The upper and lower bounds on non-real eigenvalues of indefinite Sturm-Liouville problems (Q2789854)

The regular indefinite Sturm-Liouville eigenvalue problem NEWLINENEWLINE\[NEWLINE - (p y')' + q y = \lambda w y NEWLINE\]NEWLINE NEWLINEon a bounded interval \([a, b]\) subject to the boundary conditions NEWLINENEWLINE\[NEWLINE y (a) \cos \theta_1 - p y' (a) \sin \theta_1 = 0 \quad \text{and} \quad y (b) \cos \theta_2 - p y' (b) \sin \theta_2 = 0 NEWLINE\]NEWLINE NEWLINEis considered, where \(p, q\) and \(w\) are real-valued functions satisfying \(p (x) > 0\) and \(w (x) \neq 0\) almost everywhere on \([a, b]\) as well as \(\frac{1}{p}, q, w \in L^1 (a, b)\). Moreover, \(w\) is assumed to have (at least one) sign change. The paper aims at providing upper and lower bounds for the nonreal eigenvalues of the problem under consideration. The authors provide upper bounds for the real and imaginary parts of the eigenvalues in terms of properties of the coefficients \(p, q\) and \(w\). Moreover, they prove a lower bound for the absolute values of the eigenvalues in terms of the size of the smallest positive or (in absolute value) smallest negative eigenvalue of the corresponding definite problem which arises from replacing \(w\) by \(|w|\). Also a more explicit version of this lower estimate is derived. In addition, the authors provide examples including the case that \(w\) is negative precisely on a Cantor-type set.