najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2019-10-6-608-615najo-750Research ArticleMATHEMATICSМАТЕМАТИКАExact calculation of the trace of the Birman–Schwinger operator of the one-dimensional harmonic oscillator perturbed by an attractive Gaussian potentialExact calculation of the trace of the Birman–Schwinger operator of the one-dimensional harmonic oscillator perturbed by an attractive Gaussian potentialFassariS.FassariS.

Department of Higher Mathematics, ITMO University; Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni, Univ. degli Studi Guglielmo Marconi.

St. Petersburg; PO Box 1132, CH-6601 Locarno; Via Plinio 44, I-00193 Rome

Department of Higher Mathematics, ITMO University; Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni, Univ. degli Studi Guglielmo Marconi.

St. Petersburg; PO Box 1132, CH-6601 Locarno; Via Plinio 44, I-00193 Rome

silvestro.fassari@uva.esRinaldiF.RinaldiF.

Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni, Univ. degli Studi Guglielmo Marconi.

PO Box 1132, CH-6601 Locarno; Via Plinio 44, I-00193 Rome; Naples

Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni, Univ. degli Studi Guglielmo Marconi.

PO Box 1132, CH-6601 Locarno; Via Plinio 44, I-00193 Rome; Naples

f.rinaldi@unimarconi.itITMO University; CERFIM; Univ. degli Studi Guglielmo MarconiИталияITMO University; CERFIM; Univ. degli Studi Guglielmo MarconiItalyCERFIM; Univ. degli Studi Guglielmo Marconi; Istituto Nazionale di Fisica Nucleare, Sezione di NapoliИталияCERFIM; Univ. degli Studi Guglielmo Marconi; Istituto Nazionale di Fisica Nucleare, Sezione di NapoliItaly201913082025106608615Copyright © Fassari S., Rinaldi F., 20252025Fassari S., Rinaldi F.Fassari S., Rinaldi F.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/750

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.

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.

Gaussian potentialBirman–Schwinger operatortrace class operatorharmonic oscillatorGaussian potentialBirman–Schwinger operatortrace class operatorharmonic oscillatorThe final part of S. Fassari’s contribution to this work has been financially supported by the Government of the Russian Federation through the ITMO University Fellowship and Professorship Programme. S. Fassari would like to thank Prof. Igor Yu. Popov and the entire staff at the Departament of Higher Mathematics, ITMO University, St. Petersburg for their warm hospitality throughout his stay. F. Rinaldi wishes to thank Prof. Igor Yu. Popov for inviting him to the Pierre Duclos Workshop over the years as well as for his interest in our research workThe final part of S. Fassari’s contribution to this work has been financially supported by the Government of the Russian Federation through the ITMO University Fellowship and Professorship Programme. S. Fassari would like to thank Prof. Igor Yu. 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The authors declare that there are no conflicts of interest present.