Abstract
This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper (Remling, The absolutely continuous spectrum of Jacobi matrices, http://arxiv.org/abs/0706.1101, 2007). The treatment of the continuous case in the present paper depends on the same basic ideas.
Similar content being viewed by others
References
Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, Texts and Monographs in Physics. Springer, New York (1988)
Atkinson, F.V.: On the location of the Weyl circles. Proc. R. Soc. Edinb. Sect. A Math 88, 345–356 (1981)
Ben Amor, A., Remling, C.: Direct and inverse spectral theory of Schrödinger operators with measures. Integr. Equ. Oper. Theory 52, 395–417 (2005)
Bessaga, C., Pelczynski, A.: Selected Topics in Infinite-Dimensional Topology, Mathematical Monographs, vol. 58. Polish Scientific, Warsaw (1975)
Brasche, J.F., Exner, P., Kuperin, Y.A., Seba, P.: Schrödinger operators with singular interactions. J. Math. Anal. Appl. 184, 112–139 (1994)
Brasche, J.F., Figari, R., Teta, A.: Singular Schrödinger operators as limits of point interaction Hamiltonians. Potential Anal. 8, 163–178 (1998)
Breimesser, S.V., Pearson, D.B.: Asymptotic value distribution for solutions of the Schrödinger equation. Math. Phys. Anal. Geom. 3, 385–403 (2000)
Breimesser, S.V., Pearson, D.B.: Geometrical aspects of spectral theory and value distribution for Herglotz functions. Math. Phys. Anal. Geom. 6, 29–57 (2003)
Clark, S., Gesztesy, F., Holden, H., Levitan, B.M.: Borg-type theorems for matrix-valued Schrödinger operators. J. Differ. Equ. 167, 181–210 (2000)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)
Denisov, S.: On the continuous analog of Rakhmanov’s theorem for orthogonal polynomials. J. Funct. Anal. 198, 465–480 (2003)
Denisov, S.: On Rakhmanov’s theorem for Jacobi matrices. Proc. Am. Math. Soc. 132, 847–852 (2004)
Gesztesy, F., Simon, B.: A new approach to inverse spectral theory. II. General real potentials and the connection to the spectral measure. Ann. Math. 152, 593–643 (2000)
Hinton, D.B., Klaus, M., Shaw, J.K.: Series representation and asymptotics for Titchmarsh-Weyl m-functions. Differ. Integral Equ. 2, 419–429 (1989)
Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999)
Last, Y., Simon, B.: The essential spectrum of Schrödinger, Jacobi, and CMV operators. J. Anal. Math. 98, 183–220 (2006)
Rakhmanov, E.A.: The asymptotic behavior of the ratio of orthogonal polynomials II (Russian). Mat. Sb. (N.S.) 118(160), 104–117 (1982)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I. Functional Analysis. Academic, New York (1980)
Remling, C.: The absolutely continuous spectrum of Jacobi matrices. http://arxiv.org/abs/0706.1101 (2007)
Rybkin, A.: Some new and old asymptotic representations of the Jost solution and the Weyl m-function for Schrödinger operators on the line. Bull. Lond. Math. Soc. 34, 61–72 (2002)
Sims, R., Stolz, G.: Localization in one-dimensional random media: a scattering theoretic approach. Commun. Math. Phys. 213, 575–597 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Remling, C. The Absolutely Continuous Spectrum of One-dimensional Schrödinger Operators. Math Phys Anal Geom 10, 359–373 (2007). https://doi.org/10.1007/s11040-008-9036-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11040-008-9036-9