Bound and scattering states for supersingular potentials

Supersingular potentials are defined as potentials involving derivatives of the delta function of order higher than one. In this work we solve the Schrodinger equation for supersingular potentials. For the bound states, we calculate their energy and the corresponding wave functions (when they exist). For the scattering state we calculate the reflection and transmission coefficients in addition to the wave function of these states. We study the case of the delta double prime potential in details, showing that we may have none, one or two bound states, depending on the relations between the values of the coupling constants. In general the wave function in the position space has discontinuity at the point of support of the supersingular potential. (c) 2021 Published by Elsevier Inc.