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 (oo, oo). Let A + (A -) denote the subspace of the solution space of Ly=O such that y E A+ (y E A-) iff Dky eL2[0, oo) (Dky E L2(-oo, 0]) for k=0, 1, . . ., 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.