On the spectrum of the one-dimensional schrödinger hamiltonian perturbed by an attractive gaussian potential

Silvestro Fassari, Manuel Gadella, Luis Miguel Nieto, Fabio Rinaldi

On the spectrum of the one-dimensional schrödinger hamiltonian perturbed by an attractive gaussian potential

Číslo: 6/2017
Periodikum: Acta Polytechnica
DOI: 10.14311/AP.2017.57.0385

Klíčová slova: Schrödinger equation, Gaussian potential, Birman-Schwinger method, trace class operators, Fredholm determinan, Schrödingerova rovnice, Gaussův potenciál, metoda Birman-Schwinger, provozovatelé trasových tříd, Fredholmův determinant

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Anotace: 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.