AM  Vol.5 No.15 , August 2014
Relationships among Three Multiplicities of a Differential Operator’s Eigenvalue
ABSTRACT
In this paper, the algebraic, geometric and analytic multiplicities of an eigenvalue for linear differential operators are defined and classified. The relationships among three multiplicities of an eigenvalue of the linear differential operator are given, and a fundamental fact that the algebraic, geometric and analytic multiplicities for any eigenvalue of self-adjoint differential operators are equal is proven.

Cite this paper
Fu, S. and Wang, Z. (2014) Relationships among Three Multiplicities of a Differential Operator’s Eigenvalue. Applied Mathematics, 5, 2185-2194. doi: 10.4236/am.2014.515211.
References
[1]   Liouville, J. and Sturm, J.C.F. (1837) Extrait d’une méemoire sur le développement des fonctions en serie. Journal de Mathématiques Pures et Appliquées, 2, 220-223.

[2]   Sturm, J.C.F. (1836) Mémoire sur les équations différentielles linéaires du second ordre. Journal de Mathématiques Pures et Appliquées, 1, 106-186.

[3]   Sturm, J.C.F. (1837) Mémoire sur une classe d’équations différentielles partielles. Journal de Mathématiques Pures et Appliquées, 2, 373-444.

[4]   Weyl, H. (1910) üeber gewöhnliche differentialgleichungen mit singularitäten und die zugehörigen entwichlungen willkürlicher funktionen. Mathematische Annalen, 68, 220-269.
http://dx.doi.org/10.1007/BF01474161

[5]   Naimark, M.A. (1968) Linear Differential Operators. Ungar, New York.

[6]   Everitt, W.N. and Markus, L. (1997) The Glazman-Krein-Naimark Theorem for Ordinary Differential Operators, New Results in Operator Theory and Its Applications. Operator Theory: Advances and Applications, 98, 118-130.

[7]   Cao, X. and Wu, H. (2004) Geomtric Aspects of High-Order Eigenvalue Problems I. Structures on Spaces of Boundary Conditions. International Journal of Mathematics and Mathematical Sciences, 13, 647-678.
http://dx.doi.org/10.1155/S0161171204303522

[8]   Coddington, E. and Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York.

[9]   Weidmann, J. (1987) Spectral Theory of Ordinary Differential Operator, Lecture Notes in Mathematics, Vl 1258. Springer-Verlag, Berlin.

[10]   Reed, M. and Simon, B. (1972) Methods of Modern Mathematical Physics I. Functional Analysis. Academic Press, Waltham.

[11]   Kong, Q., Wu, H. and Zettl, A. (1997) Dependence of Eigenvalues on the Problem. Mathematische Nachrichten, 188, 173-201. http://dx.doi.org/10.1002/mana.19971880111

[12]   Kong, Q., Wu, H. and Zettl, A. (1999) Dependence of the n-th Sturm-Liouville Eigenvalue on the Problem. Journal of Differential Equations, 156, 328-354.
http://dx.doi.org/10.1006/jdeq.1998.3613

[13]   Kong, Q., Wu, H. and Zettl, A. (2000) Geometric Aspects of Sturm-Liouville Problems, I. Structures on Spaces of Boundary Conditions. Proceedings of the Royal Society of Edinburgh Section A, 130, 561-589.

[14]   Zettl, A. (2005) Sturm-Liouville Theory. Mathematical Surveys and Monographs, Volume 121. American Mathematical Society.

[15]   Gohberg, I.C. and Krein, M.G. (1969) Introduction to the Theory of Linear Non-Self-Adjoint Operator. Translation of Mathematical Monographs 18, American Mathematical Society, Providence.

[16]   Keldysh, M.V. (1951) On Eigenvalues and Eigenfunctions of Some Classes of Non Self-Adjoint Equations. Doklady Akademii Nauk SSSR, 77, 11-14.

[17]   Marcenko, V.A. (1963) Expansion in Eigenfuctions of Non-Self-Adjoint Singular Differential Operators of Second Order. American Mathematical Society Translations, 25, 77-130.

[18]   Eastham, M., Kong, Q., Wu, H. and Zettl, A. (1999) Inequalities among Eigenvalues of Sturm-Liouville Problems. Journal of Inequalities and Applications, 3, 25-43.

[19]   Kong, Q., Wu, H. and Zettl, A. (1999) Inequalities among Eigenvalues of Singular Sturm-Liouville Problems. Dynamic Systems and Applications, 8, 517-531.

[20]   Kong, Q., Wu, H. and Zettl, A. (2001) Sturm-Liouville Problems with Finite Spectrum. Journal of Mathematical Analysis and Applications, 263, 748-762.
http://dx.doi.org/10.1006/jmaa.2001.7661

[21]   Kong, Q., Wu, H. and Zettl, A. (2004) Multiplicity of Sturm-Liouville Eigenvalues. Journal of Computational and Applied Mathematics, 171, 291-309.
http://dx.doi.org/10.1016/j.cam.2004.01.036

[22]   Wang, Z. and Wu, H. (2005) Equality of Multiplicities of a Sturm-Liouville Eigenvalue. Journal of Mathematical Analysis and Applications, 306, 540-547.
http://dx.doi.org/10.1016/j.jmaa.2004.10.041

[23]   Shi, D. and Huang, Z. (2010) Relationship of Multiplicities of a High-Order Ordinary Differential Operator Eigenvalue. Acta Mathematica Sinica, Chinese Series, 53, 763-772.

[24]   Wang, Z. and Wu, H. (2009) Sturm-Liouville Problems with Limit-Circle End Points. Pacific Journal of Applied Mathematics, 1, 421-447.

 
 
Top