Pseudomeasure character of the ultraspherical semigroups

We prove that the diffusion semigroups generated by the second order differential ultraspherical (Gegenbauer) operator are pseudomeasure operators.     

Sur l'existence des opérateurs d'onde pour l'équation de Wigner dans les espaces L2,pExistence of wave operators for the Wigner equation in L2,p spaces

Basing on the formalism established by Markovich, we show the completeness of wave operators for the Wigner equation in L². In the second part, using estimations proved by Castella and Perthame on the one hand, and the L^p→L^q estimations for the Schrödinger group on the other hand, we prove the existence of the wave operators in L^2,p spaces. To cite this article: H. Emamirad, P. Rogeon, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 811–816.     

Smooth distribution group and Schrödinger equation in Lp(Rn)

In College de France, Seminaire E.D.P. II, Nov. 1963–May 1964, Peetre has introduced the smooth distribution semi-group which is discussed here. This notion is redefined by introducing a functional space $\text{T}$, which measures the regularity of such a distribution. This allows us to give a spectral characterization of such groups. We show that the iterated resolvent of the infinitesimal generator of a smooth distribution group satisfies the relation

$\text{6(λ?A)}{}^{\mathrm{?q}}\text{6L(X)?cq}{}^{\mathrm{k}}\text{|λ|}{}^{\mathrm{k}}\text{|}$

, Re λ ≠ 0. The application of this notion is illustrated by the study of the Schrödinger equation in L^p($\text{R}$ⁿ).     

Chaotic weighted shifts in Bargmann space

This paper deals with the unilateral backward shift operator T on a Bargmann space F(C). This space can be identified with the sequence space ?²(N). We use the hypercyclicity criterion of Bès, Chan, and Seubert and the program of K.-G. Grosse-Erdmann to give a necessary and sufficient condition in order that T be a chaotic operator. The chaoticity of differentiation which correspond to the annihilation operator in quantum radiation field theory is in view, since the Bargmann space is an infinite-dimensional separable complex Hilbert space.     

Pseudomeasure Character of the Ultraspherical Semigroups

We prove that the diffusion semigroups generated by the second order differential ultraspherical (Gegenbauer) operator are pseudomeasure operators.     

On the existence of a periodic solution of nth-order nonlinear differential equations

This note is a generalization of one of a paper by Mehri and Hamedani. Under suitable conditions of ?, the existence of periodic solutions of the nthorder differential equation $\text{x}{}^{\mathrm{(n)}}\text{+ ?(t, x, x′,…, x}{}^{\mathrm{(n\; ?\; 1)}}\text{) = 0}$ is established.     

Solution operators of three time variables for fractional linear problems

It is well known that the time fractional equation where is the fractional time derivative in the sense of Caputo of u does not generate a dynamical system in the standard sense. In this paper, we study the algebraic properties of the solution operator T(t,s,τ) for that equation with u(s) = v. We apply this theory to linear time fractional PDEs with constant coefficients. These equations are solved by the Fourier multiplier techniques. It appears that their solution exhibits some singularity, which leads us to introduce a new kind of solution for abstract time fractional problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Chaotic Tensor Product Semigroups

Suppose {G₁(t)}_{t ≥ 0} and {G₂(t)_{t ≥ 0} be two semigroups on an infinite dimensional separable reflexive Banach space X. In this paper we give sufficient conditions for tensor product semigroup G(t): X → G₂(t)X G₁(t) to become chaotic in L with the strong operator topology and chaotic in the ideal of compact operators on X with the norm operator topology.