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/

Global: Zendy Plus annual

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

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

An important result by Agmon implies that an eigenfunction of aSchr\"{o}dinger operator in $\mathbb{R}^n$ with eigenvalue $E$ below the bottomof the essential spectrum decays exponentially if the associated classicallyallowed region $\{x \in \mathbb{R}^n~:~ V(x) \leq E \}$ is compact. We extendthis result to a class of Schr\"{o}dinger operators with eigenvalues, for whichthe classically allowed region is not necessarily compactly supported: We showthat integrability of the characteristic function of the classically allowedregion with respect to an increasing weight function of bounded logarithmicderivative leads to $L^2$-decay of the eigenfunction with respect to the sameweight. Here, the decay is measured in the Agmon metric, which takes intoaccount anisotropies of the potential. In particular, for a power law (or,respectively, exponential) weight, our main result implies that power law (or,respectively, exponential) decay of "the size of the classically allowedregion" allows to conclude power law (or, respectively, exponential) decay, inthe Agmon metric, of the eigenfunction.

Agmon-type decay of eigenfunctions for a class of Schr\"{o}dinger operators with non-compact classically allowed region