Generalized Gelfand–Yaglom formula for a discretized quantum mechanic system

Abstract

The Gelfand–Yaglom formula relates the regularized determinant of a differential operator to the solution of an initial value problem. Here, we develop a generalized Gelfand–Yaglom formula for a Hamiltonian system with Lagrangian boundary conditions in the discrete and continuous settings. Later, we analyze the convergence of the discretized Hamilton–Jacobi operator and propose a lattice regularization for the determinant.