Sturmian basis functions for the harmonic oscillator

Abstract

We point out that the discrete Sturmian set for the one-dimensional harmonic oscillator, constructed recently by Antonsen ͓Phys. Rev. A 60, 812 ͑1999͔͒, is incomplete in L x 2 2 (R) and thus does not form a basis in that Hilbert space. We show that for EϾ0, the spectrum of the Sturm-Liouville problem defining the Sturmian functions is mixed and consists of an infinite number of discrete positive eigenvalues, coinciding with those found by Antonsen, and the continuum of eigenvalues covering the negative real semiaxis. The discrete eigenvalues are simple while continuous eigenvalues are doubly degenerate. A basis in L x 2 2 (R) comprises all Sturmian functions generated by that problem, i.e., those associated with discrete eigenvalues as well as those associated with continuum eigenvalues. We consider also the defining Sturm-Liouville problem in the case Eр0 and find that then the discrete part of its spectrum is absent: the spectrum is purely continuous, doubly degenerate, and covers the negative real semiaxis; the associated Sturmian functions form a noncountable basis in L x 2 2 (R).

Describing resonances in a discrete basis

“…The method of continuum discretization coupled channels [2] discretizes the continuum by means of taking fixed intervals, or bins, of k values in the continuum states. A Sturmian basis is obtained when one uses bound states of scaled potentials which are orthogonal when weighted with the potentials [3][4][5]. The Gaussian expansion method takes a nonorthogonal basis composed of Gaussian functions in geometric progression [6].…”

Continuum discretization for weakly bound nuclei

“…The description of the continuum is difficult and there several procedures have been developed to substitute it approximately by a finite basis of normalizable states. A conceptually simple procedure is to confine the weakly bound nucleus within a sphere (box) of a certain radius [1], but the use of a Sturmian basis [2] and of Gamow states [3], among other methods, has also been proposed. One of the most popular methods in Nuclear Physics is the continuum discretization coupled channels (CDCC) [4].…”

Variational Sturmian approximation: A nonperturbative method of solving time-independent Schrödinger equation

“…They satisfy certain orthonormality conditions which we shall derive below. We should, however, note that the square-integrable Sturmians do not generally constitute a complete set of basis vectors of the Hilbert space [6]. There are certain potentials V 0 , such as the Coulomb potential, that lead to a complete set of square-integrable Sturmians [8].…”

“…Here n ∈ {0, 1, 2, · · ·} and E 0 stands for the ground state. Now, consider the Sturmian basis vectors associated with a harmonic oscillator [5,6],…”

“…Starting form the late 1950's, physical chemists and nuclear physicists have explored the use of what is called the Sturmian basis functions in solving this equation for a variety of potentials arising in molecular and atomic physics [3,4]. Recently, Antonsen [5] and Szmytkowski and Zywicka-Mozeiko [6] have studied the harmonic oscillator Sturmian functions. The purpose of the present article is to outline a general variationally improved Sturmian approximation scheme that provides a nonperturbative method of solving time-independent Schrödinger equation.…”