A hierarchy of Sturm–Liouville problems

Abstract

SUMMARYSturm-Liouville equations will be considered where the boundary conditions depend rationally on the eigenvalue parameter. Such problems apply to a variety of engineering situations, for example to the stability of rotating axles. Classes of these problems will be isolated with a rather rich spectral structure, for example oscillation, comparison and completeness properties analogous to those of the 'usual' Sturm-Liouville problem which has constant boundary conditions. In fact it will be shown how these classes can be converted into each other, and into the 'usual' Sturm-Liouville problem, by means of transformations preserving all but ÿnitely many eigenvalues.

Elliptic boundary value problems with λ-dependent boundary conditions

“…For ordinary second order differential operators in L 2 (I), I ⊂ R, and scalar rational Nevanlinna functions in the boundary condition a solution operator of similar form in L 2 (I) ⊕ C m as in the next result can be found in [12], see also [13,14]. in L 2 (∂Ω) m .…”

“…Theorem 4.6 and Corollary 4.7 for the λ-linear problem. We point out that an analogous selfadjoint solution operator in L 2 (I) ⊕ C m of a Sturm-Liouville problem on a bounded interval I ⊂ R with a scalar variant of (1.5) in the boundary condition was constructed in [12].…”

Linearizations of a class of elliptic boundary value problems

“…A special case of Theorem 4.6 below was announced in [4]. For ordinary second order differential operators in L 2 (I), I ⊂ R, and scalar rational Nevanlinna functions in the boundary condition a solution operator of similar form in L 2 (I) ⊕ C m as in the next result can be found in [10], see also [11,12].…”

“…Theorem 4.6 and Corollary 4.7 for the λ-linear problem. We point out that an analogous selfadjoint solution operator in L 2 (I)⊕C m of a Sturm-Liouville problem on a bounded interval I ⊂ R with a scalar variant of (1.5) in the boundary condition was constructed in [10]. The paper is organized as follows.…”

Inverse eigenvalue problems for Sturm–Liouville equations with spectral parameter linearly contained in one of the boundary conditions

“…Numerical techniques are discussed in [22]. In [4,9] so-called 'almost isospectral' transformations (i.e., transformations preserving all but finitely many eigenvalues) are studied and using these transformations many direct and inverse results for problems with the spectral parameter in one of the boundary conditions are derived from those for classical Sturm-Liouville problems.…”