Scattering Theory

Abstract

Scattering theory studies the comparison between evolution obeying "free dynamics" and evolution obeying some "perturbed dynamics". The asymptotic nature of free and perturbed evolution are compared to determine properties of the perturbation. A brief introduction to scattering theory and inverse scattering problems is given in Chapter 3, after covering some relevant analysis concepts and the construction of Laplacian and Dirichlet to Neumann operators in Chapters 1 and 2. The spectral duality result of [EP95] is as follows. Theorem ([EP95], Main Theorem). The following are equivalent: 1. - D has an M-fold degenerate eigenvalue k2 0 2. As k " k0, exactly M eigenphases j(k) of SD(k) converge to from below. In Chapter 4 this result is explained both mathematically and intuitively, and the signi cance of the form of the second equivalent statement is examined. Possible generalisations of the main theorem of [EP95] to potential scattering are then investigated in Chapter 5. The Cayley transform generalisation, as explained in Section 5.3, suggests studying an object which is given the name transmission operator. The transmission operator displays qualities similar to the scattering matrix and so is investigated further in Chapter 6. Many results about the spectrum of the transmission operator are proven under certain physically reasonable conditions on the considered potential for scattering.