HERMITE OPERATOR AND SUBELLIPTIC OPERATORS IN Hn

In this note, we compute the fundamental solution for the Hermite operator with singularity at arbitrary point y ∈ R. We also apply this result to obtain the fundamental solution for the sub-Laplacian Lα = − ∑n j=1(X 2 j +X 2 j+n)− iαT on the Heisenberg group. In this note, we first derive the fundamental solution of the Hermite operator n ∑ j=1 ( λjx 2 j − ∂2 ∂xj ) in Rn, i.e., we are looking for a distribution K(x,y) such that (1) n ∑ j=1 ( λjx 2 j − ∂2 ∂xj ) K(x,y) = δ(x− y). Hermite operator has been studied by mathematicians and physicists for a few generation (see e.g., [1], [11]). Its fundamental solution and the heat kernel have been known for a long time. The method in this paper is apparently new. Our separate the derivation in two parts, we compute the fundamental solution with singularity at the origin in section 1 and with singularity at any point y in section 2. The reason of dividing is that we use different method to sum up the infinite series involved. Then in section 3, we will apply results from section 1 to obtain the fundamental soultion for the sub-Laplacian on the Heisenberg group Hn: Lα = n ∑ j=1 (Xj +X 2 j+n)− iαT = − 2n ∑ j=1 ∂2 ∂xj − 1 4  n ∑ j=1 aj (x 2 j + x 2 j+n  ∂2 ∂t2 + 1 2 n ∑ j=1 aj ( xj ∂ ∂xj+n − xj+n ∂ ∂xj ) ∂ ∂t − iα ∂ ∂t , with α / ∈ Eα =  n ∑ j=1 (2kj + 1)aj , (k1, . . . , kn) ∈ (Z+)  and a1, · · · , an > 0. Here Xj = ∂ ∂xj − 1 2 ajxj+n ∂ ∂t , Xj+n = ∂ ∂xj+n + 1 2 ajxj ∂ ∂t , j = 1, . . . , n and T = ∂ ∂t form a basis of the Lie algebra of the group (see section 3). The research is partially supported by a William Fulbright Research Grant and a Competitive Research Grant at Georgetown University.