An inverse eigenvalue problem with rotational symmetry. (English) Zbl 0687.35108

Seidman, Thomas I.

Inverse Probl. 4, No. 4, 1093-1115 (1988).

Author’s abstract: We consider convergence of an approximation method for the recovery of a rotationally symmetric potential \(\Psi\) from the sequence of eigenvalues. In order to permit the consideration of ‘rough’ potentials \(\Psi\) (having essentially \(H^{-1}(0,1)\) regularity), we first indicate the appropriate interpretation of \(-\Delta +\Psi\) (with boundary conditions) as a selfadjoint, densely defined operator on \({\mathcal H}:=L^ 2(\Omega)\) and then show a suitable continuous dependence on \(\Psi\) for the relevant eigenvalues. The approach to the inverse problem is by the method of ‘generalized interpolation’ and, assuming uniqueness, it is shown that one has convergence to the correct potential (strongly, for an appropriate norm) for a sequence of computationally implementable approximations \((P_{cN})\).

Reviewer: J.R.Kuttler

MSC:

35R30	Inverse problems for PDEs
35P10	Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35J10	Schrödinger operator, Schrödinger equation
65Z05	Applications to the sciences

Keywords:

radial symmetry; approximate reconstruction of the potential; generalized interpolation