一类具多项式系数偏微分方程的Cauchy问题

The Cauchy Problem for a Class of Linear Partial Differential Equations with Polynomial Coefficients

本文应用B.Simon对缓增分布空间的Hermite表示讨论了一类变系数微分方程Cauchy问题解的存在性、唯一性及光滑性,作为本文的一个应用,导出了关于量子力学谐和振子Schrodinger方程的基本解和Mehler公式。

In this paper the existence, uniqueness and smoothness of solutionsto the Cauchy problem for a class of linear partial differential equationswith polynomial cofficients are discussed by using B. Simon's Hermiteexpressions about tempered distribution space. As simple applications,the fundamental solution to the Schrodinger equation u/t = iHu for thequantum mechanical harmonic oscillator and Mehler's famous formulafor the fundamental solution to u/t=-Hu are derived in this paper.