 (z−ξ ) , l ∈ Z0 , and ln(1−∗ m=1 m , (z−αk )−(ξ −αk ) −αk ) j+1 k) |∗∗| % 1, ⎛ s−1 (n−1)K+k 1 ⎝(κnk −1) κnk kq − + F(z) = − (ξ −αk )−1 ψV (ξ ) dξ z→αk iπ n(z−αk ) n(α pq −αk ) n J q=1
⎛ s−1
−⎝ q=1
⎛ s−1
−⎝ q=1
⎞ (n−1)K+k −2 − (ξ −αk ) ψV (ξ ) dξ⎠(z−αk ) n n(α pq −αk )2 J κnk kq
⎞ (n−1)K+k 2 − (ξ −αk )−3 ψV (ξ ) dξ⎠(z−αk ) n n(α pq −αk )3 J κnk kq
⎛
s−1
−⎝
q=1
κnk kq
(n−1)K+k − n n(α pq −αk )4
⎞
J
3 (ξ −αk )−4 ψV (ξ ) dξ ⎠ (z−αk )
⎞ + O ((z−αk )4 )⎠ ,
and F(z) = − z→αk
s−1 κnk 1 (κnk − 1) 1 kq + +(−1)N (αk ) Q0 iπ n(z − αk ) iπ n(α pq − αk ) q=1
⎛ ×⎝
N+1 $
⎞1/2 |bj−1 − αk ||aj − αk |⎠
⎛ +⎝
s−1 κnk 1 kq iπ n(α pq −αk )2 q=1
j=1
-
+ (−1)N (αk ) Q1 +α Q0
.
⎛ ⎝
N+1 $ j=1
⎞1/2 ⎞ ⎟ |bj−1 −αk ||aj −αk |⎠ ⎠ (z−αk )
338
K. T.-R. McLaughlin et al.
⎛
s−1 κnk kq ⎜1 N (αk ) 1 2 Q Q +⎝ +(−1) +α Q + β + (α ) 2 1 0 iπ 2 n(α pq −αk )3 q=1
⎛ ×⎝
N+1 $
⎞1/2 ⎞ ⎟ |bj−1 −αk ||aj −αk |⎠ ⎠ (z−αk )2 + O ((z−αk )3 ),
j=1
where (recall that α ps := αk ) N (αk ) is defined in Remark 3.2,
α := −
N 1 1 1 + , 2 bj−1 −αk aj −αk j=1
! N 1 1 1 , β := − ( )2 + 4 (aj −αk )2 bj−1 −αk j=1
Qj
⎛ ⎞ ⎞ s−1 κ
(ξ ) 2 ( V κ −1) nk kq nk ⎠+ ⎠ := ⎝ ⎝ + n(ξ −α pq ) iπ J iπ n(ξ −αk )
⎛
q=1
−1/2
×
(R(ξ ))+ dξ , (ξ −αk ) j+1 2π i
j = 0, 1, 2;
and (2) for ξ ∈ J and z ∈ / J such that |ξ/z| % 1 (e.g., |z| & max{maxq=1,2,...,s {|α pq |}, j l+1 1 max j=1,2,...,N+1 {|bj−1 − aj|}}), via the expansions ξ −z = − lj=0 zξj+1 + zl+1ξ(ξ −z) , ∗m ∗) = − ∞ ∗| % 1, l ∈ Z+ m=1 m , |∗ 0 , and ln(1−∗ ⎛ s−1 κ
1 ⎝ κnk − 1 1 (n − 1)K + k nk kq − αk + α pq − iπ nz iπ n n n
F(z) =
z→∞
q=1
⎞
× J
−
+
ξ ψV (ξ ) dξ ⎠
s−1 κ
1 1 ⎝ κnk − 1 nk kq 2 − (α ) + (α pq )2 k 2 iπ n n z q=1
(n − 1)K + k n
s−1 κ
nk kq q=1
⎛
n
⎞
J
ξ 2 ψV (ξ ) dξ ⎠
(αpq )3 −
(n − 1)K + k n
⎛
1 1 ⎝ κnk − 1 − (αk )3 iπ n z3 ⎞
J
ξ 3 ψV (ξ ) dξ ⎠
1 + O (z−5 ), z4
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
339
and, using the following N+2 moment equations (cf. (the proof of) Lemma 3.7) ⎛
ξj J
1/2
(R(ξ ))+
⎛ ⎞ ⎞ s−1 (ξ ) κ
2 κ −1) ( V nk k q ⎝ ⎝ nk ⎠+ ⎠ dξ = 0, + iπ n(ξ −αk ) n(ξ −α pq ) iπ q=1
j= 0, 1, . . . , N, ⎛ ⎞ ⎞ s−1 (ξ ) V ξ N+1 ⎝ 2 ⎝(κnk −1) κnk kq ⎠+ ⎠ dξ + 1/2 iπ n(ξ −α ) n(ξ −α pq ) iπ k J (R(ξ ))+ ⎛
q=1
= −2
(n−1)K+k , n
one arrives at
F(z) =
z→∞
+
⎛ ⎛ (n − 1)K + k α 1 1 κnk − 1 ⎝ ⎝ − Q0 − + αk iπ nz n iπ iπ n
s−1 κ
nk kq q=1
n
. ⎛ β + 1 (α )2 2 1 (n − 1)K + k α pq ⎠⎠ 2 − ⎝ Q1 + α Q0 − n iπ z ⎞⎞
⎛ ⎞⎞ s−1 κ
1 ⎝ κnk − 1 1 nk k q + (αk )2 + (α pq )2 ⎠⎠ 3 − Q2 + α Q1 iπ n n z q=1
γ + α β + . (n − 1)K + k + β + 12 (α )2 Q0 − n iπ
1 3 3! (α )
.
⎛ ⎞⎞ s−1 κ
1 ⎝ κnk − 1 nk kq 3 3 ⎠⎠ 1 (α pq ) (αk ) + + + O (z−5 ), iπ n n z4 q=1
where α := −
Q j :=
J
. . N ( N N ) 1 1 3 +a3 , 2 +a2 , γ := − 1 bj−1 +aj , β := bj−1 bj−1 j j 2 j=1 4 j=1 3! j=1
⎛ ⎛ ⎞ ⎞ s−1 (ξ ) dξ ξ N+2+ j ⎝ 2 ⎝(κnk −1) κnk V kq ⎠+ ⎠ + , j= 0, 1, 2. 1/2 iπ n(ξ −αk ) n(ξ −α pq ) iπ 2π i (R(ξ ))+ q=1
Equating coefficients of like powers of (z − αk )i , i = −1, 0, 1, 2, and z− j, j = 1, 2, 3, 4, in the above asymptotic expressions for F(z) as z → αk , k = 1, 2, . . . , K, and as z → ∞, respectively, one arrives at, for n ∈ N and
340
K. T.-R. McLaughlin et al.
k = 1, 2, . . . , K, the following expressions moments of the equilibrium measure:
J
(−1)N (αk ) ψV (ξ ) dξ = ξ −αk
for
the
‘first
three’
- N+1
.1/2 −1/2 |bj−1 −αk ||aj −αk | (R(ξ ))+
2(ℵ /n) J (ξ −αk )
j=1
⎛ ⎞ ⎞ s −1 κnk
(ξ ) i V κ −1) ( 2i kq nk ⎠+ ⎠dξ, + ×⎝ ⎝ π n(ξ −αk ) n(ξ −α pq ) π ⎛
q=1
J
(−1)N (αk ) ψV (ξ ) dξ = 2 (ξ −αk )
-
.1/2 ⎛ −1/2 (R(ξ ))+ ⎝ 2 J (ξ −αk )
N+1 j=1 |bj−1 −αk ||aj −αk |
2(ℵ /n)
⎛
⎛ ⎞ ⎞ s −1 (ξ ) 2i ⎝ (κnk −1) κnk iV kq ⎝ ⎠ ⎠dξ + × + π n(ξ −αk ) n(ξ −α pq ) π q=1
1 − 2
J
−1/2
(R(ξ ))+ (ξ −αk )
⎛
⎛ ⎞ s −1 κnk
κ −1) 2i ( kq nk ⎝ ⎝ ⎠ + π n(ξ −αk ) n(ξ −α pq ) q=1
⎞
⎞ N+1
(ξ ) 1 1 iV ⎠ dξ ⎠, + + π bm−1 −αk am −αk m=1
J
(−1)N (αk ) ψV (ξ ) dξ = (ξ −αk )3
- N+1
.1/2 ⎛ −1/2 |bj−1 −αk ||aj −αk | (R(ξ ))+ ⎝
3 2(ℵ /n) J (ξ −αk )
j=1
⎛ ⎞ ⎞ s −1 κ
(ξ ) ( κ −1) 2i i V nk kq nk ⎠+ ⎠ dξ + × ⎝ ⎝ π n(ξ −αk ) n(ξ −α pq ) π ⎛
q=1
1 − 2
J
⎛
⎛ ⎞ s −1 −1/2 κnk
(R(ξ ))+ κ −1) 2i ( kq nk ⎝ ⎝ ⎠ + π n(ξ −αk ) n(ξ −α pq ) (ξ −αk )2 q=1
⎞
N+1 −1/2
(ξ ) (R(ξ ))+ 1 1 iV ⎠ dξ + + + π b m−1 −αk am −αk J (ξ −αk ) m=1
⎛
⎛ ⎞ ⎞ s −1 (ξ ) iV 2i ⎝ (κnk −1) κnk kq ⎝ ⎠ ⎠ dξ × + + π n(ξ −αk ) n(ξ −α pq ) π q=1
⎛ 1 ×⎝ 8
1 − 4
N+1
m=1 N+1
m=1
1 1 + b m−1 −αk am −αk
!2
1 1 + (bm−1 −αk )2 (am −αk )2
!2
⎞⎞ ⎠⎠ ,
non-trivial
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
1 ξ ψV (ξ ) dξ = 2(ℵ /n) J
⎛
J
341
⎛ ⎞ s −1 κnk
2i ( κ − 1) kq nk ⎝ ⎝ ⎠ + π n(ξ − αk ) n(ξ − αpq )
ξ N+2 1/2
(R(ξ ))+
q=1
⎞
+
N+1
(ξ ) iV ⎠ dξ − 1 (bj−1 + aj ), π 2 j=1
⎛ ⎛ ⎛ ⎞ s −1 N+3 κnk
1 ξ κ −1) 2i ( kq nk ⎝ ⎝ ⎝ ⎠ + ξ ψV (ξ ) dξ = 1/2 2(ℵ /n) π n(ξ −αk ) n(ξ −α pq ) J J (R(ξ ))+ q=1
2
⎞ ⎞ ⎛ N+1 (ξ ) 1 ⎝ ξ N+2 iV ⎠ (bj−1 +aj )⎠ dξ + + 1/2 π 2 J (R(ξ ))+ j=1 ⎛ ⎞ ⎞ ⎞ s −1 (ξ ) iV 2i ⎝ (κnk −1) κnk kq ⎠ ⎠ dξ ⎠ ⎝ + + × π n(ξ −αk ) n(ξ −α pq ) π ⎛
q=1
⎛ ⎞2 N+1 . 1 N+1
1 - 2 2 bj−1 +aj + ⎝ (bj−1 +aj )⎠ , − 4 8 j=1
j=1
1 ξ ψV (ξ ) dξ = 2(ℵ /n) J 3
⎛ ⎝ J
⎞
+
⎛ ξ N+4 1/2
(R(ξ ))+ ⎛
⎛ ⎞ s −1 κnk
κ −1) 2i ( kq nk ⎝ ⎝ ⎠ + π n(ξ −αk ) n(ξ −α pq ) q=1
⎞
(ξ ) iV ⎠ dξ − 1 ⎝ (bj−1 +aj )⎠ π 2 N+1 j=1
J
ξ N+3 1/2
(R(ξ ))+
⎛ ⎞ ⎞ s −1 κnk
(ξ ) i V κ −1) ( 2i kq nk ⎠+ ⎠ dξ + ×⎝ ⎝ π n(ξ −αk ) n(ξ −α pq ) π ⎛
q=1
⎛ ⎜1 − ⎝ 4
N+1
⎛ 2 (bj−1 +aj2 )−
j=1
1⎝ 8
N+1
⎞2 ⎞ ⎟ (bj−1 +aj )⎠ ⎠
J
j=1
ξ N+2 1/2
(R(ξ ))+
⎛ ⎞ ⎞ ⎞ s −1 κnk
(ξ ) ( i V 2i κ −1) kq nk ⎠+ ⎠ dξ ⎠ + × ⎝ ⎝ π n(ξ −αk ) n(ξ −α pq ) π ⎛
q=1
⎞3 N+1 N+1 . 1 - 3 1 ⎝ (bj−1 +aj )⎠ − − bj−1 +aj3 8 · 3! 3! ⎛
j=1
+
j=1
⎞
⎛
! N+1 N+1
1 ⎝ (bj−1 +aj )⎠ (b 2m−1 +a2m ) , 8 j=1
m=1
where ℵ := ((n−1)K+k)/n. Note that, for n ∈ N and k = 1, 2, . . . , K, all of the ‘moment’ integrals above are real valued (since, for ξ ∈ J, with J ∩ {α1 , α2 , . . . , α K } = ∅, 1/2 (R(ξ ))+ = i(|R(ξ )|)1/2 ∈ iR) and bounded (since, for j= 1, 2, . . . , N+1, (R(ξ ))1/2 =ξ ↓bj−1 O ((ξ −bj−1 )1/2 ) and (R(ξ ))1/2 =ξ ↑aj O ((a j −ξ )1/2 ), that is, there
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are integrable singularities at the end-points of the support of the equilibrium measure).
R \ {α1 , α2 , . . . , α K } → R satisfy conLemma 3.8 Let the external field V: is regular. For n ∈ N and ditions (6)–(8). Suppose, furthermore, that V k = 1, 2, . . . , K, let the associated equilibrium measure, μV , and its support, supp(μV ) =: J = ∪ N+1 j=1 [bj−1 , aj ] (⊂ R \ {α1 , α2 , . . . , α K }), be as described in Lemma 3.7, and let there exist : N × {1, 2, . . . , K} → R, the associated variational constant, such that 2
(n−1)K+k n
−2
s −1
q=1
2
κnk kq
n
(n−1)K+k n s −1
−2
q=1
κnk kq
n
J
' ' z−ξ ln '' ξ −α
k
ln|z−α pq | + 2
' ' ' dμ (ξ )−2 κnk −1 ln|z−αk | V ' n
s −1
q=1
J
' ' z−ξ ln '' ξ −α
k
κnk kq
n
ln|α pq −αk |− V(z)− = 0,
z ∈ J,
' ' ' dμ (ξ )−2 κnk −1 ln|z−αk | V ' n
s −1 κ
nk kq
ln|z−α pq | + 2
q=1
n
ln|α pq −αk |− V(z)− 0,
(81)
z ∈ R \ J,
regular, the inequality in the second of (81) is strict. Then, for where, for V − n ∈ N and k = 1, 2, . . . , K : (i) g+ (z)+g− (z)− P+ 0 − P0 − V(z)− = 0, z ∈ J, where ± P0 are defined in Lemma 3.4, and g± (z) := limε↓0 g(z±iε); (ii) g+ (z) + g− (z)− − P+ 0 − P0 − V(z)− 0, z ∈ R \ J, where equality holds for at most a finite number regular, the inequality is strict; (iii) of points, and, for V ⎧ 2π iG (z), z ∈ (bj−1 , aj), j= 1, 2, . . . , N+1, ⎪ ⎪ ⎪ ⎨2π iG (z), z ∈ (a , b ), i = 1, 2, . . . , N, i i + g+ (z)−g− (z)+ P− 0 − P0 = ⎪ 2π iG (z), z ∈ (a N+1 , +∞), ⎪ ⎪ ⎩ 2π iG (z), z ∈ (−∞, b 0 ), where
G (z) :=
(n−1)K+k n
−
q∈Δ(k;z)
G (z) :=
−
κnk kq
n
(n−1)K+k n
q∈Δ(k;z)
κnk kq
n
a N+1 z
ψV (ξ ) dξ − κnk kq
+
q∈Δˆ 2 (k)
a N+1 bi
+
n
−
q∈Δˆ 2 (k)
κnk kq
n
−
J∩R> α
ψV (ξ ) dξ
k
κnk −1 1R<α (z), k n
ψV (ξ ) dξ −
(n−1)K+k n
(n−1)K+k n
J∩R> α
κnk −1 1R<α (z), k n
k
ψV (ξ ) dξ
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
G (z) := −
J∩R> α
ψV (ξ ) dξ −
k
κnk kq
q∈Δ(k;z)
n
κnk −1 1R<α (z), k n
(n−1)K+k (n−1)K+k ψV − (ξ ) dξ − n n J∩R> α k
+
+
− G (z) :=
(n−1)K+k n
q∈Δˆ 2 (k)
κnk kq
n
−
κnk kq
n
q∈Δˆ 2 (k)
343
κnk kq
q∈Δ(k;z)
n
κnk −1 1R<α (z), k n
with Δˆ 2 (k) = { j∈ {1, 2, . . . , s −1}; α p j > αk }, and Δ(k; z) = { j∈ {1, 2, . . . , s −1}; z < α p j }; and (iv) κnkkq (n−1)K+k − + 1R<α (z)+2π i q∈Δ(k;z) i g+ (z)−g− (z)+ P0 − P0 +2π i k n n ⎧ ⎪ ⎨2π (n−1)K+k ψ (z) 0, z ∈ J, V n = ⎪ ⎩0, z ∈ R \ J, regular, equality where denotes differentiation with respect to z, and, for V holds for at most a finite number of points.
Proof For n ∈ N and k = 1, 2, . . . , K, set supp(μV ) =: J = ∪ N+1 j=1 J j , where Jj = [bj−1 , aj] (the jth band), with (cf. Lemma 3.7) N ∈ Z+ and finite, Jj ∩ {α1 , 0 j= 1, 2, . . . , N+1, Ji ∩ J j = ∅, i = j∈ {1, 2, . . . , N+1}, α2 , . . . , α K } = ∅, and −∞ < b0 < a1 < b1 < a2 < · · · < bN < a N+1 < +∞, and {bj−1 , aj} N+1 satisfy j=1 the locally solvable system of 2(N+1) real moment equations given in item (1) of Lemma 3.7. Present R as follows: R = (−∞, b0 ) ∪ (a N+1 , +∞)∪ (∪ N+1 j=1 (aj , bj )) ∪ J. Consider the following cases: (1) z ∈ J j , j= 1, 2, . . . , N+1; (2) z ∈ (ai , bi ), i = 1, 2, . . . , N; (3) z ∈ (a N+1 , +∞); and (4) z ∈ (−∞, b0 ). (1) Recall the definition of g(z) given in Lemma 3.4: ⎛ κnkn −1 (z−ξ ) 1 ln ⎝(z−ξ ) n g(z) := (z−αk )(ξ −αk ) J
×
s −1 $
q=1
(z−ξ ) (z−α pq )(ξ −α pq )
κnknkq
⎞ ⎠ ψV (ξ ) dξ,
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where the representation (cf. Lemma 3.7) dμV (x) = ψV (x) dx, x ∈ J, was used. For z ∈ J j = [bj−1 , aj], j= 1, 2, . . . , N+1, since J j ∩ {α1 , α2 , . . . , α K } = ∅, one shows from the above expression for g(z) that (n−1)K+k ln(|z−ξ |)ψV (ξ ) dξ g± (z) = n J κnk −1 κnk −1 − ln(|ξ −αk |)ψV (ξ ) dξ − iπ ψV (ξ ) dξ n n J J∩R< α k
−
s−1
κnkkq
q=1
−
n
s−1
κnkkq
q=1
∓ iπ
n
J
ln(|ξ −α pq |)ψV (ξ ) dξ − iπ
q∈Δj (k;z)
n
q=1
ln|z−α pq | ± iπ
s−1
κnkkq
κnkkq
n
−
(n−1)K+k n
J∩R< α pq
a N+1 z
ψV (ξ ) dξ
ψV (ξ ) dξ
. κnk −1 ln|z−αk |±iπ 1R<α (z) , k n
where g± (z) := limε↓0 g(z±iε), Δj(k; z) := {i ∈ {1, 2, . . . , s −1}; α pi > z ∈ J j}, and, since card(Δj(k; z)) is finite, the sum kq /n is finite for every q∈Δj (k;z) κnk ordered three-tuple (n, k, q) ∈ N × {1, 2, . . . , K} × Δj(k; z). (Note: the set Δ j(k; z) accounts for the fact that some of the real poles may lie to the right of J j z, j= 1, 2, . . . , N+1, in which case Δj(k; z) = ∅ and card(Δj(k; z)) = 0; if, however, there are no real poles to the right of J j z,j= 1, 2, . . . , N+1, then Δj(k; z) = ∅ and card(Δj(k; z)) = 0, in which ± case kq /n := 0.) Via the definition of P0 given in Lemma 3.4 q∈Δj (k;z) κnk and the above expression for g± (z), one arrives at, for n ∈ N and k = 1, 2, . . . , K, a N+1 ) 1 ( (n−1)K+k + − P ψV (ξ ) dξ g+ (z) − g− (z) + P− = 0 0 2π i n z (n−1)K+k − ψV (ξ ) dξ n J∩R> α k
−
q∈Δj (k;z)
−
κnkkq
n
+
κnkkq q∈Δˆ 2 (k)
n
κnk − 1 1R<α (z), k n
where Δˆ 2 (k) := {i ∈ {1, 2, . . . , s −1}; α pi > αk }, which shows that g+ (z)− + − + g− (z)+ P− 0 − P0 ∈ iR and Re(g+ (z)−g− (z)+ P0 − P0 ) = 0; moreover, via the
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
345
Fundamental Theorem of Calculus and the fact that ψV (x) 0, x ∈ J, one shows that, for z ∈ J j, j= 1, 2, . . . , N+1, i
+ g+ (z)−g− (z)+ P− 0 − P 0 +2π i
= 2π
κnk κnk −1 kq < 1Rα (z)+2π i q∈Δj (k;z) k n n
(n−1)K+k ψV (z) 0. n
Furthermore, using the first of (the variational) conditions (81), one shows that, for n ∈ N and k = 1, 2, . . . , K, + g+ (z)+g− (z)− P− 0 − P 0 − V(z)− = 2
+2
(n−1)K+k n
s−1 κ
nk kq q=1
−2
n
J
' ' z−ξ ln '' ξ −α
k
ln|α pq −αk | − 2
' ' ' ψ (ξ ) dξ ' V
s−1 κ
nk kq
ln|z − α pq |
n
q=1
κnk −1 ln|z−αk |− V(z)− = 0, n
(82)
which gives the formula for the variational constant : N × {1, 2, . . . , K} → R, which is the same [63] (see, also, Section 7 of [62]) for each compact interval J j, j= 1, 2, . . . , N+1; in particular, =2
+2
(n − 1)K + k n
s−1
κnkkq
q=1
n
J
'⎞ ⎛' 1 ' ' ' (bN + a N+1 ) − ξ ' ⎜' 2 '⎟ ln ⎝' '⎠ ψV (ξ ) dξ ' ' ξ − αk ' '
ln|α pq − αk | − 2
s−1
κnkkq
q=1
n
' ' '1 ' ' ln ' (bN + a N+1 ) − α pq '' 2
' ' '1 ' κnk − 1 ' 1 (bN + a N+1 ) . ln ' (bN + a N+1 ) − αk '' − V −2 n 2 2
(2) For n ∈ N and k = 1, 2, . . . , K, one shows that, for z ∈ (aj, bj), j = 1, 2, . . . , N, g± (z) = −
−
(n−1)K+k n κnk −1 n
n
J
ln(|z−ξ |)ψV (ξ ) dξ
J
s−1 κ
nk kq q=1
J
ln(|ξ −αk |)ψV (ξ ) dξ − iπ
ln(|ξ −α pq |)ψV (ξ ) dξ − iπ
κnk −1 n
ψV (ξ ) dξ
J∩R< α
k
s−1 κ
nk kq q=1
n
J∩R< αp
q
ψV (ξ ) dξ
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K. T.-R. McLaughlin et al.
−
s−1
κnkkq
q=1
∓ iπ
n
ln|z−α pq | ± iπ
q∈Δj (k;z)
κnkkq
n
−
(n−1)K+k n
a N+1
bj
ψV (ξ ) dξ
. κnk −1 ln|z−αk |±iπ 1R<α (z) , k n
whence, via the definition of P± 0 given in Lemma 3.4, one arrives at a N+1 ) (n−1)K+k 1 ( + g+ (z)−g− (z)+ P− = − P ψV (ξ ) dξ 0 0 2π i n bj −
(n−1)K+k n
−
κnkkq
q∈Δj (k;z)
−
n
J∩R> α
ψV (ξ ) dξ
k
+
κnkkq n
q∈Δˆ 2 (k)
κnk −1 1R<α (z), k n
+ which shows that g+ (z)−g− (z)+ P− 0 − P0 = i const ., where const . ∈ R, and − + Re(g+ (z)−g− (z)+ P0 − P0 ) = 0; moreover, for z ∈ (aj, bj), j= 1, 2, . . . , N,
⎛
+ i ⎝g+ (z)−g− (z)+ P− 0 − P 0 +2π i
κnk −1 1R<α (z)+2πi k n
κnk kq
q∈ j (k;z)
n
⎞
⎠ = 0.
Furthermore, for n ∈ N and k = 1, 2, . . . , K, one shows that, for z ∈ (aj, bj), j= 1, 2, . . . , N, + g+ (z)+g− (z)− P− 0 − P 0 − V(z)− = 2
−2
+2
(n−1)K+k n (n−1)K+k n
s−1 κ
nk kq q=1
−2
n
J
ln(|z−ξ |)ψV (ξ ) dξ
J
ln(|ξ −αk |)ψV (ξ ) dξ
ln|α pq −αk | − 2
s−1 κ
nk kq q=1
n
ln|z−α pq |
κnk −1 ln|z−αk |− V(z)−. n
(83)
Recalling that (cf. the proof of Lemma 3.7) H : LM2 2 (C) (∗) → LM2 2 (C) (∗), f → f (ξ ) dξ (H f )(z) := lim |z−ξ |>ε z−ξ (the Hilbert transform of f ), one shows that, for π ε↓0
z ∈ (aj, bj), j= 1, 2, . . . , N, ln(|z−ξ |)ψV (ξ ) dξ = π J
z
aj
(H ψV )(ξ ) dξ +
J
ln(|aj −ξ |)ψV (ξ ) dξ.
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
347
The problem, however, is that there may be real poles between aj and z z ∈ (aj, bj), in which case, the integral aj (H ψV )(ξ ) dξ has to handled care-
fully. Towards this end, and for z ∈ (aj, bj), j= 1, 2, . . . , N, let Δj := {α pq , q ∈ {1, 2, . . . , s}; aj < α pq < z}, j= 1, 2, . . . , N, with card(Δ j ) =: K j , and where, for simplicity, the α pq ’s are enumerated so that aj < α p1 < α p2 < · · · < α pK < z: if j
Δj = ∅, then K j = 0; otherwise, if Δj = ∅, then K j = 0 (and finite). For Δj = ∅, j= 1, 2, . . . , N, set, for α pq ∈ Δj , q ∈ {1, 2, . . . , K j }, Uδ (α pq ) := {x ∈ q
R; |x−α pq | < δq }, where δq is some arbitrarily fixed, sufficiently small positive real number chosen so that, for i = q ∈ {1, 2, . . . , K j }, Uδ (α pi ) ∩ Uδ (α pq ) = ∅. q
i
0 With the understanding that, if Δj = ∅ (thus K j = 0), then ∪q=1 Uδ (α pq ) := ∅, q
one writes, for z ∈ (aj, bj), j= 1, 2, . . . , N,
⎛ z
aj
(H ψV )(ξ ) dξ = ⎝
Kj
(aj ,z)\∪q=1 U (α pq )
+
j ⎜ =⎝
+
U (α pq )
q=1
δq
K j −1
K
δq
α pi+1 −δi+1
i=1
α pi+δi
× (H ψV )(ξ ) dξ ⎛ Kj K j −1
⎜ + =⎝ q=1
U (α pq ) δq
i=1
× lim ε↓0
{ω∈J; |ξ −ω|>ε}
⎠ (H ψV )(ξ ) dξ
Kj
∪q=1 U (α pq )
δq
⎛
⎞
+
α pi+1 −δi+1
α pi +δi
α p1 −δ1
+
+
α p1 −δ1
+
aj
⎟ ⎠
α p +δ Kj Kj
aj
⎞ z
⎞ z
αp
+δ
Kj
⎟ ⎠
Kj
ψV (ω) dω : ξ −ω π
now, setting ε j := min{ε, maxq=1,2,...,K {δq }}, j= 1, 2, . . . , N, and considering j the ε j → 0 limit, one evaluates, via Fubini’s theorem, the above integrals (giving rise to integrable logarithmic singularities) to arrive at ⎛ ⎜ ⎝
α p1
aj
K j −1
+
i=1
α pi+1
α pi
+
⎞ z
αp
Kj
⎟ ⎠ (H ψV )(ξ ) dξ =
z
aj
(H ψV )(ξ ) dξ ;
whence, for z ∈(aj, bj), j= 1, 2, . . ., N, it follows that the relation J ln(|z− z ξ |)ψV (ξ ) dξ = π aj (H ψV )(ξ ) dξ + J ln(|aj −ξ |)ψV (ξ ) dξ holds true. Substitut-
348
K. T.-R. McLaughlin et al.
ing the latter relation into (83), one shows that, for n ∈ N and k = 1, 2, . . . , K, with z ∈ (aj, bj), j= 1, 2, . . . , N,
− g+ (z)+g− (z)− P+ 0 − P0 − V(z)− = 2π
(n−1)K+k n
(n−1)K+k n
+2
s −1
+2
κnk kq
n
q=1
= 2π +2 −2
κnk kq
n
κnk −1 −2 n s −1
q=1
= 2π z
aj
−2
ln(|aj −ξ |)ψV (ξ ) dξ
J
ln(|ξ −αk |)ψV (ξ ) dξ
q=1
z
aj
J
ln(|aj −ξ |)ψV (ξ ) dξ
J
ln(|ξ −αk |)ψV (ξ ) dξ
j ) − − ln|α pq −αk |− V(a
aj
κnk kq
ln|aj −α pq |+
n
(n−1)K+k n
z
aj
z
aj
z
(ξ ) dξ V
aj
1 dξ ξ −αk
!
1 dξ ξ −α pq
!
(H ψ V )(ξ ) dξ
κnk kq n
z
aj
(n−1)K+k 1 dξ + 2 ξ −α pq n
ln(|aj −ξ |)ψV (ξ ) dξ −2
J
z
ln|aj −αk |+
×
ln|z−α pq |
n
(H ψ V )(ξ ) dξ
J
κnk kq
z 1 (ξ ) dξ − 2 κnk −1 V dξ n aj ξ −αk
s −1
×
s −1
q=1
(n−1)K+k n
(H ψ V )(ξ ) dξ
ln|α pq −αk | − 2
(n−1)K+k n
q=1
−
J
(n−1)K+k n
s −1
−2
aj
κnk −1 ln|z−αk |− V(z)− n
−2
+2
z
(n−1)K+k n
−2
ln(|ξ −αk |)ψV (ξ ) dξ + 2
(n−1)K+k n
s −1
q=1
κnk kq
n
ln|α pq −αk |
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
349
s −1 κ
κnk −1 nk kq ln|aj −αk |−2 ln|aj −α pq | n n
−2
q=1
j )− , − V(a
which, via (82), simplifies to − g+ (z)+g− (z)− P+ 0 − P 0 − V(z)− =
⎛ z
aj
⎝2π
(n−1)K+k (ξ ) (H ψV )(ξ )− V n
⎞ s−1 κnk
2(κnk −1) kq ⎠ dξ. − −2 n(ξ −αk ) n(ξ −α pq )
(84)
q=1
From (the proof of) Lemma 3.7, (55), one gets that, for z ∈ C \ (J ∪ {α p1 , . . . , α ps−1 , αk }), ⎛ ⎞ s−1
κ 1 (κnk −1) (n−1)K+k nkkq ⎠, (H ψV )(z) = iF(z)+ ⎝ + n π n(z−αk ) q=1 n(z−α pq ) and, from (the proof of) Lemma 3.7, (65), and the integral representation for hV (z) given in item (2) of Lemma 3.7, one gets that (z) 1 (n−1)K+k V − (R(z))1/2 hV (z), F(z) = 2π i 2π i n whence, one shows that, for z ∈ (aj, bj), j= 1, 2, . . . , N, 2π
s−1
κnkkq (n−1)K+k (z)+ 2(κnk −1) +2 (H ψV )(z) = V n n(z−αk ) n(z−α pq ) q=1 (n−1)K+k − (R(z))1/2 hV (z). n
(85)
(In fact, the latter (85) is true for all z ∈ C \ (J ∪ {α p1 , . . . , α ps−1 , αk }), and not just for z ∈ (aj, bj), j= 1, 2, . . . , N; see, also, below.) Substituting (85) into (84), one arrives at, for n ∈ N and k = 1, 2, . . . , K, with z ∈ (aj, bj), j= 1, 2, . . . , N, z (n−1)K+k − g+ (z)+g− (z)− P+ − P − V(z)− = − (R(ξ ))1/2 hV (ξ ) dξ 0 0 n aj
(< 0).
Since (in the regular case) hV : R \ {α1 , α2 , . . . , α K } → R is real analytic (and clearly not identically zero), it follows that one has equality only at points z ∈ ∪ Nj=1 (aj, bj) for which hV (z) = 0, which are finitely denumerable, that is, 1/2
card({z ∈ ∪ Nj=1 (aj, bj); hV (z) = 0}) < ∞. (Note that, for z ∈ ∪ Nj=1 (aj, bj), (R(z))+ = 1/2
(R(z))− = (R(z))1/2 .) One also shows, using (83), that, for n ∈ N and k = 1, 2, . . . , K, ξ ∈ J, z ∈ (aj, bj), j=1, 2, . . . , N, and |(z−αk )/(ξ−αk )|%1 (e.g., 0 < |z− αk | % min{min j=1,2,...,N+1 {||bj−1 −aj|−αk |}, minq=1,2,...,s−1 {|α pq −αk |}}), via the
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j (z−αk )l+1 + 1 k) ∗) = expansions (z−αk )−(ξ = − lj=0 (ξ(z−α + (ξ −α l+1 (z−ξ ) , l ∈ Z0 , and ln(1−∗ −αk ) −αk ) j+1 k) ∞ ∗ m − m=1 m , |∗∗| % 1, since J ∩ {α1 , α2 , . . . , α K } = ∅, κnk −1 − −2 g+ (z)+g− (z)− P+ − P − V(z)− = − V(z)− ln(|z − α | +1) k 0 0 z→αk n +O (1), R \ {α1 , α2 , . . . , α K } → R is which, via condition (8) and the fact that V: + − real analytic, shows that g+ (z)+g− (z)− P0 − P0 − V(z)− < 0, z ∈ (aj, bj), j= 1, 2, . . . , N. (3) For n ∈ N and k = 1, 2, . . . , K, one shows that, for z ∈ (a N+1 , +∞),
g± (z) = −
−
(n−1)K+k n κnk −1 n
−
n
ln(|z−ξ |)ψV (ξ ) dξ
J
n
s−1 κ
nk kq q=1
J
s−1 κ
nk kq q=1
J
ln(|ξ −αk |)ψV (ξ ) dξ − iπ
ln(|ξ −α pq |)ψV (ξ ) dξ − iπ
κnk −1 n
κnk kq
ln|z−α pq |∓ iπ
n
q∈Δ(k;z)
ψV (ξ ) dξ
J∩R< α
k
s−1 κ
nk kq n
q=1
−
ψV (ξ ) dξ
J∩R< αp
q
κnk −1 (ln|z−αk | n
± iπ1R<α (z)), k
where Δ(k; z) := { j∈ {1, 2, . . . , s −1}; α p j > z} (if Δ(k; z) = ∅, that is, card(Δ(k; z)) = 0, then q∈Δ(k;z) κnkkq /n := 0), whence, via the definition of P± 0 given in Lemma 3.4, one arrives at (n−1)K+k 1 (g+ (z)−g− (z)+ P− − P+ )=− ψV (ξ ) dξ 0 0 2πi n J∩R> α k
κnk kq
−
q∈Δ(k;z)
n
+
κnk kq
n
q∈Δˆ 2 (k)
−
κnk −1 1R<α (z), k n
+ which shows that g+ (z)−g− (z)+ P− 0 − P0 = i const ., where const . ∈ R, and Re(g+ − + (z)−g− (z)+ P0 − P0 ) = 0; moreover,
⎛
κnk −1 + 1R<α (z)+2πi i ⎝g+ (z)−g− (z)+ P− 0 − P0 +2πi k n
q∈Δ(k;z)
κnk kq
n
⎞
⎠ = 0,
z ∈ (a N+1 , +∞).
Furthermore, for z ∈ (a N+1 , +∞),
n∈N
and
+ g+ (z)+g− (z)− P− 0 − P 0 − V(z)− = 2
k = 1, 2, . . . , K,
(n−1)K+k n
one
shows
that,
J
ln(|z−ξ |)ψV (ξ ) dξ
for
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal...
−2
+2
(n−1)K+k n
s−1 κ
nk kq
n
q=1
−2
351
J
ln(|ξ −αk |)ψV (ξ ) dξ
ln|α pq −αk | − 2
s−1 κ
nk kq q=1
n
ln|z−α pq |
κnk −1 ln|z−αk |− V(z)−. n
(86)
Following through with the analogue of the (removable logarithmic singularity) analysis in case (2) above, one shows that, for z ∈ (a N+1 , +∞),
J
ln(|z−ξ |)ψV (ξ ) dξ = π
z
a N+1
(H ψV )(ξ ) dξ +
J
ln(|a N+1 −ξ |)ψV (ξ ) dξ,
whence, via (86) and the latter relation, one shows that, for n ∈ N and k = 1, 2, . . . , K, with z ∈ (a N+1 , +∞), − g+ (z) + g− (z) − P+ 0 − P 0 − V(z)− = 2π
(n−1)K+k n
(n−1)K+k n
+2
(n−1)K+k n
−2
s−1 κ
nk kq
+2
n
q=1
= 2π
z
(H ψV )(ξ ) dξ
J
ln(|a N+1 −ξ |)ψV (ξ ) dξ
J
ln(|ξ −αk |)ψV (ξ ) dξ
ln|α pq −αk | − 2
s−1 κ
nk kq q=1
n
ln|z−α pq |
z
a N+1
(H ψV )(ξ ) dξ
z 1 (ξ ) dξ − 2 κnk −1 dξ V n a N+1 ξ −αk
s−1 κ
nk kq q=1
+
a N+1
(n−1)K+k n
a N+1
−2
z
κnk −1 ln|z−αk |− V(z)− n
−2
−
n
z
a N+1
1 dξ ξ −α pq
(n−1)K+k 2 ln(|a N+1 −ξ |)ψV (ξ ) dξ n J
−2
(n−1)K+k n
J
ln(|ξ −αk |)ψV (ξ ) dξ
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nk kq
+2
n
q=1
ln|α pq −αk | − 2
× ln|a N+1 −αk | − 2
s−1 κ
nk kq q=1
n
κnk −1 n
ln|a N+1 −α pq |
N+1 )− , − V(a which, via (82), simplifies to − g+ (z)+g− (z)− P+ 0 − P 0 − V(z)−
=
z
2π
a N+1
(n−1)K+k (ξ ) (H ψ V )(ξ ) − V n
⎞ s−1 κnk
2(κnk −1) kq ⎠ dξ. − −2 n(ξ −αk ) n(ξ −α pq )
(87)
q=1
Via (85), the latter equation simplifies to, for z ∈ (a N+1 , +∞), z (n−1)K+k − − P − V(z)− = − (R(ξ ))1/2 hV g+ (z)+g− (z)− P+ (ξ ) dξ 0 0 n a N+1
(< 0).
Since (in the regular case) hV : R \ {α1 , α2 , . . . , α K } → R is real analytic (and clearly not identically zero), it follows that one has equality only at points z˜ ∈ (a N+1 , +∞) for which hV which are finitely denumerable, that is, card({˜z ∈ (˜z) = 0, (a N+1 , +∞); hV N and k = 1, 2, . . . , K, (˜z) = 0}) < ∞. Furthermore, recalling that, for n∈ m M 1 (R), in particular, J (ξ −αk )−m ψV μV ∈ (ξ ) dξ < ∞ and J ξ ψV (ξ ) dξ < ∞, m ∈ N, with J ψV (ξ ) dξ = 1, one derives the following asymptotic expansions: (i)
for ξ ∈ J, z ∈ (a N+1 , +∞), and |(z−αk )/(ξ −αk )| % 1 (e.g., 0 < |z−αk | % min{min j=1,2,...,N+1 {||bj−1 −aj|−αk |},minq=1,2,...,s−1 {|α pq −αk |}}), via the expanl (z−αk ) j (z−αk )l+1 + 1 ∗) = sions (z−α )−(ξ j=0 (ξ −αk ) j+1 + (ξ −αk )l+1 (z−ξ ) , l ∈ Z0 , and ln(1−∗ −αk ) = − k ∞ ∗ m − m=1 m , |∗∗| % 1, and (86), κnk −1 − −2 +1) − P − V(z)− = − V(z)− | g+ (z)+g− (z)− P+ ln(|z−α k 0 0 z→αk n + O (1),
which, via condition (8), shows that, for n ∈ N and k = 1, 2, . . . , K, g+ (z)+g− (z)− − P+ 0 − P 0 − V(z)− < 0, z ∈ (a N+1 , +∞); (ii) for ξ ∈ J, z ∈ (a N+1 , +∞), and |ξ/z| % 1 (e.g., |z| & max{max j=1,2,...,N+1 {|bj−1− j l+1 1 = − lj=0 zξj+1 + zl+1ξ (ξ −z) , l ∈ Z+ aj|}, maxq=1,2,...,s {|α pq |}}), via the expansions ξ −z 0, ∗m ∗ and ln(1−∗∗) = − ∞ , |∗ | % 1, and (86), m=1 m . − 1 2 g+ (z)+g− (z)− P+ = − V(z)− 0 − P 0 − V(z)− n ln(z +1) z→+∞
+O (1), which, via condition (7), shows that, for n ∈ N and k = 1, 2, . . . , K, g+ (z)+g− (z)− − P+ 0 − P 0 − V(z)− < 0, z ∈ (a N+1 , +∞).
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(4)
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For n ∈ N and k = 1, 2, . . . , K, one shows that, for z ∈ (−∞, b0 ), (n−1)K+k g± (z) = ln(|z−ξ |)ψV (ξ ) dξ n J −
−
κnk −1 n
J
s−1 κ
nk kq q=1
−
n
s−1 κ
nk kq q=1
n
J
ln(|ξ −αk |)ψV (ξ ) dξ − iπ
κnk −1 ψV (ξ ) dξ n J∩R< α k
ln(|ξ −α pq |)ψV (ξ ) dξ − iπ
s−1 κ
nk kq q=1
(n−1)K+k n
ln|z−α pq |±iπ
n
q
κnk kq
∓ iπ
ψV (ξ ) dξ
J∩R< αp
n
q∈Δ(k;z)
κnk −1 (ln|z−αk |±iπ1R<α (z)) k n (if Δ(k; z) = ∅, that is, card(Δ(k; z)) = 0, then q∈Δ(k;z) κnk kq /n := 0), whence, via the −
definition of P± 0 given in Lemma 3.4, one arrives at (n−1)K+k (n−1)K+k 1 ( +) g+ (z)−g− (z)+ P− = − − P ψV (ξ ) dξ 0 0 2πi n n J∩R> α k
−
κnk kq
q∈Δ(k;z)
n
+
κnk kq
q∈Δˆ 2 (k)
n
−
κnk −1 1R<α (z), k n
+ which shows that g+ (z)−g− (z)+ P− 0 − P 0 = i const ., where const . ∈ R, and Re(g+ (z)− − + g− (z)+ P0 − P0 ) = 0; moreover, κnk κnk −1 kq + < i g+ (z)−g− (z)+ P− 1 − P +2π i (z)+2πi = 0, Rα q∈Δ(k;z) 0 0 k n n
z ∈ (−∞, b0 ). Furthermore, for n ∈ N and k = 1, 2, . . . , K, one shows that, for z ∈ (−∞, b 0 ), ' ' ' z−ξ ' (n−1)K+k + ' ' g+ (z)+g− (z)− P− − P − V(z)− = 2 ln (ξ ) dξ 0 0 ' ξ −α ' ψV n k J +2
s−1 κ
nk kq q=1
n
ln|α pq −αk | − 2
s−1 κ
nk kq q=1
κnk −1 ln|z−αk |− V(z)−. −2 n
n
ln|z−α pq |
(88)
Following through with the analogue of the (removable logarithmic singularity) analysis in case (2) above, one shows that, for z ∈ (−∞, b0 ), b0 ln(|z−ξ |)ψV (H ψ V ln(|b 0 −ξ |)ψV (ξ ) dξ = −π )(ξ ) dξ + (ξ ) dξ, J
z
J
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whence, via (88), (82), and (85), and this latter relation, one shows that, for n ∈ N and k = 1, 2, . . . , K, with z ∈ (−∞, b0 ), b0 (n−1)K+k − g+ (z)+g− (z)− P+ − P − V(z)− = (R(ξ ))1/2 hV (ξ ) dξ (< 0) : 0 0 n z equality holds only on the set {ˆz ∈ (−∞, b0 ); hV (ˆz) = 0}, for which card({ˆz ∈ (−∞, b0 ); hV (ˆz) = 0}) < ∞. Furthermore, as in case (3) above, one derives the following asymptotic expansions: (i)
for ξ ∈ J, z ∈ (−∞, b 0 ), and |(z − αk )/(ξ − αk )| % 1 (e.g., 0 < |z − αk | % min {min j=1,2,...,N+1 {||bj−1 −aj|−αk |},minq=1,2,...,s−1 {|α pq −αk |}}), via the expansions l (z−αk )l+1 (z−αk ) j + 1 ∗m ∗) = − ∞ j=0 (ξ −α ) j+1 + (ξ −α )l+1 (z−ξ ) , l ∈ Z0 , and ln(1−∗ m=1 m , (z−α )−(ξ −α ) = − k
k
|∗∗| % 1, and (88),
k
k
κnk −1 − −2 +1) g+ (z)+g− (z)− P+ ln(|z−α − P − V(z)− = − V(z)− | k 0 0 z→αk n +O (1),
which, via condition (8), shows that, for n ∈ N and k = 1, 2, . . . , K, g+ (z)+g− (z)− − P+ 0 − P 0 − V(z)− < 0, z ∈ (−∞, b0 ); (ii) for ξ ∈ J, z ∈ (−∞, b0 ), and |ξ/z| % 1 (e.g., |z| & max{max j=1,2,...,N+1 {|bj−1 −aj|}, j l+1 1 = − lj=0 zξj+1 + zl+1ξ (ξ −z) , l ∈ Z+ maxq=1,2,...,s {|α pq |}}), via the expansions ξ −z 0 , and ∞ ∗ m ln(1−∗∗) = − m=1 m , |∗∗| % 1, and (88), 1 − 2 ln(z g+ (z)+g− (z)− P+ − P − V(z)− = − V(z)− +1) + O (1), 0 0 z→−∞ n which, via condition (7), shows that, for n ∈ N and k = 1, 2, . . . , K, g+ (z)+g− (z)− − P+ 0 − P 0 − V(z)− < 0, z ∈ (−∞, b 0 ). This concludes the proof.
4 The Monic FPC ORF Family of Model RHPs In this section, the monic FPC ORF family of K auxiliary matrix RHPs formulated in Lemma 3.4 is augmented, by means of a sequence of contour deformations and transformations à la Deift–Venakides–Zhou [57–59], into simpler, ‘model’ matrix RHPs which, as n → ∞, can be solved explicitly in terms of Riemann theta functions (associated with the underlying family of K genus-N hyperelliptic Riemann surfaces) and Airy functions. R \ {α1 , α2 , . . . , α K } → R satisfy condiLemma 4.1 Let the external field V: is regular. For n ∈ N and k = tions (6)–(8). Suppose, furthermore, that V 1, 2, . . . , K, let the associated equilibrium measure, μV , and its compact support, supp(μV ) =: J = ∪ N+1 j=1 [b j−1 , a j ] (⊂ R \ {α1 , α2 , . . . , α K }), be as described in Lemma 3.7, and, along with the corresponding variational constant, , satisfy the variational conditions stated in Lemma 3.8, (81); moreover, let the
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corresponding conditions (i)–(iv) stated in Lemma 3.8 be valid. For n ∈ N and k = 1, 2, . . . , K, let Y : N × {1, 2, . . . , K} × C \ R → SL2 (C) solve the RHP (Y(n, k; z) := Y(z), V(z), R) formulated in Lemma 3.4, and set M(z) =
Y(z)E−σ3 , Y(z)Eσ3 ,
z ∈ C+ , z ∈ C− ,
where !
E := exp iπ((n−1)K+k)
J∩R> α
ψV (ξ ) dξ .
k
Then, for n ∈ N and k = 1, 2, . . . , K, M : N × {1, 2, . . . , K} × C \ R → SL2 (C) solves the following RHP: (i) M(n, k; z) := M(z) is holomorphic for z ∈ C \ R; (ii) the boundary values M± (z) := limε↓0 M(z±iε) satisfy the jump condition M+ (z) = M− (z)VM (z),
z ∈ R,
where, for j= 1, 2, . . . , N+1 and i = 1, 2, . . . , N, VM : N × {1, 2, . . . , K} × R → GL2 (C), (n, k, z) → VM (n, k; z) := VM (z) = ⎧ ! a N+1 ψV (ξ ) dξ e−2π i((n−1)K+k) z 1 ⎪ ⎪ , z ∈ (b j−1 , a j ), a ⎪ N+1 ⎪ ψV (ξ ) dξ ⎪ 0 e2π i((n−1)K+k) z ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
⎞ ⎛ a − −2π i((n−1)K+k) b N+1 ψV (ξ ) dξ n(g+ (z)+g− (z)−P + i 0 −P 0 − V(z)−) e e ⎠ ⎝ 2π i((n−1)K+k)
e
0 +
−
a N+1 bi
ψV (ξ ) dξ
I+en(g+ (z)+g− (z)−P 0 −P 0 −V(z)−) σ+ ,
, z ∈ (ai , bi ), z ∈ (−∞, b 0 ) ∪ (a N+1 , +∞),
with g(z), g± (z), and P± 0 defined in Lemma 3.4, ± Re i
a N+1 z
ψV (ξ ) dξ > 0,
z ∈ C± ∩ (∪ N+1 j=1 U j ),
where, for j= 1, 2, . . . , N+1, U j := {z ∈ C \ {α1 , α2 , . . . , α K }; infq∈(b j−1 ,a j ) |z− q|
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R\ {α1 , α2 , . . . , α K } → R, μV , and as described therein: (1) for j= for V: 1, 2, . . . , N+1 and i = 1, 2, . . . , N, + g+ (z)−g− (z)+ P− 0 − P0 = 2πi a N+1 (n−1)K+k (n−1)K+k ψV ψV (ξ ) dξ − (ξ ) dξ n n z J∩R> α
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
κnk
κnk ⎪ κnk −1 kq kq ⎪ ⎪ − + − 1R < (z), z ∈ [b j−1 , a j ], ⎪ ⎪ αk n n n ⎪ ⎪ q∈Δ(k;z) ⎪ ˆ 2 (k) q∈ Δ ⎪ ⎪ a ⎪ N+1 ⎪ (n−1)K+k (n−1)K+k ⎪ ⎪ ψ (ξ ) dξ − ψV ⎪ (ξ ) dξ ⎪ V ⎪ n n bi J∩R> ⎪ αk ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
κnk
κnk ⎪ κnk −1 kq kq ⎪ ⎪ − + − 1R < (z), z ∈ (ai , bi ), ⎪ αk ⎪ n n n ⎪ ⎨ q∈Δ(k;z) q∈Δˆ 2 (k)
κnk
κnk (n−1)K+k kq kq ⎪ ⎪ + − ψV ⎪ (ξ ) dξ − ⎪ > ⎪ n n n J∩Rα ⎪ ⎪ q∈Δ(k;z) k q∈Δˆ 2 (k) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ κnk −1 ⎪ ⎪ − (z), z ∈ (a N+1 , +∞), 1R < ⎪ ⎪ αk ⎪ n ⎪ ⎪ ⎪ κ ⎪
(n−1)K+k (n−1)K+k nk kq ⎪ ⎪ ⎪ ψV − (ξ ) dξ − ⎪ ⎪ > n n n ⎪ J∩ R αk ⎪ q∈Δ(k;z) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
κnk ⎪ κnk −1 kq ⎪ ⎪ − 1R < + (z), z ∈ (−∞, b0 ), ⎪ ⎪ αk ⎩ n n q∈Δˆ 2 (k)
where Δ(k; z) and Δˆ 2 (k) are defined in Lemma 3.8; and (2) for j= 1, 2, . . . , N+ 1 and i = 1, 2, . . . , N, − g+ (z)+g− (z)− P+ 0 − P0 − V(z)− =
⎧ 0, ⎪ ⎪ ⎪ ⎪ (n − 1)K + k z ⎪ 1/2 h (ξ ) dξ (< 0), ⎪ − ⎪ V a j (R(ξ )) ⎪ n ⎪ ⎨ (n − 1)K + k z 1/2 h (ξ ) dξ (< 0), ⎪− ⎪ V a N+1 (R(ξ )) ⎪ n ⎪ ⎪ ⎪ ⎪ ⎪ b0 ⎪ 1/2 h (ξ ) dξ (< 0), ⎩ (n − 1)K + k V z (R(ξ )) n
z ∈ [b j−1 , a j ], z ∈ (ai , bi ), z ∈ (a N+1 , +∞), z ∈ (−∞, b0 ).
Recall, also, the formula for the ‘jump matrix’ given in Lemma 3.4: −n(g (z)−g (z)+P− −P+ ) n(g (z)+g (z)−P+ −P− −V(z)−) + − − 0 0 0 0 e e + V(z) = . − + 0 en(g+ (z)−g− (z)+P0 −P0 ) Presenting R as above, and recalling that, for n ∈ N and k = 1, 2, . . . , K, κnk ∈ N and κnkkq ∈ Z+ 0 , q ∈ {1, 2, . . . , s −1}, via the definition of M(z) in terms of Y(z) stated in the Lemma, one obtains the formula for VM (z) stated in the Lemma, thus item (ii); furthermore, via the definition of M(z) in terms of Y(z) stated in the Lemma, items (iii)–(v) are the corresponding restatements of the respective items (iii)–(v) of Lemma 3.4. It remains, therefore,
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a to establish that ± Re(i z N+1 ψV (ξ ) dξ ) > 0 for z ∈ C± ∩ (∪ N+1 j=1 U j ), where, for j= 1, 2, . . . , N+1, U j := {z ∈ C \ {α1 , α2 , . . . , α K }; infq∈(b j−1 ,a j ) |z−q|
P− − P+ z ∈ ∪ N+1 0 (since, j=1 [b j−1 , a j ], implies that the quantity g+ (z)−g− (z)+ 0 by piecewise continuity, exp(2π i(κnk −1)1R<α (z)) = exp(2π i q∈Δ(k;z) κnkkq ) = k UV , of 1) has an analytic continuation, G(z), to an open neighbourhood, J, where UV := ∪ N+1 U , with U , j= 1, 2, . . . , N+1, defined above, with the j j j=1 property that ± Re(G(z)) > 0, z ∈ C± ∩ UV .
R \ {α1 , α2 , . . . , α K } → R satisfying conditions Remark 4.1 Recalling that V: (6)–(8) is regular, that is, for n ∈ N and k = 1, 2, . . . , K, hV (z) = 0 for z ∈ ∪ N+1 j=1 [b j−1 , a j ] and the second inequality of (81) is strict, and g+ (z) + − N g− (z) − P+ 0 − P0 − V(z) − < 0, z ∈ (−∞, b0 ) ∪ (a N+1 , +∞) ∪ (∪i=1 (ai , bi )), it follows that VM (z) =
n→∞
−(2πi((n−1)K+k)
e I+o(1)σ+ ,
a N+1 bi
ψV (ξ ) dξ )σ3
(I+o(1)σ+ ),
z ∈ (ai , bi ), i = 1, 2, . . . , N, z ∈ (−∞, b0 ) ∪ (a N+1 , +∞),
where, here and below, o(1) denotes terms that are exponentially small (as n → ∞). R \ {α1 , α2 , . . . , α K } → R satisfy condiLemma 4.2 Let the external field V: is regular. For n ∈ N and k = tions (6)–(8). Suppose, furthermore, that V 1, 2, . . . , K, let the associated equilibrium measure, μV , and its compact support, supp(μV ) =: J = ∪ N+1 j=1 [b j−1 , a j ] (⊂ R \ {α1 , α2 , . . . , α K }), be as described in Lemma 3.7, and, along with the corresponding variational constant, , satisfy the variational conditions stated in Lemma 3.8, (81); moreover, let the corresponding conditions (i)–(iv) stated in Lemma 3.8 be valid. For n ∈ N and k = 1, 2, . . . , K, let M : N × {1, 2, . . . , K} × C \ R → SL2 (C) solve the RHP (M(n, k; z) := M(z), VM (z), R) formulated in Lemma 4.1, and let the deformed (and oriented) contour := R ∪ (∪ N+1 j=1 (J j ∪ J j )) be as in Fig. 1 below, with
Fig. 1 Oriented and deformed contour := R ∪ (∪ N+1 j=1 (J j ∪ J j ))
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N+1 ∪ N+1 j=1 (Ω j ∪ Ω j ∪ J j ∪ J j ) ⊂ ∪ j=1 U j , where U j , j= 1, 2, . . . , N+1, are defined in Lemma 4.1. For n ∈ N and k = 1, 2, . . . , K, set
m (z) =
⎧ M(z), ⎪ ⎪ ⎪ . a N+1 ⎨ ψV (ξ ) dξ σ M(z) I−e−2π i((n−1)K+k) z − , ⎪ ⎪ . a N+1 ⎪ ⎩ ψV (ξ ) dξ σ M(z) I+e2π i((n−1)K+k) z − ,
. z ∈ C \ ∪ (∪ N+1 j=1 (' j ∪ ' j )) , . z ∈ C+ ∩ ∪ N+1 j=1 ' j , . z ∈ C− ∩ ∪ N+1 j=1 ' j .
Then, for n ∈ N and k = 1, 2, . . . , K, m : N × {1, 2, . . . , K} × C \ → SL2 (C) solves the following equivalent RHP: (i) m (n, k; z) := m (z) is holomorphic for z ∈ C \ ; (ii) the boundary values m± (z) := lim z → z ∈ m (z ) satisfy the z ∈ ± side of
jump condition
m+ (z) = m− (z)v (z),
z ∈ ,
where, for j= 1, 2, . . . , N+1 and i = 1, 2, . . . , N, v : N × {1, 2, . . . , K} × → GL2 (C), (n, k, z) → v (n, k; z) := v (z) =
⎧ z ∈ (b j−1 , a j ), iσ2 , ⎪ ⎪ a ⎪ ⎪ −2π i((n−1)K+k) z N+1 ψV ⎪ (ξ ) dξ σ , ⎪ I+e z∈ Jj , − ⎪ ⎪ a N+1 ⎪ ⎪ ψV (ξ ) dξ σ , ⎨I+e2π i((n−1)K+k) z z∈ Jj , − ⎞ ⎛ a N+1 + − −2π i((n−1)K+k) b ψV (ξ ) dξ ⎪ i ⎪ en(g+ (z)+g− (z)−P 0 −P 0 −V(z)−) ⎠ ⎪⎝e ⎪ a N+1 , z ∈ (ai , b i ), ⎪ ⎪ 2π i((n−1)K+k) b ψV (ξ ) dξ ⎪ i ⎪ 0 e ⎪ ⎪ + − ⎩ z ∈ (−∞, b 0 ) ∪ (a N+1 , +∞), I+en(g+ (z)+g− (z)−P 0 −P 0 −V(z)−) σ+ ,
with Re(i
a N+1 z
a N+1
ψV (ξ ) dξ ) > 0 (resp., Re(i
z
ψV (ξ ) dξ ) < 0), z ∈ Ω j (resp., z ∈
Ω j ); (iii) for l ∈ {+, −}, with t(+) :=, t(−) :=, r(+) := −1, and r(−) := +1, m (z)
(I+ O (z−αk ))Er(l)σ3 ;
=
z→αk
t(l) t(l) z ∈ Cl \∪ N+1 (J ∪Ω ) j=1 j j
(iv) m (z)
=
z→∞
O (1);
t(l) t(l) z ∈ Cl \∪ N+1 ∪Ω j ) j=1 (J j
(v) for q ∈ {1, 2, . . . , s −1}, m (z)
=
z→α pq
O (1).
t(l) t(l) z ∈ Cl \∪ N+1 ∪Ω j ) j=1 (J j
Proof For n ∈ N and k = 1, 2, . . . , K, items (i), (iii), (iv), and (v) in the formulation of the RHP for m (z) follow from the definition of m (z) in terms of M(z) given in the Lemma and the respective items (i), (iii), (iv), and (v) for the RHP for M(z) stated in Lemma 4.1; it remains, therefore, to verify
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item (ii), namely, the formula for v (z). Recall from item (ii) of Lemma 4.1 that, for n ∈ N, k = 1, 2, . . . , K, and z ∈ (b j−1 , a j), j= 1, 2, . . . , N+1, M+ (z) = a M− (z)VM (z), where VM (z) = exp(−(2π i((n−1)K+k) z N+1 ψV (ξ ) dξ )σ3 )+σ+ : noting the matrix factorisation ! a N+1 e−2π i((n−1)K+k) z ψV (ξ ) dξ 1 a N+1 = 0 e2π i((n−1)K+k) z ψV (ξ ) dξ
1
e2π i((n−1)K+k)
a N+1 z
ψV (ξ ) dξ
0 1
0 1 −1 0
1 e−2π i((n−1)K+k)
a z
N+1
ψV (ξ ) dξ
0 , 1
it follows that, for z ∈ (b j−1 , a j), j= 1, 2, . . . , N+1, 1 0 a m+ (z) N+1 −e−2π i((n−1)K+k) z ψV (ξ ) dξ 1
= m− (z)
1
e2π i((n−1)K+k)
a N+1 z
ψV (ξ ) dξ
0 iσ2 . 1
a It was shown in Lemma 4.1 that, for n ∈ N and k = 1, 2, . . . , K, ± Re(i z N+1 ψV (ξ ) dξ ) > 0, z ∈ C± ∩ U j, j= 1, 2, . . . , N+1. The terms ±2π i((n−1)K+k) a · z N+1 ψV (ξ ) dξ , which are pure imaginary for z ∈ R, and corresponding to a which exp(±2π i((n−1)K+k) z N+1 ψV (ξ ) dξ ) are undulatory, are continued analytically to C± ∩ (∪ N+1 j=1 U j ), respectively, corresponding to which exp(∓ a N+1 2π i((n−1)K+k) z ψV (ξ ) dξ ) are exponentially decreasing as n → ∞. As per the DZ non-linear steepest-descent method [57, 58] (see, also, the extension [59]), one now ‘deforms’ the original (and oriented) contour R to the deformed, or extended, (and oriented) contour/skeleton := R∪ (∪ N+1 j=1 (J j ∪ J j )) (Fig. 1) in such a way that the upper (resp., lower) ‘lips’
of the ‘lenses’ J j (resp., J j ), j= 1, 2, . . . , N+1, which are the boundaries of
' j (resp., ' j ), j= 1, 2, . . . , N+1, respectively, lie within the domain of ana+ lytic continuation of g+ (z)−g− (z)+ P− 0 − P0 (cf. proof of Lemma 4.1), that
N+1 is, ∪ N+1 j=1 (' j ∪ ' j ∪ J j ∪ J j ) ⊂ ∪ j=1 U j ; in particular, each (oriented and
bounded) open interval (b j−1 , a j), j= 1, 2, . . . , N+1, in the original (and oriented) contour R is ‘split’, or branched, into three, and the new (and oriented) contour is the old contour (R) together with the (oriented) boundary of N+1 lens-shaped regions, one region surrounding each (bounded and oriented) open interval (b j−1 , a j). Now, recalling the definition of m (z) in terms of M(z) given in the Lemma, and the expression for VM (z) given in Lemma 4.1, one arrives at, for n ∈ N and k = 1, 2, . . . , K, the formula for v (z) given in item (ii) of the Lemma. Recalling from Lemma 4.1 that, for n ∈ N and k = 1, 2, . . . , K, g+ (z)+ N g− (z)− P+ − P− b ) ∪ (a N+1 , +∞) ∪ (∪i=1 (ai , bi )), and 0 − V(z)− < 0, z ∈ (−∞, a N+10 a N+1 0 Re(i z ψV (ξ ) dξ ) > 0 (resp., Re(i z ψV (ξ ) dξ ) < 0), z ∈ J j (resp., z ∈ J j ),
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j= 1, 2, . . . , N+1, one arrives at the following large-n behaviour for the jump matrix v (z); for j= 1, 2, . . . , N+1 and i = 1, 2, . . . , N: v (z) =
n→∞
⎧ iσ2 , ⎪ ⎪ ⎪ ⎪ ⎪ I+ O (e−nc|z| )σ− , ⎪ ⎪ a ⎪ ⎪ ⎨e−(2π i((n−1)K+k) bi N+1 ψV (ξ ) dξ )σ3 (I+ O (e−nc|z−ai | )σ ) , + . a −(2π i((n−1)K+k) b N+1 ψV (ξ ) dξ )σ3 −nc|z−αk |−1 ⎪ i O (e )σ+ , e I+ ⎪ ⎪ ⎪ ⎪ ⎪I+ O (e−nc|z| )σ , ⎪ + ⎪ ⎪ −1 ⎩ I+ O (e−nc|z−αk | )σ+ ,
z ∈ (b j−1 , a j), z∈ Jj ∪ Jj , z ∈ (ai , bi ) \ Uδk (αk ), z ∈ (ai , bi ) ∩ Uδk (αk ), z ∈ J \ Uδk (αk ), z ∈ J ∩ Uδk (αk ),
where c is some positive constant, J := (−∞, b 0 ) ∪ (a N+1 , +∞), and, for k = 1, 2, . . . , K, Uδk (αk ) := {z ∈ C; |z−αk | < δk }, where (0, 1) δk are some arbitrarily fixed, sufficiently small positive real numbers chosen so that, ∀ r1 = r2 ∈ {1, 2, . . . , K}, Uδr1 (αr1 ) ∩ Uδr2 (αr2 ) = ∅. The above convergences are normal, that is, uniform in the respective compact subsets. Recall from Lemma 2.56 of [57] that, for an oriented skeleton in C on which the jump matrix of an RHP is defined, one may always choose to add or delete a portion of the skeleton on which the jump matrix equals I without altering the RHP in the operator sense; hence, neglecting those jumps on tending exponentially quickly (as n → ∞) to I, and removing the corresponding oriented skeletons from , it becomes more or less transparent how to construct a parametrix (an approximate solution) for the family of K RHPs (m (n, k; z) := m (z), v (z), ) stated in Lemma 4.2, that is, the large-n solution of the family of K RHPs for m : N × {1, 2, . . . , K} × C \ → SL2 (C) formulated in Lemma 4.2 should be ‘close to’, in some appropriately defined operator-theoretic sense, the solution of the following family of K ‘limiting’, or ‘model’, RHPs. R \ {α1 , α2 , . . . , α K } → R satisfy condiLemma 4.3 Let the external field V: is regular. For n ∈ N and k = tions (6)–(8). Suppose, furthermore, that V 1, 2, . . . , K, let the associated equilibrium measure, μV , and its compact support, supp(μV ) =: J = ∪ N+1 j=1 [b j−1 , a j ] (⊂ R \ {α1 , α2 , . . . , α K }), be as described in Lemma 3.7, and, along with the corresponding variational constant, , satisfy the variational conditions stated in Lemma 3.8, (81); moreover, let the corresponding conditions (i)–(iv) stated in Lemma 3.8 be valid. Then, for n ∈ N and k = 1, 2, . . . , K, : N × {1, 2, . . . , K} × C \ J → SL2 (C), where J := N (ai , bi )), solves the following RHP: (i) (n, k; z) := (z) is holomorphic J ∪ (∪i=1
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for z ∈ C \ J; (ii) the boundary values ± (z) := lim z →z∈J (z ) satisfy the jump z ∈± side of J
condition + (z) = − (z)υ (z),
z ∈ J,
where υ : N × {1, 2, . . . , K} × J → GL2 (C), (n, k, z) → υ (n, k; z) := υ (z) iσ2 , z ∈ (b j−1 , a j), j= 1, 2, . . . , N+1, = −(2π i((n−1)K+k) a N+1 ψ (ξ ) dξ )σ3 V bi e , z ∈ (ai , b i ), i = 1, 2, . . . , N; (iii) (z) =C+ z→αk (I+ O (z−αk ))E−σ3 and (z) =C− z→αk (I+ O (z−αk ))Eσ3 ; (iv) (z) =C± z→∞ O (1); and (v) for q ∈ {1, 2, . . . , s −1}, (z) =C± z→α pq O (1). It is well known that the family of K model RHPs ((n, k; z) := (z), υ (z), J) formulated in Lemma 4.3 is explicitly solvable in terms of Riemann theta functions associated with the family of K two-sheeted genus-N hyper elliptic Riemann surfaces {(y, z); y2 = R(z) = N+1 j=1 (z−b j−1 )(z−a j )}; see, for example, [73] (see, also, [66]): this will be considered elsewhere. Acknowledgements K. T.-R. McLaughlin was supported, in part, by National Science Foundation Grant Nos. DMS-0200749 and DMS-0451495, as well as a NATO Collaborative Linkage Grant ‘Orthogonal Polynomials: Theory, Applications, and Generalizations’, Ref. No. PST.CLG.979738. A. H. Vartanian was supported, in part, by a College of Charleston (CofC) Summer Research Stipend. X. Zhou was supported, in part, by National Science Foundation Grant No. DMS-0300844.
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Math Phys Anal Geom (2008) 11:365–379 DOI 10.1007/s11040-008-9047-6
Spectrum of the Lichnerowicz Laplacian on Asymptotically Hyperbolic Surfaces Erwann Delay
Received: 8 October 2007 / Accepted: 31 July 2008 / Published online: 12 September 2008 © Springer Science + Business Media B.V. 2008
Abstract We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian L contains the ray 14 , +∞ . If moreover the scalar curvature is constant then −2 and 0 are infinite dimensional eigenvalues. If, in addition, the inequality u, u L2 14 ||u||2L2 holds for all smooth compactly supported function u, then there is no other value in the spectrum. Keywords Asymptotically hyperbolic surfaces · Lichnerowicz Laplacian · Symmetric 2-tensor · Essential spectrum · Asymptotic behavior Mathematics Subject Classifications (2000) 35P15 · 58J50 · 47A53
1 Introduction This article is a complement of the papers [7, 8] where the study of the Lichnerowicz Laplacian L is given in dimension n greater than 2. We refer the reader to those papers for all the motivations. In the preceding papers, the spectrum was only given for n 3 because of the natural relation to the prescribed Ricci curvature problem. In dimension 2 this study does not appear because the corresponding problem is conform. The present paper, firstly given for completeness, appears to be particulary interesting because of the quite big differences with the other dimensions.
E. Delay (B) Laboratoire d’analyse non linéaire et géométrie, Faculté des Sciences, 33 rue Louis Pasteur, 84000 Avignon, France e-mail: [email protected] URL: http://www.math.univ-avignon.fr/Delay
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For instance on the hyperbolic space, when n 3 the spectrum of L on trace free symmetric two tensors is the ray (n − 1)(n − 9) , +∞ . 4 This spectrum is essentially characterized by non trivial trace free tensors on the boundary at infinity. In dimension 2 (so 1 at infinity) those tensors do not exist, and the situation is very different. Also, in dimension two, the cohomology of the manifold appears naturally in the spectrum. This situation was already noticed by Avez [2, 3] and Buzanca [5, 6]. The principal result is the following Theorem 1.1 Let (M, g) be an asymptotically hyperbolic surface. The essential spectrum of L on trace free symmetric two tensors contains the ray [1/4, +∞[. If moreover g has constant scalar curvature R = −2 then −2 and 0 are also in the essential spectrum. Moreover their eigenspaces are in one to one correspondence with the space of harmonic one forms respectively in L4 and in L2 (in particular they are infinite dimensional). Finally, if in addition, as for the hyperbolic plane, for all smooth compactly supported function u, u, u L2 14 ||u||2L2 , then the spectrum of L is 1 {−2} ∪ {0} ∪ [ , +∞[. 4 Along the paper we also obtain some relative results on more general surfaces, with or without constant scalar curvature.
2 Definitions, Notations and Conventions Let M be a smooth, compact surface with boundary ∂ M. Let M := M\∂ M be a non-compact surface without boundary. In our context the boundary ∂ M will play the role of a conformal boundary at infinity of M. Let g be a Riemannian metric on M. The manifold (M, g) is conformally compact if there exists on M a smooth defining function ρ for ∂ M (that is ρ ∈ C∞ (M), ρ > 0 on M, ρ = 0 on ∂ M and dρ is nowhere vanishing on ∂ M) such that g := ρ 2 g is a C2,α (M) ∩ C∞ (M) Riemannian metric on M . We will denote by g the metric induced on ∂ M. Now if |dρ|g = 1 on ∂ M, it is well known (see [12] for instance) that g has asymptotically sectional curvature −1 near its boundary at infinity. In this case we say that (M, g) is asymptotically hyperbolic. Along the paper, it will be assumed sometimes than (M, g) has constant scalar curvature : then the asymptotic hyperbolicity enforces the normalisation R(g) = −2 , where R(g) is the scalar curvature of g.
(2.1)
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The basic asymptotically hyperbolic surface is the real hyperbolic Poincaré disc. In this case M is the unit disc of R2 , with the hyperbolic metric g0 = ω−2 δ , (2.2) 1 2 δ is the Euclidean metric, ω(x) = 2 1 − |x|δ . q We denote by T p the set of rank p covariant and rank q contravariant tensors. When p = 2 and q = 0, we denote by S2 the subset of symmetric ◦
tensors, and by S 2 the subset of S2 of trace free symmetric tensors. We use the summation convention, indices are lowered with gij and raised with its inverse gij. The Laplacian is defined as = −tr∇ 2 = ∇ ∗ ∇, where ∇ ∗ is the L2 formal adjoint of ∇. In dimension 2, the Lichnerowicz Laplacian acting on trace free symmetric covariant 2-tensors is L = + 2R, where R is the scalar curvature of g. For u a covariant 2-tensorfield on M we define the divergence of u by (div u)i = −∇ juji . If u is a symmetric covariant 2-tensorfield on M, it can be seen as a one form with values in the cotangent bundle. Thus we can define its exterior differential with ∇ d u ijk := ∇i ujk − ∇juik , which is a two form with values the cotangent bundle. For ω, a one form on M, we define its divergence d∗ ω = −∇ i ωi , the symmetric part of its covariant derivative : (Lω)ij =
1 (∇i ω j + ∇jωi ), 2
(note that L∗ = div) and the trace free part of that last tensor : ◦ 1 1 ∗ L ω = (∇i ω j + ∇jωi ) + d ωgij. ij 2 2 The well known [4] Weitzenböck formula for the Hodge-De Rham Laplacian on 1-forms, in dimension 2, reads H ωi = ∇ ∗ ∇ωi + Ric(g)ik ωk = ∇ ∗ ∇ωi +
R ωi . 2
We recall also the Weitzenböck formula ∗ K := d∇ d∇ + div∗ div = + R = L − R.
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For a one form ω, we will consider the trace free symmetric covariant two tensor defined by ◦ |ω|2 gij. S ω = ωi ωj − ij 2 A TT-tensor (Transverse Traceless tensor) is by definition a symmetric divergence free and trace free covariant 2-tensor. L2 denotes the usual Hilbert space of functions or tensors with the product (resp. norm)
1 u, v L2 =
u, vdμg resp.|u| L2 = M
|u|2 dμg
2
,
M
where u, v (resp. |u|) is the usual product (resp. norm) of functions or tensors relative to g, and the measure dμg is the usual measure relative to g (we will omit the term dμg ). For k ∈ N, H k will denote the Hilbert space of functions or tensors with k-covariant derivative in L2 , endowed with its standard product and norm. We will first work near the infinity of M, so it is convenient to define for small ε > 0, the manifold Mε = {x ∈ M, ρ(x) < ε}. It is well know that near infinity, we can choose the defining function ρ to be the g-distance to the boundary. Thus, if ε is small enough, Mε can be identified with (0, ε) × ∂ M equipped with the metric g = ρ −2 dρ 2 + g(ρ)dθ 2 , where { g(ρ)}ρ∈(0,ε) is a family of smooth, positive functions on ∂ M, with g(0) = g. Let P be an uniformly degenerate elliptic operator of order 2 on some tensor bundle over M (see [9] for more details). We recall here a criterion for P to be semi-Fredholm. We first need the Definition 2.1 We say that P satisfies the asymptotic estimate Pu, u L2 ∞ C||u||2L2 resp. ||Pu|| L2 ∞ C||u|| L2 if for all ε > 0, there exists δ > 0 such that, for all smooth u with compact support in Mδ , we have Pu, u L2 (C − ε)||u||2L2 resp. ||Pu|| L2 (C − ε)||u|| L2 . Proposition 2.2 below is standard in the context of non-compact manifolds (see [7] for instance). It shows that the essential spectrum is characterized near infinity.
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Proposition 2.2 Let P : H 2 −→ L2 . Then P is semi-Fredholm (ie. has finite dimensional kernel and closed range) if and only if P satisfies an asymptotic estimate ||Pu|| L2 ∞ c||u|| L2 for some c > 0. This proposition will be used to compute the essential spectrum of L which is, by definition, the closed set σe ( L ) = {λ ∈ R, L − λId is not semi-Fredholm}.
3 Commutators of Some Natural Operators Lemma 3.1 On one forms, we have ◦ 1 R 1 div ◦L = − = ( H − R). 2 2 2 ◦
Proof In local coordinates, 2 div ◦L(ω) is equal to : − ∇ i ∇i ωj + ∇jωi − ∇ k ωk gij = ω j − ∇ k ∇jωk + ∇j∇ k ωk = ( − Ric)ω j R = − ω j. 2
Recall that in dimension 2 (see Corollary 3.2 of [8] for instance) we have: Lemma 3.2 Let (M, g) be a Riemannian surface with Levi-Civita connexion ∇. Then the following equality holds for trace free symmetric covariants two tensors: div ◦ L = H ◦ div . So we obtain Corollary 3.3 If h is a trace free symmetric covariant two tensor with L h = λh then H div h = λ div h. Lemma 3.4 On a Riemannian surface with Levi-Civita connexion ∇, we have ◦
L ◦ L = L ◦ H − S(dR, .), ◦
where S(dR, ξ )ij = 12 (∇j Rξi + ∇i Rξ j − ∇ p Rξ p gij). In particular ◦
◦
◦
L ◦ L = L ◦ H − S(dR, .).
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Moreover R is constant iff ◦
◦
L ◦ L = L ◦ H so L ◦ L = L ◦ H . Proof The first part comes from [8] lemma 3.3 where here Ric(g) = ◦
R(g) g. 2
Now, if L ◦ L = L ◦ H then for any one form ξ , S(dR, ξ ) = 0. At any point x ∈ M, we take an orthonormal basis (e1 , e2 ) on Tx∗ M, and choose ξ = e1 . We ◦ a b , where (a, b ) are then see that the matrix of S(dR, ξ ) has the form b −a the coordinates of dR. We finally deduce that dR = 0.
4 Some Decompositions of Trace Free Symmetric Two Tensors In this section, we recall two well known natural decompositions. We give their simple proofs for completeness. Lemma 4.1 For all k ∈ N, ◦
◦
H k+1 (M, S 2 ) = ker div ⊕ Im L, where the decomposition is orthogonal in L2 . Proof For ω ∈ Cc∞ (M), we have ◦ < L(ω), h >= M
< ω, div h > . M
◦ ◦ ⊥ Thus L∗ = div, which gives Im L = Ker div.
Lemma 4.2 For all k ∈ N, H k+1 (M, T1 ) = ker H ⊕ Im d ⊕ Im(∗d), where the decomposition is orthogonal in L2 . Remark 4.3 Recall that, from the definition of H , we have: ker H = ker d ∩ ker d∗ . Proof First, from the definition of d∗ , it is clear that (Im d)⊥ = ker d∗ , and so H k+2 (M, T1 ) = ker d∗ ⊕ Im d. For all H 1 function u and all H 1 one forms ω, we have ∗ ∗du, ω = d ∗ u, ω = ∗u, dω = u, ∗dω. M
M
M
M
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As a consequence, if ∗du, ω L2 = 0 for all u ∈ Cc∞ (M), then dω = 0, and if in addition d∗ ω = 0 then H ω = 0. This shows that ker d∗ = Im(∗d) ⊕ ker H . From Lemma 4.2, any one form ω in H 1 can be decomposed in a unique way with ω = η + du + ∗dv,
(4.1)
where H η = 0.
5 The Spectrum on TT-tensors ◦ Lemma 5.1 Let M be any Riemannian surface. If h ∈ C2 M, S2 , then the following properties are equivalent:
(i) div h = 0, (ii) d∇ h = 0, ◦
(iii) h = S(ω), where ω is a harmonic one form. They imply (iv) L h = Rh. Moreover, if h ∈ L2 , then (iv) implies (i ), (ii) and (iii). Proof The first part is due to Avez ([3] Lemma A and Lemma C). The second part is simply due to the following Weitzenböck formula [10]:
d∇
∗
d∇ + div∗ div = K = L − R,
and the fact that if h ∈ L2 solves (iv) weakly, then elliptic regularity gives h ∈ H ∞ ⊂ C∞ .
Corollary 5.2 There exists a non trivial eigen-TT-tensor of L iff R is constant. In this case any TT-tensor is an eigentensor with eingenvalue R. Proof The “if” part is clear. For the “only if” direction, assume that h is a non trivial eigen-TT-tensor of L , so that div h = 0 and L h = λh hold for some λ ∈ R. From Lemma 5.1, ( L − R)h = 0 and then (R − λ)h = 0. If R = λ near a point, then h has to be trivial near this point, so from the unique continuation property, h is trivial. This contradicts the assumption on h and proves the result.
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6 Spectrum on Im L If R is constant then from Lemma 3.4 and the fact that H preserves the ◦
decomposition (4.1), it suffices to study the spectrum of L on Im L, restricted successively to Ker H , Im d and Im(∗d). ◦
Lemma 6.1 When R is constant then L(Ker H ) is in the kernel of L . If R is ◦
moreover negative, L(Ker H ) is in one to one correspondence with Ker H . ◦
Proof If h = Lη, with η ∈ Ker H , then from Lemma 3.4 L h = 0. Now from Lemma 3.1 we have ◦ R 2 div ◦L(η) = − η. 2 ◦
Thus if R < cte < 0, then L is injective on H 2 .
◦
We are now interested in the spectrum on Im L ◦ d. We begin with a lemma. ◦
Lemma 6.2 If h = L ω, with ω ∈ H 1 then: 1 R 2 2 2 ||h|| L2 = ||ω|| H1 − + 1 |ω| . 2 2 M In particular, if R = −2 we obtain 2||h||2 = ||ω||2H1 . Proof Using Lemma 3.4 we compute: ◦ ◦ |h|2 = L ω, L ω 2 M
L
= div L ω, ω 2 L 1 R = |∇ω|2 − |ω|2 2 M 2 R 1 2 2 = ||ω|| H1 − + 1 |ω| . 2 2 ◦
Corollary 6.3 On an A.H. surface, for all ε > 0, there exists δ0 > 0 small such that, for all δ ∈ (0, δ0 ) and all one forms ω with compact support in Mδ , if ◦
h = L ω then ||ω||2H1 2(1 − ε)||h||2L2 .
Spectrum of L on A.H. Surfaces
Proof
||ω||2H1 −
M
373
R O(ρ)|ω|2 + 1 |ω|2 = ||ω||2H1 − 2 Mδ ||ω||2H1 + Cδ||ω||2L2 (1 + Cδ) ||ω||2H1 ,
where C is a positive constant. Lemma 6.2 concludes the proof.
Let us recall a well known lemma. Lemma 6.4 Let u be a smooth compactly supported function. If u, u L2 c||u||2L2 , then H du, du L2 c||du||2L2 and H (∗du), (∗du) L2 c|| ∗ du||2L2 . Proof H du, du = dd∗ du, du = d∗ du, d∗ du = ||u||2 c||u||||u|| cu, u = c||du||2 . H (∗du), (∗du) = d∗ d(∗du), ∗du = d ∗ du, d ∗ du = ||u||2 c||u||||u|| cu, u = c||du||2 = c|| ∗ du||2 .
We would like an equivalent to this lemma when substituting one forms to functions. This is achieved by the following lemma and its corollary. Lemma 6.5 Let ω be a smooth compactly supported one form. If H ω, ω L2 c||ω||2L2 then
◦ ◦ c 1 R L L ω, L ω ||ω||2H1 + c + 1 |ω|2 2 2 M 2 ◦ ◦ R − + 1 H ω, ω − S(dR, ω), L ω 2 . L 2 M
374
Proof ◦ ◦ L L ω, L ω
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L2
◦ ◦ = L H ω, L ω
L2
◦ = H ω, div L ω
◦ ◦ − S(dR, ω), L ω
L2
L2
◦ ◦ − S(dR, ω), L ω
L2
◦ ◦ − S(dR, ω), L ω
1 H ω, ( H − R)ω L2 L2 2 ◦ ◦ 1 1 = || H ω||2L2 − R H ω, ω L2 − S(dR, ω), Lω 2 L 2 2 ◦ ◦ 1 1 c|| H ω|| L2 ||ω|| L2 − R H ω, ω L2 − S(dR, ω), Lω 2 L 2 2 ◦ ◦ 1 1 c H ω, ω L2 − R H ω, ω L2 − S(dR, ω), Lω 2 L 2 2 1 R 1 1 c||∇ω||2L2 + c |ω|2 − R H ω, ω L2 2 2 M 2 2 ◦ ◦ − S(dR, ω), Lω 2 L 1 1 R 2 c||∇ω||2L2 + c |ω| + H ω, ω L2 2 2 M 2 ◦ ◦ 1 − (R + 2) H ω, ω L2 − S(dR, ω), Lω 2 L 2 1 1 R 2 c||∇ω||2L2 + c |ω| + c||ω||2L2 2 2 M 2 ◦ ◦ 1 − (R + 2) H ω, ω L2 − S(dR, ω), Lω 2 L 2 1 1 R c||ω||2H1 + c + 1 |ω|2 2 2 M 2 ◦ ◦ 1 − (R + 2) H ω, ω L2 − S(dR, ω), Lω 2 . L 2 =
Remark 6.6 Under the assumptions of Lemma 6.4, the assumptions of Lemma 6.5 are satisfied by ω = du or ω = ∗du. Proposition 6.5 together with Lemma 6.2 give: Corollary 6.7 If R = −2 and H ω, ω L2 c||ω||2L2 then ◦ 2 ◦ ◦ L Lω, Lω 2 cLω 2 . L
L
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In the A.H. setting we have Corollary 6.8 On an A.H. surface, for ω ∈ Im d or ω ∈ Im ∗d, ◦ ◦ L Lω, Lω
∞
L2
1 ◦ 2 Lω 2 . L 4
Proof It is well know that on A.H. surfaces, u, u ∞ 41 ||u||2L2 holds. Then (see Lemma 6.4 for instance) H ω, ω L2 ∞ 41 |ω|2L2 . We will show that the three terms in the right-hand side of Lemma 6.5 do not contribute at infinity. We work with a one form ω compactly supported in Mδ with small δ. We recall that R + 2 = O(ρ) and ||d(R + 2)|| = O(ρ). Let us begin with M
R R R R + 1 H ω, ω = + 1 ω, ω + +1 ||ω||2 2 2 2 2 M M R R = ω j∇ i + 1 ∇i ω j + + 1 ||∇ω||2 2 2 M M R R + +1 ||ω||2 . 2 2 M
The twolast terms are clearly bounded in absolute value by C1 δ||ω||2H1 . Let A(ω) = M ω j∇ i (R + 2)∇i ω j. Then: |A(ω)| ||∇ω|| L2 ||d(R + 2)ω|| L2 1/2 2 2 ||∇ω|| L2 ||d(R + 2)|| ||ω|| M
1/2
C2 δ||∇ω|| L2
||ω||2 M
C2 δ||ω||2H1 . 2
We so get: R + 1 H ω, ω C3 δ||ω||2H1 . 2 M ◦ ◦ The term S(dR, ω), Lω L2 proceed in a manner similar to A(ω) to obtain the same estimate, perhaps with a different constant. Finally, the term M R2 + 2 1 |ω| is clearly bounded in absolute value by C4 δ||ω||2H1 . The conclusion follows from Lemma 6.5, the triangular inequality and Corollary 6.3.
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Proposition 6.9 Let λ 14 and C > 0. Let P be the operator L − λId : H 2 −→ L2 . There is no asymptotic estimate |Pu| L2 ∞ C|u| L2 , ◦ for P on Im L ◦ d . Proof Let λ 14 , and μ := λ − 14 . The idea of the proof is to construct a ◦ ◦ family of tensors {h R } = L(df R ) = Hess f R with compact support in Me−R/2 such that |Ph R | L2 (M) goes to zero when R goes to infinity but |h R | L2 (M) goes to infinity when R goes to infinity. It is well known (see [11, lemma 5.1] for example) that we can change the defining function ρ into a defining function r such that the metric takes the form g = r−2 g = r−2 dr2 + g(r) , on Mδ =]0, δ[×∂∞ M (reducing δ if necessary), where g(r) is a metric on {r} × ∂∞ M. The non trivial Christoffel symbols of g = r−2 [dr2 + g(r)dθ 2 ] are r = −r−1 , rr
1 r2 g + r−2 g = r−1 g− −2r−3 g , 2 2 1 1 −1 g = −r−1 + g , −2r−1 + g = g −1 2 2
r θθ =− θ θr
where the primes denote r-derivatives. If f is a “radial” function, ie f = f (r), we compute: 1 g+ Hess f = f + r−1 f dr2 + −r−1 g f dθ 2 . 2 We deduce
f = −r
2
1 −1 g f . g f + 2
We also have ◦ 1 1 −1 2 −1 g f dr − Hess f = f +r f − g gdθ 2 =: F f (r) dr2 − gdθ 2 . 2 4 This tensor is in the set V2 of [9] page 201: substitute f there by F f here, q there by dr2 − gdθ 2 here, ρ there by r here and the dimension n + 1 there by 2 here. Recall that the Lichnerowicz Laplacian in our context is L = + K, where K = −4 + O(r). Thus, from [9] Lemma 2.9 page 202, we obtain ( L − λ)(F(r)q) = I2 (F(r))q + r X(F),
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where I2 (F) = −r2 F − 4rF − 2 f F has for characteristic exponents s1 , s2 =
√ 1 −3 ± 1 − 4λ , 2
2
d d d and X = ar2 dr 2 + br dr + c is a second order operator polynomial in r dr with g-bounded coefficients depending on g and q. √ In particular, if λ 14 and f (r) = r[a cos(μ ln(r)) + b sin(μ ln(r))], where μ = λ− 14 , then F f (r) = r−3/2 [A cos(μ ln(r))+ B sin(μ ln(r))]+ O r−1/2 and (A, B) = 0 if (a, b ) = 0. Thus we obtain I2 (F f (r)) = O r−1/2 .
Let us now define the function f R (r) = f (r) R (r), where R is as in Lemma 8.1. A simple calculation shows that F f R (r) = R (r)F f (r) + O R−1 O r−3/2 . Therefore:
I2 (F f R (r)) = O r−1/2 + O R−1 O r−3/2 ,
and
Then:
r X(F f R (r)) = O r−1/2 . ( L − λ)(F f R (r)q) = O r−1/2 + O(R−1 )O(r−3/2 ).
We deduce that
||( L − λ)(F f R (r)q)||2L2 = O R−1 .
On the other hand, we have ||(F f R (r)q)||2L2 cR, where c is a positive constant. Letting R going to infinity, this concludes the proof of the proposition.
7 Conclusion From Proposition 6.9 and Corollary 6.8, the essential spectrum of L restricted ◦ to Im L ◦ d is 1 [ , +∞[. 4
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In particular this ray is in the essential spectrum of L . If R is constant then the A. H. condition forces R = −2. Lemma 6.1 shows ◦ that any tensor in L(ker H ) is in the kernel of L . The eigenspace for 0 is then ◦
infinite dimensional as ker H (recall that L is injective if R < 0). From Lemma 5.1, any TT-tensor h is an eigentensor for the eigenvalue −2 ◦
and there is a harmonic one form omega such that h = S (ω). Moreover ω is in L4 iff h is in L2 . Assume now that u, u L2
1 ||u|| L2 4
holds for all smooth compactly supported functions u. Then Lemma 6.4 and Corollary 6.7 give, when ω = df or ω = ∗df , ◦ ◦ 1 ◦ L Lω, Lω 2 ||Lω|| L2 . L 4 This proves that there are no eigentensors with eigenvalue less than 14 ◦ ◦ in Im L ◦ d nor in Im L ◦ (∗d) . Recall that, when R is a constant, the Lichnerowicz Laplacian commute with the Hodge Laplacian, and also that the ◦
Hodge Laplacian preserves the decomposition (4.1). We so get that on Im L, the essential spectrum of L is 1 {0} ∪ [ , +∞[. 4 This concludes the proof of the main Theorem 1.1. Acknowledgements I am grateful to N. Yeganefar for discussions on forms and to F. Gautero for his comments on the original manuscript.
Appendix : A Family of Cutoff Functions In this appendix, we give a family of cutoff functions. Standard in the A.H. context, they can be found in [1], Definition 2.1 p.1362 for instance. Lemma 8.1 Let (M, g, ρ) be an asymptotically hyperbolic manifold. For R ∈ R large enough, there exits a cutoff function R : M → [0, 1] depending only on ρ, supported in the annulus {e−8R < ρ < e−R }, equal to 1 in {e−4R < ρ < e−2R } and which satisfies for R large : k d R Ck dρ k (ρ) Rρ k , for all k ∈ N\{0}, where Ck is independent of R.
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Proof Let χ : R −→ [0, 1] be a smooth function equal to 1 on ] − ∞, 1] and 0 on [2, +∞[. We define ln(ρ(x)) χ R (x) := χ , −R we then have χ R : M −→ [0, 1] is equal to 1 on ρ e−R and 0 on ρ e−2R . Now we define R := χ4R (1 − χ R ) which satisfies the announced properties.
References 1. Andersson, L.: Elliptic systems on manifolds with asymptotically negative curvature. Indiana Univ. Math. J. 42(4), 1359–1388 (1993) 2. Avez, A.: Le laplacien de Lichnerowicz. Rend. Sem. Mat. Univ. Politec. Torino (35), 123–127 (1976–1977) 3. Avez, A.: Le laplacien de Lichnerowicz sur les tenseurs. C. R. Acad. Sci. Paris Sér. A (284), 1219–1220 (1977) 4. Besse, A.L.: Einstein manifolds. Ergebnisse d. Math. 3. folge, vol. 10. Springer, Berlin (1987) 5. Buzzanca, C.: Le laplacien de Lichnerowicz sur les surfaces à coubure négative constante. C. R. Acad. Sci. Paris Sér. A (285), 391–393 (1977) 6. Buzzanca, C.: Il laplaciano di Lichnerowicz sui tensori. Boll. Un. Mat. Ital. 6(3-B), 531–541 (1984) 7. Delay, E.: Essential spectrum of the Lichnerowicz laplacian on two tensor on asymptotically hyperbolic manifolds. J. Geom. Phys. 43, 33–44 (2002) 8. Delay, E.: TT-eigentensors for the Lichnerowicz laplacian on some asymptotically hyperbolic manifolds with warped products metrics. Manuscripta Math. 123(2), 147–165 (2007) 9. Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87, 186–225 (1991) 10. Koiso, N.: On the second derivative of the total scalar curvature. Osaka J. Math. 16(2), 413–421 (1979) 11. Lee, J.M.: The spectrum of an asymptotically hyperbolic Einstein manifold. Comm. Anal. Geom. 3, 253–271 (1995) 12. Mazzeo, R.: The Hodge cohomology of a conformally compact metric. J. Differ. Geom. 28, 309–339 (1988)
Math Phys Anal Geom (2008) 11:381–398 DOI 10.1007/s11040-008-9048-5
Some Examples of Graded C∗ -Algebras Athina Mageira
Received: 15 July 2008 / Accepted: 15 September 2008 / Published online: 26 October 2008 © Springer Science + Business Media B.V. 2008
Abstract We apply the theory of C∗ -algebras graded by a semilattice to crossed products of C∗ -algebras. We establish a correspondence between the spectrum of commutative graded C∗ -algebras and the spectrum of their components. This will allow us to compute the spectrum of some commutative examples of graded C∗ -algebras. Keywords C∗ -algebras · Semilattices · Crossed products · Spectrum of C∗ -algebras Mathematics Subject Classifications (2000) 46L05 · 47A10 · 47L65
1 Introduction This paper is a continuation of a previous work ([12]), where we studied graded C∗ -algebras by a semilattice and some of their properties. In the present paper, we study crossed products of graded C∗ -algebras and we construct C∗ -algebras graded by semilattices of closed subgroups. We investigate the relation of the spectrum of commutative graded C∗ -algebras and those of its components. Then follows in particular the study of two commutative examples. C∗ -algebras graded by semilattices appear in the work of A. Boutet de Monvel and V. Georgescu in [2, 3] and [5], M. Damak and V. Georgescu
A. Mageira (B) Institut de Mathématiques de Jussieu–Université Paris-Diderot (Paris 7), 175, rue du Chevaleret, 75013 Paris, France e-mail: [email protected], [email protected]
382
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(cf. [6, 7]) and also V. Georgescu and A. Iftimovici (cf. [8, 9] and [10]) in connection with the quantum N body problem. The Hamiltonian of a N body system is of the form H = − + V where is 3N the Laplace-Beltrami operator on X = R and V is the interaction potential. This V has the form V = Vjk where Vjk ∈ C0 (X/ Xjk ) with X jk = {x ∈ 1≤ j
X; x j = xk } which describes the interaction of the particles j and k. The C∗ -algebra A generated by these potentials is a commutative C∗ algebra graded by the subsemilattice of vector subspaces of X generated by the X jk . The resolvent of H lives then in A X which was also shown to be graded. This was used to describe some spectral properties of hamiltonians of this form and to prove the Mourre estimate for N body systems (see also [4]). We refer to [12] for more references on the use of graded C∗ -algebras in the N body problem. The main goal of this paper is a) To show that the full and reduced crossed product of a graded C∗ -algebra (by a group action preserving the grading) is also graded. This is due to the fact that crossed products are compatible with inductive limits and split exact sequences. b) To study the spectrum of some commutatives graded C∗ -algebras. i) If A is a commutative C∗ -algebra graded by a good semilattice (in the sense of Definition 3.1 below), we show that its spectrum is the disjoint union of the spectra of its components. ii) When the semilattice G is formed by closed subgroups of a locally compact group G, we give the necessary and sufficient conditions to have structure morphisms and therefore to construct a G -graded commutative C∗ -algebra A with components the C∗ -algebras C0 (G/H), H ∈ G . Moreover, we study the injectivity of the morphism ϕ : A → Cb (G). iii) We explicitly compute the spectra of two commutative graded C∗ -algebras. •
•
For the first one we consider a good semilattice. We take a subsemilattice L of the lattice (for the inclusion as order relation) formed by vector subspaces of a finite dimensional vector space X and whose components are AY = C0 (X/Y) (with Y ∈ L). In particular, we study the case where X is a plane P and L = {{0}, δ1 , ..., δn , P } where δ1 , . . . , δn are n vector lines in P . We use the theory of normal forms of a Riemann surface (see [11] and [13]) to show that the spectrum of the associated graded C∗ -algebra is homeomorphic to the torus Tg with g holes where g = n2 if n is even and if n is odd this spectrum is homeomorphic to a pinched torus with g = E n2 holes (i.e. we identify two of its points). For the second example we take an arbitrary semilattice L and all the components Ai equal to C. We identify then the spectrum of the corresponding L-graded C∗ -algebra A with the set of non empty final subsemilattices of L.
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In particular, if L = Q we show that the spectrum of A is in bijection with the set R Q {−∞}. Here is a summary of the paper. • • •
• •
We show in Section 2 that crossed products of C∗ -algebras are compatible with graded C∗ -algebras. Section 3 is devoted to a study of the spectrum of commutative graded C∗ -algebras. In Section 4 we consider a particular case of semilattices. Here L will be a subsemilattice of the lattice G which is formed by closed subgroups of a locally compact group G. We examine which conditions give rise to a L-graded C∗ -algebra. Section 5 consists of applications of the previous sections to vector spaces. We study in particular the spectrum of a commutative C∗ -algebra with a grading by vector lines of a plane. Finally, we give a quite different commutative example of graded C∗ -algebras by a semilattice (which is not a good semilattice) which is generated by copies of C and we study its spectrum. The case where L = Q is investigated.
Let (L, ) be a partially ordered set. Recall that (L, ) is a semilattice if for all k, ∈ L the set {m ∈ L; m k and m } has a greatest element noted k ∧ . An initial segment of L is a subset M of L such that for all a ∈ M we have {b ∈ L; b a} ⊂ M. Every initial segment of L is a subsemilattice of L. A final subsemilattice of L is a subsemilattice M which is a final segment i.e. such that, for all a ∈ M, we have {b ∈ L; a b } ⊂ M. A L-graded C∗ -algebra (or a C∗ -algebra graded by L) is a C∗ -algebra A equipped with a linearly independent and total family (Ai )i∈L of C∗ subalgebras of A such that Ai A j ⊂ Ai∧ j for all i, j ∈ L. We will denote by (A, (Ai )i∈L ) a L-graded C∗ -algebra A and we will call the Ai ’s the components of A. For an arbitrary subset M of L denote by AM the sum of the Ai ’s for i ∈ M. We also denote by FL the set of finite subsemilattices of L.
2 Crossed Products of Graded C∗ -Algebras Theorem 2.1 Let L be a semilattice and (A, (Ai )i∈L ) a L-graded C∗ -algebra, with a continuous action α of a locally compact group G. Assume that for all i ∈ L the C∗ -algebra Ai is G-invariant. Then the full crossed product (Aα G, (Ai α G)i∈L ) and the reduced crossed product (Ar,α G, (Ai r,α G)i∈L ) are also L-graded C∗ -algebras. Proof We will only show this for the full or reduced crossed product. The same proof works equally well for both.
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Let k ∈ L, put Lk = { j ∈ L|k j } and L k = { j ∈ L|k j}. Then we have a split exact sequence (cf. [12] proposition 1.12) 0 → AL k → A ALk → 0 and since crossed products are compatible with split exact sequences we also have 0 → AL k α G − → Aα G ALk α G → 0. We will show that : i) Ak α G → Aα G is an injective morphism for all k ∈ L. ii) The family of C∗ -algebras (Ai α G)i∈L is linearly independent, iii) (A i α G)(A jα G) ⊂ Ai∧ jα G for all i, j ∈ L and iv) AF α G = Ai α G is dense in Aα G. F ∈FL
i∈L
i) Since Ak is an ideal of ALk , Ak α G is also an ideal of ALk α G and we have / AL α G Ak α G k JJ JJ JJ JJ JJ % Aα G.
ii) Let f ∈ Aα G such that f =
i∈L
fi = 0 with fi = 0 only for a finite num-
ber of i ∈ L. Denote by = { j ∈ L : f j = 0} the support of f . Assume that = ∅ and take m ∈ a maximal element then fm = 0 ⇒ || fm || = 0. ∗ Since Lm ∩ = {m} and Aα G → ALm α G is a morphism of C algebras || fm || || fi ||. In other words fm = 0, which contradicts the i∈L
assumption. iii) Let i, j ∈ L and let fi ∈ Cc (G, Ai ) , f j ∈ Cc (G, A j). By the definition of the product of the ∗-algebra Cc (G, A), for all s ∈ G we have ( fi ∗ f j)(s) ∈ Ai∧ j. Then Cc (G, Ai )Cc (G, A j) ⊂ Cc (G, Ai∧ j) and since the product is continuous we can extend this inclusion to crossed products (Ai α G)(A jα G) ⊂ (Ai∧ jα G). iv) It is clear since crossed products preserve the inductive limits. Remark 2.2 Let i, j ∈ L with i j. Denote by ϕi, j : A j → M(Ai ) the structure morphisms of A. We deduce morphisms ϕi,α j : A jα G → M(Ai α G) and ϕi,r,αj : A jr,α G → M(Ai r,α G). It is clear that these are the structure morphisms of (Aα G, (Ai α G)i∈L ) and (Ar,α G, (Ai r,α G)i∈L ) respectively.
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Moreover, if L has a least element 0 and the structure morphisms ϕi, j of A satisfy ϕi,−1j (Ai ) = {0} then by [12], proposition 1.18 the morphism ϕ0 : A → M(A0 ) is injective. Therefore, the morphism ϕr,α is injective. 0 3 Spectrum of Commutative Graded C∗ -Algebras Let L be a semilattice and (A, (Ai )i∈L ) a C∗ -algebra graded by L. We saw in [12] that A is commutative if and only if for any i ∈ L the component Ai is commutative. Definition 3.1 We say that L is a good semilattice if any totally ordered nonempty part of L is well-ordered (i.e. any nonempty subset of this part has a least element). Proposition 3.2 Let L be a semilattice. The following are equivalent: a) b) c) d)
The set L is a good semilattice. Every totally ordered nonempty part of L has a least element. Every nonempty subsemilattice of L has a least element. Every final nonempty subsemilattice of L has a least element.
Proof It’s easy to see that a) ⇔ b ). We show now that b ) ⇒ c). For any nonempty subsemilattice F of L, by Zorn’s lemma, there exists a maximal totally ordered subset A of F . We denote by a its least element and we take x ∈ F . Since F is a subsemilattice we have a ∧ x ∈ F . The set {a ∧ x} ∪ A is totally ordered and A is maximal, so a ∧ x ∈ A and therefore a ≤ a ∧ x, so a ≤ x. This means that a is the least element of F . Clearly c) ⇒ b ) and c) ⇒ d). To see that d) ⇒ c) we consider a subsemilattice I of L and we set J = { j ∈ L; ∃i ∈ I; i ≤ j }. Note that J is a final subsemilattice of L so by assumption it has a least element that we denote by j0 . Since j0 ∈ J there is i0 ∈ I such that i0 ≤ j0 . But we also have I ⊂ J, so i0 ∈ J and j0 ≤ i0 . In other words i0 = j0 is the least element of I. We will show now that the spectrum of commutative graded C∗ -algebras is related with the spectrum of their components. Remark 3.3 Let (A, (A )∈L ) be a commutative graded C∗ -algebra. Take i ∈ L and χi a character of Ai . Denote by χ i its extension to the multiplier algebra M(Ai ). Then, we can define in a unique way a character of A by χ = χ i ◦ πi where πi = ϕi ◦ pi is the morphism A → M(Ai ). We get then a continuous injective map ψi : Sp(Ai ) → Sp(A). Let χ be a character of A and assume that there exists i, j ∈ L such that χ = χ i ◦ πi = χj ◦ π j. Assume also that i ≤ j. Then πi| A j = 0 and we have χ i ◦ πi | A j = 0
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which is absurd since χj ◦ π j| A = χ j = 0. This means actually that the ψi ’s have j
disjoint images. Proposition 3.4 Let (A, (A )∈L ) be a commutative graded C∗ -algebra. If L is a good semilattice then Sp(A) =
i∈L
ψi (Sp(Ai )).
Proof Let χ ∈ Sp(A). We consider the set I = {i ∈ L; χ| Ai = 0}. Note that I is a subsemilattice of L since for i, j ∈ I and ai ∈ Ai , a j ∈ A j such that χ(ai ) = 0 and χ(a j) = 0, we have χ(ai a j) = χ(ai )χ(a j) = 0 i.e. χ| Ai∧ j = 0. It follows that the set I has a least element denoted by i0 (Proposition 3.2). We set now χi0 = χ| Ai , then χi0 = 0. We will show that χ = ψi0 (χi0 ). By linearity and continuity 0
we just need to show that these characters coincide on A j for any j ∈ L. So, let j ∈ L and a j ∈ A j. If i0 j, since i0 is the least element of I, then j∈ / I and therefore χ| A j = 0. We also have ψi0 (χi0 )(a j) = χ i0 ◦ πi0 (a j ) = 0. If i0 ≤ j, since χi0 = 0 there exists ai0 ∈ Ai0 such that χi0 (ai0 ) = χ(ai0 ) = 1 and since πi0 (ai0 ) = ai0 then ψi0 (χi0 )(ai0 ) = 1. Then, for a j ∈ A j we have ψi0 (χi0 )(a j) = ψi0 (χi0 )(a j)ψi0 (χi0 )(ai0 ) = ψi0 (χi0 )(a jai0 ) = χi0 (a jai0 ) = χ(a jai0 ) = χ(a j)χ(ai0 ) = χ(a j) Remark 3.5 Let L be a good semilattice and (A, (A )∈L ) a commutative graded C∗ -algebra. Assume that the structure morphisms ϕi, j of A are nondegenerate for all i, j ∈ L such that i ≤ j. Let M be a subsemilattice of L. Assume that AM A = A, then the inclusion ι : AM → A implies a continuous map : Sp(A) → Sp(AM ) which is given by χ→ χ ◦ ι. We will study this map with the help of the decompositions Sp(A) = ψi (Sp(Ai )) and Sp(AM ) = ρm (Sp(Am )) (the second one is due to the
i∈L
m∈M
fact that a subsemilattice of a good semilattice is itself a good semilattice). Take χ ∈ Sp(A). There exists a unique i ∈ L such that χ = ψi (χi ) ∈ ψi (Sp(Ai )). Set Li = { j ∈ L; i ≤ j}. Since χ| Ai = 0 and for all j ∈ L such that i ≤ j the morphisms ϕi, j are non-degenerate, χ is not zero over A j Ai hence χ| A j = 0. Therefore one has Li = { j ∈ L; i ≤ j} = { j ∈ L; χ| A j = 0}. Set Mi = M ∩ Li = {m ∈ M; i ≤ m} = { j ∈ M; χ| A j = 0}. Then Mi is a final subsemilattice of M and since L is a good semilattice M has a least element (Proposition 3.2). Denote by m(i) = inf Mi , then χ| Am(i) = χm(i) = 0. Note that the morphism ϕi,m(i) : Am(i) → M(Ai ) gives rise to a continuous map i : Sp(Ai ) → Sp(Am(i) ) which is given by i (χi ) = χ i ◦ ϕi,m(i) . If we
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denote by ϕ i,m(i) the extension of ϕi,m(i) to the multiplier algebra M(Am(i) ), one sees (we only need to check it on the components of AM ) that the diagram AM
/ A
M(Am(i) )
/ M(Ai )
commutes. In consequence we have commuting diagrams Sp(Ai )
i
ψi
Sp(A)
/ Sp(Am(i) )
ρm(i)
/ Sp(AM )
which show that the map is entirely described by the map ρm(i) and the maps i .
4 Semilattices of Closed Subgroups Let G be a locally compact group. Note that the set G of closed subgroups of G is a complete lattice for the inclusion: if (Hi )i∈I is a family of closed subgroups of G, it is clear that the greatest element of {H ∈ G ; ∀i ∈ I, H ⊂ Hi } is the intersection i∈I Hi ; the least element of {H ∈ G ; ∀i ∈ I, Hi ⊂ H} is the closure of the subgroup generated by the Hi ’s. For H ∈ G , set A H = C0 (G/H). If H, K ∈ G are such that K ⊂ H, every left equivalent class x ∈ G/K is contained in a left equivalent class p K,H (x) ∈ G/H. This will induce a continuous map p K,H : G/K → G/H and therefore a homomorphism ϕ K,H : A H = C0 (G/H) → Cb (G/K) = M(A K ). Lemma 4.1 Let H, K ∈ G . The following conditions are equivalent: (i) We have ϕ H∩K,H (A H )ϕ H∩K,K (A K ) ⊂ A H∩K . (ii) The map q : x → ( p H∩K,H (x), p H∩K,K (x)) from G/(H ∩ K) to G/H × G/K is closed. (iii) The subset HK = {xy; x ∈ H, y ∈ K} is closed in G and the product map H × K → HK, (x, y) → xy is open. (iv) p H (K) is closed (in G/H) and the restriction map K → p H (K) of p H is open. (v) p K (H) is closed (in G/K) and the restriction map H → p K (H) of p K is open.
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Proof To prove this lemma we will use some general topological results that can be found in [1]. Let us show that we have (ii) ⇒ (i). Since q is continuous, injective and closed, it is proper. Thus we have a map q∗ : C0 (G/H × G/K) → C0 (G/(H ∩ K)) and for f ∈ C0 (G/H), g ∈ C0 (G/K) we have q∗ ( f ⊗ g) = ( f ⊗ g) ◦ q = f ( p H∩K,H )g( p H∩K,K ) = ϕ H∩K,H ( f )ϕ H∩K,K (g) = fg. The assertion now follows since q∗ ( f ⊗ g) ∈ C0 (G/(H ∩ K)). For the converse (i) ⇒ (ii) if C is a compact subset of G/H × G/K, there exists compact subsets C1 of G/H and C2 of G/K such that C ⊂ C1 × C2 . Since q is continuous, q−1 (C) is a closed set of q−1 (C1 × C2 ). Take now f ∈ C0 (G/H), g ∈ C0 (G/K) such that f = 1 on C1 and g = 1 on C2 . Then S = {x ∈ G/(H ∩ K) : ( fg)(x) = 1} is a compact subset of G/(H ∩ K) and since q−1 (C1 × C2 ) ⊂ S then q−1 (C1 × C2 ) is compact and q is proper. We show now that (ii) ⇔ (iii). Define a map r : G → G/H × G/K by x → ( p H (x), p K (x)) and take the quotient map p : G → G/(H ∩ K). Since q is injective, p is open and r = q ◦ p, we can replace condition (ii) by the equivalent (ii) : r(G) is a closed set and r : G → r(G) is open. Take the open map t : G × G → G/H × G/K defined by (x, y) → ( p H (x), p K (y)) and consider the following diagram / G×G G OO OOO OOO OOO p t OOO r ' / G/H × G/K. G/(H ∩ K) q
Since r(G) = q(G/(H ∩ K)) = {(x, y) ∈ G/H × G/K; ∃g ∈ G : x = gH, y = gK}, we have t−1 (r(G)) = {(g1 , g2 ) ∈ G × G; ∃g ∈ G, ∃h ∈ H,
k ∈ K : g1 = gh, g2 = gk}
= (g1 , g2 ) ∈ G × G; ∃h ∈ H, k ∈ K : g1−1 g2 = h−1 k
= (g1 , g2 ) ∈ G × G : g1−1 g2 ∈ HK = HK.
Thus HK is closed if and only if t−1 (r(G)) is closed in G × G, therefore if and only if r(G) is closed. The fiber product of G by t−1 (r(G)) over r(G) is given by G×r(G) t−1 (r(G)) = G×G/H×G/K (G × G) = {(x, y, z) ∈ G × G × G; r(x) = t(y, z)} = {(x, y, z) ∈ G × G × G; x−1 y ∈ H, x−1 z ∈ K}.
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Take (g, h, k) ∈ G× H× K and set s(g, h, k) = g and ϕ(g, h, k) = (gh, gk) ∈ t−1 (r(G)). Then the map ψ : G × H × K → G×r(G) t−1 (r(G)) defined by (g, h, k) → (g, gh, gk) is a homeomorphism and gives rise to a cartesian square φ
/ t−1 (r(G)) G× H×K MMM MMM tr(G) MMM s MM& / r(G). G r
Since the maps of this diagram are surjective, tr(G) is open (since t is open) and G × H × K is the fiber product G×r(G) t−1 (r(G)) then r is open if and only if φ is open. If we denote by T : t−1 (r(G)) → G × HK the map (g1 , g2 ) → (g1 , g1−1 g2 ) which is a homeomorphism then φ is open if and only if T ◦ φ : G × H × K → G × HK where T ◦ φ(g, h, k) = (g, hk) is open. In other words if and only if H × K → HK is open. The final equivalence (iii) ⇔ (iv) follows by the fact that p−1 H p H (K) = K H therefore if we pass to the inverse elements of K H we have that p H (K) is closed if and only if HK is closed and also by the cartesian square: K×H
/ KH
K
/ p H (K)
where the homeomorphism K × H → K× p H (K) K H is defined by (k, h) → (k, kh). Take now a subsemilattice L of G and assume that every H, K ∈ L satisfy the equivalent conditions of Lemma 4.1. We have then the following proposition (cf. [12] Theorem 2.2). Proposition 4.2 There exists a L-graded C∗ -algebra (A, (A H ) H∈L ) whose structure morphisms are the ϕ K,H ’s defined as above. The multiplier algebra of C0 (G) is the algebra of continuous bounded functions on G, Cb (G). Denote by p H : G → G/H the quotient map for H ∈ L. We can then identify C0 (G/H) as a C∗ -subalgebra of Cb (G) via the application f → f ◦ p H . We have then morphisms ϕ H which are nondegenerate for all H ∈ L and we can then extend them to morphisms ϕ H defined on the multiplier algebra Cb (G/H). It is easy to see that ϕ H = ϕ K ◦ ϕ K,H for all H, K ∈ L such that K ⊂ H. This implies that ϕ H ( f )ϕ K (g) = ϕ H∩K ( fg) for all K, H ∈ L and f ∈
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C0 (G/H), g ∈ C0 (G/K) and gives rise to a homomorphism ϕ : A → Cb (G) (cf. [12], proposition 1.10). Proposition 4.3 The homomorphism ϕ : A → Cb (G) is injective if and only if for any H, K ∈ L satisfying K ⊂ H and H = K the quotient space H/K is not compact. We will need the following lemma: Lemma 4.4 Let K, H be two subgroups of G with K ⊂ H. a) If H/K is compact we have ϕ K,H (C0 (G/H)) ⊂ C0 (G/K). b) If H/K is not compact we have ϕ K,H (C0 (G/H)) ∩ C0 (G/K) = {0}. For this we will show the following lemma: Lemma 4.5 Let G be a locally compact group and let H and K be closed subgroups of G with K ⊂ H. a) If H/K is compact, p K,H is proper. b) If H/K is not compact, for all a ∈ G/H the subset p−1 K,H ({a}) of G/K is not relatively compact. Proof a) Let C be a compact subset of G/H, since G is locally compact and p H is open and surjective, there exists a compact subset A of G such that p H (A) = C. We have p−1 K,H (C) = {ax; a ∈ A, x ∈ H/K}, which is the image of the compact set A × H/K by the continuous map G × G/K → G/K. Thus p−1 K,H (C) is a compact subset of G/K. b) Take g ∈ G whose class is a, the application x → gx is a homeomorphism of G/K that maps H/K into p−1 K,H ({a}). But since H/K is closed in G/K and it is not compact then it is not relatively compact. Proof of Lemma 4.4 a) Let f ∈ C0 (G/H). For every ε > 0, {x ∈ G/K; | f ◦ p K,H (x)| ε} = p−1 K,H ({x ∈ G/H; | f (x)| ε}) is a compact set, therefore f ◦ p K,H ∈ C0 (G/K). b) Let f ∈ C0 (G/K) ∩ ϕ K,H (C0 (G/H)). For all a ∈ G/H, the map f which is constant on p−1 K,H ({a}) is then zero. Proof of Proposition 4.3 If there exists H, K with K ⊂ H, H = K and H/K compact, then for f ∈ C0 (G/H) we have ϕ( f − ϕ K,H ( f )) = ϕ H ( f ) − ϕ K (ϕ K,H ( f )) = 0, therefore ϕ is not injective.
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Conversely assume that for all H, K ∈ L such that K ⊂ H and H = K the quotient space H/K is not compact. We have three cases: a) If {e} ∈ L. This follows immediately by applying Lemma 4.4 and by proposition 1.18 in [12]. b) Assume that neither of H ∈ L is compact, if we set L = L ∪ {{e}} we get to the first case. c) If there exists H ∈ L with H compact, then for all K ∈ L, the space H/(H ∩ K) is compact. But this is possible only if H ∩ K = H. Therefore H is the least element of L. Then π H : A → Cb (G/H) ⊂ Cb (G) is injective (proposition 1.18 in [12]). Remark 4.6 Assume that we are given a semilattice of closed subgroups that satisfy the conditions of Lemma 4.1 and Proposition 4.3. Thanks to Theorem 2.1 the family of C∗ -subalgebras C0 (G/H)red G ⊂ B(L2 (G)) form a graded C∗ -subalgebra of B(L2 (G)).
5 Examples If H is a Hilbert space then B(H) and K(H) will denote the C∗ -algebras of all bounded and compact operators on H respectively. 5.1 Now let us reconsider the crossed product studied by M. Damak and V. Georgescu (cf. [6]) and by V. Georgescu and A. Iftimovici (cf. [8], theorem 3.12). Recall that X will be a finite dimensional vector space and G the lattice of all vector subspaces of X for the inclusion. Since in finite dimension, every vector subspace is closed and every linear surjective map is open, condition (iii) of Lemma 4.1 is satisfied and for all Y, Z ∈ G we have C0 (X/Y)C0 (X/Z ) ⊂ C0 (X/(Y ∩ Z )). Thanks to Proposition 4.2, for every subsemilattice L of G we have a L-graded C∗ -algebra A= C0 (X/Y). Y∈L
Moreover, since every vector space is not compact, condition b ) of Lemma 4.5 is also satisfied and we can then realize A as a C∗ -subalgebra of Cb (X) = M(C0 (X)) ⊂ B(L2 (X)). We take now the crossed product A X where X acts continuously by translation. Then Proposition 2.1 implies that this crossed product is also a L-graded C∗ -algebra i.e. A X = (C0 (X/Y) X). Y∈L
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By Remark 2.2, A X ⊂ M(C0 (X) X)
= M(K(L2 (X))) = B(L2 (X)).
In other words A X is a graded C∗ -algebra of operators defined on the Hilbert space L2 (X). 5.2 We take a plane P and n vector lines δi , i = 1, ..., n of P . Then we can form a lattice of vector spaces F = {{0}, δ1 , ..., δn , P }. The previous example shows that we have a F -graded C∗ -algebra: A= C0 (P /Y) ⊂ Cb (P ) Y∈F
which is commutative and has a unit. Our goal is to compute the spectrum of this algebra. We will start by studying the case n = 1. Then F = {{0}, δ1 , P } and A = C0 (P )⊕C0 (P /δ1 )⊕C. Consider the sublattice of F , F1 = {{0}, δ1 }. Then A1 = C0 (P )⊕C0 (P /δ1 ) is a F1 -graded C∗ -subalgebra of Cb (P ) and moreover it is an ideal for A. The Gelfand–Naimark theorem now implies that A1 C0 (Sp B1 ). Identify P with R2 and δ1 with the “axis of x” (i.e. {(x, y); y = 0}). Then we can consider an element of C0 (P /δ1 ) as the function (x, y) → g(y) where g ∈ C0 (R). There exists a unique injective homomorphism ψ : A1 → C0 (T × R) whose restriction to C0 (P ) and C0 (P /δ1 ) is the natural inclusion ψ0 : C0 (P ) → C0 (T × R) and ψδ1 : C0 (P /δ1 ) → C0 (T × R) respectively ([12], proposition 1.10). Note that the product fg is given by the formula ( fg)(x, y) = f (x, y)g(y) for all (x, y) ∈ T × R. 1 thus the morWe get the algebra A by adjoining a unit to A1 , i.e. A = A phism ψ has a unique extension to a isomorphism ψ : A → C((T × R) ∪ {∞}) where (T × R) ∪ {∞} is the one point compactification which is homeomorphic topologically to a pinched sphere. For n 2, consider in the first place the vector space E = Rn and the canonical basis (e1 , ..., en ). Let I be a subset of {1, ..., n}. Denote by P I the vector subspace generated by (ei )i∈I . Let M be the set {P I ; I ⊆ {1, ..., n}}, then M is a lattice whose least element is {0}. Note that the map I → P I is an isomorphism of lattices. We get then a M-graded C∗ -algebra (B, (C0 (E/H)) H∈M ) whose structure morphisms are the ϕ K,H ’s defined as above and we have B= C0 (E/H) C(Sp(B)) ⊂ Cb (E). H∈M
Thus Sp(B) is a compactification of E which is homeomorphic to the n dimensional torus, Tn . Indeed, take H ∈ F then we can write H = P I where I ⊆ J. The quotient maps E → E/H and the natural inclusions of the quotients
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E/H into their compactifications T{1,...,n}\I deduce the following commuting diagrams
Cb (E).
ψH
/ C(Tn ) t t tt tt t y t t
C0 (E/H)
Note that the morphism C(Tn ) → Cb (E) is injective since Tn is a compactification of E which implies that the ψ H ’s are injective. Therefore there exists a unique and injective homomorphism ψ : B → C(Tn ) whose restriction on C0 (E/H) is ψ H ([12], proposition 1.10). Moreover by the Stone-Weiestrass theorem ψ is surjective. We go back now to the case of A. Define n linear forms fi , i = 1, ..., n of P with kernels equal to the lines δi , i = 1, ..., n respectively. To the proper inclusion ( f1 , ..., fn ) : P → E corresponds a surjective homomorphism Cb (E) → Cb (P ) which induces together with the inclusion B ⊂ Cb (E) a morphism : B → Cb (P ). We will show now that (B) = A. For all H ∈ M, let H be the restriction of on the components of B, C0 (E/H). If dim H n − 2 we have P → E/H since P ∩ H = {0}. This implies that the image of C0 (E/H) by H in Cb (P ) is C0 (P ). If we denote by Hi = P{1,...,n}\{i} , i = 1, ..., n the hyperplans of E and H = Hi , i = 1, ..., n then for any i, E/Hi is homeomorphic to P /δi since δi = P ∩ Hi for i = 1, ..., n and we have an isomorphism Hi : C0 (E/Hi ) → C0 (P /δi ). Denote by P the closure of P in Tn and consider C(P ) as a subalgebra of Cb (P ), then is the restriction of C(Tn ) to P . Thus Sp(A) = P . We will show that it is homeomorphic to the torus Tg with g holes where g = E n2 which is pinched, i.e. we force the identification of two of its points, when n is odd. Denote by Q the closure of P in (R)n . Then the closure P of P in Tn is equal to ρ(Q) where ρ is the canonical surjection (R)n → Tn . With our first assumptions, the plane P will have the following form:
P = (x1 , ..., xn ) ∈ Rn ; xk = Aijk xi + Bijk x j , where i, k, j ∈ {1, ..., n}, i, j are fixed, k ∈ / {i, j} the real coefficients Aijk , Bijk are non zero and they satisfy Bijk Aij = Aijk Bij for all k, ∈ / {i, j} with k = . This form of P implies that the only possible limits in Q are vectors of whom a coefficient is finite and the others are infinite i.e. of the form (∞, ..., ∞, ai , ∞, ..., ∞) with i ∈ {1, ..., n}. We will call them the infinite lines and since i covers the set {1, ..., n} they are 2n. We will show that the number of intersection points of these lines, which we will call infinite points, is also 2n. In consequence Q will have the form of a polygon with 2n sides. We can see this by choosing in the first place an orientation of P and a point n δi . We then choose to call the first line that we meet, counterA ∈ P \ ∪i=1 clockwise, δ1 , the second one δ2 until the nth δn . We choose now the n linear forms fi , i = 1, ..., n so as to have fi (A) = 1 for all i. Then, inside the sector
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formed by the lines δ1 and δn which contain the point A, the signs of the infinite point will all be positive. If we pass the line δ1 to the sector formed by δ1 and δ2 above the point A, the values of f1 become negative, thus the first sign will change to negative i.e. the infinite point will be (−∞, +∞, ..., +∞). With the same procedure we find only 2n possibilities of signs that have the form (+, ..., +, −, ..., −) and (−, ..., −, +, ..., +). If we pass now from (R)n to Tn , we identify +∞ and −∞, which means that a) we identify each side with the opposite one b) we identify all vertices.
The first identification, gives rise to a Riemann surface Yg of whom the canonical form is α1 α2 ...αn α1−1 α2−1 ...αn−1 (cf. for example [11, 13]) which is homeomorphic to −1 −1 α1 α2 α1−1 α2−1 ...αn−1 αn αn−1 αn
if n is even and −1 −1 α1 α2 α1−1 α2−1 ...αn−2 αn−1 αn−2 αn−1 αn αn−1
if n is odd. It will be then a torus with g holes where g = E n2 . We index the vertices by Z/2nZ. By doing the identification of the edges we also identify the jth vertex with the j + n − 1th one for all j ∈ Z/2nZ. In other words two vertices are identified if and only if they are in the same orbit for the addition of (n − 1) in Z/2nZ. But if n is even GCD(n − 1, 2n) = 1 and if n is odd GCD(n − 1, 2n) = 2.
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Therefore the map Yg → Sp(A) is a homeomorphism if n is even. If n is odd, we get from Yg to Sp(A) by identifying those two points, the spectrum Sp(A) is then homeomorphic to a pinched surface of genus g. Note that if we take the lattice F minus a line, for example let us choose F = F \{δn }, then for the corresponding graded C∗ -algebras AF and AF we have AF ⊂ AF and therefore we have a map : Sp(AF ) → Sp(AF ). If we denote by A H = C0 (P /H) for all H ∈ F , since F and F are finite, they are good lattices and Proposition 3.4 implies that Sp(AF ) = Sp(A H ) and H∈F Sp(AF ) = Sp(A K ) . K∈F
Take χ ∈ Sp(AF ). There exists a unique H ∈ F such that χ = χ H ∈ Sp(A H ). If we apply Remark 3.5 we get that: If H ∈ F , the image of χ H by the map is itself. If H ∈ F \F i.e. H = δn , then inf{K ∈ F ; δn ⊆ K} = P and the image of χδn by is in Sp(AP ) i.e. it is the point at infinity. In other words, one obtains the spectrum of AF from the spectrum of AF by contracting the line P /δn to one point: the point at infinity. 6 Example of a C∗ -Algebra Associated to a Semi Lattice Let L be a semilattice and (Ai )i∈L the family of C∗ -algebras defined by Ai = C for all i ∈ L. For i, j ∈ L such that i ≤ j set ϕi, j = IdC . These morphisms satisfy properties a) and b ) of proposition 1.16 in [12], so there exists up to isomorphism a unique L-graded C∗ -algebra (A, (Ai )i∈L ) whose structure morphism are the ϕi, j’s ([12], theorem 2.2). We want to compute its spectrum Sp(A). We will show that the set of nonempty final subsemilattices of L is in bijection with the set of characters of A. First of all, denote by ei , i ∈ L the image of 1 ∈ Ai in A. It is easy to see that ei is an idempotent for all i ∈ L therefore if χ is a character of A we have χ(ei ) ∈ {0, 1}. Moreover, by the definition of the product in A we have ei e j = ei∧ j for all i, j ∈ L. Let χ be a character of A. Set Mχ = {i ∈ L; χ(ei ) = 1}. It is a subsemilattice of L since for i, j ∈ Mχ we have χ(ei∧ j) = χ(ei e j) = χ(ei )χ(e j) = 1. Moreover it is a final subsemilattice because for i ∈ Mχ and j ∈ L such that i ≤ j we have χ(e j) = χ(ei )χ(e j) = χ(ei e j) = χ(ei ) = 1 i.e. j ∈ Mχ . Let M be a non zero final subsemilattice of L. There exists a unique homomorphism χM : A → C satisfying χM (λei ) =
λ, i ∈ M 0, i ∈ /M
for all λ ∈ C and
all i ∈ L. Indeed, since M is a final subsemilattice we have χM (λei )χM (μe j) = χM (λμei∧ j) then apply proposition 1.10 from [12]. Since M = ∅, χM is a character of A. Note that since each character is determined by its value on ei (i ∈ L) the maps χ → Mχ and M → χM give the wanted bijection. Finally, observe that since Ai = C, the spectrum Sp(Ai ) has only one point IdC . We have ψi (IdC )(e j) =
1, i j 0, i j
i.e. ψi (IdC ) = χLi . This will mean that a
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character χ of A is in
ψi (Sp(Ai )) if and only if Mχ has the form of Li i.e. if
i∈L
and only if it has a least element. In other words Sp(A) = only if L is a good semilattice. We will study now a particular case of this situation.
ψi (Sp(Ai )) if and
i∈L
Example 6.1 Let us consider the set Q with its order. Note that Q being totally ordered, it is a semilattice which is not a good semilattice. We will study the spectrum of the associated graded C∗ -algebra (A, (Ai )i∈Q ). Denote by Bb (R) the C∗ -algebra of Borel bounded complex functions defined on R. For any i ∈ Q we define a map τi : Ai → Bb (R) by τi (λei ) = λ1]−∞,i] where 1 is the characteristic function. On can see that τi is an injective morphism of C∗ -algebras satisfying the equality τi∧ j(λei μe j) = τi (λei )τ j(μe j) for all i, j ∈ Q and λ, μ ∈ C. Therefore there exists a unique morphism τ : A → Bb (R) whose restriction to Ai is τi for all i ∈ Q. To see that τ is injective we need to show that τF , the restriction of τ to AF , is injective for any finite subsemilattice F of Q ([12], proposition 1.9). Let F ∈ FQ . Assume that F has n elements. Since it is totally ordered we can also assume that F = {i1 , ..., in } with i1 < i2 < ... < in . Take x ∈ ker τF . Then x= λi ei and λi 1]−∞,i] (s) = 0 for all s ∈ R. If s is such that in−1 < s in i∈F
i∈F
we find λn = 0. If s is taken such that in−2 < s ≤ in−1 we have λn−1 + λn = 0 therefore λn−1 = 0. In the same way we show that λi = 0 for all i ∈ F and the assertion follows. We identify now A with its image by τ and we will show that τ (A) = A where A is the space of bounded Borel complex functions defined on R that are regulated, left-continuous at all points of R, continuous on R\Q, zero at +∞ and that have a limit at −∞. Note that A is a C∗ -subalgebra of Bb (R) since it is closed. This implies together with the fact that AL is dense in A and τ (AL ) ⊂ A that τ (A) ⊂ A. Take f ∈ A. Assume that the limit of f at −∞ is equal to λ, then let ε > 0, there exists a ∈ Q such that for all x a we have | f (x) − λ| < ε. On the other hand f is zero at +∞ therefore there exists A ∈ Q such that for all x ≥ A we have | f (x)| < ε. This means that for all x ∈] − ∞, a]∪]A, +∞[ we have | f (x) − λ1]−∞,a] (x)| < ε. Since f is a regulated function, it is uniformly approximated on [a, A] by a step function g. There exists then a subdivision i0 , i1 , ..., is of [a, A] such that i0 = a and is = A and g is equal to a constant cr over ]ir−1 , ir [ for all r ∈ {1, ..., s}. By assumption f is left-continuous at ir for all r ∈ {0, ..., s} then | f (ir ) − cr | = lim | f (t) − g(t)| ε. t→ir −
/ Q, the function f is continuous at ir therefore there exists jr ∈ If ir ∈ ]ir , ir+1 [∩Q such that for all t ∈ [ir , jr ] we have | f (t) − f (ir )| < ε. Then for t ∈ [ir , jr ] we have | f (t) − cr | | f (t) − f (ir )| + | f (ir ) − cr | < 2ε.
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If ir ∈ Q, set jr = ir . We have now a new subdivision j0 , ..., js of [a, A]. Let g be the function defined by ⎧ 0, t > A ⎪ ⎪ ⎨ g(t) = λ, t a ⎪ ⎪ ⎩ cr , t ∈] jr−1 , jr ] with r ∈ {1, ..., s}. In other words g = λ1]−∞,a] +
s
cr 1] jr−1 , jr ] = λτ (ea ) +
r=1
s
cr τ (e jr − e jr−1 ) ∈
r=1
τ (AF ) where F = { j0 , ..., js } ∈ FQ . Since || f − g||∞ = sup | f (t) − g(t)| < 2ε t∈R
and τ (A) is closed then f ∈ τ (A). We have Sp(A) = Sp(τ (A)) = Sp(A). Since there is a bijection between the characters χ of A and the nonempty final subsemilattices M of Q which have the only possible forms: M = Q or M = Q ∩ [t, +∞[≡ Lt or M = Q∩]t, +∞[≡ Mt with t ∈ Q then we have a natural bijection between χ and {Q} ∪ {Lt ; t ∈ R} ∪ {Mt ; t ∈ Q}. Identify now the character which corresponds to each of these final subsemilattices through the isomorphism τ : A → A described as above. We find the following cases: •
χQ (a) = lim τ (a)(s),
• •
χLt (a) = τ (a)(t) for all t ∈ R, χMt (a) = lim τ (a)(s) for all t ∈ Q,
s→−∞
s→t+
(we only have to check it on the generators ei (i ∈ L)) that give the bijection of the spectrum of A with the set R Q {−∞}. Acknowledgements I am much indebted to Georges Skandalis for his valuable advice and useful comments during the preparation of the paper.
References 1. Bourbaki, N.: Eléments des mathématiques, Topologie Générale, Chapitres 1 à 4, Hermann, Paris (1971) 2. Boutet de Monvel-Berthier, A., Georgescu, V.: Graded C∗ -algebras in the N-body problem. J. Math. Phys. 32, 3101–3110 (1991) 3. Boutet de Monvel-Berthier, A., Georgescu, V.: Graded C∗ -algebras and many-body perturbation theory. I. The N-body problem. C. R. Acad. Sci. Paris Sér. I Math. 312(6), 477–482 (1991) 4. Boutet de Monvel-Berthier, A., Georgescu, V.: Graded C∗ -algebras and many-body perturbation theory. II. The Mourre estimate. Astérisque 6–7(210), 75–96 (1992) 5. Boutet de Monvel-Berthier, A., Georgescu, V.: Graded C∗ -algebras associated to symplectic spaces and spectral analysis of many channel Hamiltonians. In: Dynamics of Complex and Irregular Systems (Bielefeld 1991), Bielefeld Encount. Math. Phys., vol. 8, pp. 22–66. World Science, River Edge (1993)
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6. Damak, M., Georgescu, V.: C∗ -cross products and a generalized mechanical N-body problem. Mathematical physics and quantum field theory. Electron. J. Differential Equations 4, 51–69 (2000) 7. Damak, M., Georgescu, V.: C∗ -algebras related to the N-body problem and the self-adjoint operators affiliated to them, preprint 99–482. http://www.ma.utexas.edu/mparc/ 8. Georgescu, V., Iftimovici, A.: C∗ -algebras of energy observables. I. General theory and bumps algebras, preprint 00-521. http://www.ma.utexas.edu/mparc/ 9. Georgescu, V., Iftimovici, A.: C∗ -algebras of energy observables. II. Graded symplectic algebras and magnetic Hamiltonians, preprint 01-99. http://www.ma.utexas.edu/mparc/ 10. Georgescu, V., Iftimovici, A.: Crossed products of C∗ -algebras and spectral analysis of quantum hamiltonians. Comm. Math. Phys. 228, 519–560 (2002) 11. Massey, W.S.: Algebraic Topology: An Introduction. Springer, New York (1967) 12. Mageira, A.: Graded C∗ -algebras. J. Funct. Anal. 254, 1683–1701 (2008) 13. Springer, G.: Introduction to Riemann Surfaces. Addison-Wesley, Reading (1957)
