23 research outputs found
A Taylor-like Expansion of a Commutator with a Function of Self-Adjoint, Pairwise Commuting Operators
Let be a -vector of self-adjoint, pairwise commuting operators and
a bounded operator of class . We prove a Taylor-like expansion
of the commutator for a large class of functions
f\colon\mathbm{R}^\nu \to \mathbm{R}, generalising the one-dimensional result
where 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 and a simple sufficient Muckenhoupt condition
Boundedness for a class of projection operators, which includes the coordinate projections, on matrix weighted -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 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 'th iterated commutator
of arbitrary order between a Hamiltonian
and a conjugate operator
, where
is the operator of multiplication with the real analytic
function which depends real analytically on the parameter ,
and the operator is the operator of convolution with the
(sufficiently nice) function , and is some vector field
determined by . Under certain assumptions, which are satisfied for
the Yukawa potential, we then prove estimates of the form
where is some constant which depends continuously on .
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