ON THE SPECTRUM OF THE ONE-DIMENSIONAL SCHRÖDINGER HAMILTONIAN PERTURBED BY AN ATTRACTIVE GAUSSIAN POTENTIAL

Authors

  • Silvestro Fassari Department of Theoretical, Atomic and Optical Physics University of Valladolid Valladolid
  • Manuel Gadella Department of Theoretical, Atomic and Optical Physics University of Valladolid Valladolid
  • Luis Miguel Nieto Department of Theoretical, Atomic and Optical Physics University of Valladolid Valladolid
  • Fabio Rinaldi Dip. di Fisica Nucleare, Subnucleare e delle Radiazioni Università degli Studi G. Marconi Roma

DOI:

https://doi.org/10.14311/AP.2017.57.0385

Keywords:

Schrödinger equation, Gaussian potential, Birman-Schwinger method, trace class operators, Fredholm determinan

Abstract

We propose a new approach to the problem of finding the eigenvalues (energy levels) in the discrete spectrum of the one-dimensional Hamiltonian with an attractive Gaussian potential by using the well-known Birman-Schwinger technique. However, in place of the Birman-Schwinger integral operator we consider an isospectral operator in momentum space, taking advantage of the unique feature of this potential, that is to say its invariance under Fourier transform. 
Given that such integral operators are trace class, it is possible to determine the energy levels in the discrete spectrum of the Hamiltonian as functions of the coupling constant with great accuracy by solving a finite number of transcendental equations. We also address the important issue of the coupling constant thresholds of the Hamiltonian, that is to say the critical values of λ for which we have the emergence of an additional bound state out of the absolutely continuous spectrum. 

Downloads

Download data is not yet available.

Downloads

Published

2017-12-30

Issue

Vol. 57 No. 6 (2017)

Section

Articles

License

Authors who publish with this journal agree to the following terms:

1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.

3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).

4. ddd

How to Cite

Fassari, S., Gadella, M., Nieto, L. M., & Rinaldi, F. (2017). ON THE SPECTRUM OF THE ONE-DIMENSIONAL SCHRÖDINGER HAMILTONIAN PERTURBED BY AN ATTRACTIVE GAUSSIAN POTENTIAL. Acta Polytechnica, 57(6), 385-390. https://doi.org/10.14311/AP.2017.57.0385