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

Abstract

In this note we consider a one-dimensional quantum mechanical particle constrained by a parabolic well perturbed by a Gaussian potential. As the related Birman-Schwinger operator is trace class, the Fredholm determinant can be exploited in order to compute the modified eigenenergies which differ from those of the harmonic oscillator due to the presence of the Gaussian perturbation. By taking advantage of Wang's results on scalar products of four eigenfunctions of the harmonic oscillator, it is possible to evaluate quite accurately the two lowest-lying eigenvalues as functions of the coupling constant $\lambda$.

arXiv comment15 pages, 4 figures

References in corpus (3)

• Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations
• Root system of singular perturbations of the harmonic oscillator type operators
• The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity

Cited by in corpus (2)

• Quantum Tunnelling and Thermally Driven Transitions in a Double Well Potential at Finite Temperature
• On the eigenvalues of the harmonic oscillator with a Gaussian perturbation