Multidimensional Almost-Periodic Schrödinger Operators with Cantor Spectrum

Abstract

We construct multidimensional almost-periodic Schrödinger operators whose spectrum has zero lower box counting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure.

On the Spectra of Separable 2D Almost Mathieu Operators

“…(b) R. Han and S. Jitomirskaya [18] proved that periodic (not necessarily separable) multidimensional discrete Schrödinger operators have interval spectra when the norm of the potential is small and at least one period is odd. This is the discrete analog of the L. Parnovski [35] (c) D. Damanik, J. Fillman, and A. Gorodetski [15] proved that there exists a dense subset B of 1D limit-periodic potentials such that B-type operators have Cantor spectra with zero (lower) box-counting dimension. Furthermore, separable multidimensional discrete Schrödinger operators generated by B-type operators have Cantor spectra with zero (lower) boxcounting dimension.…”

“…(a) We establish a single-interval-characterization for the spectra of separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic analytic operators H k = Δ+V k with Diophantine-frequencydependent sufficiently small norms V k . The same characterization can be said about periodic V k [18] and cannot be said about limit-periodic V k [15]. (b) R. Han and S. Jitomirskaya [18] proved that periodic (not necessarily separable) multidimensional discrete Schrödinger operators have interval spectra when the norm of the potential is small and at least one period is odd.…”

“…We conclude this section with a few words on related mathematical results and questions. The definitions of limit-periodic and almost-periodic potential can be found within [15] and [36], respectively. It suffices to say that the collection of all almost-periodic potentials is a broad class of potentials which contains the periodic and limit-periodic and quasiperiodic potentials.…”

Bloch wave approach to almost periodic homogenization and approximations of effective coefficients

“…The radius δ R of the neighborhood U R depends on the gap between the first and second Bloch eigenvalues of the operator A R . The limit operator of A R is the almost periodic operator A which often has a Cantor-like spectrum [19]. Hence, we expect the spectral gap to vanish in the limit R → ∞.…”

“…) which is reflected in the inverse identity (19). For simplicity, take Ω = R d and consider the equation…”