Scattering theory for elliptic operators of arbitrary order

Abstract

We prove the main theorems of scattering theory for selfadjoint elliptic partial differential operators of arbitrary order. Under various hypotheses we show that the wave operators exist and are complete, that the intertwining relations hold, and that the invariance principle holds. One of our main hypotheses is that each lower order coefficientq(x) satisfies.

$$(1 + \left| x \right|)^\alpha \int\limits_{\left| {x - y} \right|< a} {\left| {q(y)} \right|dy \in L^p (E^n )}$$

for some α≥0,a>0 and forp≤∞ such that

$$\alpha > 1 - \frac{{2n}}{{(n + 1)p}}$$