Spectral estimations for Laplace operator for the discrete Heisenberg group

K. P. Kokhas' ; A. Suvorov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part III, Tome 256 (1999), pp. 129-144

Résumé

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 $$ m([-1,-1+t]\cup [1-t,1])>\mathrm{const}\,t^{2+\alpha} $$ for any positive alpha.