Almost invariant subspaces in the topologically non-trivial setting: a pedestrian approach
Abstract
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 ℓ2(ℤ2,ℂn). More precisely, for rational values of b0/(2π) we describe an iterative procedure that takes a ``locally gapped" projection P0 (possibly topologically non-trivial) which commutes with Hb0, into a projection Pε that commutes with Hb0+ε up to an error of order ε∞. Our construction insures that the Chern number of Pε equals the Chern number of P0 as long as Pε exists. Moreover, its integrated density of states varies linearly with ε (the ``Středa formula"). We also show that any spatially localized Parseval frame of P0 can be mapped into to a localized Parseval frame of Pε. 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.
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BibTeX
@misc{cornean2026almost,
title = {{Almost invariant subspaces in the topologically non-trivial setting: a pedestrian approach}},
author = {Horia Cornean and Gabriele Peluso and Mikkel Hviid Thorn},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16393},
eprint = {2610.16393},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16393},
}
RIS
TY - UNPB
TI - Almost invariant subspaces in the topologically non-trivial setting: a pedestrian approach
AU - Horia Cornean
AU - Gabriele Peluso
AU - Mikkel Hviid Thorn
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16393
UR - https://arxiv.org/abs/2610.16393
AB - 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 ℓ^2(ℤ^2,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_ε that commutes with H_b_0+ε up to an error of order ε^∞. Our construction insures that the Chern number of P_ε equals the Chern number of P_0 as long as P_ε exists. Moreover, its integrated density of states varies linearly with ε (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_ε. 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.
ER -