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Discretization of the Schrödinger spectral problem. (English) Zbl 1087.39020

Summary: Exact discretization of the Schrödinger operator in connection with one-dimensional inverse scattering theory and the Riemann-Hilbert problem are discussed. By comparing the discrete and the continuous Green functions, qualitative estimates for the discrete spectral problem are obtained. Birational automorphisms of transparent potentials are described.

MSC:

39A12 Discrete version of topics in analysis
39A70 Difference operators
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
34A55 Inverse problems involving ordinary differential equations
34L25 Scattering theory, inverse scattering involving ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
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