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.

翻译：切尔诺夫逼近方法是泛函分析中一种强大而灵活的工具，在许多情况下能够通过线性微分算子L的变系数来表达exp(tL)。本文证明了一个定理，使得我们可以应用该方法来求解L的预解式。我们的定理表明：$C_0$-半群的切尔诺夫逼近的拉普拉斯变换收敛于该半群生成元的预解式。我们以变系数二阶微分算子为例演示了所提出的方法。作为推论，我们得到了二阶非齐次线性常微分方程解的一种新表示形式，该表示以方程系数函数（作为问题参数）直接表达解。对于基于移位算子的切尔诺夫函数，我们给出了逼近解收敛速度的估计。