On the solutions of a class of linear selfadjoint differential equations

Abstract:

Let $L$ be a linear selfadjoint ordinary differential operator with coefficients which are real and sufficiently regular on $( - \infty ,\infty )$. Let ${A^ + }({A^ - })$ denote the subspace of the solution space of $Ly = 0$ such that $y \in {A^ + }(y \in {A^ - })$ iff ${D^k}y \in {L^2}[0,\infty )({D^k}y \in {L^2}( - \infty ,0])$ for $k = 0,1, \ldots ,m$ where $2m$ is the order of $L$. A sufficient condition is given for the solution space of $Ly = 0$ to be the direct sum of ${A^ + }$ and ${A^ - }$. This condition which concerns the coefficients of $L$ reduces to a necessary and sufficient condition when these coefficients are constant. In the case of periodic coefficients this condition implies the existence of an exponential dichotomy of the solution space of $Ly = 0$.