Oscillation results for Sturm--Liouville problems with an indefinite weight function
Let \(q\) and \(r\) be real-valued functions in \(L^1(0,l)\) with \(| r| >0\) a.e.\ and consider the Sturm-Liouville problem \[ -y''(x) + q(x) y(x) = \lambda r(x) y(x) \qquad\text{ on}\qquad (0,l), \qquad y'(0) = y'(l) = 0. \] When \(r>0\) a.e., denote the eigenvalues by \(\lambda_1<\lambda_2<\dots\) and the corresponding eigenfunctions by \(y_n\). Then by the classical Sturm oscillation theory, the number of zeros (or the oscillation count) of \(y_n\) in \((0,l)\) is equal to \(n-1\). When the weight \(r\) is indefinite (i.e., takes both signs on sets of positive measure), the spectrum consists of two sequences of real eigenvalues tending to \(-\infty\) and \(+\infty\), and, possibly, finitely many pairs of complex conjugate nonreal eigenvalues; moreover, a finite number of eigenvalues might be nonsimple. The main aim of the paper is to establish a formula for the oscillating count \(\omega_n\) of the eigenfunction \(y_n\) corresponding to the \(n\)th positive eigenvalue \(\lambda_n\) of the above Sturm-Liouville problem in the indefinite weight case. The oscillation count \(\omega_n\) is shown to depend on the algebraic multiplicities of the eigenvalues and their signatures. The proof exploits the Titchmarsh-Weyl \(m\)-function and the Prüfer angle method.
scientific article; zbMATH DE number 2085586