Bound states of discrete Schrödinger operators with super-critical inverse square potentials
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- by David Damanik and Gerald Teschl PDF
- Proc. Amer. Math. Soc. 135 (2007), 1123-1127 Request permission
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 $E$ as this energy tends to the bottom of the essential spectrum.References
- 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, DOI 10.1007/s00220-003-0868-7
- 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, DOI 10.1016/S0022-1236(03)00070-3
- David Damanik and Rowan Killip, Half-line Schrödinger operators with no bound states, Acta Math. 193 (2004), no. 1, 31–72. MR 2155031, DOI 10.1007/BF02392550
- David Damanik, Rowan Killip, and Barry Simon, Schrödinger operators with few bound states, Comm. Math. Phys. 258 (2005), no. 3, 741–750. MR 2172016, DOI 10.1007/s00220-005-1366-x
- D. Damanik and C. Remling, Schrödinger operators with many bound states, to appear in Duke Math. J.
- 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, DOI 10.4007/annals.2003.158.253
- 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, DOI 10.1016/0003-4916(88)90248-5
- 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, DOI 10.1080/10236190310001641227
- 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
- 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, DOI 10.1007/s002200050822
- Gerald Teschl, Oscillation theory and renormalized oscillation theory for Jacobi operators, J. Differential Equations 129 (1996), no. 2, 532–558. MR 1404392, DOI 10.1006/jdeq.1996.0126
- Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 1711536, DOI 10.1090/surv/072
Additional Information
- David Damanik
- Affiliation: Mathematics 253–37, California Institute of Technology, Pasadena, California 91125
- MR Author ID: 621621
- 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
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 1123-1127
- MSC (2000): Primary 47B36, 81Q10; Secondary 39A11, 47B39
- DOI: https://doi.org/10.1090/S0002-9939-06-08550-9
- MathSciNet review: 2262914