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Schrödinger operators with nonlocal potentials. (English) Zbl 1289.34237

The authors describe classes of first- and second-order differential operators on \(L^2(\mathbb R^n)\) with nonlocal terms possessing selfadjoint restrictions. A typical example is the first-order operator \[ A_{\text{max}}\psi (x)=i\frac{d\psi (x)}{dx}+v_1(x)[\psi (x_0- 0)+\frac{i}2(\psi ,v_1)]+v_2(x)[\psi (x_0+0)-\frac{i}2(\psi ,v_2)]. \] The restrictions are determined by boundary conditions.
For the equation \[ i\psi'(x)+v(x)[\psi (+0)-\psi (-0)]=\lambda \psi (x),\quad -\pi <x<\pi, \] the direct and inverse spectral problems are studied, as well as the scattering problem on the whole axis.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L05 General spectral theory of ordinary differential operators
34L25 Scattering theory, inverse scattering involving ordinary differential operators
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
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