STURM–LIOUVILLE PROBLEMS WITH BOUNDARY CONDITIONS RATIONALLY DEPENDENT ON THE EIGENPARAMETER. I

Abstract

Abstract We consider the Sturm–Liouville equation $$ -y''+qy=\lambda y\quad\text{on }[0,1], $$ subject to the boundary conditions $$ y(0)\cos\alpha=y'(0)\sin\alpha,\quad\alpha\in[0,\pi), $$ and $$\frac{y'}{y}(1)=a\lambda+b-\sum_{k=1}^N\frac{b_k}{\lambda-c_k}. $$ Topics treated include existence and asymptotics of eigenvalues, oscillation of eigenfunctions, and transformations between such problems. AMS 2000 Mathematics subject classification: Primary 34B24; 34L20