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Sturm-Liouville problems with finite spectrum. (English) Zbl 1001.34019

The authors consider the Sturm-Liouville equation \[ -(py')'+qy=\lambda w y \;\;\text{ on} \;J=(a,b) \;\text{ with} -\infty\leq a<b \leq \infty \] together with boundary conditions. Here, \(\lambda\) is the spectral parameter, \(1/p\), \(q\), \(w\in L^1(J,\mathbb{C})\), where \(L^1(J,\mathbb{C})\) denotes the collection of Lebesgue integrable functions from \(J\) to the complex numbers \(\mathbb{C}\). Choosing \(p\) and \(w\) to be such that \(1/p\) and \(w\) are alternatively zero on consecutive subintervals, the authors prove that for any (integer) \(n\) there exists a Sturm-Liouville problem having exactly \(n\) (finite) eigenvalues.

MSC:

34B24 Sturm-Liouville theory
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
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References:

[1] Atkinson, F. V., Discrete and Continuous Boundary Value Problems (1964), Academic Press: Academic Press New York/London · Zbl 0117.05806
[2] Eastham, M. S.P.; Kong, Q.; Wu, H.; Zettl, A., Inequalities among eigenvalues of Sturm-Liouville problems, J. Inequal. Appl., 3, 25-43 (1999) · Zbl 0927.34017
[3] Everitt, W. N.; Race, D., On necessary and sufficient conditions for the existence of Caratheodory solutions of ordinary differential equations, Quaest. Math., 3, 507-512 (1976) · Zbl 0392.34002
[4] Kong, Q.; Wu, H.; Zettl, A., Dependence of the \(n\) th Sturm-Liouville eigenvalue on the problem, J. Differential Equations, 156, 328-354 (1999) · Zbl 0932.34081
[5] Kong, Q.; Wu, H.; Zettl, A., Dependence of eigenvalues on the problem, Math. Nachr., 188, 173-201 (1997) · Zbl 0888.34017
[6] Kong, Q.; Zettl, A., Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131, 1-19 (1996) · Zbl 0862.34020
[7] Zettl, A., Sturm-Liouville Problems, (Hinton, D.; Schaefer, P., Spectral Theory and Computational Methods of Sturm-Liouville Problems. Spectral Theory and Computational Methods of Sturm-Liouville Problems, Lecture Notes in Pure and Applied Mathematics, 191 (1997), Dekker: Dekker New York), 1-104 · Zbl 0882.34032
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