# New Marie Skłodowska-Curie Fellowship

We are delighted to announce that our team has been awarded another Marie Skłodowska-Curie Fellowship. Frank Rösler, with the supervision of Jonathan Ben-Artzi, has been successful with his proposal entitled “Computational Complexity in Quantum Mechanics” (COCONUT).

The short description provided in the proposal states: “The goal of this project is to improve our understanding of how to perform computations in quantum mechanics and classify their complexity. This will be achieved by using modern methods from spectral approximation theory in conjunction with the recently introduced Solvability Complexity Index.”

The total value of the award is €212,934.

# New paper: Uniform convergence in von Neumann’s ergodic theorem in the absence of a spectral gap

Jonathan Ben-Artzi and Baptiste Morisse recently submitted a paper entitled Uniform convergence in von Neumann’s ergodic theorem in absence of a spectral gap.

Von Neumann’s ergodic theorem states that “time” averages converge to “spatial” avergaes: given a one-parameter family of unitary maps $U_t:\mathcal{H}\to\mathcal{H},\,t\in\mathbb{R},$ the average $\frac{1}{2T}\int_{-T}^TU_tf\,\mathrm{d}t$ converges to the projection of $f$ onto the space of functions invariant under $U_t$ as $T\to+\infty$.

Generally there is no rate. However, if the generator of $U_t$ has a spectral gap, the rate is $T^{-1}$. In the present paper, it is shown that even in the absence of a spectral gap one can obtain a rate, albeit on a subspace of $\mathcal{H}$, and with a rate worse than $T^{-1}$. This is done by obtaining a suitable estimate for the density of the spectrum near zero (low frequencies).

# New paper: Weak Poincaré inequalities in the absence of spectral gaps

*** Update: this paper is now published in Ann. Henri Poincaré ***

Jonathan Ben-Artzi recently uploaded a new paper entitled Weak Poincaré inequalities in absence of spectral gaps, co-authored with Amit Einav.

For Markov semigroups it is well-known that the following are equivalent:

• The generator has a spectral gap,
• The generator satisfies a Poincaré inequality,
• Solutions decay exponentially

In this paper, they study semigroups which lack a spectral gap (such as the heat semigroup in $\mathbb{R}^d$) and try to see how much of the above theorem remains true. They prove that an estimate on the density of the spectrum near 0 leads to a weak Poincaré inequality, which in turn leads to an algebraic decay rate.

This is applied to the heat semigroup, where the optimal decay rate $t^{-d/2}$ is recovered. In this case, the weak Poincaré inequality is no more than the Nash inequality. This is done for the fractional Laplacian as well, with similar results.

# Research journey to Lyon

Thanks to his LMS Scheme 3 grant worth £1200, Baptiste enjoyed a two weeks stay in Lyon from November 10 to 25. Baptiste worked with his collaborator Francesco Fanelli on a new project that aims to explore the question of the inviscid limit for some hydrodynamical equations – as the Euler equation with variable density – in the wake of the important work of Jacob Bedrossian and Nader Masmoudi on Euler equations. This project lies mathematically at the intersection of Baptiste’s line of research on Gevrey regularity and Francesco’s knowledge on hydrodynamical systems.

Baptiste was also invited to the Analysis Seminar of Besançon and gave a talk on weakly hyperbolic systems. The audience was very interested by this subject and raised many interesting questions. It may be the sign that Baptiste will go back to Besançon in the near future!

# Talks & Travels

In his second month here at Cardiff Frank Rösler has been invited to the University of Cambridge and Imperial College London to give seminar talks. These visits provided the opportunity to discuss his research with a wider audience and make progress on open questions from his recent research project on spectral approximation.

On 14th December he will attend the next meeting of the South Wales Analysis & Probability Seminar in Swansea.

# New team member: Frank Rösler

On 1 October 2018 we were joined by a new team member, Dr Frank Rösler. Frank completed his PhD at Durham University and worked as a Research Assistant in Freiburg (Germany).

He is interested in the spectral theory of non-selfadjoint operators and other operator-theoretic questions in PDE theory. His past projects involved pseudospectra of non-normal Schrödinger Operators and more general resolvent norm estimates of partial differential operators. More recently, he studied problems in Asymptotic Analysis and Homogenisation from an operator-theoretic perspective.

Welcome Frank!