Geometric analysis on generalized Hermite operators

In this paper, we shall, by using Hamiltonian and Lagrangian formalism study the generalized Hermite operator: L=−∑j=1n∂2∂xj2+∑j,k=1nbjkxjxk. Given two points x0 and x in Rn, we count the number of “geodesics” connecting these two points. Here geodesics are defined as the projection of solutions of the Hamiltonian system onto the x-space. Then we construct the action function. Using the famous Van Vleckʼs formula, one may construct the heat kernel for the operator ∂∂t+L.