Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

Abstract

First, we reconsider the magnetic pseudodifferential calculus and show that for a large class of non-decaying symbols, their corresponding magnetic pseudodifferential operators can be represented, up to a global gauge transform, as generalized Hofstadter-like, bounded matrices. As a by-product, we prove a Calderón-Vaillancourt type result. Second, we make use of this matrix representation and prove sharp results on the spectrum location when the magnetic field strength b varies. Namely, when the operators are self-adjoint, we show that their spectrum (as a set) is at least 1/2-Hölder continuous with respect to b in the Hausdorff distance. Third, when the magnetic perturbation comes from a constant magnetic field we show that their spectral edges are Lipschitz continuous in b. The same Lipschitz continuity holds true for spectral gap edges as long as the gaps do not close.

Regularity properties of bulk and edge current densities at positive temperature

“…Combining (10) with the estimates (47) and (49) gives that for λ > 0 sufficiently large, there exist C, δ > 0 such that…”

“…In order to prove Theorem 1.1(ii), we first show (2) for (h E b + λ) −1 , i.e. in the case when W = 0, using the explicit structure of (h E b + λ) −1 given by (10) and the estimate on the integral kernels provided in Appendix B. Then, the general case is obtained using the second resolvent identity.…”

“…Since λ and b are fixed, we use the shorthand notation S b (−λ) ≡ S and T b (−λ) ≡ T for the remaining part of this proof. By (10) we can express the resolvent of the edge Hamiltonian in terms of S and T , that is…”

“…Hence by arguing as in the proof of Proposition 2.3 and applying Lemma B.3 and (52) we obtain i[H E b , X j ]S b (−λ) = 2P j S b (−λ). Combining this with (10)…”

“…However, (H E b + λ) −1 is smooth with respect to b in the strong operator topology (cf. [12, Corollary 1.2]) By using the magnetic pseudodifferential calculus [33,26,27,11,10], it is possible to prove smoothing properties for the resolvent of the bulk Hamiltonian, i.e. for λ > 0 sufficiently large and any m ∈ N, there exists N large enough such that (H b + λ) −N maps L 2 (R 2 ) continuously to C m (R 2 ).…”