@article{THESIS,
      recid = {12408},
      author = {Chen, Binglu},
      title = {Partial Differential Equations Arising from Topological  Insulators},
      publisher = {University of Chicago},
      school = {Ph.D.},
      address = {2024-06},
      number = {THESIS},
      pages = {189},
      abstract = {In this thesis, we develop a scattering theory for the  asymmetric transport observed at interfaces separating  two-dimensional topological insulators. Starting from the  spectral decomposition of an unperturbed confining  Hamiltonian, we present a limiting absorption principle and  construct a generalized eigenfunction expansion for  perturbed systems. We then relate the interface  conductivity, a current observable quantifying the  transport asymmetry, to the scattering matrix associated to  the generalized eigenfunctions. In particular, we show that  the interface conductivity is concretely expressed as a  difference of transmission coefficients and is stable  against perturbations. We apply the theory to systems of  perturbed Dirac equations with asymptotically linear domain  wall.

In the presence of random perturbations in the  Hamiltonians, the limiting behavior of the scattering  matrix entries as the thickness L of the random medium  increases gives rise to a second order diffusion operator  by the diffusion approximation theory. We call such  diffusion operator a mixed type generalized Kimura  diffusion operator. We model the operator andprovide the  degenerate Hölder space-type estimates for model operators.  With the analysis of perturbation term we establish the  existence of solutions. We also give proofs of the  existence and regularity of the global heat kernel.

We  also concern the long-time asymptotics of this degenerate  diffusion operators with mixed linear and quadratic  degeneracies. In one space dimension, we characterize all  possible invariant measures for such a class of operators  and in all cases show exponential convergence of the  Green’s kernel to such invariant measures. We generalize  the results to a class of two dimensional operators  including those used in the analysis of topological  insulators. Several numerical simulations illustrate our  theoretical findings.},
      url = {http://knowledge.uchicago.edu/record/12408},
      doi = {https://doi.org/10.6082/uchicago.12408},
}