Rudi Weikard

A Borg–Levinson theorem for trees

We prove that the Dirichelet-to-Neumann map for a Schrödinger operator on a finite simply connected tree determines uniquely the potential on that tree.

Picard potentials and Hill's equation on a torus

Hills equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness. Of particular importance in the last 20 years is its connection with the KdV hierarchy and hence with integrable systems. We show in this paper that regarding the independent variable as a complex variable yields a breakthrough for the problem of an efficient characterization of all elliptic finitegap potentials, a major open problem in the field. Specifically, we show that elliptic finitegap potentials of Hills equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variable, complementing a classical theorem of Picard. The intimate connection between Picards theorem and elliptic finite-gap solutions of completely integrable systems is established in this paper for the first time. In addition, we construct the hyperelliptic Riemann surface associated with a finitegap potential (not necessarily elliptic), i.e., determine its branch and singular points from a comparison of the geometric and algebraic multiplicities of eigenvalues of certain auxiliary operators associated with Hills equation. These multiplicities are intimately correlated with the pole structure of the diagonal Greens function of the operator H = d2/dx2+q(x) in L2(R). Our construction is new in the present general complex-valued periodic finite-gap case. Before describing our approach in some detail, we shall give a brief account of the history of the problem involved. This theme dates back to a 1940 paper of Ince [43] who studied what is presently called the Lam6-Ince potential

Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies—an analytic approach

We provide an overview of elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard.

Treibich-Verdier potentials and the stationary (m)KDV hierarchy.

Here .~(x) _= .~ (x ; col, co3) denotes the elliptic Weierstrass function with fundamental periods (f.p.) 2a~l, 2co3, Im(co3/wl)5 / 0 (see [1], Ch. 18). In the special case where col is real and co3 is purely imaginary the potential q is real-valued and Inces striking result [31], in modern spectral theoretic terminology, yields the fact that the self-adjoint operator L associated with the differential expression d 2 + q in L2(IR) has a finite-gap spectrum of the type

On the leading correction of the Thomas-Fermi model: Lower bound

SummaryWe prove that the quantum mechanical ground state energy of an atom with nuclear chargeZ can be bounded from below by the sum of the Thomas-Fermi energy of the problem plusq/8Z2 plus terms of ordero(Z2).

On the Inverse Resonance Problem

An ew technique is presented which gives conditions under which perturbations of certain base potentials are uniquely determined from the location of eigenvalues and resonances in the context of a Schr¨ odinger operator on a half line. The method extends to complex-valued potentials and certain potentials whose first moment is not integrable.

Weak stability for an inverse Sturm-Liouville problem with finite spectral data and complex potential*

It is well known that knowing the Dirichlet–Dirichlet eigenvalues and the Dirichlet–Neumann eigenvalues determines uniquely the potential of a one-dimensional SchrA¶dinger equation on a finite interval. We investigate here how well a potential may be approximated if only N of each type of eigenvalues are known to within an error Iµ.

A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy

Here ~9(x):p(x; wl,w3) denotes the elliptic Weierstrass function with fundamental periods 2wl and 2w3 (Im(w3/Wl)#0). In the special case where Wl is real and w3 is purely imaginary, the potential q(x) in (1.1) is real-valued and Inces striking result [51], in modern spectral-theoretic terminology, yields that the spectrum of the unique self-adjoint operator associated with the differential expression L2=d2/dx 2 +q(x) in L2(R) exhibits finitely many bands (and gaps, respectively), that is,

Spectral deformations and soliton equations

Publisher Summary This chapter describes the construction of new solutions V of the Korteweg-deVries (KdV) hierarchy of equations by deformations of a given finite-gap solution V0. To describe the nature of these deformations the chapter assumes a moment that the given real-valued quasi-periodic finite-gap solution V0 is described in terms of the Its–Matveev formula. The chapter also gives brief account of the KdV hierarchy using a recursive approach ans describes real-valued quasi-periodic finite-gap solutions and the underlying Its–Matveev formula in some detail. To describe the hierarchy of KdV equations the chapter first recalls the recursive approach to the underlying Lax pairs. In addition, the chapter introduces isospectral and non-isospectral deformations in a systematic way by alluding to single and double commutation techniques.

The inverse resonance problem for Jacobi operators

We consider the class of CMV operators with super-exponentially decaying Verblunsky coecients. For these we dene the concept of a reso- nance. Then we prove the existence of Jost solutions and a uniqueness theo- rem for the inverse resonance problem: Given the location of all resonances, taking multiplicities into account, the Verblunsky coecients are uniquely de- termined.