Recovery of a potential from the ratio of reflection and transmission coefficients. (English) Zbl 1062.81147

Summary: For the one-dimensional Schrödinger equation, the analysis is provided to recover the potential from the data consisting of the ratio of a reflection coefficient to the transmission coefficient. It is investigated whether such data uniquely constructs a reflection coefficient, the number of bound states, bound-state energies, bound-state norming constants, and a corresponding potential. In all three cases when there is no knowledge of the support of the potential, the support of the potential is confined to a half-line, and the support is confined to a finite interval, various uniqueness and nonuniqueness results are established, the precise criteria are provided for the uniqueness and the nonuniqueness and the degree of nonuniqueness, and the recovery is illustrated with some explicit examples.


81U40 Inverse scattering problems in quantum theory
34A55 Inverse problems involving ordinary differential equations
34L25 Scattering theory, inverse scattering involving ordinary differential operators
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