Chernoff approximations as a method for finding the resolvent of a linear operator and solving a linear ODE with variable coefficients

TL;DR

This paper introduces a theorem applying Chernoff approximation to find the resolvent of linear operators, enabling new solutions for variable coefficient ODEs with convergence rate estimates.

Contribution

It proves a theorem linking Chernoff approximations to the resolvent of linear operators and demonstrates this on second-order differential operators with variable coefficients.

Findings

Convergence of Chernoff approximations to the resolvent of the generator.

New representation of solutions for nonhomogeneous second-order linear ODEs.

Rate of convergence estimate for Chernoff approximations based on shift operators.

Abstract

The Chernoff approximation method is a powerful and flexible tool of functional analysis, which allows in many cases to express exp(tL) in terms of variable coefficients of a linear differential operator L. In this paper, we prove a theorem that allows us to apply this method to find the resolvent of L. Our theorem states that the Laplace transforms of Chernoff approximations of a $C_0$-semigroup converge to the resolvent of the generator of this semigroup. We demonstrate the proposed method on a second-order differential operator with variable coefficients. As a consequence, we obtain a new representation of the solution of a nonhomogeneous linear ordinary differential equation of the second order in terms of functions that are coefficients of this equation, playing the role of parameters of the problem. For the Chernoff function, based on the shift operator, we give an estimate for the rate of convergence of approximations to the solution.