Duke Mathematical Journal

Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains

A. Kiselev and Y. Last

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J. Volume 102, Number 1 (2000), 125-150.

First available in Project Euclid: 17 August 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35P05: General topics in linear spectral theory
Secondary: 35J10: Schrödinger operator [See also 35Pxx] 47B38: Operators on function spaces (general) 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47) 47N50: Applications in the physical sciences 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis


Kiselev, A.; Last, Y. Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains. Duke Math. J. 102 (2000), no. 1, 125--150. doi:10.1215/S0012-7094-00-10215-3. http://projecteuclid.org/euclid.dmj/1092749258.

Export citation


  • S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1979), 151--218.
  • S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math. 30 (1976), 1--38.
  • M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209--273.
  • Bateman Manuscript Project, Higher Transcendental Functions, 2, McGraw-Hill, New York, 1953--1955.
  • J. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monogr. 17, Amer. Math. Soc., Providence, 1968.
  • M. Birman and M. Solomyak, Spectral Theory of Selfadjoint Operators in Hilbert Space, Trans. S. Khrushchëv and V. Peller, Math. Appl. (Soviet Ser.), D. Rédel Publishing Co., Dordrecht, 1987.
  • M. Christ and A. Kiselev, Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results, J. Amer. Math. Soc. 11 (1998), 771--797.
  • J. M. Combes, ``Connections between quantum dynamics and spectral properties of time-evolution operators'' in Differential Equations with Applications to Mathematical Physics, Math. Sci. Engrg. 192, Academic Press, Boston, 1993, 59--68.
  • D. Damanik, $\alpha$-continuity properties of one-dimensional quasicrystals, Comm. Math. Phys. 192 (1998), 169--182.
  • F. Delyon, B. Simon, and B. Souillard, From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincaré Phys. Théor. 42 (1985), 283--309.
  • R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69 (1996), 153--200.
  • L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, Amer. Math. Soc., Providence, 1998.
  • D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer-Verlag, Berlin, 1977.
  • D. Gilbert, On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 213--229.
  • D. Gilbert and D. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987), 30--56.
  • I. Guarneri, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95--100.
  • --. --. --. --., On an estimate concerning quantum diffusion in the presence of a fractal spectrum, Europhys. Lett. 21 (1993), 729--733.
  • S. Jitomirskaya, in preparation.
  • S. Jitomirskaya and Y. Last, Dimensional Hausdorff properties of singular continuous spectra, Phys. Rev. Lett. 76 (1996), 1765--1769.
  • --------, Power law subordinacy and singular spectra. I. Half-line operators, preprint.
  • --------, Power law subordinacy and singular spectra. II. Line operators, in preparation.
  • T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math. 12 (1959), 403--425.
  • R. Ketzmerick, K. Kruse, S. Kraut, and T. Geisel, What determines the spreading of a wave-packet?, Phys. Rev. Lett. 79 (1997), 1959--1963.
  • A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials, Comm. Math. Phys. 179 (1996), 377--400.
  • --------, Absolutely continuous spectrum of perturbed Stark operators, to appear in Trans. Amer. Math. Soc.
  • A. Kiselev, Y. Last, and B. Simon, Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators, Comm. Math. Phys. 194 (1998), 1--45.
  • S. Kotani and N. Ushiroya, One-dimensional Schrödinger operators with random decaying potentials, Comm. Math. Phys. 115 (1988), 247--266.
  • P. Kuchment, Bloch solutions of periodic partial differential equations, Funktsional Anal. i Prilozhen 14 (1980), 65--66.
  • Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), 406--445.
  • Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), 329--367.
  • S. Mizohata, The Theory of Partial Differential Equations, Trans. K. Miyahara, Cambridge University Press, New York, 1973.
  • S. N. Naboko and A. B. Pushnitskii, Point spectrum on a continuous spectrum for weakly perturbed Stark type operators, Funct. Anal. Appl. 29 (1995), 248--257.
  • M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, New York, 1978.
  • F. Rellich, Jber das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u =0$ in unendlichen Gebieten, Über. Deutsch. Math. Verein 53 (1943), 57--65.
  • C. Remling, The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials, Comm. Math. Phys. 193 (1998), 151--170.
  • C. A. Rogers, Hausdorff Measures, Cambridge University Press, London, 1970.
  • C. A. Rogers and S. J. Taylor, Additive set functions in Euclidean space. II, Acta Math. 109 (1963), 207--240.
  • I. Sch'nol, On the behavior of the Schrödinger equation, Mat. Sb. 42 (1957), 273--286.
  • B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447--526.
  • --. --. --. --., The Neumann Laplacian of a jelly roll, Proc. Amer. Math. Soc. 114 (1992), 783--785.
  • --. --. --. --., Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc. 124 (1996), 3361--3369.
  • G. Stolz, Bounded solutions and absolute continuity of Sturm-Liouville operators, J. Math. Anal. Appl. 169 (1992), 210--228.
  • R. S. Strichartz, Fourier asymptotics of fractal measures, J. Funct. Anal. 89 (1990), 154--187.
  • N. Vilenkin, Special functions and the theory of group representations (in Russian), ``Nauka,'' Moscow, 1991.