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 0-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;-(n2×2)as n → ∞.
CITATION STYLE
Fassari, S., Albeverio, S., Rinaldi, F., Gadella, M., & Nieto, L. M. (2019). The birman-schwinger operator for a zero-thickness layer in the presence of an attractive gaussian impurity. Frontiers in Physics, 7(JULY). https://doi.org/10.3389/fphy.2019.00097
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