The uncertainty principle lemma under gravity and the discrete spectrum of Schr\"odinger operators

TL;DR

This paper generalizes the uncertainty principle lemma from Euclidean spaces to large classes of complete noncompact manifolds, providing criteria for the discreteness of the spectrum of Schrödinger operators.

Contribution

It extends the uncertainty principle lemma to non-Euclidean manifolds and establishes new criteria for the discrete spectrum of Schrödinger operators.

Findings

Generalized the lemma to hyperbolic and other noncompact manifolds

Provided criteria for the finiteness of the discrete spectrum

Connected geometric properties with spectral behavior

Abstract

The uncertainty principle lemma for the Laplacian on Euclidean spaces shows the borderline-behavior of a potential for the following question : whether the Schr\"odinger operator has a finite or infinite number of the discrete pectrum. In this paper, we will give a generalization of this lemma on Euclidean spaces to that on large classes of complete noncompact manifolds. Replacing Euclidean spaces by some specific classes of complete noncompact manifolds, including hyperbolic spaces, we also establish some criterions for the above-type question.