Journal of the American Mathematical Society

Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results

Author(s): Michael Christ; Alexander Kiselev
Journal: J. Amer. Math. Soc. 11 (1998), 771-797.
MSC (1991): Primary 34L40, 81Q05, 42B20; Secondary 81Q15, 42B25
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Abstract: The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectra of free and periodic Schrödinger operators are preserved under all perturbations

satisfying $|V(x)|\leq C(1+|x|)^{-\alpha},$ $\alpha >\frac{1}{2}.$ This result is optimal in the power scale. Slightly more general classes of perturbing potentials are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on maximal function and norm estimates, and on almost everywhere convergence results for certain multilinear integral operators.

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Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 34L40, 81Q05, 42B20, 81Q15, 42B25

Michael Christ
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: mchrist@math.berkeley.edu

Alexander Kiselev
Affiliation: Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637
Email: kiselev@math.uchicago.edu

DOI: 10.1090/S0894-0347-98-00276-8
PII: S 0894-0347(98)00276-8
Keywords: Schr\"odinger operator, absolutely continuous spectrum, a.e. convergence, decaying potential, WKB asymptotics, norm estimates
Received by editor(s): June 30, 1997
Additional Notes: The first author's work was partially supported by NSF grant DMS96-23007
The second author's work at MSRI was partially supported by NSF grant DMS 902140
Copyright of article: Copyright 1998, American Mathematical Society

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ISSN: 1088-6834(e) ISSN: 0894-0347(p)

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