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Wave scattering in one dimension with absorption. (English) Zbl 1001.34074

Summary: Wave scattering is analyzed in a one-dimensional nonconservative medium governed by the generalized Schrödinger equation \(d^2\psi/dx^2+k^2\psi=[ikP(x)+Q(x)]\psi\), where \(P(x)\) and \(Q(x)\) are real, integrable potentials with finite first moments. Various properties of the scattering solutions are obtained. The corresponding scattering matrix is analyzed, and its small-\(k\) and large-\(k\) asymptotics are established. The bound states, which correspond to the poles of the transmission coefficient in the upper-half complex plane, are studied in detail. When the medium is not purely absorptive, i.e., unless \(P(x)\leq 0\), it is shown that there may be bound states at complex energies, degenerate bound states, and singularities of the transmission coefficient imbedded in the continuous spectrum. Some explicit examples are provided illustrating the theory.

MSC:

34L25 Scattering theory, inverse scattering involving ordinary differential operators
81U05 \(2\)-body potential quantum scattering theory
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35P25 Scattering theory for PDEs
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[1] DOI: 10.1007/BF01645775
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