On the asymptotic distribution of the eigenvalue branches of the Schrödinger operator HW in a spectral gap of H.

Hempel, Rainer

doi:10.1515/crll.1989.399.38

Cited by 13 publications

(1 citation statement)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A variety of articles is concerned with asymptotics of gap eigenvalues, e.g. with the asymptotic number of eigenvalues in a given subinterval of the gap as the coupling constant c (when the perturbation s has the form s(x) = cs 0 (x)) tends to infinity (Deift & Hempel 1986;Hempel 1989;Birman , 1991Sobolev 1991) or as an endpoint of the subinterval tends to an endpoint of the gap, considering in particular the question of accumulation of eigenvalues at this gap edge (Schmidt 2000;Krüger & Teschl 2008, 2009. For fast-decaying perturbations s, it is shown by Gesztesy & Simon (1993) that sufficiently high gaps contain at most two or precisely one gap eigenvalue.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators

2011

View full text Add to dashboard Cite

Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at ±∞. It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinstein's bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account.

show abstract