Fundamental solutions on partial differential operators of second order with application to matrix Riccati equations

摘要

In this paper, we study the geometry associated with Schr?dinger operator via Hamiltonian and Lagrangian formalism. Making use of a multiplier technique, we construct the heat kernel with the coefficient matrices of the operator both diagonal and non-diagonal. For applications, we compute the heat kernel of a Schr?dinger operator with terms of lower order, and obtain a globally closed solution to a matrix Riccati equations as a by-product. Besides, we finally recover and generalise several classical results on some celebrated operators.