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On the asymptotics of solutions for some Schrödinger equations with dissipative perturbations of rank one. (English) Zbl 1078.35099

The authors study the spectral and scattering theories for the Schrödinger operator on the line with a dissipative perturbation of rank one, given by a delta function with a complex coupling constant. They characterize the different parts of the spectrum of the Schrödinger operator, they prove the existence of the wave operators, they construct the generalized Fourier transforms and they prove a generalized Parseval formula. Moreover, they classify the asymptotics of the solutions for large times in terms of properties of the initial data. Namely, they give necessary and sufficient conditions on the large time asymptotics of the solution for the component of the initial data on the singular subspace of the Schrödinger operator to be non-zero, and also necessary and sufficient conditions for the component of the initial data on the singular subspace of the Schrödinger operator to be zero.

MSC:

35Q40 PDEs in connection with quantum mechanics
35P25 Scattering theory for PDEs
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q15 Perturbation theories for operators and differential equations in quantum theory
81U05 \(2\)-body potential quantum scattering theory
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