An official website of the United States government
This article shows that for a large class of discrete periodic Schrödinger operators, most wavefunctions resemble Bloch states. More precisely, we prove quantum ergodicity for a family of periodic Schrödinger operators H on periodic graphs. This means that most eigenfunctions of H on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our results cover the adjacency matrix on Z^d, the triangular lattice, the honeycomb lattice, Cartesian products, and periodic Schrödinger operators on Z^d. The theorem applies more generally to any periodic Schrödinger operator satisfying an assumption on the Floquet eigenvalues.
more »
« less
Boshernitzan’s condition, factor complexity, and an application
Boshernitzan gave a decay condition on the measure of cylinder sets that implies unique ergodicity for minimal subshifts. Interest in the properties of subshifts satisfying this condition has grown recently, due to a connection with discrete Schrödinger operators, and of particular interest is how restrictive the Boshernitzan condition is. While it implies zero topological entropy, our main theorem shows how to construct minimal subshifts satisfying the condition, and whose factor complexity grows faster than any pre-assigned subexponential rate. As an application, via a theorem of Damanik and Lenz, we show that there is no subexponentially growing sequence for which the spectra of all discrete Schrödinger operators associated with subshifts whose complexity grows faster than the given sequence have only finitely many gaps.
more »
« less
- Award ID(s):
- 2054643
- PAR ID:
- 10400268
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society, Series B
- Volume:
- 9
- Issue:
- 11
- ISSN:
- 2330-1511
- Page Range / eLocation ID:
- 95 to 110
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract We initiate the study of Schrödinger operators with ergodic potentials defined over circle map dynamics, in particular over circle diffeomorphisms. For analytic circle diffeomorphisms and a set of rotation numbers satisfying Yoccoz’s $${{\mathcal{H}}}$$ arithmetic condition, we discuss an extension of Avila’s global theory. We also give an abstract version and a short proof of a sharp Gordon-type theorem on the absence of eigenvalues for general potentials with repetitions. Coupled with the dynamical analysis, we obtain that, for every $$C^{1+BV}$$ circle diffeomorphism, with a super Liouville rotation number and an invariant measure $$\mu $$, and for $$\mu $$-almost all $$x\in{{\mathbb{T}}}^1$$, the corresponding Schrödinger operator has purely continuous spectrum for every Hölder continuous potential $$V$$.more » « less
-
On Gaps in the Spectra of Quasiperiodic Schrödinger Operators with Discontinuous Monotone PotentialsWe show that, for one-dimensional discrete Schrödinger operators, stability of Anderson localization under a class of rank one perturbations implies absence of intervals in spectra. The argument is based on well-known results of Gordon and del Rio–Makarov–Simon, combined with a way to consider perturbations whose ranges are not necessarily cyclic. The main application of the results is showing that a class of quasiperiodic operators with sawtooth-like potentials, for which such a version of stable localization is known, has Cantor spectra. We also obtain several results on gap filling under rank one perturbations for some general (not necessarily monotone) classes of quasiperiodic operators with discontinuous potentials.more » « less
-
In this paper, we show that one-dimensional discrete multifrequency quasiperiodic Schrödinger operators with smooth potentials demonstrate ballistic motion on the set of energies on which the corresponding Schrödinger cocycles are smoothly reducible to constant rotations. The proof is performed by establishing a local version of strong ballistic transport on an exhausting sequence of subsets on which reducibility can be achieved by a conjugation uniformly bounded in the Cℓ-norm. We also establish global strong ballistic transport under an additional integral condition on the norms of conjugation matrices. The latter condition is quite mild and is satisfied in many known examples.more » « less
-
null (Ed.)Abstract We consider separable 2D discrete Schrödinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies, we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator and utilizes the Newhouse Gap Lemma on sums of Cantor sets.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/bproc/90
