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Explicit expressions for Sturm-Liouville operator problems

Published online by Cambridge University Press:  20 January 2009

Lucas Jodar
Lucas Jodar
Affiliation:
Department of Applied Mathematics, Polytechnical University of Valencia, P.O. Box 22. 012, Valencia, Spain
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Throughout this paper H will denote a complex separable Hilbert space and L(H) denotes the algebra of all bounded linear operators on H. If T lies in L(H), its spectrum σ(T) is the set of all complex numbers z such zI–T is not invertible in L(H) and its compression spectrum σcomp(T) is the set of all complex numbers z such that the range (zI-T)(H) is not dense in H ([3, p. 240]). This paper is concerned with the Sturm–Liouville operator problem

where λ is a complex parameter and X(t), Q, Ei, Fi for i = l,2, and t∈[0,a], are bounded operators in L(H). For the scalar case, the classical Sturm-Liouville theory yields a complete solution of the problem, see [4], and [7]. For the finite-dimensional case, second order operator differential equations are important in the theory of damped oscillatory systems and vibrational systems ([2, 6]). Infinite-dimensional differential equations occur frequently in the theory of stochastic processes, the degradation of polymers, infinite ladder network theory in engineering [1, 17], denumerable Markov chains, and moment problems [10, 20]. Sturm-Liouville operator problems have been studied by several authors and with several techniques ([12, 13, 14, 15, 16]).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 2 , June 1987 , pp. 301 - 309
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

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