Le papillon de Hofstadter, revisité. (Hofstadter’s butterfly, revised). (French) Zbl 0732.44004

Helffer, B.; Kerdelhué, P.; Sjöstrand, J.

Mém. Soc. Math. Fr., Nouv. Sér. 43, 87 p. (1990).

The spectrum of the Harper operator \(\cos (hD)+\cos (x)\) is studied in dependence of h. Qualitative properties of this “Hofstadter butterfly” are derived from the conjecture that spectral gaps only close for rational values of h.

Reviewer: J.Asch (Berlin)

MSC:

44A45	Classical operational calculus
35Q40	PDEs in connection with quantum mechanics
35S99	Pseudodifferential operators and other generalizations of partial differential operators
35P99	Spectral theory and eigenvalue problems for partial differential equations
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory

Keywords:

Harper operator; Hofstadter butterfly; spectral gaps

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