On Hermite Functions, Integral Kernels, and Quantum Wires

Silvestro Fassari, Manuel Gadella, Luis M. Nieto () and Fabio Rinaldi
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Silvestro Fassari: Centro Europeo Ricerche in Fisica Matematica (CERFIM), P.O. Box 1132, CH-6601 Locarno, Switzerland
Manuel Gadella: Departamento de Física Teórica, Atómica y Óptica, and IMUVA, Universidad de Valladolid, 47011 Valladolid, Spain
Luis M. Nieto: Departamento de Física Teórica, Atómica y Óptica, and IMUVA, Universidad de Valladolid, 47011 Valladolid, Spain
Fabio Rinaldi: Centro Europeo Ricerche in Fisica Matematica (CERFIM), P.O. Box 1132, CH-6601 Locarno, Switzerland

Mathematics, 2022, vol. 10, issue 16, 1-11

Abstract: In this note, we first evaluate and subsequently achieve a rather accurate approximation of a scalar product, the calculation of which is essential in order to determine the ground state energy in a two-dimensional quantum model. This scalar product involves an integral operator defined in terms of the eigenfunctions of the harmonic oscillator, expressed in terms of the well-known Hermite polynomials, so that some rather sophisticated mathematical tools are required.

Keywords: Gaussian potential; Birman–Schwinger operator; Hilbert–Schmidt operator; contact interaction (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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