Geometric analysis on generalized Hermite operators (Q719786)

In this article, the generalized Hermite operator defined as \(L=-\sum_{j=1}^n\frac{\partial^2}{\partial x_j^2}+\sum_{j,k=1}^nb_{jk}x_jx_k\) is studied by using a Hamiltonian and Lagrangian formalism. For two points situated in the \(n\)-dimensional space the number of geodesics connecting these two points is counted, where geodesics are projections of solutions of the Hamiltonian system onto the \(x\)-space. Note that, without loss of generality, the two-dimensional case is considered. Based on these results, the authors construct the action function distinguishing eight different cases. Finally, the heat kernel for the operator \(\frac{\partial}{\partial t}+L\) is constructed using Van Vleck's formula.