Operators with singular continuous spectrum, V. Sparse potentials

Authors:
B. Simon and G. Stolz

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2073-2080

MSC (1991):
Primary 34L40, 34B24

DOI:
https://doi.org/10.1090/S0002-9939-96-03465-X

MathSciNet review:
1342046

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: By presenting simple theorems for the absence of positive eigenvalues for certain one-dimensional Schrödinger operators, we are able to construct explicit potentials which yield purely singular continuous spectrum.

**[1]**R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon,*Operators with singular continuous spectrum, IV. Hausdorff dimension, rank one perturbations, and localization*, preprint.**[2]**R. Del Rio, S. Jitomirskaya, N. Makarov, and B. Simon,*Singular continuous spectrum is generic*, Bull. Amer. Math. Soc. (N.S.)**31**(1994), no. 2, 208–212. MR**1260519**, https://doi.org/10.1090/S0273-0979-1994-00518-X**[3]**R. del Rio, N. Makarov, and B. Simon,*Operators with singular continuous spectrum, II. Rank one operators*, Commun. Math. Phys.**165**(1994), 59--67. CMP**95:02****[4]**R. del Rio, B. Simon, and G. Stolz,*Stability of spectral types for Sturm-Liouville operators*, Math. Research Lett.**1**(1994), 437--450. CMP**95:03****[5]**A. Ya. Gordon, S. A. Molchanov, and B. Tsagani,*Spectral theory of one-dimensional Schrödinger operators with strongly fluctuating potentials*, Funktsional. Anal. i Prilozhen.**25**(1991), no. 3, 89–92 (Russian); English transl., Funct. Anal. Appl.**25**(1991), no. 3, 236–238 (1992). MR**1139884**, https://doi.org/10.1007/BF01085500**[6]**A. Hof, O. Knill, and B. Simon,*Singular continuous spectrum for palindromic Schrödinger operators*, Commun. Math. Phys.**174**(1995), 149--159.**[7]**S. Jitomirskaya and B. Simon,*Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators*, Commun. Math. Phys.**165**(1994), 201--205. CMP**95:02****[8]**W. Kirsch, S. Kotani, and B. Simon,*Absence of absolutely continuous spectrum for some one-dimensional random but deterministic Schrödinger operators*, Ann. Inst. H. Poincaré Phys. Théor.**42**(1985), no. 4, 383–406 (English, with French summary). MR**801236****[9]**S. Molchanov,*Lectures on the Random Media*, Summer School in Probability Theory, Saint-Flour, France, 1992.**[10]**D. B. Pearson,*Singular continuous measures in scattering theory*, Comm. Math. Phys.**60**(1978), no. 1, 13–36. MR**484145****[11]**B. Simon,*Operators with singular continuous spectrum, I. General operators*, Ann. of Math.**141**(1995), 131--145. CMP**95:07****[12]**------,*norms of the Borel transform and the decomposition of measures*, Proc. Amer. Math. Soc.**123**(1995), 3749--3755. CMP**94:13****[13]**------,*Operators with singular continuous spectrum, VI. Graph Laplacians and Laplace-Beltrami operators*, Proc. Amer. Math. Soc. (to appear). CMP**95:05****[14]**Barry Simon and Thomas Spencer,*Trace class perturbations and the absence of absolutely continuous spectra*, Comm. Math. Phys.**125**(1989), no. 1, 113–125. MR**1017742****[15]**G. Stolz,*Spectral theory for slowly oscillating potentials, II. Schrödinger operators*, Math. Nachrichten (to appear).

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Additional Information

**B. Simon**

Affiliation:
Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125-0001

Email:
bsimon@caltech.edu

**G. Stolz**

Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170

Email:
stolz@vorteb.math.uab.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03465-X

Received by editor(s):
January 9, 1995

Additional Notes:
This material is based upon work supported by the National Science Foundation under grant no. DMS-9101715. The government has certain rights to this material.

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1996
B. Simon and G. Stolz