Investigation of the Asymptotics of the Eigenvalues of a Second-Order Quasidifferential Boundary Value Problem

Abstract

We construct the asymptotics of the eigenvalues for a quasidifferential Sturm–Liouville boundary value problem on eigenvalues and eigenfunctions considered on a segment \(J = [a,b]\), with the boundary conditions of type I on the left and right, that is, for a problem of the form (in the explicit notation) \({{p}_{{22}}}(t)\left( {{{p}_{{11}}}(t)\left( {{{p}_{{00}}}(t)x(t)} \right){\kern 1pt} '\; + {{p}_{{10}}}(t)\left( {{{p}_{{00}}}(t)x(t)} \right)} \right){\kern 1pt} '\; + {{p}_{{21}}}(t)\left( {{{p}_{{11}}}(t)\left( {{{p}_{{00}}}(t)x(t)} \right){\kern 1pt} '\; + {{p}_{{10}}}(t)\left( {{{p}_{{00}}}(t)x(t)} \right)} \right)\) \( + \;{{p}_{{20}}}(t)\left( {{{p}_{{00}}}(t)x(t)} \right) = - \lambda \left( {{{p}_{{00}}}(t)x(t)} \right)\;\;(t \in J = [a,b]),\) \({{p}_{{00}}}(a)x(a) = {{p}_{{00}}}(b)x(b) = 0.\) The requirements for smoothness of the coefficients (that is, functions \({{p}_{{ik}}}( \cdot ):J \to \mathbb{R}\), \(k \in 0:i\), \(i \in 0:2\)) in the equation are minimal, namely, these are as follows: the functions \({{p}_{{ik}}}( \cdot ):J \to \mathbb{R}\) are such that the functions \({{p}_{{00}}}( \cdot )\) and \({{p}_{{22}}}( \cdot )\) are measurable, nonnegative, almost every finite, and almost everywhere nonzero and the functions \({{p}_{{11}}}( \cdot )\) and \({{p}_{{21}}}( \cdot )\) also are nonnegative on the segment \(J,\) and, in addition, the functions \({{p}_{{11}}}( \cdot )\) and \({{p}_{{22}}}( \cdot )\) are essentially bounded on \(J,\) the functions \(\frac{1}{{{{p}_{{11}}}( \cdot )}},\;\;\frac{{{{p}_{{10}}}( \cdot )}}{{{{p}_{{11}}}( \cdot )}},\;\;\frac{{{{p}_{{20}}}( \cdot )}}{{{{p}_{{22}}}( \cdot )}},\;\;\frac{{{{p}_{{21}}}( \cdot )}}{{{{p}_{{22}}}( \cdot )}},\;\;\frac{1}{{\min \{ {{p}_{{11}}}(t){{p}_{{22}}}(t),1\} }}\) are summable on the segment \(J.\) The function \({{p}_{{20}}}( \cdot )\) acts as a potential. It is proved that under the condition of nonoscillation of a homogeneous quasidifferential equation of the second order on \(J\), the asymptotics of the eigenvalues of the boundary value problem under consideration has the form \({{\lambda }_{k}} = {{(\pi k)}^{2}}\left( {D + O({\text{1/}}{{k}^{2}})} \right)\) as \(k \to \infty ,\) where \(D\) is a real positive constant defined in some way.