No 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).