Spectral properties of the two-dimensional Schrödinger Hamiltonian with various solvable confinements in the presence of a central point perturbation

Abstract

We study three solvable two-dimensional systems perturbed by a point interaction centered at the origin. The unperturbed systems are the isotropic harmonic oscillator, a square pyramidal potential and a combination thereof. We study the spectrum of the perturbed systems. We show that, while most eigenvalues are not affected by the point perturbation, a few of them are strongly perturbed. We show that for some values of one parameter, these perturbed eigenvalues may take lower values than the immediately lower eigenvalue, so that level crossings occur. These level crossings are studied in some detail.

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

“…However, as shown in [22], in two dimensions the harmonic oscillator perturbed by a point interaction exhibits level crossings even in the repulsive case, and such level crossings are located at different values of the coupling parameter. As attested in [28], the same spectral features appear even more spectacularly when the harmonic confinement gets replaced by the square pyramidal one or by a mixture of the type 1 2 (x 2 + |y|). Although they have been studied to a far lesser extent than quantum models with point interactions, other potentials/interactions leading to solvable or quasi-solvable models have been considered in the relevant literature.…”

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

“…In two dimensions the bound state keeps existing even if the contact interaction is repulsive and the dependence becomes exponential (see [1]). The latter behavior is confirmed even when a confinement potential is present in addition to the contact potential, which physically mimics the presence of impurities or thin barriers in the material inside which the quantum particle is moving [29,30].…”

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

“…As is well known, ψ 2n (x 0 ) behaves like n −1/4 as n → +∞ for any fixed x 0 (see [16,17,18,19,20,21]), so that the strictly positive sequence inside the sum decays even more rapidly than exponentially as n → +∞, which guarantees the fast convergence of the positive series. It is worth pointing out that the latter series contains 2 2n in the denominator of its sequence, differently from the series appearing in various models involving point perturbations of the harmonic oscillator (see [16,17,18,19,20,21,22,23,24,25,26,27]). As a result, we cannot expect to express the series in terms of a ratio of Gamma functions, as was done in the abovementioned papers.…”

“…Motivated to a certain extent by [11], it might be worth pointing out that the method can be extended to the 2D/3D analogues of our model, even though, given that the 2D/3D counterparts of our Birman-Schwinger operator are no longer nuclear but only Hilbert-Schmidt integral operators (see [3,19,24,25,26,27]), the Fredholm determinant will have to be replaced by its modified version used for such operators (see [29]). Work in this direction is in progress.…”