The radial curvature of an end that makes eigenvalues vanish in the essential spectrum II

作者：Kumura Hironori^*

来源：Bulletin of the London Mathematical Society, 2011, 43: 985-1003.

DOI：10.1112/blms/bdr039

In quantum mechanics, the non-existence of L(2)-eigenfunction of the Schrodinger operator implies that particles recede to infinity as time tends to infinity or minus infinity in the sense of time average, that is, the non-existence of L(2)-eigenfunction expresses the scattering state. In this paper, we shall consider an analogue of this fact on Riemannian manifolds with a specific end: we assume that the radial curvature of one of its ends tends to zero at infinity with some quadratic order. Then, we shall derive growth estimates of solutions to the eigenvalue equation of the Laplacian and show the absence of eigenvalues. Compared to the case that the radial curvature of one of its ends tends to a negative constant at infinity, this case that the radial curvature of one of its ends tends to zero at infinity is much more difficult. The consequence of this paper will have an important role in the proof of the limiting absorption principle and absolute continuity.

• 出版日期2011-10