Philippe Briet

Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide i.e., a domain of the form Ωθ: = r θ ω × ℝ where ω ⊂ ℝ2 is a bounded domain, and r θ = r θ(x 3) is a rotation by the angle θ(x 3) depending on the longitudinal variable x 3. We are interested in the spectral analysis of the Dirichlet Laplacian H acting in L 2(Ωθ). We suppose that the derivative of the rotation angle can be written as (x 3) = β − ϵ(x 3) with a positive constant β and ϵ(x 3) ∼ L|x 3|−α, |x 3| → ∞. We show that if L > 0 and α ∈ (0,2), or if L > L 0 > 0 and α = 2, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.

Locating the spectrum for magnetic Schrödinger and Dirac operators

ABSTRACT Some spectral properties of magnetic Schrödinger and Dirac operators perturbed by long range magnetic fields are investigated. If the intensity of the field is small enough, a better location of the perturbed spectrum is given. In particular, if the unperturbed spectrum is discrete, we show that the perturbed eigenvalues are given in terms of an absolutely convergent series with respect to a magnetic parameter, from which the usual asymptotic expansion can be derived.

Diamagnetic expansions for perfect quantum gases

In this work we study the diamagnetic properties of a perfect quantum gas in the presence of a constant magnetic field of intensity B. We investigate the Gibbs semigroup associated with the one particle operator at finite volume, and study its Taylor series with respect to the field parameter ω≔eB∕c in different topologies. This allows us to prove the existence of the thermodynamic limit for the pressure and for all its derivatives with respect to ω (the so-called generalized susceptibilities).

Diamagnetism of quantum gases with singular potentials

We consider a gas of quasi-free quantum particles confined to a finite box, subjected to singular magnetic and electric fields. We prove in great generality that the finite volume grand-canonical pressure is analytic with respect to the chemical potential and the intensity of the external magnetic field. We also discuss the thermodynamic limit.

Diamagnetic expansions for perfect quantum gases II: Uniform bounds

Consider a charged, perfect quantum gas, in the effective mass approximation, and in the grand-canonical ensemble. We prove in this paper that the generalized magnetic susceptibilities admit the thermodynamic limit for all admissible fugacities, uniformly on compacts included in the analyticity domain of the grand-canonical pressure. The problem and the proof strategy were outlined in MPRF 11 (2005), 177-188. In J. Math. Phys. 47 (2006), 083511 we proved in detail the pointwise thermodynamic limit near z = 0. The present paper is the last one of this series, and contains the proof of the uniform bounds on compacts needed in order to apply Vitalis Convergence Theorem.

Sojourn time for rank one perturbations

We consider a self-adjoint, purely absolutely continuous operator M. Let P be a rank one operator Pu=⟨φ,u⟩φ such that for β0 Hβ0≔M+β0P has a simple eigenvalue E0 embedded in its absolutely continuous spectrum, with corresponding eigenvector ψ. Let Hω be a rank one perturbation of the operator Hβ0, namely, Hω=M+(β0+ω)P. Under suitable conditions, the operator Hω has no point spectrum in a neighborhood of E0, for ω small. Here, we study the evolution of the state ψ under the Hamiltonian Hω, in particular, we obtain explicit estimates for its sojourn time τω(ψ)=∫−∞∞∣⟨ψ,e−iHωtψ⟩∣2dt. By perturbation theory, we prove that τω(ψ) is finite for ω≠0, and that for ω small it is of order ω−2. Finally, by using an analytic deformation technique, we estimate the sojourn time for the Friedrichs model in Rn.

Do Bosons Condense in a Homogeneous Magnetic Field

It has been known since the paper(26) and then due to a rigorous result(3) that the answer to the question in the title is negative for a three-dimensional “ideal gas of charged bosons”. The present paper adds a new rigorous result in this direction. We show that the answer to the question becomes positive, if this “ideal gas of charged bosons” is simultaneously embedded in an appropriate periodic external potential. We prove that it is true for the Perfect Bose Gas (PBG), as well as for the Imperfect Bose Gas with a Mean-Field repulsive particle interaction.

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.

Bender–Wu formula for the Zeeman effect

We prove the Bender–Wu formula in the case of magnetic Schrodinger operators under general considerations on the potential. We also give a bound on the behavior of the perturbation series coefficients for the ground state energy.

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.