Absolutely continuous spectrum for onedimensional Schrödinger operators with slowly decaying potentials: Some optimal results
Authors:
Michael Christ and Alexander Kiselev
Journal:
J. Amer. Math. Soc. 11 (1998), 771797
MSC (1991):
Primary 34L40, 81Q05, 42B20; Secondary 81Q15, 42B25
MathSciNet review:
1621882
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Abstract: The absolutely continuous spectrum of onedimensional 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 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 onedimensional 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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 V.S. Buslaev and V.B. Matveev, Wave operators for Schrödinger equation with a slowly decreasing potentials, Theor. Math. Phys. 2 (1970), 266274. MR 57:13246
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 A. Kiselev, Stability of the absolutely continuous spectrum of Schrödinger equation under perturbations by slowly decreasing potentials and a.e. convergence of integral operators, Duke Math. J. Vol. 95 (1998), 128.
 15.
 A. Kiselev, Interpolation theorem related to a.e. convergence of integral operators, Proc. Amer. Math. Soc. (to appear). CMP 98:03
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 A. Kiselev, Y. Last and B. Simon, Modified Prüfer and EFGP transforms and the spectral analysis of onedimensional Schrödinger operators, Commun. Math. Phys. (to appear).
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 A. Kiselev, C. Remling and B. Simon, Effective perturbation methods for onedimensional Schrödinger operators, submitted.
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 V.P. Maslov, S.P. Molchanov, and A.Ya. Gordon, Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture, Russian J. Math. Phys. 1 (1993), 71104. MR 95a:81059
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Additional Information
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:
http://dx.doi.org/10.1090/S0894034798002768
PII:
S 08940347(98)002768
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 DMS9623007
The second author’s work at MSRI was partially supported by NSF grant DMS 902140
Article copyright:
© Copyright 1998
American Mathematical Society
