Mouez Dimassi

Quantum electrodynamics of relativistic bound states with cutoffs

We consider a Hamiltonian with ultraviolet and infrared cutoffs, describing the interaction of relativistic electrons and positrons in the Coulomb potential with photons in Coulomb gauge. The interaction includes both interaction of the current density with transversal photons and the Coulomb interaction of charge density with itself. We prove that the Hamiltonian is self-adjoint and has a ground state for sufficiently small coupling constants.

Semiclassical asymptotics in magnetic Bloch bands

This paper gives a simple construction of wave packets localized near semiclassical trajectories for an electron subject to external electric and magnetic fields. We assume that the magnetic and electric potentials are slowly varying perturbations of the potential of a constant magnetic field and a periodic lattice potential, respectively.

SPECTRAL SHIFT FUNCTION FOR OPERATORS WITH CROSSED MAGNETIC AND ELECTRIC FIELDS

We obtain a representation formula for the derivative of the spectral shift function ξ(λ; B, ∊) related to the operators and H(B, ∊) = H0(B, ∊) + V(x, y), B > 0, ∊ > 0. We establish a limiting absorption principle for H(B, ∊) and an estimate for ξ′(λ; B, ∊), provided λ ∉ σ(Q), where .

Développements asymptotiques des perturbations lentes de l'opérateur de Schrödinger périodique

In the semi-classical regime we study the eigenvalues of the operators where V is periodic with respect to a lattice and is bounded with all its derivatives. A(x) is a magnetic potential such that all derivates of non-vanishing order are bounded. We obtain an esymptotic expansion in powers of h of tr where I is an interval disjoint from the essential spectrum. In the case of a simple band we give explicitly the coefficients of this expansion.

Resonances for magnetic Stark Hamiltonians in two-dimensional case

We study the resonances of the two-dimensional Schrodinger operator P1(B;β)=(Dx−By)2+Dy2+βx+V(x,y), B > 0, β > 0, with constant magnetic and electric fields. We define the resonances of P 1 (B; β) and the spectral shift function ξ(λ) related to P 1 (B; β) and P 0 (B; β) = P 1 (B; β) − V(x, y) without any restriction on B and β. For strong magnetic fields (B → ∞) we obtain a representation of the derivative of ξ(λ), a trace formula for tr(f(P1(B;β))−f(P0(B;β))) and an upper bound for the number of the resonances lying in {z∈3/8:|ℜz−(2n−1)B|≤αB, Imz≥μImθ}, 0 0 is a constant independent of B and n ∈ ℕ* = ℕ \ {0}.

The quantum electrodynamics of relativistic bound states with cutoffs. I.

In this note, we consider a Hamiltonian with ultraviolet and infrared cutoffs describing the interaction of relativistic electrons and positrons in a Coulomb potential with transversal photons in Coulomb gauge. We prove that the Hamiltonian is self-adjoint in the Fock space and has a ground state for a sufficiently small coupling constant.

Trace asymptotics via almost analytic extensions

In this work we give a stationary approach to results on trace asymptotics with small remainder estimates, which in this degree of generality are due to Ivrii [I], who followed the more traditional method (for getting sharp estimates) of studying the trace of an associated unitary evolution group. The originality of Ivrii’s results is that he does not require any parametrix constructions, and this permitted him to get very general results. For earlier results with parametrix constructions see Hormander [H], Chazarain [C], Helfer-Robert [HeR]. That stationary methods can also produce sharp remainder estimates is known. See Metivier [M].

DÉVELOPPEMENTS ASYMPTOTIQUES DE L'OPÉRATEUR DE SCHRÖDINGER AVEC CHAMP MAGNÉTIQUE FORT

We consider the Schrödinger operator P B (V) = P B + V, in L 2 (R 2d ) where P B = (i ∇ + BA(x))2. Here V is a decreasing potential and B is a large constant. We assume that the magnetic field “dA(x)” is constant. For f ∈ C 0 ∞ (I), where I is an open interval noncontaining the origin, we obtain an asymptotic expansion in powers of B −1, of tr (f(P B (V) − BΛ)), when B → + ∞. Here B Λ is a fixed Landau level of σ (P B ). We give plicitly the two leading terms. Hence, we get precise remainder estimates for the counting function of eigenvalues of P B (V) near BΛ. We apply these results to the Pauli operator in the two-dimensional case.