1,459 research outputs found
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 -conjecture can be established very
simply.Comment: 4 page
Semiclassical resolvent bounds for short range potentials with singularities at the origin
We consider, for , resolvent estimates for the semiclassical
Schr\"odinger operator . The potential takes the form , where is a long range potential which is Lipschitz with
respect to the radial variable, while as
for some . Both and may behave like
as , provided . We find
that, as , the resolvent bound is of the form for some . The -dependence of the bound
improves if 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
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