Almost invariant subspaces in the topologically non-trivial setting: a pedestrian approach
We present a self-contained, algorithmic construction of almost invariant subspaces [Panati et al. (2003), De Nittis and Panati (2010), Freund and Teufel (2016)] for Hofstadter-like Hamiltonians $H_b $ acting on $\ell^2(\mathbb{Z}^2,\mathbb{C}^n)$. More precisely, for rational values of $b_0/(2π)$ we describe an iterative procedure that takes a ``locally gapped" projection $P_{0}$ (possibly topologically non-trivial) which commutes with $H_{b_0}$, into a projection $P_{\varepsilon}$ that commutes with $H_{b_0+\varepsilon}$ up to an error of order $\varepsilon^\infty$. Our construction insures that the Chern number of $P_{\varepsilon}$ equals the Chern number of $P_0$ as long as $P_\varepsilon$ exists. Moreover, its integrated density of states varies linearly with $\varepsilon$ (the ``Středa formula"). We also show that any spatially localized Parseval frame of $P_{0}$ can be mapped into to a localized Parseval frame of $P_{\varepsilon}$. We choose to work with a discrete model in order to keep the technicalities to a minimum level, with no need for advanced pseudo-differential calculus. Nevertheless, we are confident that most of our results can be extended to (unbounded) magnetic (pseudo)differential operators, but the technical burden would be much heavier. This is left for future work.