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.


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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