Continuous spectra of second-order differential operators
Don B Hinton · Pacific Journal of Mathematics · 1970
We consider the differential operator l(y) = y" + qy, where q is a positive, continuously differentiable function defined on a ray [α, oo).The operator I determines, with appropriate restrictions, self-ad joint operators defined in the hilbert space J2^[α, oo) of quadratically summable, complexvalued functions on [a, oo).In this note, we prove that if L is such a selfadjoint operator, then the conditions q(t)->oo and q'(t)q(t)~l l2 -*0 as t -> oo are sufficient for the continuous spectrum C(L) of L to cover the entire real axis.Similar results are well-known; however, monotonicity conditions on q and q r are usually required.For example, in [1], p. 116, it is proved that if q tends monotonically to co as t -> co, preserving the direction of convexity for large ί, then the condition q'(t)q(t)~l β -+Q as £-• oo is sufficient to imply C(L) = (-oo, co) for every self-adjoint operator L determined by I. THEOREM. // q(t) -• oo as t -> oo, q\t)q{t)~l β-* 0 as t -* oo,