In this paper, we analyze conditions under which the spectrum of Sturm–Liouville operator on a certain smooth curve is localized near countable number rays. case where potential piecewise analytical, an asymptotic eigenvalues found for each series corresponding ray. The result obtained allows one to generalize well-known formula distribution function stated by Davies in finite localization

In this paper, we handle an impulsive Sturm–Liouville equation with complex potential on the semi axis. The objective of work is to examine some spectral properties equation. By help a transfer matrix B, obtain Jost solution problem. Furthermore, using solution, find Green function and resolvent operator Finally, consider two unperturbated operators. We eigenvalues singularities these problems.

We revisit the Krein-von Neumann extension in case where underlying symmetric operator is strictly positive and apply this to derive explicit form of for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, boundary conditions minimal are explicitly expressed terms generalized values adapted (possible) singularity structure coefficients ne...

The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient λq(s), s ∈ [s0,∞) is investigated, where λ ∈ R and q(s) is a nondecreasing step function tending to ∞ as s → ∞. Let S denote the set of those λ’s for which the corresponding differential equation has a solution not tending to 0. It is proved that S is an additive group. Four examples are given with S = {0}, S =...

Since it became clear that the band structure of the spectrum of periodic Sturm–Liouville operators t = − (d/dr2) + q(r) does not survive a spherically symmetric extension to Schrödinger operators T = −1+V with V (x) = q(|x|) for x ∈ R , d ∈ N\ {1}, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential s...