Long Range Scattering Theory

Summary

In Chapter XIV we studied the spectral properties of short range perturbations of fairly general differential operators P_o(D) with constant coefficients in ℝⁿ . The short range condition imposed on the perturbing differential operator V(x, D) was designed to allow a study of the resolvent of the perturbed operator by compactness arguments. Roughly speaking it required that the coefficients of V decrease as fast as an integrable function of |x| and that for x frozen at x_o the operator V(x_o,D) is compact with respect to P₀(D) in the sense that $\tilde V\left( {x_0 ,\xi } \right)/\tilde P_0 (\xi ) \to 0$ . In this section we shall relax the hypotheses on V (x, ξ) both when x is large and when ξ is large. However, to bring out the main points as simply as possible we shall assume that P₀ is elliptic, of order m. Our first condition on V is then that V is of order m, that the coefficients of the terms of order m are continuous and that P₀ (D) + V (x, D) is also elliptic. Furthermore, we assume that V = V^s(x, ξ)+ V^L(x,ξ) where as in Chapter XIV the coefficients of the short range term V^s decrease as fast as an integrable function of |x|, while the long range term V^L has the bound

$$ \left| {D_x^\alpha V^L \left( {x,\xi } \right)} \right| \leqq C(1 + \left| x \right|)^{ - \left| \alpha \right| - \varepsilon } (1 + \left| \xi \right|)^m ,\quad \left| \alpha \right| \leqq 2, $$

((30.1))

for some ε>0. A more precise discussion of these conditions is given in Section 30.1. A sufficient condition for (30.1) is of course that V^L is homogeneous in x of degree -ε for |x|>1; the Coulomb potential for