Note on differential operators with a purely continuous spectrum
Fadwa Odeh · Proceedings of the American Mathematical Society · 1965
In [l ], Kreith gave an example of a Sturm-Liouville operator with positive coefficients,which has a purely continuous spectrum.The novelty of the example lies in the relatively weak assumptions on the potential q.Thus, in the case p = r=l, one need not assume that q is integrable at infinity -compare [2, Chapter 9, Problem 4]-but it is sufficient to assume q to be monotonically decreasing.In this note a similar theorem is given which holds in any number of dimensions.The proof, which applies to Kreith's case also, shows that the nonexistence of eigenfunctions may be ascribed to two different reasons depending on the asymptotic behavior of q(x).In one simple case it is due to the boundary condition while in the other, and more important, case it is a consequence of the behavior of q at infinity.For simplicity the proof is restricted to the case of Schroedinger's equation in three dimensions, defined in the exterior X of a closed smooth surface T. Hence, we consider the eigenvalue problem, (la) Lu = -A« + qu = Am, subject to the boundary conditions