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On the direct and inverse transmission eigenvalue problems for the Schrödinger operator on the half line. (English) Zbl 1459.34068

The following transmission boundary value problem is considered: \[ -\psi''+q(x)\psi=\lambda \psi,\; -\psi_0''=\lambda\psi_0,\; 0<x<1, \] \[ \psi'(0)-h\psi(0)=\psi'_0(0)-h\psi_0(0)=0,\; \psi_0(1)=\psi(1),\; \psi_0'(1)=\psi'(1). \] The eigenvalue asymptotics is established, and the inverse problem of recovering \(q\) from the spectrum is studied. Under additional assumptions the uniqueness theorem is proved for this inverse problem.

MSC:

34A55 Inverse problems involving ordinary differential equations
34L25 Scattering theory, inverse scattering involving ordinary differential operators
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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