Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices

Journal of Mathematical Analysis and Applications

Volumn 340, Issue 2, 2008, Pages 892-900

  Hundertmark, Dirk ^a   Simon, Barry ^b  

^a UNIVERSITY OF BIRMINGHAM   (United Kingdom)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Eigenvalue bounds; Jacobi matrices; Schr dinger operators

Indexed keywords

EID: 38049117390     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2007.08.059     Document Type: Article

References (56)

1. Alama S., Avellaneda M., Deift P.A., and Hempel R. On the existence of eigenvalues of a divergence-form operator A + λ B in a gap of σ (A). Asymptot. Anal. 8 (1994) 311-344
2. Alama S., Deift P.A., and Hempel R. Eigenvalue branches of the Schrödinger operator H - λ W in a gap of σ (H). Comm. Math. Phys. 121 (1989) 291-321
3. Antony A.J., and Krishna M. Almost periodicity of some Jacobi matrices. Proc. Indian Acad. Sci. Math. Sci. 102 (1992) 175-188
4. Aptekarev A.I. Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains. Mat. Sb. (N.S.) 125 167 (1984) 231-258 (in Russian)
5. Bargmann V. On the number of bound states in a central field of force. Proc. Natl. Acad. Sci. USA 38 (1952) 961-966
6. Birman M.Sh. On the spectrum of singular boundary-value problems. Mat. Sb. (N.S.) 55 97 (1961) 125-174 translated in:
7. Birman M.Sh. On the spectrum of singular boundary-value problems. Amer. Math. Soc. Transl. 53 (1966) 23-80
8. Birman M.Sh. Discrete spectrum in the gaps of the continuous one in the large-coupling-constant limit. Order, Disorder and Chaos in Quantum Systems. Dubna, 1989. Oper. Theory Adv. Appl. vol. 46 (1990), Birkhäuser, Basel 17-25
9. Birman M.Sh. Discrete spectrum in a gap of perturbed periodic operator at large coupling constants. Rigorous Results in Quantum Dynamics. Liblice, 1990 (1991), World Sci. Publishing, River Edge, NJ 16-24
10. Birman M.Sh. Discrete spectrum in the gaps of a continuous one for perturbations with large coupling constant. Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations. Leningrad, 1989-1990. Adv. Soviet Math. vol. 7 (1991), Amer. Math. Soc., Providence, RI 57-73
11. Birman M.Sh. On a discrete spectrum in gaps of a second-order perturbed periodic operator. Funct. Anal. Appl. 25 (1991) 158-161 Russian original:
12. Birman M.Sh. On a discrete spectrum in gaps of a second-order perturbed periodic operator. Funktsional. Anal. i Prilozhen. 25 (1991) 89-92
13. Birman M.Sh. The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. I. Regular perturbations. Boundary Value Problems, Schrödinger Operators, Deformation Quantization. Math. Top. vol. 8 (1995), Akademie Verlag, Berlin 334-352
14. Birman M.Sh. The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential. St. Petersburg Math. J. 8 (1997) 1-14 Russian original:
15. Birman M.Sh. The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential. Algebra i Analiz 8 (1996) 3-20
16. Birman M.Sh. The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. II. Nonregular perturbations. St. Petersburg Math. J. 9 (1998) 1073-1095 Russian original:
17. Birman M.Sh. The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. II. Nonregular perturbations. Algebra i Analiz 9 (1997) 62-89
18. Birman M.Sh., Laptev A., and Suslina T.A. The discrete spectrum of a two-dimensional second-order periodic elliptic operator perturbed by a decreasing potential. I. A semi-infinite gap. St. Petersburg Math. J. 12 (2001) 535-567 Russian original:
19. Birman M.Sh., Laptev A., and Suslina T.A. The discrete spectrum of a two-dimensional second-order periodic elliptic operator perturbed by a decreasing potential. I. A semi-infinite gap. Algebra i Analiz 12 (2000) 36-78
20. Birman M.Sh., and Pushnitskiǐ A.B. The discrete spectrum in the gaps of the perturbed pseudo-relativistic magnetic Hamiltonian. J. Math. Sci. (N.Y.) 101 (2000) 3437-3447 Russian original:
21. Birman M.Sh., and Pushnitskiǐ A.B. The discrete spectrum in the gaps of the perturbed pseudo-relativistic magnetic Hamiltonian. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 249 (1997)
22. Birman M.Sh., and Pushnitskiǐ A.B. The discrete spectrum in the gaps of the perturbed pseudo-relativistic magnetic Hamiltonian. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 29 (1995) 102-117 315
23. Birman M.Sh., and Raǐkov G.D. Discrete spectrum in the gaps for perturbations of the magnetic Schrödinger operator. Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations. Leningrad, 1989-1990. Adv. Soviet Math. vol. 7 (1991), Amer. Math. Soc., Providence, RI 75-84
24. Birman M.Sh., and Weidl T. The discrete spectrum in a gap of the continuous one for compact supported perturbations. Mathematical Results in Quantum Mechanics. Blossin, 1993. Oper. Theory Adv. Appl. vol. 70 (1994), Birkhäuser, Basel 9-12
25. J. Christiansen, B. Simon, M. Zinchenko, Finite gap Jacobi matrices, I. The isospectral torus, in preparation
26. Craig W. The trace formula for Schrödinger operators on the line. Comm. Math. Phys. 126 (1989) 379-407
27. Cwikel M. Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. of Math. (2) 106 (1977) 93-100
28. D. Damanik, R. Killip, B. Simon, Perturbations of orthogonal polynomials with periodic recursion coefficients, preprint
29. Deift P.A. Applications of a commutation formula. Duke Math. J. 45 (1978) 267-310
30. Deift P.A., and Hempel R. On the existence of eigenvalues of the Schrödinger operator H - λ W in a gap of σ (H). Comm. Math. Phys. 103 (1986) 461-490
31. Dubrovin B.A., Matveev V.B., and Novikov S.P. Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties. Uspekhi Mat. Nauk 31 1 (187) (1976) 55-136 (in Russian)
32. R. Frank, B. Simon, T. Weidl, Eigenvalue bounds for perturbations of Schrödinger operators and Jacobi matrices with regular ground states, Comm. Math. Phys., in press
33. Gesztesy F., Gurarie D., Holden H., Klaus M., Sadun L., Simon B., and Vogl P. Trapping and cascading of eigenvalues in the large coupling limit. Comm. Math. Phys. 118 (1988) 597-634
34. Gesztesy F., and Simon B. On a theorem of Deift and Hempel. Comm. Math. Phys. 116 (1988) 503-505
35. Hempel R. On the asymptotic distribution of the eigenvalue branches of the Schrödinger operator H ± λ W in a spectral gap of H. J. Reine Angew. Math. 399 (1989) 38-59
36. Hempel R. Eigenvalues in gaps and decoupling by Neumann boundary conditions. J. Math. Anal. Appl. 169 (1992) 229-259
37. Hempel R. On the asymptotic distribution of eigenvalues in gaps. Quasiclassical Methods. Minneapolis, MN, 1995. IMA Vol. Math. Appl. vol. 95 (1997), Springer, New York 115-124
38. Hundertmark D., and Simon B. Lieb-Thirring inequalities for Jacobi matrices. J. Approx. Theory 118 (2002) 106-130
39. Kirsch W., and Simon B. Comparison theorems for the gap of Schrödinger operators. J. Funct. Anal. 75 (1987) 396-410
40. Klaus M. Some applications of the Birman-Schwinger principle. Helv. Phys. Acta 55 (1982-1983) 49-68
41. Levendorskiǐ S.Z. Lower bounds for the number of eigenvalue branches for the Schrödinger operator H - λ W in a gap of H: The case of indefinite W. Comm. Partial Differential Equations 20 (1995) 827-854
42. Lieb E.H. Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Amer. Math. Soc. 82 (1976) 751-753
43. Lieb E.H., and Thirring W. Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35 (1975) 687-689
44. Phys. Rev. Lett. 35 (1975) 1116 Erratum
45. Lieb E.H., and Thirring W. Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann (1976), Princeton University Press, Princeton, NJ 269-303
46. Peherstorfer F., and Yuditskii P. Asymptotic behavior of polynomials orthonormal on a homogeneous set. J. Anal. Math. 89 (2003) 113-154
47. Reed M., and Simon B. Methods of Modern Mathematical Physics, IV: Analysis of Operators (1978), Academic Press, New York
48. Rozenblum G.V. Distribution of the discrete spectrum of singular differential operators. Soviet Math. (Izv. VUZ) 20 (1976) 63-71 Russian original:
49. Rozenblum G.V. Distribution of the discrete spectrum of singular differential operators. Izv. Vyssh. Uchebn. Zaved. Mat. 1 164 (1976) 75-86
50. Safronov O.L. The discrete spectrum in the gaps of the continuous one for non-signdefinite perturbations with a large coupling constant. Comm. Math. Phys. 193 (1998) 233-243
51. Safronov O.L. The discrete spectrum in the spectral gaps of semibounded operators with non-sign-definite perturbations. J. Math. Anal. Appl. 260 (2001) 641-652
52. Safronov O.L. The discrete spectrum of selfadjoint operators under perturbations of variable sign. Comm. Partial Differential Equations 26 (2001) 629-649
53. Schwinger J. On the bound states of a given potential. Proc. Natl. Acad. Sci. USA 47 (1961) 122-129
54. Simon B. Analysis with weak trace ideals and the number of bound states of Schrödinger operators. Trans. Amer. Math. Soc. 224 (1976) 367-380
55. Sodin M., and Yuditskii P. Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7 (1997) 387-435
56. Widom H. Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3 (1969) 127-232