We consider the Schrödinger operator L = −d2 /dx 2 +q on the interval [0,1] depending on an L 2-potential q and endowed with periodic or anti-periodic boundary conditions. We prove results about correspondencies between the asymptotic behaviour of the spectral gaps of L and the regularity of q in the Gevrey case, among others. The proofs are based on a Fourier block decomposition due to Kappeler &Mityagin, and a novel application of the implicit function theorem.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
P. DJAKOV & B. MITYAGIN , Smoothness of Schr ödinger operator potential in the case of Gevrey type asymptotics of the gaps. J. Funct. Anal. 195 (2002), 89-128.
P. DJAKOV & B. MITYAGIN , Spectral triangles of Schr ödinger operators with omplex potentials. Selecta Math. (N.S.) 9 (2003), 495-528.
P. DJAKOV & B. MITYAGIN , Instability zones of one-dimensional periodic Schr ödinger and Dirac operators. (Russian) Uspekhi Mat. Nauk 61 (2006), 77-182; translation in Russian Math. Surveys 61 (2006).
M. G. GASYMOV, Spectral analysis of a class of second order nonselfadjoint differential operators. Funct. Anal. Appl. 14 (1980), 14-19.
H. HOCHSTADT, Estimates on the stability interval’s for the Hill’s equation. Proc. AMS 14 (1963),930-932.
T. KAPPELER & B. MITYAGIN , Gap estimates of the spectrum of Hill’s equation and action variables for KdV. Trans. Amer. Math. Soc. 351 (1999), 619-646.
T. KAPPELER & B. MITYAGIN , Estimates for periodic and Dirichlet eigenvalues of the Schr ödinger operator. SIAM J. Math. Anal. 33 (2001), 113-152.
V. A. MARCˇ ENKO & I. O. OSTROWSKĬ, A characterization of the spectrum of Hill’s operator. Math. USSR Sbornik 97 (1975), 493-554.
G. PÓLYA & G. SZEG ö , Problems and Theorems in Analysis I. Springer, New York, 1976.
J. P öSCHEL , Hill’s potentials in weihtes Sobolev spaces and their spectral gaps. Preprint 2004, www.poschel.de/pbl.html.
. J. J. SANSUC & V. TKACHENKO , Spectral properties of non-selfadjoint Hill’s operators with smooth potentials. Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993),371-385, Kluwer, 1996.
E. TRUBOWITZ , The inverse problem for periodic potentials. Comm. Pure Appl. Math. 30 (1977),321-342.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science + Business Media B.V
About this paper
Cite this paper
Pöschel, J. (2008). Spectral gaps of potentials in weighted Sobolev spaces. In: Craig, W. (eds) Hamiltonian Dynamical Systems and Applications. NATO Science for Peace and Security Series. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-6964-2_17
Download citation
DOI: https://doi.org/10.1007/978-1-4020-6964-2_17
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-6962-8
Online ISBN: 978-1-4020-6964-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)