Generalized Gelfand-Yaglom Formula for a Discretized Quantum Mechanic System

TL;DR

This paper extends the Gelfand-Yaglom formula to discretized quantum systems with Hamiltonian dynamics, providing a new approach for calculating determinants via lattice regularization and analyzing convergence.

Contribution

It introduces a generalized Gelfand-Yaglom formula applicable to discrete and continuous Hamiltonian systems with boundary conditions, including a lattice regularization method.

Findings

Derived a generalized Gelfand-Yaglom formula for Hamiltonian systems

Analyzed convergence of discretized Hamilton-Jacobi operators

Proposed a lattice regularization for determinants

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.