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Journal of Physics A Mathematical and General

Analytic properties of energy levels in models with delta-function potentials

A.G. Ushveridze

Published Feb 21, 1988• 22 citations
Publisher

Abstract

The analytic properties of energy levels as the functions of coupling parameters in one-dimensional quantum mechanical models described by the potentials of delta-function type are investigated. The following models are considered: (i) the rectangular potential well with delta-function barrier, (ii) two delta-function wells in the unbounded space, (iii) the Kronig-Penney model with periodic delta-function potential, and (iv) the Bose gas with delta-function interaction between particles.

DOI
10.1088/0305-4470/21/4/023

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The analytic properties of energy levels as the functions of coupling parameters in one-dimensional quantum mechanical models described by the potentials of delta-function type are investigated. The following models are considered: (i) the rectangular potential well with delta-function barrier, (ii) two delta-function wells in the unbounded space, (iii) the Kronig-Penney model with periodic delta-function potential, and (iv) the Bose gas with delta-function interaction between particles.
An application of the quantum wave impedance method for the analytical study of quantum-mechanical systems containing a set of delta-wells and delta-barriers is considered in this study.The algorithm for determining both transmission and reflection coefficients for a set of delta-barriers was proposed.Similarly, the scheme for analytically deriving the expressions for energy levels and wave functions of bound states for a set of delta-wells was developed.In both cases, the solution for the quantum wave impedance function in the region with a constant potential [1] and the matching conditions at the origins of deltafunctions were used.As an illustration of the application of the proposed approach, the quantum-mechanical systems with symmetric double-barrier potential and double-well potential were investigated.In the case of a single delta-well, the eigenenergy value as well as the expression for the eigenfunction were obtained directly using the non-linear differential equation for the quantum wave impedance function.The results demonstrate the effectiveness of the quantum wave impedance method in handling quantum-mechanical systems with deltalike potentials, offering a robust framework for further studies in this area.
It is well-known that a delta potential well in 1D has only one bound state but that in 3D it supports an {\it infinite} number of bound states with {\it infinite} binding energy for the lowest level. We show how this also holds for the less familiar 2D case, and then discuss why this makes 3D delta potential wells unphysical as models of interparticle interactions for condensed-matter many-body systems. However, both 2D and 3D delta wells can be ``regularized'' to support a single bound level which in turn renders them conveniently simple single-parameter interactions, e.g., for modeling the pair-forming dynamics of quasi-2D superconductors such as the cuprates, or in 3D of other superconductors and of neutral-fermion superfluids such as ultra-cold trapped Fermi gases.
Considering a gas of independent fermions in the presence of an attractive localized potential, one can show that the properties of the system as a whole are smooth (analytic) functions of the strength parameter of the potential, even at those values where new single-particle bound states appear. Thus for the system as a whole, the transition from ``unbound'' to ``bound'' states is continuous and the concept of a bound state cannot be made precise. This is illustrated here for a simple mathematically soluble model—noninteracting spinless fermions moving in the presence of a delta-function potential in one dimension. Some related physical ideas are also presented.
A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, $\ensuremath{\gamma}$. When $\ensuremath{\gamma}$ is small, Bogoliubov's perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of $\ensuremath{\gamma}$ and show that it is analytic for all $\ensuremath{\gamma}$, except $\ensuremath{\gamma}=0$. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum---not one as is ordinarily supposed.