Wave scattering in 1-D nonconservative media. (English) Zbl 0903.34073

Ramm, Alexander G. (ed.), Spectral and scattering theory. Proceedings of the 1st international congress of the International Society for Analysis, Applications and Computing (ISAAC), University of Delaware, Newark, DE, USA, June 3–7, 1997. New York, NY: Plenum Press. 1-18 (1998).
The generalized Schrödinger equation \[ d^2 \psi/dx^2 +k^2 \psi= \bigl[ik P(x)+ Q(x)\bigr] \psi \] is considered, where \(P(x)\) and \(Q(x)\) are real, integrable potentials with finite first moments. Scattering solutions and bound state solutions are studied, the scattering coefficients and their small-\(k\) and large-\(k\) asymptotics are analyzed. 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 for real \(k\). Some illustrative examples are provided.
For the entire collection see [Zbl 0890.00039].


34L25 Scattering theory, inverse scattering involving ordinary differential operators
81U99 Quantum scattering theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)