We consider an infinite Jacobi matrix with off-diagonal entries dominated by the diagonal entries going to infinity. The corresponding self-adjoint operator J has discrete spectrum and our purpose is to present results on the approximation of eigenvalues of J by eigenvalues of its finite submatrices.
@article{bwmeta1.element.doi-10_2478_s11533-013-0348-z,
author = {Anne Monvel and Lech Zielinski},
title = {Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {445-463},
zbl = {06271167},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0348-z}
}
Anne Monvel; Lech Zielinski. Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices. Open Mathematics, Tome 12 (2014) pp. 445-463. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0348-z/
[1] Boutet de Monvel A., Naboko S., Silva L.O., Eigenvalue asymptotics of a modified Jaynes-Cummings model with periodic modulations, C. R. Math. Acad. Sci. Paris, 2004, 338(1), 103–107 http://dx.doi.org/10.1016/j.crma.2003.12.001[Crossref] | Zbl 1037.47019
[2] Boutet de Monvel A., Naboko S., Silva L.O., The asymptotic behavior of eigenvalues of a modified Jaynes-Cummings model, Asymptot. Anal., 2006, 47(3–4), 291–315 | Zbl 1139.47024
[3] Boutet de Monvel A., Zielinski L., Explicit error estimates for eigenvalues of some unbounded Jacobi matrices, In: Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, Berlin, July 12–16, 2010, Oper. Theory Adv. Appl., 221, Birkhäuser/Springer, Basel, 2012, 187–217
[4] Boutet de Monvel A., Zielinski L., Asymptotic behavior of large eigenvalues of a modified Jaynes-Cummings model, preprint available at http://www.math.uni-bielefeld.de/~bibos/preprints/12-07-409.pdf | Zbl 1320.47032
[5] Cojuhari P.A., Janas J., Discreteness of the spectrum for some unbounded Jacobi matrices, Acta Sci. Math. (Szeged), 2007, 73(3–4), 649–667 | Zbl 1211.47059
[6] Davies E.B., Spectral Theory and Differential Operators, Cambridge Stud. Adv. Math., 42, Cambridge University Press, Cambridge, 1995 http://dx.doi.org/10.1017/CBO9780511623721[Crossref]
[7] Helffer B., Sjöstrand J., Équation de Schrödinger avec champ magnétique et équation de Harper, In: Schrödinger Operators, Sønderborg, August 1–12, 1988, Lecture Notes in Phys., 345, Springer, Berlin, 1989, 118–197 | Zbl 0714.34130
[8] Janas J., Malejki M., Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices, J. Comput. Appl. Math., 2007, 200(1), 342–356 http://dx.doi.org/10.1016/j.cam.2005.09.033[WoS][Crossref] | Zbl 1110.47020
[9] Janas J., Naboko S., Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal., 2004, 36(2), 643–658 http://dx.doi.org/10.1137/S0036141002406072[Crossref] | Zbl 1091.47026
[10] Janas J., Naboko S., Stolz G., Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices, Int. Math. Res. Not. IMRN, 2009, 4, 736–764 [WoS] | Zbl 1175.47029
[11] Malejki M., Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices, Linear Algebra Appl., 2009, 431(10), 1952–1970 http://dx.doi.org/10.1016/j.laa.2009.06.035[Crossref][WoS]
[12] Malejki M., Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices, Cent. Eur. J. Math., 2010, 8(1), 114–128 http://dx.doi.org/10.2478/s11533-009-0064-x[WoS] | Zbl 1197.47046
[13] Volkmer H., Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation, Constr. Approx., 2004, 20(1), 39–54 http://dx.doi.org/10.1007/s00365-002-0527-9[Crossref] | Zbl 1063.33028