We consider the Cauchy problem for a Schrdinger equation whose Hamiltonian is the difference of the operator of multiplication by the potential and the operator of taking the second derivative. Here the potential is a real differentiable function of a real variable such that this function and its derivative are bounded. This equation has been studied since the advent of quantum mechanics and is still a good model case for various methods of solving partial differential equations. We find solutions of the Cauchy problem in the form of quasi-Feynman formulae by using Remizov’s theorem. Quasi-Feynman formulae are relatives of Feynman formulae containing multiple integrals of infinite multiplicity. Their proof is easier than that of Feynman formulae but they give longer expressions for the solutions. We provide detailed proofs of all theorems and deliberately restrict the spectrum of our results to the domain of classical mathematical analysis and elements of real analysis trying to avoid general methods of functional analysis. As a result, the paper is long but accessible to readers who are not experts in the field of functional analysis.

我们考虑一个薛定inger方程的柯西问题，该方程的哈密顿量是乘势乘以二阶导数和算子的差。在此，势是实变量的实微分函数，因此该函数及其导数是有界的。自从量子力学问世以来，已经对该方程进行了研究，并且对于解决偏微分方程的各种方法仍然是一个很好的模型案例。通过使用雷米佐夫定理，我们以准费曼公式的形式找到了柯西问题的解决方案。拟Feynman公式是包含多重多重性无限积分的Feynman公式的相对形式。他们的证明比费曼公式更容易，但是它们给出的解更长。我们提供所有定理的详细证明，并有意将结果的范围限制在经典数学分析和实际分析的范围之内，从而避免使用泛函分析的一般方法。结果，该论文篇幅较长，但对于那些不是功能分析领域专家的读者来说却可以访问。