The Birman-Schwinger operator for a zero-thickness layer in the presence of an attractive Gaussian impurity

In this note we are concerned with the limiting case of a zero-thickness layer with harmonic confinement along one of the two available dimensions. We investigate the Birman-Schwinger operator for such a model assuming the presence of a Gaussian impurity inside the layer and prove that such an integral operator is Hilbert-Schmidt, which allows the use of the modified Fredholm determinant in order to compute the impurity bound states. Furthermore, we consider the Hamiltonian H0 −λ√πδ(x)e−y2, that is to say the energy operator with the interaction term having a point interaction in place of the Gaussian along the x-direction, and prove that such an operator is self-adjoint as well as that it is the limit in the norm resolvent sense of the sequence H0 −λne−(n2x2+y2) as n →∞.