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Multidimensional inverse spectral problem for the equation \(-\Delta \psi -(v(x)-Eu(x))\psi =0\). (English. Russian original) Zbl 0689.35098

Funct. Anal. Appl. 22, No. 4, 263-272 (1988); translation from Funkts. Anal. Prilozh. 22, No. 4, 11-22 (1988).
I. M. Gelfand in 1954 and A. P. Calderon in 1980 posed a multidimensional inverse problems of the following type. Let the equation \[ (*)\quad - \Delta \psi +(v(x)-EU(x))\psi =0 \] be valid in the bounded domain \(D\subset {\mathbb{R}}^ n,\) \(n\geq 2\). (Let v(x) and u(x) be bounded in \(\bar D,\) and \(\partial D\) be smooth.) Let \({\hat \phi}\)(E) be the operator, determined by v(x) and u(x), such that \((\frac{\partial}{\partial v}\psi |_{\partial D})={\hat \phi}(E)(\psi |_{\partial D})\) for every solution \(\psi\) of the equation (*), where v is the outward normal of \(\partial D\). The problem is to reconstruct the unknown potentials v(x) and u(x) from the known operator \({\hat \phi}\)(E). Such problem arises in several applied fields.
The main results of this work are the following. 1) The operator \({\hat \phi}\)(E) uniquely determines the potential \(v(x)-EU(x)\) at fixed energy E (in the case \(n=2\) under some additional assumptions about the potential E). 2) This potential can be reconstructed from the operator \({\hat \phi}\)(E), \(E=const\) by solving a linear Fredholm’s integral equation of second type. 3) The reconstruction of the potentials v(x) and u(x) (where \(u(x)\geq u_ 0>0)\) from the spectrum and boundary values of the normal derivatives of the eigenfunctions the equation (*) with Dirichlet boundary conditions is given. 4) The reconstruction of the potential \(v(x)-EU(x)\) (where v(x)\(\equiv 0\) and u(x)\(\equiv 1\) outside D) from the scattering amplitude at fixed energy is given by solving linear integral equations.
Reviewer: R.G.Novikov

MSC:

35R30 Inverse problems for PDEs
35J10 Schrödinger operator, Schrödinger equation
45B05 Fredholm integral equations
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