Persistence of gaps in the spectrum of certain almost periodic operators

DOI10.4310/ATMP.2012.V16.N2.A7zbMATH Open1269.81227arXivmath/0504088MaRDI QIDQ1951477

Norbert Riedel

Publication date: 6 June 2013

Published in: Advances in Theoretical and Mathematical Physics (Search for Journal in Brave)

Abstract: It is shown that for any irrational rotation number and any admissible gap labelling number the almost Mathieu operator (also known as Harper's operator) has a gap in its spectrum with that labelling number. This answers the strong version of the so-called "Ten Martini Problem". When specialized to the particular case where the coupling constant is equal to one, it follows that the "Hofstadter butterfly" has for any quantum Hall conductance the exact number of components prescribed by the recursive scheme to build this fractal structure.

Full work available at URL: https://arxiv.org/abs/math/0504088

Mathematics Subject Classification ID

Almost and pseudo-almost periodic solutions to functional-differential equations (34K14) Selfadjoint operator theory in quantum theory, including spectral analysis (81Q10) Many-body theory; quantum Hall effect (81V70) Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) (34B30)

Related Items (2)

Almost Mathieu operators with completely resonant phases ⋮ Gap labelling for discrete one-dimensional ergodic Schrödinger operators

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