New Classes of Toda Soliton Solutions

We provide a detailed investigation of limits of N–soliton solutions of the Toda lattice as N tends to infinity. Our principal results yield new classes of Toda solutions including, in particular, new kinds of soliton–like (i.e., reflectionless) solutions. As a byproduct we solve an inverse spectral problem for one–dimensional Jacobi operators and explicitly construct tri–diagonal matrices that yield a purely absolutely continuous spectrum in (-1,1) and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of . Received: 16 October 1995/Accepted: 23 July 1996     

Connectedness of the Isospectral Manifold for One-Dimensional Half-Line Schrödinger Operators

Journal of Statistical Physics - Let V 0be a real-valued function on [0,∞) and V∈L 1([0,R]) for all R&;gt;0 so that H(V 0)=? $\frac{{d^2 }}{{dx^2 }}$ +V 0in L 2([0,∞))...     

An explicit characterization of Calogero-Moser systems

Combining theorems of Halphen, Floquet, and Picard and a Frobenius type analysis, we characterize rational, meromorphic simply periodic, and elliptic KdV potentials. In particular, we explicitly describe the proper extension of the Airault-McKean-Moser locus associated with these three classes of algebro-geometric solutions of the KdV hierarchy with special emphasis on the case of multiple collisions between the poles of solutions. This solves a problem left open since the mid-1970s.

     

A generalization of the virial theorem for strongly singular potentials

Using scale transformations we prove a generalization of the virial theorem for the eigenfunctions of Schrödinger Hamiltonians which are defined as the Friedrichs extension of strongly singular differential operators. Our theorem also applies to situations where the ground state has divergent kinetic and potential energy and thus the usual version of the virial theorem becomes meaningless.     

First-order relativistic corrections and spectral concentration

Pauli Hamiltonians with first-order relativistic corrections according to Foldy and Wouthuysen are rigorously studied applying methods of singular perturbation theory. The results include a proof of first-order spectral concentration in the non-relativistic limit and a characterization of first-order pseudoeigenvalues by means of formal perturbation theory.     

Holomorphy of the scattering matrix with respect toc −2 for Dirac operators and an explicit treatment of relativistic corrections

We prove holomorphy of the scattering matrix at fixed energy with respect toc ^–2 for abstract Dirac operators. Relativistic corrections of orderc ^–2 to the nonrelativistic limit scattering matrix (associated with an abstract Pauli Hamiltonian) are explicitly determined. As applications of our abstract approach we discuss concrete realizations of the Dirac operator in one and three dimensions and explicitly compute relativistic corrections of orderc ^–2 of the reflection and transmission coefficients in one dimension and of the scattering matrix in three dimensions. Moreover, we give a comparison between our approach and the firstorder relativistic corrections according to Foldy-Wouthuysen scattering theory and show complete agreement of the two methods.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich by an E. Schrödinger Fellowship and by Project No. P7425     

Invariance of the essential spectra for perturbations with unbounded hard cores

Based on the Feynman-Kac formula and a certain variant of Weyl's theorem, we determine the essential spectra of Schrödinger operators including unbounded hard cores.     

When is a non-self-adjoint Hill operator a spectral operator of scalar type?

We derive necessary and sufficient conditions for a one-dimensional periodic Schrödinger (i.e., Hill) operator $H=?{\mathrm{d}}^2/\mathrm{d}{x}^2+V$ in $L^2(\mathbb{R})$ to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. To cite this article: F. Gesztesy, V. Tkachenko, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A Borg-Type Theorem Associated with Orthogonal Polynomials on the Unit Circle

We prove a general Borg-type result for reflectionless unitaryCMV operators U associated with orthogonal polynomials on theunit circle. The spectrum of U is assumed to be a connectedarc on the unit circle. This extends a recent result of Simonin connection with a periodic CMV operator with spectrum thewhole unit circle. In the course of deriving the Borg-type result we also use exponentialHerglotz representations of Caratheodory functions to provean infinite sequence of trace formulas connected with the CMVoperator U.