Eigenvalue Properties of Schrödinger Operators

Abstract

In Evans–Lewis [5] and Evans–Lewis–Saitō [6], [7], [8], [9] we have been discussing conditions for the finiteness and for the infiniteness of bound states of Schrödinger-type operators using geometric methods. Here the ideas and results obtained so far are summarized and presented in an expository manner. These bound states correspond to eigenvalues below the essential spectrum of the operator. After basic results are presented, Schrödinger operators of atomic type will be discussed to show how these basic results can be applied to various types of $N$-body Schrödinger operators.