Spectral estimations for the Laplace operator of the discrete Heisenberg group (Q5947528)

Let \(H\) be the 3-dimensional Heisenberg group and \(x, y, z\) its standard generators. The element \(\Delta=\frac 14(x+x^{-1}+y+y^{-1})\) of the group algebra of \(H\) is called the Laplace operator. It is well-known that the spectrum of \(\Delta\) in the regular representation of \(H\) is the interval \([-1,1]\). The authors study the spectral measure \(m(A)=(E(A)e,e)\), where \(e\) is the characteristic function of the unit element of \(H\). They prove the inequality \[ m([-1,-1+t]\cup[1-t,1])>\text{const.} t^{2+\alpha}. \] {}.