arXiv subjects
Publications and source records attributed to C. A. Marx.
We prove that the resolvent set of any, possibly singular, quasi periodic Jacobi operator is characterized as the set of all energies whose associated Jacobi cocycles induce a dominated splitting. This extends a well-known result by R. A. Johnson for Schrödinger operators.
We extend our results in \cite{hislop_marx_1} on the quantitative continuity properties, with respect to the single-site probability measure, of the density of states measure and the integrated density of states for random Schrödinger operators. For lattice models on $\mathbb{Z}^d$, with $d \geq 1$, we treat the case of non-compactly supported probability measures with finite first moments. For random Schrödinger operators on $\mathbb{R}^d$, with $d \geq 1$, we prove results analogous to those in \cite{hislop_marx_1} for compactly supported probability measures. The method of proof makes use of the Combes-Thomas estimate and the Helffer-Sjöstrand formula.
We quantify the coupling asymptotics for the Lyapunov-exponent of a one-frequency quasi-periodic Schrödinger operator with analytic potential sampling function. The result refines the well-known lower bound of the Lyapunov-exponent by Sorets and Spencer.
The extended Harper's model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic symmetry (Aubry duality). Previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space, one of which is characterized by the positivity of the Lyapunov exponent. In this paper, we complete the picture with a description of the spectral measures over the entire remaining (self-dual) region, for all irrational values of the frequency parameter (the magnetic flux in the model). Most notably, we prove that in the entire interior of this regime, the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the NNN interaction becomes symmetric. In physics literature, extensive numerical analysis had indicated such "spectral collapse," however so far not even a heuristic argument for this phenomenon could be provided. On the other hand, in the remaining part of the self-dual region, the spectral measures are singular continuous irrespective of such symmetry. The analysis requires some rather delicate number theoretic estimates, which ultimately depend on the solution of a problem posed by Erdős and Szekeres.
Quasi-periodic Schrödinger-type operators naturally arise in solid state physics, describing the influence of an external magnetic field on the electrons of a crystal. In the late 1970s, numerical studies for the most prominent model, the almost Mathieu operator (AMO), produced the first example of a fractal in physics known as "Hofstadter's butterfly," marking the starting point for the ongoing strong interest in such operators in both mathematics (several of B. Simon's problems) and physics (e.g. Graphene, quantum Hall effect). Whereas research in the first three decades was focused mainly on unraveling the unusual properties of the AMO and operators with similar structure of potential, in recent years a combination of techniques from dynamical systems with those from spectral theory has allowed for a more "global," model-independent point of view. Intriguing phenomena first encountered for the AMO, notably the appearance of criticality corresponding to purely singular continuous spectrum for a measure theoretically typical realization of the phase, could be tested for prevalence in general models. The intention of this article is to survey the theory of quasi-periodic Schrödinger-type operators attaining this "global" view-point with an emphasis on dynamical aspects of the spectral theory of such operators.
We give a criterion implying subcritical behavior for quasi-periodic Schrödinger operators where the potential sampling function is given by a trigonometric polynomial. Subcritical behavior, in the sense of Avila's global theory, is known to imply purely absolutely continuous spectrum for all irrational frequencies and all phases.
Consider a quasi-periodic Schrödinger operator $H_{α,θ}$ with analytic potential and irrational frequency $α$. Given any rational approximating $α$, let $S_+$ and $S_-$ denote the union, respectively, the intersection of the spectra taken over $θ$. We show that up to sets of zero Lebesgue measure, the absolutely continuous spectrum can be obtained asymptotically from $S_-$ of the periodic operators associated with the continued fraction expansion of $α$. This proves a conjecture of Y. Last in the analytic case. Similarly, from the asymptotics of $S_+$, one recovers the spectrum of $H_{α,θ}.$
We show how to extend (and with what limitations) Avila's global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for extended Harper's model, for all values of parameters and all irrational frequencies. In particular, this includes the self-dual regime for which even heuristic results did not previously exist in physics literature. The extension of Avila's global theory is also shown to imply continuous behavior of the LE on the space of analytic $M_2(\mathbb{C})$-cocycles. This includes rational approximation of the frequency, which so far has not been available.
We prove that the Lyapunov exponent of quasi-periodic cocyles with singularities behaves continuously over the analytic category. We thereby generalize earlier results, where singularities were either excluded completely or constrained by additional hypotheses. Applications are one-parameter families of analytic Jacobi operators, such as extended Harper's model describing crystals subject to external magnetic fields.
For ergodic 1d Jacobi operators we prove that the random singular components of any spectral measure are almost surely mutually disjoint as long as one restricts to the set of positive Lyapunov exponent. In the context of extended Harper's equation this yields the first rigorous proof of the Thouless' formula for the Lyapunov exponent in the dual regions.
We consider averages $κ$ of spectral measures of rank one perturbations with respect to a $σ$-finite measure $ν$. It is examined how various degrees of continuity of $ν$ with respect to $α$-dimensional Hausdorff measures ($0 \leq α\leq 1$) are inherited by $κ$. This extends Kotani's trick where $ν$ is simply the Lebesgue measure.