The uncertainty principle lemma under gravity and the discrete spectrum of Schrödinger operators

2008/12/26 by Kazuo Akutagawa, Akutagawa, Kazuo, Hironori Kumura +1
Mathematics · #35J10 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP) #math.DG #math.SP #msc:35J10 #msc:58J50

paper · pdf · doi:10.48550/arxiv.0812.4663

19 pages, 1 figure

arxiv created 2009/01/13 · arxiv updated 2009/12/01

Abstract

The uncertainty principle lemma for the Laplacian on Euclidean spaces shows the borderline-behavior of a potential for the following question : whether the Schrödinger 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.

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