Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Oscillation of the perturbed Hill equation
and the lower spectrum of radially
periodic Schrödinger operators in the plane


Author: Karl Michael Schmidt
Journal: Proc. Amer. Math. Soc. 127 (1999), 2367-2374
MSC (1991): Primary 34C10, 34D15, 35P15
Published electronically: April 9, 1999
MathSciNet review: 1626474
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Generalizing the classical result of Kneser, we show that the
Sturm-Liouville equation with periodic coefficients and an added perturbation term $-c^{2}/r^{2}$ is oscillatory or non-oscillatory (for $r \rightarrow \infty $) at the infimum of the essential spectrum, depending on whether $c^{2}$ surpasses or stays below a critical threshold. An explicit characterization of this threshold value is given. Then this oscillation criterion is applied to the spectral analysis of two-dimensional rotation symmetric Schrödinger operators with radially periodic potentials, revealing the surprising fact that (except in the trivial case of a constant potential) these operators always have infinitely many eigenvalues below the essential spectrum.


References [Enhancements On Off] (What's this?)

  • 1. M. Š. Birman, On the spectrum of singular boundary-value problems, Mat. Sb. (N.S.) 55 (97) (1961), 125–174 (Russian). MR 0142896 (26 #463)
  • 2. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745 (32 #6181)
  • 3. MSP Eastham: Theory of ordinary differential equations. Van Nostrand, London 1970
  • 4. M. S. P. Eastham, Results and problems in the spectral theory of periodic differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Springer, Berlin, 1975, pp. 126–135. Lecture Notes in Math., Vol. 448. MR 0404749 (53 #8549)
  • 5. Philip Hartman, Ordinary differential equations, John Wiley & Sons Inc., New York, 1964. MR 0171038 (30 #1270)
  • 6. Rainer Hempel, Ira Herbst, Andreas M. Hinz, and Hubert Kalf, Intervals of dense point spectrum for spherically symmetric Schrödinger operators of the type -Δ+cos\vert𝑥\vert, J. London Math. Soc. (2) 43 (1991), no. 2, 295–304. MR 1111587 (92f:35109), http://dx.doi.org/10.1112/jlms/s2-43.2.295
  • 7. Tosio Kato, Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 135–148 (1973). MR 0333833 (48 #12155)
  • 8. A Kneser: Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen. Math. Ann. 42 (1893) 409-435
  • 9. John Piepenbrink, Finiteness of the lower spectrum of Schrödinger operators, Math. Z. 140 (1974), 29–40. MR 0355368 (50 #7842)
  • 10. F. S. Rofe-Beketov, Expansion in eigenfunctions of systems with summable potential, Dokl. Akad. Nauk SSSR 156 (1964), 1029–1032 (Russian). MR 0170060 (30 #301)
  • 11. F. S. Rofe-Beketov, Generalization of the Prüfer transformation and the discrete spectrum in gaps of the continuous spectrum, Spectral theory of operators (Proc. Second All-Union Summer Math. School, Baku, 1975) (Russian), “Èlm”, Baku, 1979, pp. 146–153 (Russian). MR 558545 (81i:34021)
  • 12. FS Rofe-Beketov: Spectrum perturbations, the Kneser-type constants and effective masses of zones-type potentials, in: Constructive theory of functions, Sofia 1984, 757-766
  • 13. C. A. Swanson, Comparison and oscillation theory of linear differential equations, Academic Press, New York, 1968. Mathematics in Science and Engineering, Vol. 48. MR 0463570 (57 #3515)
  • 14. Joachim Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, 1987. MR 923320 (89b:47070)
  • 15. D. Willett, Classification of second order linear differential equations with respect to oscillation, Advances in Math. 3 (1969), 594–623 (1969). MR 0280800 (43 #6519)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34C10, 34D15, 35P15

Retrieve articles in all journals with MSC (1991): 34C10, 34D15, 35P15


Additional Information

Karl Michael Schmidt
Affiliation: Mathematisches Institut der Universität, Theresienstr. 39, D-80333 München, Germany
Email: kschmidt@rz.mathematik.uni-muenchen.de

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05069-8
PII: S 0002-9939(99)05069-8
Received by editor(s): October 31, 1997
Published electronically: April 9, 1999
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society