## Marchenko inversion for perturbations: I.(English)Zbl 0641.35068

[For part II see the review below.]
The Marchenko inversion method of the one-dimensional non-relativistic quantum mechanics is generalized to perturbations. The change in the potential that corresponds to a finite change in the scattering matrix is obtained by formulating a Marchenko-like integral equation. The method used here may be generalizable to higher dimensions because the inversion is formulated in terms of scattering matrices, not only in terms of reflection coefficients.

### MSC:

 35Q99 Partial differential equations of mathematical physics and other areas of application 81U20 $$S$$-matrix theory, etc. in quantum theory 35R30 Inverse problems for PDEs 35P25 Scattering theory for PDEs

Zbl 0641.35069
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