# 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 ).

## 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 ).