在歐氏空間上薛丁格算子的熱核漸進展開

Abstract

在這篇文章中，我們研究關於在歐氏空間中薛丁格算子的熱核的表現式與它的漸近展開。先從觀察簡單的常係數薛丁格算子在歐氏空間中的行為開始，並引入新的符號空間，由逼近的方法將熱核的表現式寫下來。從而證明在歐氏空間中薛丁格算子的熱核的存在唯一性，並得到其在時間趨近於零時的漸進展開。

In this thesis, we study the expression of the heat kernel of the Schrodinger operator in Euclidean space and its asymptotic expansion. We start by observing the behavior of the constant coefficient Schr'odinger operator in Euclidean space and introduce the new symbol space, and write down the expression of the heat kernel by the approximation method. It's proved the existence and uniqueness of the heat kernel for the Schr'odinger operator in Euclidean space, and the asymptotic expansion of the heat kernel at time is approach to 0.