The inverse problem of quantum mechanics for a linear potential

The applicability of the Gel'fand–Levitan method for solving the inverse problem in the case of potentials that increase unboundedly at infinity is demonstrated for the example of a linear potential. The following cases are considered: 1) change in the normalization of one of the eigenvalues; 2) complete elimination of one of the eigenstates; 3) inclusion in the spectrum of a new state with arbitrary energy. For all three cases, the asymptotic behavior of the new wave functions and the corrections to the reference (linear) potential are calculated.