Spectral estimations for the Laplace operator of the discrete Heisenberg group
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Publication:5947528
DOI10.1023/A:1012477708874zbMATH Open0982.22006arXivmath/9905133MaRDI QIDQ5947528
Author name not available (Why is that?)
Publication date: 16 October 2001
Published in: (Search for Journal in Brave)
Abstract: Let H be the discrete 3-dimensional Heisenberg group with the standard generators x, y, z. The element Delta of the group algebra for H of the form Delta= (x+x^{-1}+y+y^{-1})/4 is called the Laplace operator. This operator can also be defined as transition operator for random walk on the group. The spectrum of Delta in the regular representation of H is the interval [-1,1]. Let E(A), where A is a subset of [-1,1], be a family of spectral projectors for Delta and m(A)=(E(A)e, e) be the corresponding spectral measure. Here e is the characteristic function of the unit element of the group H. We estimate the value m([-1,-1+t] cup [1-t,1]) when t tends to 0. More precisely we prove the inequality m([-1,-1+t] cup [1-t,1]) > const t^{2+alpha} for any positive alpha.
Full work available at URL: https://arxiv.org/abs/math/9905133
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