Reviews in Mathematical Physics

We study local minimizing solutions for the Faddeev–Skyrme problem on the 3-sphere with radius R. We show that for the Hopf soliton is a local minimizing solution and all the other local minimizing solutions near the Hopf soliton are obtained as the composition of isometries of S³ with the Hopf soliton.

We prove a general theorem about the self-adjointness and domain of Pauli–Fierz type Hamiltonians. Our proof is based on commutator arguments which allow us to treat fields with non-commuting components. As a corollary, it follows that the domain of the Hamiltonian of non-relativistic QED with Coulomb interactions is independent of the coupling constant.

In the framework of asymptotically flat spacetimes, we discuss the notion of a covariant scalar free field at null infinity. We construct the associated equations of motion exploiting either the theory of induced representations or the white noise analysis. Eventually, we prove that these equations can also be derived as the minimum of a suitable Lagrangian and that a real scalar field theory at null infinity admits a Hamiltonian formulation.

We study the evolution of a one-dimensional model atom with δ-function binding potential, subjected to a dipole radiation field E(t)x with E(t) a 2π/ω-periodic real-valued function. We prove that when E(t) is a trigonometric polynomial, complete ionization occurs, i.e. the probability of finding the electron in any fixed region goes to zero as t → ∞.

For ψ(x, t = 0) compactly supported and general periodic fields, we decompose ψ(x, t) into uniquely defined resonance terms and a remainder. Each resonance is 2π/ω periodic in time and behaves like the exponentially growing Green's function near x = ±∞. The remainder is given by an asymptotic power series in t^-1/2 with coefficients varying with x.

We study a class of continuous matrix-valued Anderson models acting on L²(ℝ^d) ⊗ ℂ^N. We prove the existence of their Integrated Density of States for any d ≥ 1 and N ≥ 1. Then, for d = 1 and for arbitrary N, we prove the Hölder continuity of the Integrated Density of States under some assumption on the group G_μE generated by the transfer matrices associated to our models. This regularity result is based upon the analoguous regularity of the Lyapounov exponents associated to our model, and a new Thouless formula which relates the sum of the positive Lyapounov exponents to the Integrated Density of States. In the final section, we present an example of matrix-valued Anderson model for which we have already proved, in a previous article, that the assumption on the group G_μE is verified. Therefore, the general results developed here can be applied to this model.