Oscillation Theory and Eigenvalue Asymptotics for Sturm–Liouville Equations with Transfer Conditions

Abstract

The Sturm–Liouville equation $$-(p{y}^{\prime }{)}^{\prime }+q(x)y=\lambda ry,[-a,0)\cup (0,b]$$ is considered subject to separated boundary conditions $$y(-a)\cos \alpha =(p{y}^{\prime })(-a)\sin \alpha ,y(b)\cos \beta =(p{y}^{\prime })(b)\sin \beta$$ and a point transfer condition $$\begin{array}{r}\left[\begin{array}{c}y(0^{+})\\ (p{y}^{\prime })(0^{+})\end{array}\right]=\mathbf{T}\left[\begin{array}{c}y(0^{-})\\ (p{y}^{\prime })(0^{-})\end{array}\right],\end{array}$$ where $\mathbf{T}$ is a real $2\times 2$ matrix with $det\mathbf{T}>0$ . A formula for the asymptotics of eigenvalues, corresponding to the above problem, is determined in terms of generalized oscillation counts of the corresponding eigenfunctions.