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

Abstract

By taking advantage of Wang's results on the scalar product of four eigenfunctions of the 1D harmonic oscillator, we explicitly calculate the trace of the Birman-Schwinger operator of the one-dimensional harmonic oscillator perturbed by a Gaussian potential, showing that it can be written as a ratio of Gamma functions.

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

“…the last equality being due to [35,44]. Hence, due to (A2) and (A3), K E,n (x, x , y, y ) meets both requirements of the Lemma listed after Theor.…”

The two lowest eigenvalues of the harmonic oscillator in the presence of a Gaussian perturbation

“…As shown in full detail in [2], the above differential equation can be recast as the following integral equation…”

“…We remind the reader that the positive integral operator on the right hand side of equation (2.7) is the renowned Birman-Schwinger operator, widely used in the literature on small perturbations of the Laplacian in the sense of quadratic forms, and that the two integral operators are isospectral (see [13], [14]). The key result established in [2] is that, for any E < 1 2 , the positive isospectral integral operators…”

“…This remarkable property has stimulated researchers to study various types of models involving perturbations of the harmonic oscillator over many decades. The interested reader can find a brief review of the literature on time independent perturbations of the harmonic oscillator in [2] (see also [3] and [4]).…”

“…Although the Birman-Schwinger principle was used in [5] and [6] to investigate the spectral effects of a particular type of short range perturbation of the one-dimensional harmonic oscillator, namely a Lorentzian perturbation, the method is clearly applicable to any absolutely summable potential. Given our recent interest in various quantum models involving Gaussian perturbations (see [2], [7], [8], [9]), we have decided to make use of the above-mentioned principle in order to investigate the modifications of the discrete spectrum of the one-dimensional harmonic oscillator once a Gaussian perturbation is added to the Hamiltonian H 0 = 1 2 − d 2 dx 2 + x 2 ≥ 1 2 . Furthermore, we have been motivated to study such a model due to the lack of relevant contributions to the existing literature in theoretical/mathematical physics.…”