## Exact solutions of the Schrödinger equation and the non-uniqueness of inverse scattering on the line.(English)Zbl 0645.34019

Certain exact solutions to the Schrödinger equation on the line are given for each positive integer N such that the corresponding transmission coefficient behaves like $$O(k^ N)$$ as $$k\to 0$$. Such solutions form a one-parameter family for each N, and all the potentials in each family cause the same scattering at all energies. In each family there are infinitely many potentials that do not support any bound states.

### MSC:

 34L99 Ordinary differential operators

### Keywords:

Schrödinger equation; potentials; scattering
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