English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

In quantum mechanics, the non‐existence of  L  2 ‐eigenfunction of the Schrödinger 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.

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