Inverse problems for selfadjoint Schrödinger operators on the half line with compactly supported potentials (original) (raw)
Related papers
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.
Transmission eigenvalues for the self-adjoint Schrödinger operator on the half line
Inverse Problems, 2014
The transmission eigenvalues corresponding to the half-line Schrödinger equation with the general self-adjoint boundary condition is analyzed when the potential is real valued, integrable, and compactly supported. It is shown that a transmission eigenvalue corresponds to the energy at which the scattering from the perturbed system agrees with the scattering from the unperturbed system. A corresponding inverse problem for the recovery of the potential from a set containing the boundary condition and the transmission eigenvalues is analyzed, and a unique reconstruction of the potential is given provided one additional constant is contained in the data set. The results are illustrated with various explicit examples.
Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line
Recently A. G. Ramm (1999) has shown that a subset of phase shifts δ l , l = 0, 1,. . ., determines the potential if the indices of the known shifts satisfy the Müntz condition l =0,l∈L 1 l = ∞. We prove the necessity of this condition in some classes of potentials. The problem is reduced to an inverse eigenvalue problem for the half-line Schrödinger operators.
Inverse resonance scattering for Dirac operators on the half-line
2020
We consider massless Dirac operators on the half-line with compactly supported potentials. We solve the inverse problems in terms of Jost function and scattering matrix (including characterization). We study resonances as zeros of Jost function and prove that a potential is uniquely determined by its resonances. Moreover, we prove the following: (1) resonances are free parameters and a potential continuously depends on a resonance, (2) the forbidden domain for resonances is estimated, (3) asymptotics of resonance counting function is determined, (4) these results are applied to canonical systems.
Inverse Scattering on the Half Line for the Matrix Schrodinger Equation
Zurnal matematiceskoj fiziki, analiza, geometrii
The matrix Schrödinger 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 one-toone 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 classic result by Agranovich and Marchenko from the Dirichlet boundary condition to the general selfadjoint boundary condition.
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.
Small-energy analysis for the selfadjoint matrix Schrödinger operator on the half line. II
Journal of Mathematical Physics, 2014
The matrix Schrödinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a second moment, it is shown that the corresponding scattering matrix is differentiable at zero energy. An explicit formula is provided for the derivative of the scattering matrix at zero energy. The previously established results when the potential has only the first moment are improved when the second moment exists, by presenting the small-energy asymptotics for the related Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.
Inverse scattering on the half-line for energy-dependent Schrödinger equations
Inverse Problems, 2020
In this paper, we study the inverse scattering problem for energy-dependent Schrödinger equations on the half-line with energy-dependent boundary conditions at the origin. Under certain positivity and very mild regularity assumptions, we transform this scattering problem to the one for non-canonical Dirac systems and show that, in turn, the latter can be placed within the known scattering theory for ZS-AKNS systems. This allows us to give a complete description of the corresponding scattering functions S for the class of problems under consideration and justify an algorithm of reconstructing the problem from S.
On the inverse problem in quantum scattering theory
International Journal of Quantum Chemistry, 2009
The Gel'fand-Levitan formulation of the inverse problem in quantum scattering theory is discussed with respect to completeness and analytic extensions. The classic Green's function and the associated completeness relation are analyzed within the Titchmarsh-Wcyl framework. An attractive feature of the Titchmarsh-Weyl formulation concerns the possibility to invoke complex scaling to a rather general set of potentials in order to expose resonance structures in the complex plane. In addition this procedure allow for an analytic extension of the classic Green's function and the associated completeness relation. The generalized completeness relation can be used to construct the kernels of the Gel'fand-Levitan integral equation. In addition to supplying a possibility for testing completeness properties of generalized expansions one may also find inversion formulas for potentials that exhibit analytic extensions to some sector in the complex plane. As a test we have analyzed a simple exponential potential which was found to contain a whole string of complex energy resonances with the resulting generalized spectral density being subjected to a particular deflation property.