The inverse resonance problem for perturbations of algebro-geometric potentials

We prove that a compactly supported perturbation of a rational or simply periodic algebro-geometric potential of the one-dimensional Schr\"odinger equation on the half line is uniquely determined by the location of its Dirichlet eigenvalues and resonances.Comment: 14 page

On the Two Spectra Inverse Problem for Semi-Infinite Jacobi Matrices

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg-Marchenko theorem for Schr{\"o}dinger operators in the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions.Comment: In this slightly revised version we have reworded some of the theorems, and we updated two reference

The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case

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.Comment: 26 pages. Changes in the presentation of some result

Skew-self-adjoint discrete and continuous Dirac type systems: inverse problems and Borg-Marchenko theorems

New formulas on the inverse problem for the continuous skew-self-adjoint Dirac type system are obtained. For the discrete skew-self-adjoint Dirac type system the solution of a general type inverse spectral problem is also derived in terms of the Weyl functions. The description of the Weyl functions on the interval is given. Borg-Marchenko type uniqueness theorems are derived for both discrete and continuous non-self-adjoint systems too

ON A THEOREM OF HALPHEN AND ITS APPLICATION TO INTEGRABLE SYSTEMS

We extend Halphen’s theorem which characterizes solutions of certain nth-order differential equations with rational coefficients and meromorphic fundamental systems to a first-order n × n system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the KdV hierarchy and illustrate some of the connections with completely integrable models of the Calogero–Moser type. In particular, our treatment recovers the complete characterization of the isospectral class of such rational KdV solutions in terms of a precise description of the Airault–McKean–Moser locus of their poles

ON A THEOREM OF HALPHEN AND ITS APPLICATION TO INTEGRABLE SYSTEMS

Abstract. We extend Halphen’stheorem which characterizesthe solutions of certain nth-order differential equationswith rational coefficientsand meromorphic fundamental systems to a first-order n×n system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the KdV hierarchy and illustrate some of the connectionswith completely integrable modelsof the Calogero-Moser-type. In particular, our treatment recovers the complete characterization of the isospectral class of such rational KdV solutions in terms of a precise description of the Airault-McKean-Moser locus of their poles. 1

On Gelfand-Dickey and Drinfeld-Sokolov systems

We study the connections between Gelfand–Dickey (GD) systems and their modified counterparts, the Drinfeld–Sokolov (DS) systems in the case of general matrix–valued coefficients with entries in a commutative algebra over an arbitrary field. Our main results describe auto–Bäcklund transformations for the GD hierarchy based on Miura–type transformations associated with factorizations of n-th order linear differential expressions