Asymptotics for a class of weighted eigenvalue problems

Abstract

This paper deals with the asymptotic behavior at infinity of the solutions to /(y) = λwy on [a, oo) where / is an nth order ordinary linear differential operator, λ is a nonzero complex number and w is a suitably chosen positive valued continuous functions. As an application the deficiency indices of certain symmetric differential operators in Hubert space are computed. 1* Preliminaries* Throughout the first three sections / will denote an operator of the form, (1.1) S(v) = y + Σ PkV' on [α, oo) , where each of p2, , pn is a continuous complex valued function on [α, oo). In view of the transformation indicated on p. 309 of [2] it results in no great loss of generality to take the coefficient of y~~ to be zero, and in order to simplify the exposition we shall do this. We shall be concerned with the behavior at infinity of the solutions to (1.2) s{y) = Xwy on [α, oo) where λ is a nonzero complex number and w is an appropriate weight (i.e., positive valued continuous function). For a given / we shall consider the weights w indicated by the following definition. £f(a, oo) denotes the Banach space of all complex valued measurable functions which are absolutely Lebesgue integrable on [α, oo). DEFINITION. If / is as in 1.1 the statement that w is an /-admissible weight means that (1) w is differentiate, strictly increasing, and unbounded on (2) each of [w'/w]' and [(w'/wy(l/w)] is continuous on [α, oo) and is in £f{a, oo); and (3) pόlw G £f{a, oo) for j = 2, 3, . , n. For example if s(y)(t) = y{t) ± ty{t) for t ^ 1 and w(t) = P then w will be an /-admissible weight if and only if β > 0 and β > 2(α + 1). We shall demonstrate that when w is an /-admissible weight the solutions of 1.2 have a particularly simple asymptotic behavior and