Spectral estimations for Laplace operator for the discrete Heisenberg group

Details Spectral estimations for Laplace operator for the discrete Heisenberg group Spectral estimations for Laplace operator for the discrete Heisenberg group

1999-05-20

arxiv.org/abs/math/9905133v1

Mathematics

Spectral Theory

8 pages, AMSTEX(+amsppt) + 1 page: 2 figures in one Postscript file

Scientific paper

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.

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