Date of Award
Spring 2023
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
First Advisor
Schlag, Wilhelm
Abstract
One of the classical questions in the scattering theory of quantum particles is asymptotic completeness for short-range potentials. Short-range potentials are potentials that decay fast enough in all directions, and asymptotic completeness means, in this context, that we can decompose any state into two functions: an asymptotically free one (that behaves as though there was no potential present) and one that lays in the closure of the $L^2$-eigenfunctions of our Hamiltonian.In this thesis, based on two papers, we study the scattering properties of Schrödinger operators with potentials that have short-range decay in some, but not all, directions. For these operators, we use micro-local tools to give similar results to the classic asymptotic completeness. We provide a dynamical description of the interacting states, which are states in the orthogonal complement of the asymptotically free wave functions. In other words, we can show that a similar decomposition as above holds - only we should replace the non-free depiction as the closure of eigenfunction with some dynamical description. This description implies that the interacting states may interact with the potential even at long times (hence the name). Furthermore, when the potential geometry is compatible, in a way described in the body of the work, with a combination of half-spaces, we can improve this description to a purely spatial characterization. In these cases, the interacting states escape a geometric restriction, coming from the potential's geometry, at a sub-linear rate in a suitable sense. The main idea in our proofs is a variation on the Enns method, whose geometric flavor is well suited for the settings considered in this work. A way to phrase the main idea in Enns's original argument is as follows: states localized spatially far from the potential, and have free momentum that points away from it, should be asymptotically free. Enns relied heavily on the RAGE theorem, which ties between the spectral subspaces and the dynamics of the wave functions. In our interpretation, we utilized the argument to extract the dynamic picture rather than getting a spectral one. This new view of Enss' method, combined with the micro-local tools, enables us to get the desired description of the interacting states. The first geometry we consider is for Hamiltonians with potential concentrated near a subspace of $\mathbb{R}^d$. This specific setting allows us to extract the essence of the method and introduce the main ideas in our approach. With these insights in hand, we generalize our results to get a portrayal of the interacting states where the potential decay only inside a collection of cones. We also review some relevant examples of these settings that allow different types of dynamics in the interacting subspace.
Recommended Citation
Malinovitch, Tal, "Scattering for Schrödinger Operators in the Presence of Anisotropic Potentials" (2023). Yale Graduate School of Arts and Sciences Dissertations. 1005.
https://elischolar.library.yale.edu/gsas_dissertations/1005
