We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles of random n × n Hermitian matrices Z −1 n,N | det M | 2α e −N Tr V (M ) dM with α > −1/2, where the factor | det M | 2α induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as n, N → ∞ such that n 2/3 (n/N − 1) = O(1). We use the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight |x| 2α e −N V (x) . Our main attention is on the construction of a local parametrix near the origin by means of the ψ-functions associated with a distinguished solution of the Painlevé 2000 Mathematics Subject Classification: 15A52, 33E17, 34M55
In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form Z −1 n,N | det M | 2α e −N Tr V (M ) dM with α > −1/2. The factor | det M | 2α induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we computed the limiting eigenvalue correlation kernel in the double scaling limit as n, N → ∞ such that n 2/3 (n/N − 1) = O(1) by using the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight |x| 2α e −N V (x) . Our main attention was on the construction of a local parametrix near the origin by means of the ψ-functions associated with a distinguished solution u α of the Painlevé XXXIV equation. This solution is related to a particular solution of the Painlevé II equation, which however is different from the usual Hastings-McLeod solution. In this paper we compute the asymptotic behavior of u α (s) as s → ±∞. We conjecture that this asymptotics characterizes u α and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painlevé XXXIV equation which includes u α . We identify this family as the family of tronquée solutions of the thirty fourth Painlevé equation.
ABSTRACT. The aim of this paper is to construct exact formulae for reflectionless potentials for ordinary differential operators of order four. They lead to soliton type solutions which are well known for one dimensional Schrödinger operators. Such solitons are solutions of some non-linear integrable systems appeared in [8] (see also [9]).
A non-linear functional Q [u, v] is given that governs the loss, respectively gain, of (doubly degenerate) eigenvalues of fourth order differential operators L = ∂ 4 + ∂ u ∂ + v on the line. Apart from factorizing L as A * A + E 0 , providing several explicit examples, and deriving various relations between u, v and eigenfunctions of L, we find u and v such that L is isospectral to the free operator L 0 = ∂ 4 up to one (multiplicity 2) eigenvalue E 0 < 0. Not unexpectedly, this choice of u, v leads to exact solutions of the corresponding time-dependent PDE's.
This paper is dedicated to Barry Simon on the occasion of his 60th birthday in appreciation for all that he has taught us.
AbstractIn this paper, the authors show how to use Riemann-Hilbert techniques to prove various results, some old, some new, in the theory of Toeplitz operators and orthogonal polynomials on the unit circle (OPUCs). There are four main results: the first concerns the approximation of the inverse of a Toeplitz operator by the inverses of its finite truncations. The second concerns a new proof of the 'hard' part of Baxter's theorem, and the third concerns the Born approximation for a scattering problem on the lattice Z + . The fourth and final result concerns a basic proposition of Golinskii-Ibragimov arising in their analysis of the Strong Szegö Limit Theorem.Let d be a probability measure on the unit circle = {z ∈ C : |z| = 1} and let n = z n + · · ·, n 0, be the (monic) orthogonal polynomials (OPUCs) associated with d , m (z) n (z) d = 0, m = n, m, n 0 (see [20]). Let = ( n ) n∈Z + denote the vector of Verblunsky coefficients n = − n+1 (0), n 0. By Verblunsky's theorem (see [17]), the map V : d → is a bijection from the probability measures on onto × ∞ j =0 D, where D = {z ∈ C : |z| < 1} is the (open) unit disc in C. Following Cantero et al.[5], we may, given , construct a (pentadiagonal) unitary matrix operator U = U( ) in l 2 + = l 2 (Z + ) (the so-called CMV matrix) with the following property: e 0 = (1, 0, . . . ) T is a cyclic vector for U, i.e. U k e 0 −∞
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