Schrödinger and Heat Operators

Abstract

In the first chapter we introduced the operator d+6, which can be restricted to different subspaces to give the de Rham-Hodge operator and the signature operator. Its square operator A = (d + S) 2 can be expressed as A = -(A 0 + F) by the Weizenbock formula, where Ao is Laplace-Beltrami operator defined in Definition 1.4.2 and F is an ^"(M)-linear operator. Such kind of A does not contain first order covariant derivatives, so we may give it a special name, a Schrodinger operator or an operator of Laplacian type will be named popularly.It is well-known that either d + 6 or A is an elliptic operator. But in this book we will not go into the general theory of the elliptic operators. Instead we confine ourselves to the Schrodinger operators. The most important research for the Schrodinger operator is the Hodge theorem, which can be used to solve the following equationwhere <f> and / are unknown and known functions respectively. Given a Schrodinger operator A, the operator J^ + A is called a heat operator. The central point for the heat operator theory is the existence of a fundamental solution and its applications. In 1951 Milgram-Rosenbloom [24] proposed an approach to prove the Hodge theorem by using the fundamental solution. Moreover after the sixties of twentieth century people found that the fundamental solution can also be used to give a proof of the local index theorem for some geometric operators. So all attention will be paid to the corresponding heat equation instead of the Schrodinger operator equation in this book. We will introduce basic knowledge about the fundamental solution, and use it to prove the Hodge theorem in this chapter and the local index theorem later. For the simplicity of formulations in this chapter we temporarily assume that there are many initial solutions, and leave the proof of this assumption to Chapter 3. As far as the 83 The Index Theorem and the Heat Equation Method Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 08/22/15. For personal use only. SCHRODINGER AND HEAT OPERATORSapproach to the Hodge theorem is concerned, Milgram-Rosenbloom had not shown the existence and some nice properties of the fundamental solution yet. Berger and others[5] had proved the existence. This chapter will supply all the details after consulting [15] and an opinion of Professor Wang Guan-Yin. The most important detail is the existence and regularity of the solutions to the Cauchy initial problem of the heat equation, which were of course needed by the Milgram-Rosenbloom approach.