Bound states of discrete Schrödinger operators with supercritical inverse square potentials
Authors:
David Damanik and Gerald Teschl
Journal:
Proc. Amer. Math. Soc. 135 (2007), 11231127
MSC (2000):
Primary 47B36, 81Q10; Secondary 39A11, 47B39
Published electronically:
October 4, 2006
MathSciNet review:
2262914
Fulltext PDF Free Access
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Additional Information
Abstract: We consider discrete onedimensional Schrödinger operators whose potentials decay asymptotically like an inverse square. In the supercritical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of the number of eigenvalues below a given energy as this energy tends to the bottom of the essential spectrum.
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Damanik, D.
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 1.
 D. Damanik, D. Hundertmark, R. Killip, and B. Simon, Variational estimates for discrete Schrödinger operators with potentials of indefinite sign, Commun. Math. Phys. 238, 545562 (2003). MR 1993385 (2004i:81068)
 2.
 D. Damanik, D. Hundertmark, and B. Simon, Bound states and the Szego condition for Jacobi matrices and Schrödinger operators, J. Funct. Anal. 205, 357379 (2003). MR 2017691 (2005i:81180)
 3.
 D. Damanik and R. Killip, Halfline Schrödinger operators with no bound states, Acta Math. 193, 3172 (2004). MR 2155031
 4.
 D. Damanik, R. Killip, and B. Simon, Schrödinger operators with few bound states, Comm. Math. Phys. 258, 741750 (2005). MR 2172016 (2006e:47069)
 5.
 D. Damanik and C. Remling, Schrödinger operators with many bound states, to appear in Duke Math. J.
 6.
 R. Killip and B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. 158, 253321 (2003). MR 1999923 (2004f:47040)
 7.
 W. Kirsch and B. Simon, Corrections to the classical behavior of the number of bound states of Schrödinger operators, Ann. Phys. 183, 122130 (1988). MR 0952875 (90b:35065)
 8.
 F. Luef and G. Teschl, On the finiteness of the number of eigenvalues of Jacobi operators below the essential spectrum, J. Difference Equ. Appl. 10, 299307 (2004). MR 2049680 (2005b:39017)
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 P. B. Naman, The set of isolated points of increase of the spectral function pertaining to a limitconstant Jacobi matrix (Russian), Izv. Vysš. Ucebn. Zaved. Matematika 8, 129135 (1959). MR 0131776 (24:A1624)
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 K. M. Schmidt, Critical coupling constants and eigenvalue asymptotics of perturbed periodic SturmLiouville operators, Comm. Math. Phys. 211, 465485 (2000). MR 1754525 (2001i:34147)
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 G. Teschl, Oscillation theory and renormalized oscillation theory for Jacobi operators, J. Diff. Eqs. 129, 532558 (1996). MR 1404392 (98m:47053)
 12.
 G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon. 72, Amer. Math. Soc., Rhode Island, 2000. MR 1711536 (2001b:39019)
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Additional Information
David Damanik
Affiliation:
Mathematics 253–37, California Institute of Technology, Pasadena, California 91125
Email:
damanik@caltech.edu
Gerald Teschl
Affiliation:
Faculty of Mathematics, Nordbergstrasse 15, 1090 Wien, Austria – and – International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria
Email:
Gerald.Teschl@univie.ac.at
DOI:
http://dx.doi.org/10.1090/S0002993906085509
PII:
S 00029939(06)085509
Keywords:
Discrete Schr\"odinger operators,
bound states,
oscillation theory
Received by editor(s):
September 3, 2005
Received by editor(s) in revised form:
November 9, 2005
Published electronically:
October 4, 2006
Additional Notes:
This work was supported by the National Science Foundation under Grant No.\ DMS0500910 and the Austrian Science Fund (FWF) under Grant No.\ P17762
Communicated by:
Joseph A. Ball
Article copyright:
© Copyright 2006
American Mathematical Society
