Analytic Methods for Inverse Scattering Theory

The purpose of these lectures is to provide basic analytic tools of fixed energy inverse scattering theory. As a model uses we study the inverse scattering problems for time harmonic acoustic and Schrödinger equations. Section 1 describes these two problems. In Section 2 we introduce the Hardy-Littlewood maximal function and define the Sobolev spaces in ℝ n . At the end of this Section we prove an important characterization of W p 1 (ℝ n ) due to P. Hajlasz. In the third Section we prove the continuity of (∆ + k 2)−1 for L p (Ω) to L q (Ω), for 1 ≤ p ≤ 2 ≤ q ≤ ∞ together with an appropriate norm estimate. As a special case p = q = 2 we get S. Agmon’s result that norm of (∆ + k 2)−1 in this case behaves as $$\frac{1}{k}$$ for large k.