Chernoff approximations as a method for finding the resolvent of a linear operator and solving a linear ODE with variable coefficients
Ivan D. Remizov
TL;DR
The paper develops a rigorous Chernoff-approximation framework to compute the resolvent of linear operators with variable coefficients by representing it as a Laplace transform of Chernoff approximations to the associated semigroup. It provides a direct Feynman-type integral representation for the resolvent and for solutions of second-order linear ODEs with variable coefficients, under precise regularity and positivity conditions. A key contribution is the translation-based Chernoff scheme, which yields explicit convergence rates and error bounds for both the resolvent and ODE solutions, and highlights computational advantages such as pointwise evaluation and parallelizability. These results extend the Chernoff machinery from semigroup evaluation to resolvent computation and ODE solution representations, offering new analytic and numerical tools for variable-coefficient problems.
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.