Convergence of quantum systems on grids

We discuss here the convergence of quantum systems on grids embedded in Rd and generalize the earlier results found for scalar-valued potentials to the case of matrix-valued potentials. We also discuss the essential self-adjointness of Schrödinger operators for a large class of matrix potentials and give a Feynman–Kac formula for their associated imaginary time Schrödinger semigroups when the matrix potential is positive and continuous. Furthermore, we establish an operator kernel estimate for the semigroups.