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

Abstract

In this note we consider a quantum mechanical particle moving inside an infinitesimally thin layer constrained by a parabolic well in the x-direction and, moreover, in the presence of an impurity modelled by an attractive Gaussian potential. We investigate the Birman-Schwinger operator associated to 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 bound states created by the impurity. Furthermore, we consider the case where the Gaussian potential degenerates to a δ-potential in the x-direction and a Gaussian potential in the y-direction. We construct the corresponding self-adjoint Hamiltonian and prove that it is the limit in the norm resolvent sense of a sequence of corresponding Hamiltonians with suitably scaled Gaussian potentials. Satisfactory bounds on the ground state energies of all Hamiltonians involved are exhibited.

Exact calculation of the trace of the Birman–Schwinger operator of the one-dimensional harmonic oscillator perturbed by an attractive Gaussian potential

“…0) (having taken account of the fact that the eigenfunctions of the harmonic oscillator are real-valued functions), as follows from Wang's results on such scalar products of four eigenfunctions of the harmonic oscillator (see [57,58]), one gets:…”

The Energy of the Ground State of the Two-Dimensional Hamiltonian of a Parabolic Quantum Well in the Presence of an Attractive Gaussian Impurity

“…In a previous paper [37], we have studied some technicalities that arise in a twodimensional model in which the free Hamiltonian is a free particle Hamiltonian in one variable and a harmonic oscillator in the other. Then, we have added a two-dimensional isotropic Gaussian impurity and shown, by means of the renowned Kato, Lions, Lax-Milgram, Nelson theorem [38], that the ensuing Hamiltonian is self-adjoint.…”

“…In the present article, we want to proceed further with the model studied in [37] with a rigorous approach to the study of its energy spectrum. We describe this model in Section 2, where we have considered the isotropic Gaussian potential and not the Dirac interaction studied as a limiting case in [37].…”

“…In the present article, we want to proceed further with the model studied in [37] with a rigorous approach to the study of its energy spectrum. We describe this model in Section 2, where we have considered the isotropic Gaussian potential and not the Dirac interaction studied as a limiting case in [37]. We show that its Hamiltonian has an absolutely continuous spectrum on [1/2, ∞) with degeneracy equal to n on each of the intervals of the form [(2n − 1)/2, (2n + 1)/2), plus a sequence of bound states, which in general are embedded in the continuous.…”

“…Concerning the point spectrum (bound states), we remind the reader that in [37], we have established a lower bound for the spectrum in terms of the coupling constant that multiplies the Gaussian interaction. Furthermore, we have found the asymptotic behaviour of this lower bound for higher values of the coupling constant.…”