Some differential operators related to the non-isotropic Heisenberg sub-Laplacian

Let L-alpha = -1/2 Sigma (n)(j=1) (Z(j)(Z) over bar (j) + (Z) over bar (j)Z (j)) + i alphaT be the sub-Laplacian on the non-isotropic heisenberg group H-n where Z(j), (Z) over bar (j) for j = 1, 2, ... , n and T are a basis of the Lie algebra h(n). We apply the Laguerre calculus to obtain the fundame ntal solution of the heat kernel exp{-sL(alpha)}, the Schrodinger operator exp{-isL(alpha)} and the operator Delta (lambda,alpha) = -1/2 Sigma (n)(j=1 ) lambda (j) (Z(j)(Z) over bar (j) + (Z) over bar (j)Z(j)) + i alphaT. We a lso discuss some basis properties of the wave operator.