Reviews in Mathematical Physics

We determine the representations of the "conformal" group , the restriction of which on the "Poincaré" subgroup are unitary irreducible. We study their restrictions to the "De Sitter" subgroups and (they remain irreducible or decompose into a sum of two) and the contraction of the latter to "Poincaré". Then we discuss the notion of masslessness in n dimensions and compare the situation for general n with the well-known case of 4-dimensional space-time, showing the specificity of the latter.

We consider a semidirect product G=A×′H, with A abelian, and its unitary representations of the form where x₀ is in the dual group of A, G₀ is the stability group of x₀ and m is an irreducible unitary representation of G₀∩H. We give a new selfcontained proof of the following result: the induced representation is square-integrable if and only if the orbit G[x₀] has nonzero Haar measure and m is square-integrable. Moreover we give an explicit form for the formal degree of .

We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall–Chaundy curves, Baker–Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.

A quantum version of Nekhoroshev estimates for Floquet Hamiltonians associated to quasi-periodic time dependent perturbations is developped. If the unperturbed energy operator has a discrete spectrum and under finite Diophantine conditions, an effective Floquet Hamiltonian with pure point spectrum is constructed. For analytic perturbations, the effective time evolution remains close to the original Floquet evolution up to exponentially long times. We also treat the case of differentiable perturbations.