Spectral and dynamical analysis of delocalization for random Schrödinger operators
Final Report Abstract
This research project was focused on the spectral analysis of differential and pseudodifferential operators in mathematical physics, quantum mechanics, and statistical mechanics. The aim was to study the asymptotic behavior of eigenvalues, their distribution, and related properties of the corresponding eigenfunctions with emphasize on the effects of randomness and disorder on spectral statistics and on localization and delocalization phenomena. The spectral theory of differential operators plays a crucial role in mathematical physics. A significant example is the Anderson model: Mathematically, the motion of an electron in a random background can be modeled by a random Schrödinger operator, the Anderson operator, and spectral properties, in particular spectral localization and delocalization of eigenvectors determine properties of electronic motion. The Anderson operator is considered for example in Euclidean space or on the lattice. Another example is the Anderson operator on an infinite regular tree, which is one of the earliest studied models for Anderson localization. To analyze properties of localization and delocalization for the Anderson model on the tree we studied finite random regular graphs that approximate the local geometry of infinite regular trees. For random operators it is conjectured that spectral properties of an infinite-volume operator are related to the distribution of spectral gaps of finite-volume approximations. In particular, localization and pure point spectrum in infinite volume is expected to correspond to Poisson eigenvalue statistics. Motivated by results about the Anderson model on the infinite tree I consider random Schrödinger operators on finite regular graphs. I study local spectral statistics: I analyze the number of eigenvalues in intervals with length comparable to the inverse of the number of vertices of the graph, in the limit where this number tends to infinity. I show that the random point process generated by the rescaled eigenvalues converges in certain spectral regimes of localization to a Poisson process. In these regimes the results prove the conjectured relation of localization and eigenvalue statistics. In a related work I consider a random regular graph of fixed degree with n vertices. I study spectral properties of the adjacency matrix and of random Schrödinger operators on such a graph as n tends to infinity. I prove that the integrated density of states on the graph converges to the integrated density of states on the infinite regular tree and give uniform bounds on the rate of convergence. Based on related estimates for the Green function I establish existence of spectral regimes where eigenvectors are delocalized. Another successful strategy to rigorously study models in mathematical physics – besides the inclusion of randomness – is the analysis of certain asymptotic regimes, in particular the asymptotic analysis of deterministic differential operators, especially the asymptotic distribution of eigenvalues. During this project I extended my research on asymptotic spectral properties of differential operators in the semiclassical limit. We prove precise asymptotic formulas for the sum of the eigenvalues of the Laplace operator on a bounded domain describing the energy of noninteracting fermionic particles in the domain and we show how the asymptotics depends on different boundary conditions. I also apply the developed methods to give a proof for the asymptotics of the sum of the eigenvalues for a large class of differential operators including operators with non-homogeneous symbols. In addition, we study the ground state of a magnetic polaron. A polaron describes an electron interacting with the quantized optical modes of a polar crystal and we analyze this model in the presence of a homogeneous magnetic field. The purpose of this work is to derive an effective non-linear one-dimensional equation for the ground state energy. Again we give a precise asymptotic formula, in this case for the ground state energy in the limit where the strength of the magnetic field tends to infinity.
Publications
- Semi-classical analysis of the Laplace operator with Robin boundary conditions, Bull. Math. Sci. 2 (2012), no. 2, 281–319
R. L. Frank and L. Geisinger
- Convergence of the density of states and delocalization of eigenvectors on random regular graphs, Journal of Spectral Theory 5 (2015), no. 4, 783–827
L. Geisinger
- Poisson eigenvalue statistics for random Schrödinger operators on regular graphs, Annales Henri Poincaré 16 (2015), no. 8, 1779–1806
L. Geisinger
- The ground state energy of a polaron in a strong magnetic field, Comm. Math. Phys. 338 (2015), no. 1, 1–29
R. L. Frank and L. Geisinger
