Boundary Problems of Sturmian Type on an Infinite Interval

Abstract

Two types of Sturmian boundary problems on $( - \infty ,\infty )$, with boundary conditions specifying that the proper functions are of integrable square, are considered with the aid of the principal solutions of the involved differential equation at $\infty $ and $ - \infty $, and the reduction of the given problem to an associated Sturmian problem on a finite interval. For problems of the first type it follows from classical Sturmian theory that the totality of proper values may be ordered as a simple sequence, with the proper function corresponding to the jth proper value possessing exactly $j - 1$ zeros on $( - \infty ,\infty )$. Problems of the second type involve “turning points,” and in this instance Sturmian comparison theorems are used to establish the existence of a sequence of sets of proper values such that the proper functions corresponding to parameter values in the jth set possess exactly $j - 1$ zeros on $( - \infty ,\infty )$.