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Dependence of the \(n\)th Sturm-Liouville eigenvalue on the problem. (English) Zbl 0932.34081

A selfadjoint regular Sturm-Liouville problem with positive leading coefficient and weight function, i.e. \( -(py')'+q(x)y=\lambda\text{wy}\), \(x\in(a,b)\), \[ (A|B)\left(\begin{matrix} y(a)\\ (py')(a)\\ y(b) \\ (py')(b)\end{matrix} \right)=0 \] is considered. Here, \((A|B)\) stands for the set of \((2\times 4)\)-matrices over \(\mathbb{C}\) with rank 2. It was shown by W. N. Everitt, M. Möller and A. Zettl [in: C. Bandle (ed.) et al., General inequalities 7. Proceedings. Basel: Birkhäuser. ISNM 123, 145-150 (1997; Zbl 0886.34023)] that in general the eigenvalues of the problem do not depend on the problem continuously. In particular, the lowest eigenvalue has an infinite jump when the Sturm-Liouville equation is fixed and the boundary conditions of the problem approach the Dirichlet boundary conditions in a certain way. The authors study the discontinuity set for an eigenvalue of the problem.

MSC:

34L05 General spectral theory of ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B24 Sturm-Liouville theory

Citations:

Zbl 0886.34023
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Full Text: DOI

References:

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