Equivalence of fourth order boundary value problems and matrix eigenvalue problems. (English) Zbl 1267.34036

The authors establish equivalent relations between a class of regular self-adjoint fourth-order linear boundary value problems and a class of matrix problems. More precisely, they show that
(i) for any boundary value problem in the first class, there exists a matrix problem in the second class with exactly the same spectrum;
(ii) for any matrix problem in the second class, there exists a boundary value problem in the first class with exactly the same spectrum.
This work is an extension of an earlier work by Kong, Wu, and Zettl in 2001 and that by Kong, Volkmer, and Zettl in 2009 for second-order problems, and is a follow-up of the authors’ work on fourth-order problems.


34B09 Boundary eigenvalue problems for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
15A18 Eigenvalues, singular values, and eigenvectors
Full Text: DOI


[1] Ao, J.J., Sun, J., Zettl, A.: Matrix representations of fourth order boundary value problems. Linear Algebra Appl. (2011, in press) · Zbl 1244.34019
[2] Atkinson F.V.: Discrete and Continuous Boundary Value Problems. Academic Press, New York (1964) · Zbl 0117.05806
[3] Gansterer W.N., Ward R.C., Muller R.P., Goddard W.A.: Computing approximate eigenpairs of symmetric block tridiagonal matrices. SIAM J. Sci. Comput. 25, 65–85 (2003) · Zbl 1038.65031 · doi:10.1137/S1064827501399432
[4] Hao, X., Sun, J., Zettl, A.: Canonical forms of self-adjoint boundary conditions for differential operators of order four. J. Math. Anal. Appl. 387, 1176–1187 (2012) · Zbl 1245.34018
[5] Igelnik B., Simon D.: The eigenvalues of a tridiagonal matrix in biogeography. Appl. Math. Comput. 218, 195–201 (2011) · Zbl 1255.15009 · doi:10.1016/j.amc.2011.05.054
[6] Kong Q., Wu H., Zettl A.: Sturm-Liouville problems with finite spectrum. J. Math. Anal. Appl. 263, 748–762 (2001) · Zbl 1001.34019 · doi:10.1006/jmaa.2001.7661
[7] Kong Q., Volkmer H., Zettl A.: Matrix Representations of Sturm-Liouville problems with finite spectrum. Results Math. 54, 103–116 (2009) · Zbl 1185.34032 · doi:10.1007/s00025-009-0371-3
[8] Kong Q., Zettl A.: The study of Jacobi and cyclic Jacobi matrix eigenvalue problems using Sturm-Liouville theory. Linear Algebra Appl. 434, 1648–1655 (2011) · Zbl 1210.15011 · doi:10.1016/j.laa.2010.04.035
[9] Kong Q., Zettl A.: Inverse Sturm-Liouville problems with finite spectrum. J. Math. Anal. Appl. 386, 1–9 (2012) · Zbl 1232.34023 · doi:10.1016/j.jmaa.2011.06.083
[10] Naimark, M.A.: Linear Differential Operators. Ungar, New York (1968) · Zbl 0227.34020
[11] Wang A., Sun J., Zettl A.: The classification of self-adjoint boundary conditions: separated, coupled, and mixed. J. Funct. Anal. 255, 1554–1573 (2008) · Zbl 1170.34017 · doi:10.1016/j.jfa.2008.05.003
[12] Weidmann, J.: Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics, vol. 1258. Springer, Berlin (1987) · Zbl 0647.47052
[13] Zettl, A.: Sturm-Liouville Theory. Mathematical Surveys and Monographs, vol. 121. American Mathematical Society, Providence (2005) · Zbl 1103.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.