The Flux-Across-Surfaces Theorem and Zero-Energy Resonances

In this contribution we shall first introduce the Flux Across Surfaces (FAS) theorem, placing it in the general context of the Quantum Scattering Theory. Then we shall review briefly the theory of resonances in non-relativistic Quantum Mechanics and outline a proof of the FAS theorem for non-relativistic potential scattering, which covers also the case in which there is a zero energy resonance.     

Infrared Problem for the Nelson Model on Static Space-Times

We consider the Nelson model on some static space-times and investigate the problem of existence of a ground state. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We investigate the existence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass m(x) tends to 0 at spatial infinity. We show that if m(x) ≥ C |x|⁻¹ at infinity for some C > 0 then the Nelson Hamiltonian has a ground state.     

A Note on Reflectionless Jacobi Matrices

The property that a Jacobi matrix is reflectionless is usually characterized either in terms of Weyl m-functions or the vanishing of the real part of the boundary values of the diagonal matrix elements of the resolvent. We introduce a characterization in terms of stationary scattering theory (the vanishing of the reflection coefficients) and prove that this characterization is equivalent to the usual ones. We also show that the new characterization is equivalent to the notion of being dynamically reflectionless, thus providing a short proof of an important result of Breuer et al. (Commun Math Phys 295:531–550, 2010). The motivation for the new characterization comes from recent studies of the non-equilibrium statistical mechanics of the electronic black box model and we elaborate on this connection.     

Topological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene

We investigate the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the multilayer case. Since the decrease of the Wannier functions is characterised by the structure of the Bloch eigenspaces around the Dirac points, we introduce a geometric invariant of the family of eigenspaces, baptised eigenspace vorticity. We compare it with the pseudospin winding number. For every value $n \in \mathbb {Z}$ of the eigenspace vorticity, we exhibit a canonical model for the local topology of the eigenspaces. With the help of these canonical models, we show that the single band Wannier function $w$ satisfies $|w(x)| \le {\mathrm {const}} \cdot |x|^{-2}$ as $|x| \rightarrow \infty $ , both in monolayer and bilayer graphene.     

Geometric Currents in Piezoelectricity

As a simple model for piezoelectricity we consider a gas of infinitely many non-interacting electrons subject to a slowly time-dependent periodic potential. We show that in the adiabatic limit the macroscopic current is determined by the geometry of the Bloch bundle. As a consequence we obtain the King-Smith and Vanderbilt formula up to errors smaller than any power of the adiabatic parameter.     

Symmetry and Localization for Magnetic Schrödinger Operators: Landau Levels,Gabor Frames and All That

We investigate the relation between broken time-reversal symmetry and localization of the electronic states, in the explicitly tractable case of the Landau model. We first review, for the reader’s convenience, the symmetries of the Landau Hamiltonian and the relation of the latter with the Segal-Bargmann representation of Quantum Mechanics. We then study the localization properties of the Landau eigenstates by applying an abstract version of the Balian-Low Theorem to the operators corresponding to the coordinates of the centre of the cyclotron orbit in the classical theory. Our proof of the Balian-Low Theorem, although based on Battle’s main argument, has the advantage of being representation-independent.

     

Gell-Mann and Low Formula for Degenerate Unperturbed States

The Gell-Mann and Low switching allows to transform eigenstates of an unperturbed Hamiltonian H ₀ into eigenstates of the modified Hamiltonian H ₀ + V. This switching can be performed when the initial eigenstate is not degenerate, under some gap conditions with the remainder of the spectrum. We show here how to extend this approach to the case when the ground state of the unperturbed Hamiltonian is degenerate. More precisely, we prove that the switching procedure can still be performed when the initial states are eigenstates of the finite rank self-adjoint operator P₀VP₀{\mathcal{P}_{0}V\mathcal{P}_{0}} , where P₀{\mathcal{P}_0} is the projection onto a degenerate eigenspace of H ₀.     

Effective Dynamics for Bloch Electrons: Peierls Substitution and Beyond

We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, (x), and vector potential A(x), with x ^d and 1. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of L ² ( ^d ) and an effective Hamiltonian governing the evolution inside this subspace to all orders in . To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics.     

Removal of UV Cutoff for the Nelson Model with Variable Coefficients

We consider the Nelson model with variable coefficients. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We study the removal of the ultraviolet cutoff.