Bound states of discrete Schrödinger operators with super-critical inverse square potentials

Authors:
David Damanik and Gerald Teschl

Journal:
Proc. Amer. Math. Soc. **135** (2007), 1123-1127

MSC (2000):
Primary 47B36, 81Q10; Secondary 39A11, 47B39

Published electronically:
October 4, 2006

MathSciNet review:
2262914

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider discrete one-dimensional Schrödinger operators whose potentials decay asymptotically like an inverse square. In the super-critical 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.

**1.**D. Damanik, D. Hundertmark, R. Killip, and B. Simon,*Variational estimates for discrete Schrödinger operators with potentials of indefinite sign*, Comm. Math. Phys.**238**(2003), no. 3, 545–562. MR**1993385**, 10.1007/s00220-003-0868-7**2.**David Damanik, Dirk Hundertmark, and Barry Simon,*Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators*, J. Funct. Anal.**205**(2003), no. 2, 357–379. MR**2017691**, 10.1016/S0022-1236(03)00070-3**3.**David Damanik and Rowan Killip,*Half-line Schrödinger operators with no bound states*, Acta Math.**193**(2004), no. 1, 31–72. MR**2155031**, 10.1007/BF02392550**4.**David Damanik, Rowan Killip, and Barry Simon,*Schrödinger operators with few bound states*, Comm. Math. Phys.**258**(2005), no. 3, 741–750. MR**2172016**, 10.1007/s00220-005-1366-x**5.**D. Damanik and C. Remling,*Schrödinger operators with many bound states*, to appear in Duke Math. J.**6.**Rowan Killip and Barry Simon,*Sum rules for Jacobi matrices and their applications to spectral theory*, Ann. of Math. (2)**158**(2003), no. 1, 253–321. MR**1999923**, 10.4007/annals.2003.158.253**7.**Werner Kirsch and Barry Simon,*Corrections to the classical behavior of the number of bound states of Schrödinger operators*, Ann. Physics**183**(1988), no. 1, 122–130. MR**952875**, 10.1016/0003-4916(88)90248-5**8.**Franz Luef and Gerald Teschl,*On the finiteness of the number of eigenvalues of Jacobi operators below the essential spectrum*, J. Difference Equ. Appl.**10**(2004), no. 3, 299–307. MR**2049680**, 10.1080/10236190310001641227**9.**P. B. Naĭman,*The set of isolated points of increase of the spectral function pertaining to a limit-constant Jacobi matrix*, Izv. Vysš. Učebn. Zaved. Matematika**1959**(1959), no. 1 (8), 129–135 (Russian). MR**0131776****10.**Karl Michael Schmidt,*Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators*, Comm. Math. Phys.**211**(2000), no. 2, 465–485. MR**1754525**, 10.1007/s002200050822**11.**Gerald Teschl,*Oscillation theory and renormalized oscillation theory for Jacobi operators*, J. Differential Equations**129**(1996), no. 2, 532–558. MR**1404392**, 10.1006/jdeq.1996.0126**12.**Gerald Teschl,*Jacobi operators and completely integrable nonlinear lattices*, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR**1711536**

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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/S0002-9939-06-08550-9

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. DMS-0500910 and the Austrian Science Fund (FWF) under Grant No. P17762

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2006
American Mathematical Society