Eigenvalues in spectral gaps of a perturbed periodic manifold

Mathematische Nachrichten

Volumn 261-262, Issue , 2003, Pages 141-162

  Post, Olaf ^a  

^a RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

Eigenvalues; Perturbation of periodic structures; Spectral gap

Indexed keywords

EID: 0346361628     PISSN: 0025584X     EISSN: None     Source Type: Journal    
DOI: 10.1002/mana.200310117     Document Type: Article

References (32)

1. S. Alama, M. Avellaneda, P. A. Deift, and R. Hempel, On the existence of eigenvalues of a divergence-form operator A + λB in a gap of σ (A), Asymptotic Anal. 8, 311-344 (1994).
2. S. Alama, P. A. Deift, and R. Hempel, Eigenvalue branches of the Schrödinger operator H - λW in a gap of σ (H), Commun. Math. Phys. 121 (2), 291-321 (1989).
3. M. Sh. Birman, Discrete spectrum of the periodic Schrödinger operator perturbed by a decaying potential, St. Petersbg. Math. J. 8, 1-14 (1997).
4. O. Bratteli, P. E. T. Jørgensen, and D. W. Robinson, Spectral asymptotics of periodic elliptic operators, Math. Z. 232 (4), 621-650 (1999).
5. J. Brüning and T. Sunada, On the spectrum of periodic elliptic operators, Nagoya Math. J. 126, 159-171 (1992).
6. I. Chavel and E. A. Feldman, Spectra of domains in compact manifolds, J. Funct. Anal. 30, 198-222 (1978).
7. I. Chavel, Eigenvalues in Riemannian Geometry (Academic Press, Orlando, 1984).
8. I. Chavel, Riemannian Geometry (Cambridge University Press, Cambridge, 1993).
9. E. B. Davies, Spectral Theory and Differential Operators (Cambridge University Press, Cambridge, 1996).
10. P. A. Deift and R. Hempel, On the existence of eigenvalues of the Schrödinger operator H - λW in a gap of σ (H), Commun. Math. Phys. 103, 461-490 (1986).
11. H. Donnelly and P. Li, Pure point spectrum and negative curvature for noncompact manifolds, Duke Math. J. 46, 497-503 (1979).
12. 2-Betti numbers for Abelian groups, Can. Math. Bull. 24, 91-95 (1981).
13. H. Donnelly, Asymptotic properties of eigenfunctions - the hyperbolic plane, J. Anal. Math. 58, 135-151 (1992).
14. P. Exner and P. Seba, Bound states in curved quantum waveguides, J. Math. Phys. 30 (11), 2574-2580 (1989).
15. R. Froese and I. Herbst, Realizing holonomic constraints in classical and quantum mechanics, Studies in Advanced Mathematics 16, 121-131 (2000).
16. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1977).
17. J. Garnett and E. Trubowitz, Gaps and bands of one dimensional periodic Schrödinger operators II, Comment. Math. Helv. 62, 18-37 (1987).
18. R. Hempel and A. Besch, Magnetic barriers of compact support and eigenvalues in spectral gaps, Electronic J. Diff. Equations 48, 1-25 (2003).
19. E. Hebey, Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Mathematics Vol. 1635 (Springer-Verlag, Berlin, 1996).
20. P. D. Hislop, The geometry and spectra of hyperbolic manifolds, Proc. Indian Acad. Sci. Math. Sci. 104, 715-776 (1994); Spectral and Inverse Spectral Theory (Bangalore, 1993).
21. P. D. Hislop, The geometry and spectra of hyperbolic manifolds, Proc. Indian Acad. Sci. Math. Sci. 104, 715-776 (1994); Spectral and Inverse Spectral Theory (Bangalore, 1993).
22. T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966).
23. J. Li, The problem of eigenvalue on noncompact complete Riemannian manifold, J. Partial Diff. Equations 5 (4), 87-95 (1992).
24. K. A Mitchell, Gauge fields and extrapotentials in constrained quantum systems, Phys. Rev. A 63, Nr. 042112 (2001).
25. O. Post, Periodic manifolds, spectral gaps, and eigenvalues in gaps, Ph.D. thesis, Technische Universität Braunschweig (2000).
26. O. Post, Periodic manifolds with spectral gaps, J. Diff. Equations 187, 23-45 (2003).
27. M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis (Academic Press, New York, 1980).
28. M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic Press, New York, 1978).
29. O. L. Safronov, The discrete spectrum in the spectral gaps of semibounded operators with non-sign-definite perturbations, J. Math. Anal. Appl. 260, 641-652 (2001).
30. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1 (Publish or Perish, Boston, 1970).
31. Y. Safarov and D. Vassiliev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Translations of Mathematical Monographs 155 (Providence, 1998).
32. K. Yoshitomi, Band gap of the spectrum in periodically curved quantum waveguides, J. Diff. Equations 142, 123-166 (1998).