Regularization of energy-dependent pointlike interactions in 1D quantum mechanics

Abstract

We construct a family of Hermitian potentials in 1D quantum mechanics that converges in the zero-range limit to a δ interaction with an energy-dependent coupling. It does not belong to the standard four-parameter family of pointlike interactions in 1D, obtained by requiring hermiticity. But we show that although our Hamiltonian is Hermitian for the standard inner product when the range of the potential is finite, the eigenstates become orthogonal for a different inner product in the zero-range limit. This inner product attributes a finite probability (and not probability density) for the particle to be exactly located at the position of the potential. Such pointlike interactions can then be used to construct potentials with a finite support with an energy-dependent coupling.