Trace formula and Spectral Riemann Surfaces for a class of tri-diagonal matrices

Journal of Approximation Theory

Volumn 139, Issue 1-2, 2006, Pages 293-326

  Djakov, Plamen ^a   Mityagin, Boris ^b  

^a UNIVERSITY OF SOFIA   (Bulgaria)

^b OHIO STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33645871285     PISSN: 00219045     EISSN: 10960430     Source Type: Journal    
DOI: 10.1016/j.jat.2005.09.007     Document Type: Article

References (40)

1. Bender C., and Wu T.T. An anharmonic oscillator. Phys. Rev. 184 5 (1969) 1231-1260
2. Blanch G. Numerical aspects of Mathieu's equations. Rend. Circ. Mat. Palermo 15 (1966) 51-97
3. Blanch G., and Clemm D.S. The double points of Mathieu's differential equation. Math. Comput. 23 105 (1969) 97-108
4. Boutet de Monvel A., Naboko S., and Silva L. Eigenvalue asymptotics of a modified Jaynes-Cummings model with periodic modulations. C.R. Math. Acad. Sci. Paris 338 (2004) 103-107
5. Djakov P., and Mityagin B. Simple and double eigenvalues of the Hill operator with a two-term potential. J. Approx. Theory 135 (2005) 70-104
6. Dombrowski J., and Pedersen S. Spectral measures and Jacobi matrices related to Laguerre-type systems of orthogonal polynomials. Constr. Approx. 13 (1997) 421-433
7. Fazullin Z.Yu., and Murtazin Kh.Kh. Trace formulas for nonnuclear perturbations. Dokl. Akad. Nauk 368 4 (1999) 442-444 (Russian)
8. Fazullin Z.Yu., and Murtazin Kh.Kh. The regularized trace of a two-dimensional harmonic oscillator. Mat. Sb. 192 5 (2001) 87-124 (Russian)
9. Z.Yu. Fazullin, Kh.Kh. Murtazin, The classical regularized trace formula for a multidimensional harmonic oscillator, Trudy Semin. Petrovs. (21) (2001) 298-339, 342 (Russian); translation in J. Math. Sci. 108 (4) (2002) 608-633 (New York).
10. Hunter C., and Guerrieri B. The eigenvalues of Mathieu's equation and their branch points. Stud. Appl. Math. 64 (1981) 113-141
11. Ince E.L. A linear differential equation with periodic coefficients. Proc. Lond. Math. Soc. 23 (1923) 56-74
12. Janas J., and Naboko S. Multithreshold spectral phase transition examples in a class of unbounded Jacobi matrices. Recent Advances in Operator Theory, Operation Theory Adv. Appl. vol. 124 (2001), Birkhäuser-Verlag, Basel 267-285
13. Janas J., and Naboko S. Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach. SIAM J. Math. Anal. 36 (2004) 643-658
14. Janas J., Naboko S., and Stolz G. Spectral theory for a class of periodically perturbed unbounded Jacobi matrices: elementary methods. J. Comput. Appl. Math. 171 (2004) 265-276
15. Kato T. Perturbation Theory for Linear Operators (1966), Springer, Berlin
16. Korotyaev E., and Pushnitski A. A trace formula and high-energy spectral asymptotics for the perturbed Landau Hamiltonian. J. Funct. Anal. 217 (2004) 221-248
17. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, Oxford University Press, New York, 1995.
18. Magnus W., and Winkler S. Hill's Equation (1969), Interscience Publishers, Wiley
19. Meixner J., and Schäfke F.W. Mathieusche Funktionen und Sphäroidfunktionen (1954), Springer, Berlin
20. J. Meixner, F.W. Schäfke, G. Wolf, Mathieu functions and spheroidal functions and their mathematical foundations, Lecture Notes in Mathematics, vol. 837, Springer, Berlin, 1980.
21. Murtazin Kh.Kh., and Fazullin Z.Yu. The spectrum and trace formula for the two-dimensional Schrödinger operator in a homogeneous magnetic field. Dokl. Akad. Nauk 390 6 (2003) 743-745 (Russian)
22. Sadovnichii V.A., and Podol'skii V.E. Traces of operators with a relatively compact perturbation. Mat. Sb. 193 2 (2002) 129-152 (Russian); translation in Sb. Math. 193 (1-2) (2002) 279-302
23. Savchuk A.M. The first-order regularized trace of the Sturm-Liouville operator with δ-potential. Uspekhi Mat. Nauk 55 336 (2000) 155-156 (Russian); translation in Math. Surveys 55 (2000) 1168-1169 (Russian)
24. Savchuk A.M., and Shkalikov A.A. The trace formula for Sturm-Liouville operators with singular potentials. Mat. Zametki 69 (2001) 427-442 (Russian); translation in Math. Notes 69 (2001) 387-400
25. Shanley P.E. Spectral properties of the scaled quartic anharmonic oscillator. Ann. Phys. 186 (1988) 292-324
26. Shanley P.E. Nodal properties of the scaled quartic anharmonic oscillator. Ann. Phys. 186 (1988) 325-354
27. Shanley P.E. Double wells, inverted wells, and level crossing. Phys. Lett. A 141 (1989) 331-334
28. Simon B. Coupling constant analyticity for the anharmonic oscillator. Ann. Phys. 58 (1970) 76-136
29. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, pp. 1-466; Part 2: Spectral Theory, pp. 467-1044, AMS Colloquium Publications, vol. 54, Parts 1 and 2, AMS, Providence, RI, 2005.
30. Titchmarsh E.C. The Theory of Functions (1966), Oxford University Press, Oxford
31. Tur E.A. Jaynes-Cumming model; solution without rotating wave approximation. Opt. Spectrosc. 89 (2000) 628-642 (Russian)
32. Tur E.A. Weyl's function continued fraction representation for Jacobi matrix and its applications. Proceedings of the Banach Center Research Seminar IV (2000), Warsaw, Poland 21-26
33. E.A. Tur, Jaynes-Cumming model without rotating wave approximation; Asymptotics of eigenvalues, arXiv: math-ph/0211055, vol. 1, 22 November 2002.
34. Turbiner A. Spectral Riemannian Surfaces of the Sturm-Liouville Operator and Quasi-Exactly-Solvable Problems. Funktzional Anal. i Prilozhen. 22 (1988) 92-94 (Russian); Soviet Math.-Functional Anal. Appl. 22 (1988) 163-166 (Engl. transl.)
35. Turbiner A. Quantum mechanics: the problems lying between exactly-solvable and non-solvable. Sov. Phys.-ZhETF 94 (1988) 33-45 English translation: JETP 67 (1988) 230-236
36. Urwin K.M., and Arscott F.M. Theory of the Whittaker-Hill equation. Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/71) 28-44
37. Volkmer H. Quadratic growth of convergence radii for eigenvalues of two-parameter Sturm-Liouville equations. J. Differential Equations 128 (1996) 327-345
38. Volkmer H. On the growth of convergence radii for eigenvalues of the Mathieu equation. Math. Nachr. 192 (1998) 239-253
39. Volkmer H. Convergence radii for eigenvalues of two-parameter Sturm-Liouville problems. Analysis (Munich) 20 3 (2000) 225-236
40. Volkmer H. On Riemann surfaces of analytic eigenvalue functions. Complex Var. Theory Appl. 49 (2004) 169-182