Abstract
We prove that for any one-dimensional Schrödinger operator with potentialV(x) satisfying decay condition|V(x)|≦Cx −3/4−ε, the absolutely continuous spectrum fills the whole positive semi-axis. The description of the set in ℝ+ on which the singular part of the spectral measure might be supported is also given. Analogous results hold for Jacobi matrices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Behncke, H.: Absolute continuity of Hamiltonians with von Neumann-Wigner potentials. Proc. Am. Math. Soc.111, 373–384 (1991)
Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math.116, 135–157 (1966)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. New York: McGraw-Hill, 1955, p. 92
del Rio, R., Makarov, N., Simon, B.: Operators with singular continuous spectrum, II. Rank one operators. Commun. Math. Phys.165, 59–67 (1994)
Delyon, F., Simon, B., Souillard, B.: From power pure point to continuous spectrum in disordered systems. Ann. Inst. H. Poincaré42, 283–309 (1985)
Eastham, M.S.P., Kalf, H.: Schrödinger-type Operators with Continuous Spectra. Research Notes in Mathematics65, London: Pitman Books Ltd., 1982
Eastham, M.S.P., Mc Leod, J.B.: The existence of eigenvalues imbedded in the continuous spectrum of ordinary differential operators. Proc. Roy. Soc. Edinburgh,79A, 25–34 (1977)
Gilbert, D.J., Pearson, D.B.: On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. Appl.128, 30–56 (1987)
Gordon, A.: Pure point spectrum under 1-parameter perturbations and instability of Anderson localization. Commun. Math. Phys.164, 489–505 (1994)
Harris, W.A., Lutz, D.A.: Asymptotic integration of adiabatic oscillator. J. Math. Anal. Appl.51, 76–93 (1975)
Hinton, D.B., Shaw, J.K.: Absolutely continuous spectra of second-order differential operators with short and long range potentials. SIAM J. Math. Anal.17, 182–196 (1986)
Jitomirskaya, S., Last, Y.: In preparation
Kac, I.S.: Multiplicity of the spectrum of differential operator and eigenfunction expansion. Izv. Akad. Nauk USSR27, 1081–1112 (1963) (Russian)
Khan, S., Pearson, D.B.: Subordinacy and spectral theory for infinite matrices. Helv. Phys. Acta65, 505–527 (1992)
Kotani, S., Ushiroya, N.: One-dimensional Schrödinger operators with random decaying potentials. Commun. Math. Phys.115, 247–266 (1988)
Matveev, V.B.: Wave operators and positive eigenvalues for a Schrödinger equation with oscillating potential. Theoret. and Math. Phys.15, 574–583 (1973)
Menchoff, D.: Sur les séries de fonctions orthogonales. Fund. Math.10, 375–420 (1927)
Naboko, S.N.: Dense point spectra of Schrödinger and Dirac operators. Theor. Math.68, 18–28 (1986)
Naboko, S.N., Yakovlev, S.I.: On the point spectrum of discrete Schrödinger operators. Funct. Anal.26(2), 145–147 (1992)
Pearson, D.B.: Singular continuous measures in scattering theory. Commun. Math. Phys.60, 13–36 (1978)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness. New York: Academic Press, 1975
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III. Scattering Theory. New York: Academic Press, 1979
Rudin, W.: Real and Complex Analysis. New York: McGraw-Hill, 1987
Simon, B.: Operators with singular continuous spectrum, I. General operators. Ann. Math.141, 131–145 (1995)
Simon, B.: Some Schrödinger operators with dense point spectrum. Proc. Am. Math. Soc. (to appear)
Simon, B.: Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators. Proc. Am. Math. Soc. (to appear)
Stolz, G.: Bounded solutions and absolute continuity of Sturm-Liouville Operators. J. Math. Anal. Appl.169, 210–228 (1992)
Thurlow, C.R.: The point-continuous spectrum of second-order, ordinary differential operators. Proc. Roy. Soc. Edinburgh84A, 197–211 (1979)
Weidman, J.: Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics1258, Berlin: Springer-Verlag, 1987
White, D.A.W.: Schrödinger operators with rapidly oscillating central potentials. Trans. Am. Math. Soc.275, 641–677 (1983)
Zygmund, A.: A remark on Fourier transforms. Proc. Camb. Phil. Soc.32, 321–327 (1936)
Author information
Authors and Affiliations
Additional information
Communicated by B. Simon
Rights and permissions
About this article
Cite this article
Kiselev, A. Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials. Commun.Math. Phys. 179, 377–399 (1996). https://doi.org/10.1007/BF02102594
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02102594