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Series: Math Physics Seminar

A limit-periodic function on R^d is one which lies in the L^\infty closure of the space of periodic functions. Schr\"odinger operators with limit-periodic potentials may have very exotic spectral properties, despite being very close to periodic operators. Our discussion will revolve around the transition between ``thick'' spectra and ``thin'' spectra.

Series: Math Physics Seminar

We shall consider a three-dimensional Quantum Field Theory model of an electron
bound to a Coulomb impurity in a polar crystal and exposed to a homogeneous
magnetic field of strength B > 0. Using an argument of Frank and Geisinger
[Commun. Math. Phys. 338, 1-29 (2015)] we can see that as B → ∞ the ground-
state energy is described by a one-dimensional minimization problem with a delta-
function potential. Our contribution is to extend this description also to the ground-
state wave function: we shall see that as B → ∞ its electron density in the direction
of the magnetic field converges to the minimizer of the one-dimensional problem.
Moreover, the minimizer can be evaluated explicitly.

Series: Math Physics Seminar

I will talk about what happens on the spectral transition lines for the almost Mathieu operator. This talk is based on joint works with Svetlana Jitomirskaya and Qi Zhou. For both transition lines \{\beta(\alpha)=\ln{\lambda}\} and \{\gamma(\alpha,\theta)=\ln{\lambda}\} in the positive Lyapunov exponent regime, we show purely point spectrum/purely singular continuous spectrum for dense subsets of frequencies/phases.

Series: Math Physics Seminar

We consider a class of singular ordinary differential equations,
describing systems subject to a quasi-periodic forcing term and
in the presence of large dissipation, and study the existence of
quasi-periodic solutions with the same frequency vector as the forcing term.
Let A be the inverse of the dissipation coefficient.
More or less strong non-resonance conditions on the frequency
assure different regularity in the dependence on the parameter A:
by requiring a non-degeneracy condition on the forcing term,
smoothness and analyticity, and even Borel-summability,
follow if suitable Diophantine conditions are assumed,
while, without assuming any condition, in general no more than a continuous dependence on A is obtained.
We investigate the possibility of weakening the non-degeneracy
condition and still obtaining a solution for arbitrary frequencies.

Series: Math Physics Seminar

Non-compact hyperbolic surfaces serve as a model case for quantum scattering theory with chaotic classical dynamics. In this talk I’ll explain how scattering resonances are defined in this context and discuss our current understanding of their distribution. The primary focus of the talk will be on some recent conjectures inspired by the physics of quantum chaotic systems. I will introduce these and discuss the numerical evidence as well as recent theoretical progress.

Series: Math Physics Seminar

TBA

Series: Math Physics Seminar

We generalize the Lp spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. We show that these bounds are optimal on any manifold in a very strong sense. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators and their optimality follows by semi-classical analysis.

Series: Math Physics Seminar

We introduce a quantum version of the Kac Master equation,and we explain issues like equilibria, propagation of chaos and the corresponding quantum Boltzmann equation. This is joint work with Eric Carlen and Maria Carvalho.

Series: Math Physics Seminar

We investigate a dynamical system consisting of $N$ particles moving on a $d$-dimensional torus under the action of an electric field $E$ with a Gaussian thermostat to keep the total energy constant. The particles are also subject to stochastic collisions which randomize direction but do not change the speed. We prove that in the van Hove scaling limit, $E\to 0$ and $t\to t/E^2$, the trajectory of the speeds $v_i$ is described by a stochastic differential equation corresponding to diffusion on a constant energy sphere.Our results are based on splitting the system's evolution into a ``slow'' process and an independent ``noise''. We show that the noise, suitably rescaled, converges to a Brownian motion. Then we employ the Ito-Lyons continuity theorem to identify the limit of the slow process.

Series: Math Physics Seminar

We will talk about discrete versions of the Bethe-Sommerfeld conjecture. Namely, we study the spectra of multi-dimensional periodic Schrödinger operators on various discrete lattices with sufficiently small potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we characterize those energies at which gaps may open, and we give sharp arithmetic criteria on the periods that ensure no gaps open. We will also provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero. This talk is based on a joint work with Svetlana Jitomirskaya and another work with Jake Fillman.