# Research

We are mostly interested in applications of analysis, and spectral analysis in particular, to real world problems. The spectrum of a physical system encodes many of its important physical properties (such as stability, long time behaviour, allowable energy levels….). A good understanding of the structure of the spectrum can lead to a better understanding of the corresponding system.

### Objectives

There are two main topics that we work on:

1. Obtain uniform ergodic theorems for Hamiltonian dynamical systems, in particular systems that lack a spectral gap.
2. Understand how “complex” it is to compute the spectrum starting from finite dimensional approximations. More generally, understand how one can compute infinite dimensional objects.

In addition, we are interested in applications of these topics, in particular in kinetic theory.

### Research in Detail

#### Uniform Ergodic Theorems

The roots of ergodic theory trace back to the second half of the 19th century, when scientists such as Boltzmann and Maxwell initiated the study of large physical systems from an averaged point of view, a field known as statistical mechanics today. The so-called “ergodic hypothesis” (due to Boltzmann) asserted that any typical mechanical system (for instance, gas dynamics) would eventually pass through any state on the energy surface. [For a more detailed historic treatise we refer to C. Moore’s recent survey]. This was formulated mathematically by J. von Neumann and G. Birkhoff in 1931. We are interested in von Neumann’s version of the ergodic theorem (known as the mean ergodic theorem):

Theorem (von Neumann). Let $\mathcal{H}$ be a separable Hilbert space and let $G_t:\mathcal{H}\to\mathcal{H}$ be a continuous one-parameter group of unitary transformations. Let $P$ be the orthogonal projection onto $\{v\in\mathcal{H} : G_tv=v, \forall t\}$. Then for every $f\in\mathcal{H}$ $\displaystyle \mathop{\lim}_{T\to\infty}\frac{1}{2T}\int_{-T}^TG_tf\,dt=Pf.$

The proof relies on the spectral theorem: plugging the expression $G_t=\int_{\mathbb{R}}e^{it\lambda}\,dE(\lambda)$ into the left hand side and changing the order of integration one gets $\displaystyle \mathop{\lim}_{T\to\infty}\int_{\mathbb{R}} \frac{\sin T\lambda}{T\lambda}\,dE(\lambda)$ which vanishes for all $\lambda\neq0$. We are therefore left with the contribution due to $\lambda=0$, which turns out to be precisely the projection operator $P$.

Unfortunately, this theorem does not provide a rate of convergence. Indeed, in this level of generality there is no rate: convergence can be arbitrarily slow. We are interested in obtaining rates of convergence for unitary transformations arising from incompressible flows (such as Hamiltonian flows). In general such flows also need not have a rate of convergence, but there are certain instances where it is known that there is a rate, for instance when there is a spectral gap, i.e. $0$ is an isolated point in the spectrum. This leads to an exponential rate of convergence. Interestingly, it turns out that even when there is no spectral gap it is possible to extract a rate (though it will be algebraic, not exponential, and it will only hold for an appropriate subspace of $\mathcal{H}$).

In a nutshell, to obtain such results, one needs detailed estimates of the density of the spectral measure $E(\lambda)$ near $\lambda=0$. Informally speaking, this means that we need to control how many ‘slow’ modes there are.

Publications:

• Weak Poincaré and Nash-type inequalities via density of states estimates
J. Ben-Artzi, A. Einav
Submitted (2018) | arXiv
• Instabilities in kinetic theory and their relationship to the ergodic theorem
J. Ben-Artzi
Contemp. Math., 653, 25-40 (2015) | arXiv | journal | doi
• On the spectrum of shear flows and uniform ergodic theorems
J. Ben-Artzi
J. Funct. Anal., 267, 299-322 (2014) | arXiv | journal | doi

#### Computing in Infinite Dimensions

This part of the project aims to address a question that is at the crossroads of pure, applied and computational mathematics: can we compute “infinite-dimensional objects” from finite-dimensional approximations, and, if so, how? An important specific example we have in mind is the spectrum of an operator. Of course, such questions  have received extensive attention over the years from various communities, ranging from logicians and theoretical computer scientists to applied and computational mathematicians.

However the point of view that we take is slightly different: our goal is to develop a general, rigorous theory, that at the same time addresses the concrete issues that arise in practical computations. We classify the complexity of a computation by the number of limits it requires. For instance, we have shown that if one want to compute the spectrum of a bounded infinite matrix (i.e. an element of $\mathcal{B}(\ell^2(\mathbb{N}))$) starting from finite-dimensional approximations (i.e. reading a finite section of the matrix) then three limits are required! If there is additional information this number may decrease. For instance, if the matrix is known to be self-adjoint then two limits are enough, and if it’s known to be compact then one limit suffices.

Publications:

• Can everything be computed? – On the Solvability Complexity Index and Towers of Algorithms
J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel
Submitted (2016) | arXiv
• Approximations of strongly continuous families of unbounded operators
J. Ben-Artzi, T. Holding
Commun. Math. Phys.
, 345, 615-630 (2016) | arXiv | journal | doi
• New barriers in complexity theory: On The Solvability Complexity Index and Towers of Algorithms
J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel
, 353, 931-936 (2015) | journal | doi

#### Stability of Plasmas

In kinetic theory gases and plasmas are typically modelled using a probability distribution function $f(t,x,v)$ that measures the density of particles that at time $t$ are located at the point $x$ and have momentum $v$. The function $f$ is transported in phase space by the Vlasov equation $\displaystyle \frac{\partial f}{\partial t}+v\cdot\nabla_{x}f+F\cdot\nabla_{v}f=0.$

The physics of the problem is encoded in the forcing term $F$. As we have plasmas in mind (gases of charged particles), this should be the Lorentz force $F=q \left(E+\frac{v}{c}\times B\right)$. The electromagnetic fields are governed by Maxwell’s equations $\displaystyle \nabla\cdot{E}=\rho\qquad\qquad\qquad\qquad\quad\nabla\cdot{B}=0$ $\displaystyle \nabla\times{E}=-\frac{1}{c}\frac{\partial{B}}{\partial t}\qquad\nabla\times{B}=\frac{1}{c}{j}+\frac{1}{c}\frac{\partial{E}}{\partial t}$

which also provide the coupling to the Vlasov equation through the charge density $\rho=4\pi q\int f\;\text dv$ and current density $j=4\pi q\int vf\;\text dv.$

We are interested in stability analysis of steady-state solutions to the Vlasov-Maxwell system. Asymptotic analysis of Hamiltonian dynamical systems (such as this one) is a vast field. In particular, in the context of kinetic theory, we note the important recent results on Landau damping and on the stability of galactic models. Such analysis typically involves a detailed study of the spectrum of the linearized operator.

Publications:

• Concentrating solutions of the relativistic Vlasov-Maxwell system
J. Ben-Artzi, S. Calogero, S. Pankavich
Submitted, (2018)
• Arbitrarily large solutions of the Vlasov-Poisson system
J. Ben-Artzi, S. Calogero, S. Pankavich
SIAM J. Math. Anal., 50, 4311-4326 (2018) | arXiv | journal | doi
• Instabilities of the relativistic Vlasov-Maxwell system on unbounded domains
J. Ben-Artzi, T. Holding
SIAM J. Math. Anal.
, 49, 4024-4063 (2017) | arXiv | journal | doi
• Instability of nonsymmetric nonmonotone equilibria of the Vlasov-Maxwell system
J. Ben-Artzi
J. Math. Phys.
, 52, 123703 (2011) | arXiv | journal | doi
• Instability of nonmonotone magnetic equilibria of the relativistic Vlasov-Maxwell system
J. Ben-Artzi
Nonlinearity
, 24, 3353-3389 (2011) | arXiv | journal | doi