Miklós Horváth

On the inverse spectral theory of Schrödinger and Dirac operators

We prove that under some conditions finitely many partially known spectra and partial information on the potential entirely determine the potential. This extends former results of Hochstadt, Lieberman, Gesztesy, Simon and others.

On the first two eigenvalues of Sturm-Liouville operators

Among the Schrodinger operators with single-well potentials defined on (0, π) with transition point at (π/2, the gap between the first two eigenvalues of the Dirichlet problem is minimized when the potential is constant. This extends former results of Ashbaugh and Benguria with symmetric single-well potentials. An analogous result is given for the Dirichlet problem of vibrating strings with single-barrier densities for the ratio of the first two eigenvalues.

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 1 , l = 0, 1 determines the potential if the indices of the known shifts satisfy the Miintz condition Σ 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 Schrodinger operators.

SOLUTION OF THE INVERSE SCATTERING PROBLEM AT FIXED ENERGY FOR POTENTIALS BEING ZERO BEYOND A FIXED RADIUS

Based on the relation between the m-function and the spectral function we construct an inverse quantum scattering procedure at fixed energy which can be applied to spherical radial potentials vanishing beyond a fixed radius a. To solve the Gelfand–Levitan–Marchenko integral equation for the transformation kernel, we determine the input symmetrical kernel by using a minimum norm method with moments defined by the input set of scattering phase shifts. The method applied to the box and Gauss potentials needs further practical developments regarding the treatment of bound states.

The stability of inverse scattering with fixed energy

We consider a three-dimensional inverse scattering problem with fixed energy and with a spherically symmetrical, compactly supported potential. The resulting one-dimensional radial Schrodinger operator defines the sequence of phase shifts. We give some estimates of the potential perturbation by the perturbation of the phase shifts. More precisely, an exact estimate is given for an integral norm of the potential perturbation by the forward differences of the normalized perturbation of phase shifts. Another upper bound is provided if only the first few phase shifts are available with some error.

NOTES ON THE DISTRIBUTION OF PHASE SHIFTS

We consider three-dimensional inverse scattering with fixed energy for which the spherically symmetrical potential is nonvanishing only in a ball. We give exact upper and lower bounds for the phase shifts. We provide a variational formula for the Weyl–Titchmarsh m-function of the one-dimensional Schrodinger operator defined on the half-line.

SEMI-ANALYTIC EQUATIONS TO THE COX THOMPSON INVERSE SCATTERING METHOD AT FIXED ENERGY FOR SPECIAL CASES

Solution of the Cox–Thompson inverse scattering problem at fixed energy1–3 is reformulated resulting in semi-analytic equations. The new set of equations for the normalization constants and the nonphysical (shifted) angular momenta are free of matrix inversion operations. This simplification is a result of treating only the input phase shifts of partial waves of a given parity. Therefore, the proposed method can be applied for identical particle scattering of the bosonic type (or for certain cases of identical fermionic scattering). The new formulae are expected to be numerically more efficient than the previous ones. Based on the semi-analytic equations an approximate method is proposed for the generic inverse scattering problem, when partial waves of arbitrary parity are considered.

Simplified solutions of the Cox–Thompson inverse scattering method at fixed energy

Simplified solutions of the Cox–Thompson inverse quantum scattering method at fixed energy are derived if a finite number of partial waves with only even or odd angular momenta contribute to the scattering process. Based on new formulae various approximate methods are introduced which also prove applicable to the generic scattering events.

Inverse eigenvalue problems

In this article we consider inverse eigenvalue problems for the Schrodinger operator on a finite interval. We extend and strengthen previously known uniqueness theorems. A partially known potential is identified by some sets of eigenvalues and norming constants.

On the stability in Ambarzumian theorems

We provide extensions of the classical Ambarzumian theorem for bounded C 3 domains of any dimension. The simple proof is based on classical spectral function asymptotics. We prove a stability property by showing that if the perturbation of the eigenvalues of the zero potential is small in some sense then the L 2-norm of the potential is also small. The problem is motivated by connections to a number of applications.