On the Analyticity of the Spectral Density for Semiclassical NLS

In a previous work, we have analyzed the semiclassical behavior of solutions to the focusing,completely integrable nonlinear Schroedinger equation, under the assumption of real analytic initial data (among others). We have provided global semiclassical asymptotics under the so-called ”finite gap” assumption. In a subsequent paper, we have justified the ”finite gap” assumption, again under several assumptions, the main assumption being that the limiting spectral density of the eigenvalues of the associated Dirac operator has an analytic extension in the upper half-plane. In the present article, we show that this constraint is unnecessary. In fact, analyticity of the neccessary quantities in the analysis can be recovered via the solution of a scalar Riemann-Hilbert problem

Asymptotics via Steepest Descent for an Operator Riemann-Hilbert Problem

In this paper, we take the first step towards an extension of the nonlinear steepest descent method of Deift, Its and Zhou to the case of operator Riemann-Hilbert problems. In particular, we provide long range asymptotics for a Fredholm determinant arising in the computation of the probability of finding a string of n adjacent parallel spins up in the antiferromagnetic ground state of the spin 1/2 XXX Heisenberg Chain. Such a determinant can be expressed in terms of the solution of an operator Riemann-Hilbert factorization problem

Comment on the article "Existence and Regularity for an Energy Maximization Problem in Two Dimensions" by Spyridon Kamvissis, Evguenii A. Rakhmanov

A revised version of the last appendix of the (previous) paper "Existence and Regularity for an Energy Maximization Problem in Two Dimensions" by S.Kamvissis and E.A.Rakhmanov, that appeared in the Journal of Mathematical Physics, v.46, n.8, 2005. arXiv:0907.5571Comment: 6 page