Electron. J. Diff. Eqns., Vol. 2000(2000), No. 55, pp. 1-8.

A generalization of Gordon's theorem and applications to quasiperiodic Schrodinger operators

David Damanik & Gunter Stolz

Abstract:
We present a criterion for absence of eigenvalues for one-dimensional Schrodinger operators. This criterion can be regarded as an L^1-version of Gordon's theorem and it has a broader range of application. Absence of eigenvalues is then established for quasiperiodic potentials generated by Liouville frequencies and various types of functions such as step functions, Holder continuous functions and functions with power-type singularities. The proof is based on Gronwall-type a priori estimates for solutions of Schrodinger equations.

Submitted May 12, 2000. Published July 18, 2000.
Math Subject Classifications: 34L05, 34L40, 81Q10.
Key Words: Schrodinger operators, eigenvalue problem, quasiperiodic potentials.

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David Damanik
Department of Mathematics 253-37, California Institute of Technology
Pasadena, CA 91125, USA
and Fachbereich Mathematik, Johann Wolfgang Goethe-Universitat
60054 Frankfurt, Germany
e-mail: damanik@its.caltech.edu

Gunter Stolz
Department of Mathematics, University of Alabama at Birmingham
Birmingham, AL 35294, USA
e-mail: stolz@math.uab.edu

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