Abstract: This talk will be on the spectral properties of elastic beam Hamiltonian defined on periodic hexagonal lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar valued fourth-order Schrödinger operator equipped with a real periodic symmetric potential. Unlike the second-order Schrödinger operator commonly applied in quantum graph literature, here the self-adjoint vertex conditions encode geometry of the graph by their dependence on angles at which edges are met. I will firstly consider this Hamiltonian on a special equal-angle lattice, known as graphene or honeycomb lattice. I will also discuss spectral properties for the same operator on lattices in the geometric neighborhood of graphene. This talk is based on a recent joint work with Mahmood Ettehad (University of Minnesota), https://arxiv.org/pdf/2110.05466.pdf. 

The table and the chair tiling are two aperiodic tilings of the plane that are typical examples of two-dimensional quasicrystals. One way to treat such systems in dimension one, is to approximate these systems by suitable (periodic) approximations. Based on this, we raise the following questions: Is there a general method to approximate spectral properties of a given operator by the underlying geometry or dynamics? If so, can we control the approximations and which spectral properties are preserved? During the talk, we provide a short overview over such results with a special focus on dynamicallydened operator families. We will see as how to apply those results explicitly and what they tell us about the table and the chair tiling. These results are joint works with Ram Band, Jean Bellissard, Horia Cornean, Giusseppe De Nittis, Felix Pogorzelski, Alberto Takase and Lior Tenenbaum.

We are interested in the nature of the spectrum of the one-dimensional Schr\"odinger operator
$$
  - \frac{d^2}{dx^2}-Fx + \sum_{n \in \mathbb{Z}}g_n \delta(x-n)
$$
with $F>0$ and two different choices of the coupling constants $\{g_n\}_{n\in \mathbb{Z}}$. In the first model $g_n \equiv \lambda$ and we prove that if $F\in \pi^2 \mathbb{Q}$ then the spectrum is $\mathbb{R}$ and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model $g_n$ are independent random variables with mean zero and variance $\lambda^2$. Under certain assumptions on the distribution of these random variables we prove that almost surely the spectrum is dense pure point if $F < \lambda^2/2$ and purely singular continuous if $F> \lambda^2/2$. Based on joint work with Rupert Frank.

In this talk, I will introduce a recent work in showing positivity of the Lyapunov exponent for Schr\"odinger operators with potentials generated by hyperbolic dynamics. Specifically, we showed that if the base dynamics is a subshift of finite type with an ergodic measure admitting a local product structure and if it has a fixed point, then for all nonconstant H\"older continuous potentials, the set of energies with zero Lyapunov exponent is a discrete set. If the potentials are locally constant or globally fiber bunched, then the set of zero Lyapunov exponent is finite. We also showed that for generic such potentials, we have full positivity in the general case and uniform postivity in the special cases. Such hyperbolic dynamics include expanding maps such as the doubling map on the unit circle, or Anosov diffeomorphism such as the Arnold's Cat map on 2-dimensional torus. It also can be applied to Markov chains whose special cases include the i.i.d. random variable. This is a joint with A. Avila and D. Damanik.