# Riccati Representation for Elements in H-1(T) and its Applications

Kappeler, T; Topalov, P (2003). Riccati Representation for Elements in H-1(T) and its Applications. Pliska Bulgarski Matematicheski Studii, 15:171-188.

## Abstract

The paper is concerned with the spectral properties of the Schrödinger operator Lq def= − d2/dx2 + q with periodic potential q from the Sobolev space H −1 (T1 ). We obtain asymptotic formulas and a priori estimates for the periodic and Dirichlet eigenvalues which generalize known results for the case of potentials q ∈ L 2 0 (T1 ). The key idea is to reduce the problem to a known one – the spectrum of the impedance operator – via a nonlinear analytic isomorphism of the Sobolev spaces H −1 0 (T1 ) and L2 0 (T1 ).

The paper is concerned with the spectral properties of the Schrödinger operator Lq def= − d2/dx2 + q with periodic potential q from the Sobolev space H −1 (T1 ). We obtain asymptotic formulas and a priori estimates for the periodic and Dirichlet eigenvalues which generalize known results for the case of potentials q ∈ L 2 0 (T1 ). The key idea is to reduce the problem to a known one – the spectrum of the impedance operator – via a nonlinear analytic isomorphism of the Sobolev spaces H −1 0 (T1 ) and L2 0 (T1 ).

## Citations

Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2003 18 Feb 2010 13:18 05 Apr 2016 13:25 Institute of Mathematics and Informatics. Bulgarian Academy of Sciences 0204-9805 http://www.math.bas.bg/~pliska http://www.ams.org/mathscinet-getitem?mr=2071691
Permanent URL: https://doi.org/10.5167/uzh-21883