Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-28T09:26:56.439Z Has data issue: false hasContentIssue false
  • English
  • Français

On the essential spectra of linear 2nth order differential operators with complex coefficients

Published online by Cambridge University Press:  14 November 2011

David Race
David Race
Affiliation:
Department of Mathematics, University of the Witwatersrand, Jan Smuts Avenue, Johannesburg, South Africa
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper, a formally J-symmetric, linear differential expression of 2nth order, with complex-valued coefficients, is considered. A number of results concerning the location of the essential spectrum of associated operators are obtained. These are extensions of earlier work dealing with complex Strum-Liouville operators, and include results which, in the real case, are due to Birman, Glazman and others. They lead to criteria, for the non-emptiness of the regularity field, of the corresponding minimal operator-a condition which is needed in the theory of J-selfadjoint extensions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 92 , Issue 1-2 , 1982 , pp. 65 - 75
Copyright
Copyright © Royal Society of Edinburgh 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Atkinson, F. V.. Limit-n criteria Of integral type. Proc. Roy. Soc. Edinburgh Sect. A 73 (1975), 167–198.CrossRefGoogle Scholar
2Balslev, E. and Gamelin, T. W.. The essential spectrum of a class of ordinary differential operators. Pacific J. Math. 14 (1964), 755–776.CrossRefGoogle Scholar
3Birman, M. Sh.. Perturbation of quadratic forms and the spectrum of singular boundary-value problems. (Russian). Dokl. Akad. Nauk SSSR 125 (1959), 471–474.Google Scholar
4Brinck, I.. Selfadjointness and spectra of Sturm-Liouville operators. Math. Scand. 7 (1959), 219–239.CrossRefGoogle Scholar
5Dunford, N. and Schwartz, J. T.. Linear Operators, Pt II (New York: Wiley, 1963).Google Scholar
6Evans, W. D.. On the spectra of non-self-adjoint realisations of second-order elliptic operators. Proc. Roy. Soc. Edinburgh Sect. A 90 (1981), 71–105.CrossRefGoogle Scholar
7Evans, W. D., Kwong, M. and Zettl, A.. On the location of the spectra and essential spectra of 2n-th order differential operators, preprint.Google Scholar
8Everitt, W. N. and Race, D.. On necessary and sufficient conditions for the existence of Carathéodory solutions of ordinary differential equations. Quaestiones Math. 2 (1978), 507–512.CrossRefGoogle Scholar
9Glazman, I. M.. Direct methods of qualitative spectral analysis of singular differential operators (Jerusalem: Isreal Program for Scientific Translations, 1965.)Google Scholar
10Hinton, D.. Strong limit-point and Dirichlet criteria for ordinary differential expressions of order 2n. Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 301–310.CrossRefGoogle Scholar
11Io, I.. Some questions of spectral theory for the one-dimensional nonselfadjoint Schrödinger operator with potential in L1. Soviet Math. Dokl. 21 (1980), 22–24.Google Scholar
12Kamimura, Y.. On the spectrum of an ordinary differential operator with an r-integrable complex-valued potential. J. London Math. Soc. 20 (1979), 86–100.CrossRefGoogle Scholar
13Kato, T.. Perturbation theory for linear operators (Berlin: Springer, 1966).Google Scholar
14Knowles, I.. Dissipative Sturm-Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 329–343.CrossRefGoogle Scholar
15Knowles, I.. On the boundary conditions characterizing J-selfadjoint extensions of J-symmetric operators. J. Differential Equations 40 (1981), 193–216.CrossRefGoogle Scholar
16Knowles, I. and Race, D.. On the point spectra of complex Strum-Liouville operators. Proc. Roy. Soc. Edinburgh Sect A 85 (1980), 263–289.CrossRefGoogle Scholar
17Knowles, I. and Race, D.. On the correctness of boundary conditions for certain linear differential operators. In Proc. Conf. Spectral Theory of Diff. Operators, Alabama, 1981, Eds Knowles, I. W. and Lewis, R. T., North-Holland Mathematics Studies 55, 279–287 (Amsterdam: North Holland, 1981).Google Scholar
18Naimark, M. A.. Linear differential operators, Pt II (New York: Ungar, 1968).Google Scholar
19Race, D.. On the location of the essential spectra and regularity fields of complex Sturm-Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 1–14.CrossRefGoogle Scholar
20Race, D.. The spectral theory of complex Sturm-Liouville operators (Ph.D. thesis: Univ. of the Witwatersrand, Johannesburg, 1980).Google Scholar
21Race, D.. m(λ)-functions for complex Sturm-Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 275–289.CrossRefGoogle Scholar
22Read, T. T.. A limit-point criterion for expressions with oscillatory coefficients. Pacific J. Math. 66 (1976), 243–255.CrossRefGoogle Scholar
23Zelenko, L. B.. Spectrum of Schrödinger's equation with a complex pseudoperiodic potential I, II. Differential Equations 12 (1976), 563–569, 999–1006.Google Scholar
24Zhikhar, N. A.. The theory of J-symmetric operators (Russian). Ukrain. Math. Z. 11 (4) (1959), 352–364.Google Scholar