  In this paper, we study the geometry associated with Schroedinger 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
Schroedinger 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.
