Inverse Spectral Problems for Schr" odinger-Type Operators with Distributional Matrix-Valued Potentials (original) (raw)
On spectral theory for Schr�dinger operators with strongly singular potentials
Math Nachr, 2006
We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schr\"odinger operators on [a,\infty), a\in\bbR, with a regular finite end point a and the case of Schr\"odinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2\times 2 matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schr\"odinger operators on (a,\infty) with a potential strongly singular at the end point a. We focus on situations where the potential is so singular that the associated maximally defined Schr\"odinger operator is self-adjoint (equivalently, the associated minimally defined Schr\"odinger operator is essentially self-adjoint) and hence no boundary condition is required at the finite end point a. For this case we show that the Weyl-Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schr\"odinger operators.
Journal of Mathematical Physics, 2015
For a selfadjoint Schrödinger operator on the half line with a real-valued, integrable, and compactly-supported potential, it is investigated whether the boundary parameter at the origin and the potential can uniquely be determined by the scattering matrix or by the absolute value of the Jost function known at positive energies, without having the bound-state information. It is proved that, except in one special case where the scattering matrix has no bound states and its value is +1 at zero energy, the determination by the scattering matrix is unique. In the special case, it is shown that there are exactly two distinct sets consisting of a potential and a boundary parameter yielding the same scattering matrix, and a characterization of the nonuniqueness is provided. A reconstruction from the scattering matrix is outlined yielding all the corresponding potentials and boundary parameters. The concept of "eligible resonances" is introduced, and such resonances correspond to real-energy resonances that can be converted into bound states via a Darboux transformation without changing the compact support of the potential. It is proved that the determination of the boundary parameter and the potential by the absolute value of the Jost function is unique up to the inclusion of eligible resonances. Several equivalent characterizations are provided to determine whether a resonance is eligible or ineligible. A reconstruction from the absolute value of the Jost function is given, yielding all the corresponding potentials and boundary parameters. The results obtained are illustrated with various explicit examples.
2016
For a selfadjoint Schrödinger operator on the half line with a real-valued, integrable, and compactly-supported potential, it is investigated whether the boundary parameter at the origin and the potential can uniquely be determined by the scattering matrix or by the absolute value of the Jost function known at positive energies, without having the bound-state information. It is proved that, except in one special case where the scattering matrix has no bound states and its value is +1 at zero energy, the determination by the scattering matrix is unique. In the special case, it is shown that there are exactly two distinct sets consisting of a potential and a boundary parameter yielding the same scattering matrix, and a characterization of the nonuniqueness is provided. A reconstruction from the scattering matrix is outlined yielding all the corresponding potentials and boundary parameters. The concept of "eligible resonances" is introduced, and such resonances correspond to real-energy resonances that can be converted into bound states via a Darboux transformation without changing the compact support of the potential. It is proved that the determination of the boundary parameter and the potential by the absolute value of the Jost function is unique up to the inclusion of eligible resonances. Several equivalent characterizations are provided to determine whether a resonance is eligible or ineligible. A reconstruction from the absolute value of the Jost function is given, yielding all the corresponding potentials and boundary parameters. The results obtained are illustrated with various explicit examples.
On Spectral Theory for Schrödinger Operators with Operator-Valued Potentials
2013
Given a complex, separable Hilbert space , we consider differential expressions of the type τ = - (d^2/dx^2) + V(x), with x ∈ (a,∞) or x ∈. Here V denotes a bounded operator-valued potential V(·) ∈() such that V(·) is weakly measurable and the operator norm V(·)_() is locally integrable. We consider self-adjoint half-line L^2-realizations H_α in L^2((a,∞); dx; ) associated with τ, assuming a to be a regular endpoint necessitating a boundary condition of the type sin(α)u'(a) + cos(α)u(a)=0, indexed by the self-adjoint operator α = α^* ∈(). In addition, we study self-adjoint full-line L^2-realizations H of τ in L^2(; dx; ). In either case we treat in detail basic spectral theory associated with H_α and H, including Weyl--Titchmarsh theory, Green's function structure, eigenfunction expansions, diagonalization, and a version of the spectral theorem.
Inverse Spectral Problems for Dirac Operators with Summable Matrix-Valued Potentials
Integral Equations and Operator Theory, 2012
We solve the direct and inverse spectral problems for Dirac operators on (0, 1) with matrix-valued potentials whose entries belong to L p (0, 1), p ∈ [1, ∞). We give a complete description of the spectral data (eigenvalues and suitably introduced norming matrices) for the operators under consideration and suggest a method for reconstructing the operator from the spectral data.
The Inverse Scattering Problem for the Matrix Schr\"odinger Equation
arXiv: Mathematical Physics, 2017
The matrix Schrodinger equation is considered on the half line with the general selfadjoint boundary condition at the origin described by two boundary matrices satisfying certain appropriate conditions. It is assumed that the matrix potential is integrable, is selfadjoint, and has a finite first moment. The corresponding scattering data set is constructed, and such scattering data sets are characterized by providing a set of necessary and sufficient conditions assuring the existence and uniqueness of the correspondence between the scattering data set and the input data set containing the potential and boundary matrices. The work presented here provides a generalization of the classical result by Agranovich and Marchenko from the Dirichlet boundary condition to the general selfadjoint boundary condition. The theory presented is illustrated with various explicit examples.
Inverse spectral problems for Sturm Liouville operators with singular potentials
Inverse Problems, 2003
The inverse spectral problem is solved for the class of Sturm-Liouville operators with singular real-valued potentials from the space W -1 2 (0, 1). The potential is recovered via the eigenvalues and the corresponding norming constants. The reconstruction algorithm is presented and its stability proved. Also, the set of all possible spectral data is explicitly described and the isospectral sets are characterized.