A rigorous approach to the magnetic response in disordered systems

This paper is a part of an ongoing study on the diamagnetic behavior of a 3-dimensional quantum gas of non-interacting charged particles subjected to an external uniform magnetic field together with a random electric potential. We prove the existence of an almost-sure non-random thermodynamic limit for the grand-canonical pressure, magnetization and zero- field orbital magnetic susceptibility. We also give an explicit formulation of these thermodynamic limits. Our results cover a wide class of physically relevant random potentials which model not only crystalline disordered solids, but also amorphous solids.Comment: 35 pages. Revised version. Accepted for publication in RM

Exponential decay and resonances in a driven system

We study the resonance phenomena for time periodic perturbations of a Hamiltonian $H$ on the Hilbert space $L^2(\mathbb R ^d)$. Here, resonances are characterized in terms of time behavior of the survival probability. Our approach uses the Floquet-Howland formalism combined with the results of L. Cattaneo, J.M. Graf and W. Hunziker on resonances for time independent perturbations.Comment: 16 page

Stark resonances in 2-dimensional curved quantum waveguides

In this paper we study the influence of an electric field on a two dimen-sional waveguide. We show that bound states that occur under a geometrical deformation of the guide turn into resonances when we apply an electric field of small intensity having a nonzero component on the longitudinal direction of the system. MSC-2010 number: 35B34,35P25, 81Q10, 82D77

Locally decodable codes and the failure of cotype for projective tensor products

It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product \ell_p\tp X fails to have finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then \ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3} does not have finite cotype. This is a proved via a connection to the theory of locally decodable codes

Hardy inequalities in globally twisted waveguides

We establish various Hardy-type inequalities for the Dirichlet Laplacian in perturbed periodically twisted tubes of non-circular cross-sections. We also state conjectures about the existence of such inequalities in more general regimes, which we support by heuristic and numerical arguments.Comment: 18 pages, 1 figur

A rigorous proof of the Landau-Peierls formula and much more

We present a rigorous mathematical treatment of the zero-field orbital magnetic susceptibility of a non-interacting Bloch electron gas, at fixed temperature and density, for both metals and semiconductors/insulators. In particular, we obtain the Landau-Peierls formula in the low temperature and density limit as conjectured by T. Kjeldaas and W. Kohn in 1957.Comment: 30 pages - Accepted for publication in A.H.

Violating the Shannon capacity of metric graphs with entanglement

The Shannon capacity of a graph G is the maximum asymptotic rate at which messages can be sent with zero probability of error through a noisy channel with confusability graph G. This extensively studied graph parameter disregards the fact that on atomic scales, Nature behaves in line with quantum mechanics. Entanglement, arguably the most counterintuitive feature of the theory, turns out to be a useful resource for communication across noisy channels. Recently, Leung, Mancinska, Matthews, Ozols and Roy [Comm. Math. Phys. 311, 2012] presented two examples of graphs whose Shannon capacity is strictly less than the capacity attainable if the sender and receiver have entangled quantum systems. Here we give new, possibly infinite, families of graphs for which the entangled capacity exceeds the Shannon capacity.Comment: 15 pages, 2 figure