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THE DISCRETENESS OF SPECTRUM FOR HIGHER-ORDER DIFFERENTIAL OPERATORS IN WEIGHTED FUNCTION SPACES

Part of: Ordinary differential operators

Published online by Cambridge University Press:  09 August 2012

MAOZHU ZHANG ,
JIONG SUN  and
JIJUN AO
MAOZHU ZHANG*
Affiliation:
College of Mathematics and Statistics, Taishan University, Taian 271021, PR China School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, PR China (email: zhangmaozhu2000@163.com)
JIONG SUN
Affiliation:
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, PR China (email: masun@imu.edu.cn)
JIJUN AO
Affiliation:
College of Sciences, Inner Mongolia University of Technology, Hohhot 010051, PR China
*
✉For correspondence; e-mail: zhangmaozhu2000@163.com
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Abstract

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In this paper we consider the discreteness of spectrum for higher-order differential operators in weighted function spaces. Using the method of embedding theorems of weighted Sobolev spaces Hnp in weighted spaces Ls,r, we obtain a new sufficient and necessary condition to ensure that the spectrum is discrete, which can be easily used to judge the discreteness of some differential operators.

Keywords

differential operatorsspectrumdiscreteness

MSC classification

Secondary: 34L05: General spectral theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 370 - 376
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The work of the authors is supported by the National Nature Science Foundation of China (grant number 11161030) and Shandong Education Fund of College Scientific Research (grant number J12LI57).

References

[1]Edmunds, D. E. and Evans, W. D., Spectral Theory and Differential Operators (Oxford University Press, Oxford, 1987).Google Scholar
[2]Edmunds, D. E. and Sun, J., ‘Embedding theorems and the spectra of certain differential operators’, Proc. R. Soc. Lond. A. 434 (1991), 643–656.Google Scholar
[3]Glazman, I. M., ‘Direct method of qualitative spectral analysis of singular differential operators’, Israel Programme for Scientific Translations, Jerusalem, 1965.Google Scholar
[4]Müller, P. E., Spectral Theory of Ordinary Differential Operators (Ellis Horwood, Chichester, 1981).Google Scholar
[5]Müller, P. E. and Sun, J., ‘On the discrete spectrum of ordinary differential operators in weighted function spaces’, J. Anal. Appl. 14(3) (1995), 637–646.Google Scholar
[6]Oinarov, R. and Otelbaev, M., ‘A criterion for the discreteness of the spectrum of the general Sturm–Liouville operator and embedding theorems connected with it’, Differ. Uravn. 24 (1988), 584–594.Google Scholar
[7]Sun, J., ‘On the spectrum of a class of differential operators and embedding theorems’, Acta Math. Sin. (N.S.) 10(4) (1994), 415–427.Google Scholar