Hostname: page-component-7479d7b7d-767nl Total loading time: 0 Render date: 2024-07-10T05:32:31.487Z Has data issue: false hasContentIssue false
  • English
  • Français

INVERSE SPECTRAL PROBLEMS FOR SINGULAR RANK-ONE PERTURBATIONS OF A HILL OPERATOR

Part of: General theory of linear operators General theory in ordinary differential equations

Published online by Cambridge University Press:  15 December 2009

KAZUSHI YOSHITOMI
KAZUSHI YOSHITOMI*
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan (email: yositomi@tmu.ac.jp)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate an inverse spectral problem for the singular rank-one perturbations of a Hill operator. We give a necessary and sufficient condition for a real sequence to be the spectrum of a singular rank-one perturbation of the Hill operator.

Keywords

Hill operatorℋ−2-perturbationinverse spectral problem

MSC classification

Secondary: 34A55: Inverse problems 47A75: Eigenvalue problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 3 , December 2009 , pp. 421 - 428
Copyright
Copyright © Australian Mathematical Publishing Association, Inc. 2009

References

[1]Akhiezer, N. I. and Glazman, I. M., Theory of Linear Operators in Hilbert Space (Dover Publications, New York, 1993).Google Scholar
[2]Albeverio, S., Gesztesy, F., Høegh-Krohn, R. and Holden, H., Solvable Models in Quantum Mechanics, 2nd edn, With an appendix by Pavel Exner (AMS Chelsea Publishing, Providence, RI, 2005).Google Scholar
[3]Albeverio, S. and Kurasov, P., ‘Rank one perturbations, approximations, and selfadjoint extensions’, J. Funct. Anal. 148 (1997), 152–169.CrossRefGoogle Scholar
[4]Albeverio, S. and Kurasov, P., ‘Rank one perturbations of not semibounded operators’, Integral Equations Operator Theory 27 (1997), 379–400.CrossRefGoogle Scholar
[5]Albeverio, S. and Kurasov, P., Singular Perturbations of Differential Operators. Solvable Schrödinger Type Operators, London Mathematical Society Lecture Note Series, 271 (Cambridge University Press, Cambridge, 2000).CrossRefGoogle Scholar
[6]Borg, G., ‘Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte’, Acta Math. 78 (1946), 1–96.CrossRefGoogle Scholar
[7]Exner, P. and Grosse, H., ‘Some properties of the one-dimensional generalized point interactions (a torso)’, arXiv:math-ph/9910029, 1999.Google Scholar
[8]Garnett, J. and Trubowitz, E., ‘Gaps and bands of one-dimensional periodic Schrödinger operators’, Comment. Math. Helv. 59 (1984), 258–312.CrossRefGoogle Scholar
[9]Garnett, J. and Trubowitz, E., ‘Gaps and bands of one-dimensional periodic Schrödinger operators II’, Comment. Math. Helv. 62 (1987), 18–37.CrossRefGoogle Scholar
[10]Hassi, S. and de Snoo, H., ‘On rank one perturbation of selfadjoint operators’, Integral Equations Operator Theory 29 (1997), 288–300.CrossRefGoogle Scholar
[11]Hochstadt, H., ‘On the determination of a Hill’s equation from its spectrum’, Arch. Rational Mech. Anal. 19 (1965), 353–362.CrossRefGoogle Scholar
[12]Kappeler, T. and Möhr, C., ‘Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator with singular potentials’, J. Funct. Anal. 186 (2001), 62–91.CrossRefGoogle Scholar
[13]Kiselev, A. and Simon, B., ‘Rank one perturbations with infinitesimal coupling’, J. Funct. Anal. 130 (1995), 345–356.CrossRefGoogle Scholar
[14]Korotyaev, E., ‘Characterization of the spectrum of Schrödinger operators with periodic distributions’, Int. Math. Res. Not. 37 (2003), 2019–2031.CrossRefGoogle Scholar
[15]Kurasov, P. and Larson, J., ‘Spectral asymptotics for Schrödinger operators with periodic point interactions’, J. Math. Anal. Appl. 266 (2002), 127–148.CrossRefGoogle Scholar
[16]Kuroda, S. T. and Nagatani, H., ‘Resolvent formulas of general type and its application to point interactions’, J. Evol. Equ. 1 (2001), 421–440.CrossRefGoogle Scholar
[17]Levin, B. Ja., Distribution of Zeros of Entire Functions (American Mathematical Society, Providence, RI, 1980).Google Scholar
[18]Levitan, B. M., Inverse Sturm–Liouville Problems (VNU Science Press, Utrecht, 1987).CrossRefGoogle Scholar
[19]Magnus, W. and Winkler, S., Hill’s Equation (Dover Publications, New York, 2004).Google Scholar
[20]Marchenko, V. A. and Ostrovskii, I. V., ‘A characterization of the spectrum of the Hill operator’, Math. USSR-Sb. 26 (1975), 493–554.CrossRefGoogle Scholar
[21]Posilicano, A., ‘A Krein-like formula for singular perturbations of self-adjoint operators and applications’, J. Funct. Anal. 183 (2001), 109–147.CrossRefGoogle Scholar
[22]Reed, M. and Simon, B., Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic Press, San Diego, CA, 1978).Google Scholar
[23]Ungar, P., ‘Stable Hill equations’, Comm. Pure Appl. Math. 14 (1961), 707–710.CrossRefGoogle Scholar