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Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularity

Nursultanov M., Rozenblum G.

Opuscula Mathematica

2017

Vol. 37, no. 1

109--139

We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously.

Multipoint normal differential operators of first order

Ismailov Z. I.

Opuscula Mathematica

2009

Vol. 29, no. 4

399-414

In this paper we discuss all normal extensions of a minimal operator generated by a linear multipoint differential-operator expression of first order in the Hilbert space of vector-functions on the finite interval in terms of boundary and interior point values. Later on, we investigate the structure of the spectrum, its discreteness and the asymptotic behavior of the eigenvalues at infinity for these extensions.