https://doi.org/10.1140/epjp/s13360-020-00761-6
Regular Article
The two lowest eigenvalues of the harmonic oscillator in the presence of a Gaussian perturbation
1
Department of Higher Mathematics, ITMO University, St. Petersburg, Russian Federation
2
CERFIM, PO Box 1132, 6601, Locarno, Switzerland
3
Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni, Univ. degli Studi Guglielmo Marconi, Via Plinio 44, 00193, Rome, Italy
4
Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, 47011, Valladolid, Spain
Received:
15
May
2020
Accepted:
7
September
2020
Published online:
14
September
2020
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 
.
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
