The Bulk-Edge Correspondence for Disordered Chiral Chains

We study one-dimensional insulators obeying a chiral symmetry in the single-particle picture. The Fermi level is assumed to lie in a mobility gap. Topological indices are defined for infinite (bulk) or half-infinite (edge) systems, and it is shown that for a given Hamiltonian with nearest neighbor hopping the two indices are equal. We also give a new formulation of the index in terms of the Lyapunov exponents of the zero energy Schr\"odinger equation, which illustrates the conditions for a topological phase transition occurring in the mobility gap regime.Comment: 20 pages, 3 figure

Chiral Random Band Matrices at Zero Energy

We present a special model of random band matrices where, at zero energy, the famous Fyodorov and Mirlin $\sqrt{N}$-conjecture can be established very simply.Comment: 4 page

Semiclassical resolvent bounds for short range L∞L^\infty potentials with singularities at the origin

We consider, for $h, E > 0$, resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V - E$. The potential takes the form $V = V_L+ V_S$, where $V_L$ is a long range potential which is Lipschitz with respect to the radial variable, while $V_S = O(|x|^{-1} (\log |x|)^{-\rho})$ as $|x| \to \infty$ for some $\rho > 1$. Both $|V_L|$ and $|V_S|$ may behave like $|x|^{-\beta}$ as $|x| \to 0$, provided $0 \le \beta < 2(\sqrt{3} -1)$. We find that, as $h \to 0^+$, the resolvent bound is of the form $\exp(Ch^{-2} (\log(h^{-1}))^{1 + \rho})$ for some $C > 0$. The $h$-dependence of the bound improves if $V_S$ decays at a faster rate toward infinity.Comment: 17 page

Semiclassical Resolvent Estimates and Wave Decay in Low Regularity

In this thesis, we prove weighted resolvent upper bounds for semiclassical Schr¨odinger operators. These upper bounds hold in the semiclassical limit. First, we consider operators in dimension two when the potential is Lipschitz with long range decay. We prove that the resolvent norm grows at most exponentially in the inverse semiclassical parameter, while near infinity it grows at most linearly. Both of these bounds are optimal. Second, we work in any dimension and require that the potential belong to L∞ and have compact support. Again, we find that the weighted resolvent norm grows at most exponentially, but this time with an additional loss in the exponent. Finally, we apply the resolvent bounds to prove two logarithmic local energy decay rates for the wave equation, one when the wavespeed is a compactly supported Lipschitz perturbation of unity, and the other when the wavespeed is a compactly supported L∞ perturbation of unity