A Taylor-like Expansion of a Commutator with a Function of Self-Adjoint, Pairwise Commuting Operators

Let $A$ be a $\nu$-vector of self-adjoint, pairwise commuting operators and $B$ a bounded operator of class $C^{n_0}(A)$. We prove a Taylor-like expansion of the commutator $[B,f(A)]$ for a large class of functions f\colon\mathbm{R}^\nu \to \mathbm{R}, generalising the one-dimensional result where $A$ is just a self-adjoint operator. This is done using almost analytic extensions and the higher-dimensional Helffer-Sj\"ostrand formula.Comment: To appear in Math.Scan

On the optical properties of carbon nanotubes--Part I. A general formula for the dynamical optical conductivity

This paper is the first one of a series of two articles in which we revisit the optical properties of single-walled carbon nanotubes (SWNT). Produced by rolling up a graphene sheet, SWNT owe their intriguing properties to their cylindrical quasi-one-dimensional (quasi-1D) structure (the ratio length/radius is experimentally of order of 10^3). We model SWNT by circular cylinders of small diameters on the surface of which the conduction electron gas is confined by the electric field generated by the fixed carbon ions. The pair-interaction potential considered is the 3D Coulomb potential restricted to the cylinder. To reflect the quasi-1D structure, we introduce a 1D effective many-body Hamiltonian which is the starting-point of our analysis. To investigate the optical properties, we consider a perturbation by a uniform time-dependent electric field modeling an incident light beam along the longitudinal direction. By using Kubo's method, we derive within the linear response theory an asymptotic expansion in the low-temperature regime for the dynamical optical conductivity at fixed density of particles. The leading term only involves the eigenvalues and associated eigenfunctions of the (unperturbed) 1D effective many-body Hamiltonian, and allows us to account for the sharp peaks observed in the optical absorption spectrum of SWNT.Comment: Comments: 24 pages. Revised version. Accepted for publication in J.M.

Projection operators on matrix weighted LpL^p and a simple sufficient Muckenhoupt condition

Boundedness for a class of projection operators, which includes the coordinate projections, on matrix weighted $L^p$-spaces is completely characterised in terms of simple scalar conditions. Using the projection result, sufficient conditions, which are straightforward to verify, are obtained that ensure that a given matrix weight is contained in the Muckenhoupt matrix $A_p$ class. Applications to singular integral operators with product kernels are considered

Spectral deformation for two-body dispersive systems with e.g. the Yukawa potential

We find an explicit closed formula for the $k$'th iterated commutator $\mathrm{ad}_A^k(H_V(\xi))$ of arbitrary order $k\ge1$ between a Hamiltonian $H_V(\xi)=M_{\omega_\xi}+S_{\check V}$ and a conjugate operator $A=\frac{\mathfrak{i}}{2}(v_\xi\cdot\nabla+\nabla\cdot v_\xi)$, where $M_{\omega_\xi}$ is the operator of multiplication with the real analytic function $\omega_\xi$ which depends real analytically on the parameter $\xi$, and the operator $S_{\check V}$ is the operator of convolution with the (sufficiently nice) function $\check V$, and $v_\xi$ is some vector field determined by $\omega_\xi$. Under certain assumptions, which are satisfied for the Yukawa potential, we then prove estimates of the form $\lVert\mathrm{ad}_A^k(H_V(\xi))(H_0(\xi)+\mathfrak{i})^{-1}\rVert\le C_\xi^kk!$ where $C_\xi$ is some constant which depends continuously on $\xi$. The Hamiltonian is the fixed total momentum fiber Hamiltonian of an abstract two-body dispersive system and the work is inspired by a recent result [Engelmann-M{\o}ller-Rasmussen, 2015] which, under conditions including estimates of the mentioned type, opens up for spectral deformation and analytic perturbation theory of embedded eigenvalues of finite multiplicity