The number of bound states of singular oscillating potentials

✍ Scribed by K. Chadan

• Publisher: Springer
• Year: 1976
• Tongue: English
• Weight: 264 KB
• Volume: 1
• Category: Article
• ISSN: 0377-9017
• DOI: 10.1007/bf00398482

✦ Synopsis

For a spherically symmetric potential such that r VEL ~ (a, oo), V a > 0, and is such that, if we define W = -fr V(t) dt, W belongs to L 1 (0, oo) and rW-~O as r-~0, we show that the number of bound states in any partial-wave satisfies the bound n <2 f ~ r W 2 dr. It was shown in a previous paper [ 1 ] that this class of potentials is regular from the point of view of abstract scattering theory as well as from the time-independent theory and the Jost function approach.

We show also that, for large values of the coupling constant, n(gV) has the asymptotic behaviour C+_ Igl f~ [lffr)l drasg~ +_oo.