Introduction

The theory of pseudo-differential operators (abbreviated ψDOp) shows the essential behavior of many operators appears most clearly when applied to rapidly oscillating functions, studying the behavior as the frequency tends to infinity. This provides a more flexible and transparent approach to our “finite-rank” problems than the method of integral operators of Chapter 7. It is also often simpler, for simple problems — but for this reason, we are encouraged to attack harder problems. (If you are not familiar with ψDOp, don't worry; our argument is direct. Here, the intention is only to show why we don't cite results from this theory.)

We have a ψDOp which, by hypothesis — a hypothesis we hope to contradict — has finite rank. A ψDOp of finite rank is trivial: its symbol is identically zero. So all we have to do is compute the symbol. But it must be computed in some detail. Often the (apparent) principal and subprincipal symbols both vanish identically and uselessly — we must go to the third stage, sometimes further. This is not surprising since we start with second order operators, so the coefficient of the zero-order part only begins to play a role at the third stage. We also deal with quite degenerate problems; after all, we expect to find a contradiction. The theory of ψDOp serves only as inspiration, since it ordinarily computes explicitly only the principal symbol.