In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients that do not change over time, this general equation is solved in the paper. We construct a family of translation operators in the space of square integrable functions and then use methods of functional analysis based on Chernoff product formula to prove that this family approximates the solution-giving semigroup. This leads us to some formulas that express the solution for Cauchy problem in terms of initial condition and coefficients of the equations studied.

中文翻译：

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表示动量和位置薛定ding方程的柯西问题解的公式

在本文中，我们推导了两个公式，分别表示两个Schrödinger方程的Cauchy问题解：具有多项式势的一维动量空间方程和具有局部平方可积势的多维位置空间方程。第一个方程是具有任意高阶导数和不随时间变化的可变系数的演化方程的常系数特例，该通用方程在本文中得到了解决。我们在平方可积函数的空间中构造了一个翻译算子族，然后使用基于Chernoff乘积公式的泛函分析方法来证明该族近似于给定半群。