Asymptotic formulas for eigenvalues of the indefinite Sturm-Liouville problem with finite number of turn points (Q1866876)

The author considers a spectral problem generated by the Sturm-Liouville equation \[ -y''+[\lambda^2f(x)+q(x)]y=0 \] on a finite segment \([A,B]\) with separated boundary conditions. It is assumed that \(f\) and \(g\) are functions smooth enough and \(f(x)\) has a finite number of simple turning points. The main result of the paper is as follows. The eigenvalues of the problem can be set in several series. Each series has an asymptotics with a prescribed number of terms (the number of the asymptotics terms is determined only by the smoothness of \(f\) and \(q\)). For each series the first three terms of asymptotics are determined in the explicit form.