The dressed nonrelativistic electron in a magnetic field

We consider a nonrelativistic electron interacting with a classical magnetic field pointing along the $x_{3}$-axis and with a quantized electromagnetic field. When the interaction between the electron and photons is turned off, the electronic system is assumed to have a ground state of finite multiplicity. Because of the translation invariance along the $x_{3}$-axis, we consider the reduced Hamiltonian associated with the total momentum along the $x_{3}$-axis and, after introducing an ultraviolet cutoff and an infrared regularization, we prove that the reduced Hamiltonian has a ground state if the coupling constant and the total momentum along the $x_{3}$-axis are sufficiently small. Finally we determine the absolutely continuous spectrum of the reduced Hamiltonian.Comment: typos correction

Growth of Sobolev norms for abstract linear Schrodinger equations

We prove an abstract theorem giving a (t)ϵ bound (for all ϵ > 0) on the growth of the Sobolev norms in linear Schrodinger equations of the form i Ψ = H0ψ + V(t)ψ as t → ∞. The abstract theorem is applied to several cases, including the cases where (i) H0 is the Laplace operator on a Zoll manifold and V (t) a pseudodifferential operator of order smaller than 2; (ii) H0 is the (resonant or nonresonant) harmonic oscillator in Rd and V (t) a pseudodifferential operator of order smaller than that of H0 depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of [MR17]

Multi-site study of a new approach to farm work within the framework of organic vegetable production: permanent crop beds

The mineralization rate of a commercial organic fertiliser was evaluated over the course of three years in an organic rice field in the Camargue (France). The effect of different mounts of fertiliser applied at different periods was tested. The organic fertiliser rapidly mineralised under flooded conditions. On the basis of this result, we demonstrated that an adaptation of organic fertilisation practices, similar to those employed for mineral fertilisers, would result in the optimisation of organic fertilisers, leading to improved profitability

Skew-self-adjoint discrete and continuous Dirac type systems: inverse problems and Borg-Marchenko theorems

New formulas on the inverse problem for the continuous skew-self-adjoint Dirac type system are obtained. For the discrete skew-self-adjoint Dirac type system the solution of a general type inverse spectral problem is also derived in terms of the Weyl functions. The description of the Weyl functions on the interval is given. Borg-Marchenko type uniqueness theorems are derived for both discrete and continuous non-self-adjoint systems too

Characterization of the pressure fluctuations within a Controlled-Diffusion airfoil boundary layer at large Reynolds numbers

The present investigation targets the generation of airfoil trailing-edge broadband noise that arises from the interaction of turbulent boundary layer with the airfoil trailing edge. Large-eddy simulations, carried out using a massively parallel compressible solver CharLESX, are conducted for a Controlled-Diffusion (CD) airfoil with rounded trailing edge for seven configurations, characterized with a Reynolds number, angle of attack and Mach number. An analysis of the unsteady pressure signals in the boundary layer is proposed in regard to classical trailing edge noise modelling ingredients

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ[much less-than]1,K [much greater-than] 1, s > 1, we construct smooth initial data u 0 with ||u0||Hs , so that the corresponding time evolution u satisfies u(T)Hs[greater than]K at some time T. This growth occurs despite the Hamiltonian’s bound on ||u(t)||H1 and despite the conservation of the quantity ||u(t)||L2. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems

Reducibility of Schrödinger Equation on the Sphere

In this article we prove a reducibility result for the linear Schrödinger equation on the sphere Sn with quasi-periodic in time perturbation. Our result includes the case of unbounded perturbation that we assume to be of order strictly less than 1/2 and satisfying some parity condition. As far as we know, this is one of the few reducibility results for an equation in more than one dimension with unbounded perturbations. Letus note that, surprisingly, our result does not require the use of the pseudodifferential calculus although the perturbation is unbounded

Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential

In this article, we prove a reducibility result for the linear Schrödinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less than or equal to 1/2. As far as we know, this is the first reducibility result for an unbounded perturbation on a compact manifold different from the torus

Long-Time Existence for Semi-linear Beam Equations on Irrational Tori

We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n- 1 with n≥ 3 and d≥ 2. If ε≪ 1 is the size of the initial datum, we prove that the lifespan Tε of solutions is O(ε-A(n-2)-) where A≡A(d,n)=1+3d-1 when n is even and A=1+3d-1+max(4-dd-1,0) when n is odd. For instance for d= 2 and n= 3 (quadratic nonlinearity) we obtain Tε=O(ε-6-), much better than O(ε- 1) , the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of Δ2+1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step