## 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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### References:

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