• DOI:10.3389/fphy.2019.00102
• Corpus ID: 198986643

The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity

@article{Albeverio2019TheBO,
  title={The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity},
  author={Sergio Albeverio and Silvestro Fassari and Manuel Gadella and Luis Miguel Nieto and Fabio Rinaldi},
  journal={Frontiers in Physics},
  year={2019}
}

• S. Albeverio, S. Fassari, +2 authors F. Rinaldi
• Published 30 July 2019
• Mathematics
• Frontiers in Physics

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 →∞.