Operators with singular continuous spectrum, V. Sparse potentials
Authors:
B. Simon and G. Stolz
Journal:
Proc. Amer. Math. Soc. 124 (1996), 20732080
MSC (1991):
Primary 34L40, 34B24
MathSciNet review:
1342046
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: By presenting simple theorems for the absence of positive eigenvalues for certain onedimensional 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
(95a:47015), http://dx.doi.org/10.1090/S02730979199400518X
 [3]
R. del Rio, N. Makarov, and B. Simon, Operators with singular continuous spectrum, II. Rank one operators, Commun. Math. Phys. 165 (1994), 5967. CMP 95:02
 [4]
R. del Rio, B. Simon, and G. Stolz, Stability of spectral types for SturmLiouville operators, Math. Research Lett. 1 (1994), 437450. CMP 95:03
 [5]
A.
Ya. Gordon, S.
A. Molchanov, and B.
Tsagani, Spectral theory of onedimensional 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
(93a:34097), http://dx.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), 149159.
 [7]
S. Jitomirskaya and B. Simon, Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators, Commun. Math. Phys. 165 (1994), 201205. CMP 95:02
 [8]
W.
Kirsch, S.
Kotani, and B.
Simon, Absence of absolutely continuous spectrum for some
onedimensional random but deterministic Schrödinger operators,
Ann. Inst. H. Poincaré Phys. Théor. 42
(1985), no. 4, 383–406 (English, with French summary). MR 801236
(87h:60115)
 [9]
S. Molchanov, Lectures on the Random Media, Summer School in Probability Theory, SaintFlour, France, 1992.
 [10]
D.
B. Pearson, Singular continuous measures in scattering theory,
Comm. Math. Phys. 60 (1978), no. 1, 13–36. MR 0484145
(58 #4076)
 [11]
B. Simon, Operators with singular continuous spectrum, I. General operators, Ann. of Math. 141 (1995), 131145. CMP 95:07
 [12]
, norms of the Borel transform and the decomposition of measures, Proc. Amer. Math. Soc. 123 (1995), 37493755. CMP 94:13
 [13]
, Operators with singular continuous spectrum, VI. Graph Laplacians and LaplaceBeltrami 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
(91g:81018)
 [15]
G. Stolz, Spectral theory for slowly oscillating potentials, II. Schrödinger operators, Math. Nachrichten (to appear).
 [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 spectrum is generic, Bull. Amer. Math. Soc. 31 (1994), 208212. MR 95a:47015
 [3]
 R. del Rio, N. Makarov, and B. Simon, Operators with singular continuous spectrum, II. Rank one operators, Commun. Math. Phys. 165 (1994), 5967. CMP 95:02
 [4]
 R. del Rio, B. Simon, and G. Stolz, Stability of spectral types for SturmLiouville operators, Math. Research Lett. 1 (1994), 437450. CMP 95:03
 [5]
 A. Gordon, S. Molchanov, and B. Tsagani, Spectral theory for onedimensional Schrödinger operators with strongly fluctuating potentials, Funct. Anal. Appl. 25 (1992), 236238. MR 93a:34097
 [6]
 A. Hof, O. Knill, and B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Commun. Math. Phys. 174 (1995), 149159.
 [7]
 S. Jitomirskaya and B. Simon, Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators, Commun. Math. Phys. 165 (1994), 201205. CMP 95:02
 [8]
 W. Kirsch, S. Kotani, and B. Simon, Absence of absolutely continuous spectrum for some onedimensional random but deterministic potentials, Ann. Inst. Henri Poincaré 42 (1985), 383406. MR 87h:60115
 [9]
 S. Molchanov, Lectures on the Random Media, Summer School in Probability Theory, SaintFlour, France, 1992.
 [10]
 D. Pearson, Singular continuous measures in scattering theory, Commun. Math. Phys. 60 (1978), 1336. MR 58:4076
 [11]
 B. Simon, Operators with singular continuous spectrum, I. General operators, Ann. of Math. 141 (1995), 131145. CMP 95:07
 [12]
 , norms of the Borel transform and the decomposition of measures, Proc. Amer. Math. Soc. 123 (1995), 37493755. CMP 94:13
 [13]
 , Operators with singular continuous spectrum, VI. Graph Laplacians and LaplaceBeltrami operators, Proc. Amer. Math. Soc. (to appear). CMP 95:05
 [14]
 B. Simon and T. Spencer, Trace class perturbations and the absence of absolutely continuous spectrum, Commun. Math. Phys. 125 (1989), 113126. MR 91g:81018
 [15]
 G. Stolz, Spectral theory for slowly oscillating potentials, II. Schrödinger operators, Math. Nachrichten (to appear).
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
34L40,
34B24
Retrieve articles in all journals
with MSC (1991):
34L40,
34B24
Additional Information
B. Simon
Affiliation:
Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 911250001
Email:
bsimon@caltech.edu
G. Stolz
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 352941170
Email:
stolz@vorteb.math.uab.edu
DOI:
http://dx.doi.org/10.1090/S000299399603465X
PII:
S 00029939(96)03465X
Received by editor(s):
January 9, 1995
Additional Notes:
This material is based upon work supported by the National Science Foundation under grant no. DMS9101715. The government has certain rights to this material.
Communicated by:
Palle E. T. Jorgensen
Article copyright:
© Copyright 1996 B. Simon and G. Stolz
