Abstract not found

We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a one-dimensional Schrödinger operator H = − d2 dx2 + q determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of H on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl m-function techniques and densities of zeros of a class of entire functions.

We study the pseudospectral theory of a variety of non-self-adjoint constant coefficient and variable coefficient differential operators, showing that the phenomenon of non-trivial pseudospectra is typical rather than exceptional. We prove that the pseudospectra provide more stable information about the operators under various limiting procedures than does the spectrum. AMS Subject Classification: 34L05, 35P05, 47A75, 49R99. Keywords: spectrum, pseudospectrum, norms of inverses, resolvent operators, differential operators. 1 Introduction It is well established that the spectrum of a self-adjoint operator is of crucial importance in understanding its action in various applied contexts. For highly non-self-adjoint operators, on the other hand, there is increasing evidence that the spectrum is often not very helpful, and that the pseudospectra are of more importance. We refer to [17, 18] for references to the increasing literature on this concept, and for a series of examples in which t...

Abstract. We consider the inverse spectral problem for a class of reflectionless bounded Jacobi operators with empty singularly continuous spectra. Our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum. The corresponding isospectral set of Jacobi operators is explicitly determined in terms of Dirichlet-type data. 1.

It is well known that computing the eigenvalues of a self-adjoint bounded or differential

A recently established general trace formula for one-dimensional Schrödinger operators is systematically studied in the context of short-range potentials, potentials which approach different spatial asymptotes sufficiently fast, and appropriate impurity (defect) interactions in one-dimensional solids. We prove the absolute summability of the trace formula and establish its connections with scattering quantities, such as reflection coefficients, in each case.

We provide a general framework of stationary scattering theory for one-dimensional quantum systems with nontrivial spatial asymptotics. As a byproduct we characterize reflectionless potentials in terms of spectral multiplicities and properties of the diagonal Green's function of the underlying Schrodinger operator. Moreover, we prove that single (Crum-Darboux) and double commutation methods to insert eigenvalues into spectral gaps of a given background Schrodinger operator produce reflectionless potentials (i.e., solitons) if and only if the background potential is reflectionless. Possible applications of our formalism include impurity (defect) scattering in (half)crystals and charge transport in mesoscopic quantum-interference devices.

We develop direct and inverse scattering theory for one-dimensional Schrödinger operators with steplike potentials which are asymptotically close to different finite-gap potentials on different half-axes. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.

We deal primarily with spectral analysis of an abstract self-adjoint operator. H, on a Hilbert space, X”. We propose a further refinement of the absolutely continuous subspace,;F”a,, into the transient absolutely continuous subspace, &‘a,. which is the closure of those cp with (cp, e-j/Ho) = O(t-“) for all N and the recurrent absolutely continuous subspace, 2 & = ‘qc nX&. We discuss general features of this breakup. In a subsequent paper, we construct analytic almost periodic functions, V, on (--03, 03) so that H =-d*/dx * + V(x) on L2(-co, co) has only recurrent absolutely continuous spectrum in the sense that qa, =.;Y.

Abstract. Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a half-line Schrödinger operator with square integrable potential. 1.