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Scattering theory for Jacobi operators with a steplike quasi-periodic background. (English) Zbl 1152.47023

The authors consider direct and inverse scattering theory for Jacobi operators with a steplike quasi-periodic finite-gap background in the same isospectral class, using the Marchenko approach. Steplike potentials are models in quantum mechanics with applications in mesoscopic solid-state structures. The transmission coefficients can be reconstructed from the eigenvalues and the corresponding reflection coefficients. The Gelfand–Levitan–Marchenko equation is derived and it uniquely determines the operators. Conditions are formulated for the scattering data to uniquely determine the Jacobi operator.

MSC:

47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
47A40 Scattering theory of linear operators
39A70 Difference operators
81U40 Inverse scattering problems in quantum theory
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