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

Authors:
Michael Christ and Alexander Kiselev

Journal:
J. Amer. Math. Soc. **11** (1998), 771-797

MSC (1991):
Primary 34L40, 81Q05, 42B20; Secondary 81Q15, 42B25

DOI:
https://doi.org/10.1090/S0894-0347-98-00276-8

MathSciNet review:
1621882

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Abstract | References | Similar Articles | Additional Information

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 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.

**1.**P. Alsholm and Tosio Kato,*Scattering with long range potentials*, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 393–399. MR**0385351****2.**Joseph E. Avron and Barry Simon,*Transient and recurrent spectrum*, J. Functional Analysis**43**(1981), no. 1, 1–31. MR**639794**, https://doi.org/10.1016/0022-1236(81)90034-3**3.**Horst Behncke,*Absolute continuity of Hamiltonians with von Neumann-Wigner potentials*, Proc. Amer. Math. Soc.**111**(1991), no. 2, 373–384. MR**1036983**, https://doi.org/10.1090/S0002-9939-1991-1036983-3**4.**Matania Ben-Artzi and Allen Devinatz,*Spectral and scattering theory for the adiabatic oscillator and related potentials*, J. Math. Phys.**20**(1979), no. 4, 594–607. MR**529723**, https://doi.org/10.1063/1.524128**5.**V. S. Buslaev and V. B. Matveev,*Wave operators for the Schrödinger equation with slowly decreasing potential*, Teoret. Mat. Fiz.**2**(1970), no. 3, 367–376 (Russian, with English summary). MR**473580****6.**René Carmona and Jean Lacroix,*Spectral theory of random Schrödinger operators*, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR**1102675****7.**M. Christ and A. Kiselev,*A maximal inequality*, preprint.**8.**M. Christ, A. Kiselev and C. Remling,*The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials*, Math. Research Lett.,**4**(1997), 719-723. CMP**98:05****9.**E.A. Coddington and N. Levinson,*Theory of Ordinary Differential Equations*, McGraw-Hill, New York, 1955. MR**16:1022b****10.**W. A. Harris Jr. and D. A. Lutz,*Asymptotic integration of adiabatic oscillators*, J. Math. Anal. Appl.**51**(1975), 76–93. MR**369840**, https://doi.org/10.1016/0022-247X(75)90142-0**11.**D. B. Hinton and J. K. Shaw,*Absolutely continuous spectra of second order differential operators with short and long range potentials*, SIAM J. Math. Anal.**17**(1986), no. 1, 182–196. MR**819222**, https://doi.org/10.1137/0517017**12.**Lars Hörmander,*The existence of wave operators in scattering theory*, Math. Z.**146**(1976), no. 1, 69–91. MR**393884**, https://doi.org/10.1007/BF01213717**13.**A. Kiselev,*Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials*, Comm. Math. Phys.**179**(1996), no. 2, 377–400. MR**1400745****14.**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), 1-28.**15.**A. Kiselev,*Interpolation theorem related to a.e. convergence of integral operators*, Proc. Amer. Math. Soc. (to appear). CMP**98:03****16.**A. Kiselev, Y. Last and B. Simon,*Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators*, Commun. Math. Phys. (to appear).**17.**A. Kiselev, C. Remling and B. Simon,*Effective perturbation methods for one-dimensional Schrödinger operators*, submitted.**18.**S. Kotani and N. Ushiroya,*One-dimensional Schrödinger operators with random decaying potentials*, Comm. Math. Phys.**115**(1988), no. 2, 247–266. MR**931664****19.**Victor P. Maslov, Stanislav A. Molchanov, and Alexander Ya. Gordon,*Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture*, Russian J. Math. Phys.**1**(1993), no. 1, 71–104. MR**1240494****20.**V.B. Matveev,*Wave operators and positive eigenvalues for Schrödinger equation with oscillating potential,*Theor. Math. Phys.**15**(1973), 574-583.**21.**S. Molchanov,*in preparation*.**22.**S. N. Naboko,*On the dense point spectrum of Schrödinger and Dirac operators*, Teoret. Mat. Fiz.**68**(1986), no. 1, 18–28 (Russian, with English summary). MR**875178****23.**Michael Reed and Barry Simon,*Methods of modern mathematical physics. III*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR**529429****24.**C. Remling,*Some Schrödinger operators with power-decaying potentials and pure point spectrum*, Commun. Math. Phys. Vol.**186**(1997), 481-493. CMP**97:16****25.**C. Remling,*The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials*, Commun. Math. Phys.**4**(1997), no. 5, 719-723. CMP**98:05****26.**Barry Simon,*Some Schrödinger operators with dense point spectrum*, Proc. Amer. Math. Soc.**125**(1997), no. 1, 203–208. MR**1346989**, https://doi.org/10.1090/S0002-9939-97-03559-4**27.**Barry Simon,*Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators*, Proc. Amer. Math. Soc.**124**(1996), no. 11, 3361–3369. MR**1350963**, https://doi.org/10.1090/S0002-9939-96-03599-X**28.**Elias M. Stein and Guido Weiss,*Introduction to Fourier analysis on Euclidean spaces*, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR**0304972****29.**Günter Stolz,*Bounded solutions and absolute continuity of Sturm-Liouville operators*, J. Math. Anal. Appl.**169**(1992), no. 1, 210–228. MR**1180682**, https://doi.org/10.1016/0022-247X(92)90112-Q**30.**J. von Neumann and E.P. Wigner,*Über merkwürdige diskrete eigenwerte*, Z. Phys.**30**(1929), 465-467.**31.**Joachim Weidmann,*Spectral theory of ordinary differential operators*, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, 1987. MR**923320****32.**Denis A. W. White,*Schrödinger operators with rapidly oscillating central potentials*, Trans. Amer. Math. Soc.**275**(1983), no. 2, 641–677. MR**682723**, https://doi.org/10.1090/S0002-9947-1983-0682723-7

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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:
https://doi.org/10.1090/S0894-0347-98-00276-8

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

Article copyright:
© Copyright 1998
American Mathematical Society