The Heat Equation for the Generalized Hermite and the Generalized Landau Operators

Abstract

We give a formula for the one-parameter strongly continuous semigroups e −tL λ and e −tà , t > 0 generated by the generalized Hermite operator L λ , λ ∈ R\{0} respectively by the generalized Landau operator A. These formula are derived by means of pseudo-differential operators of the Weyl type, i.e. Weyl transforms, Fourier-Wigner transforms and Wigner transforms of some orthonormal basis for L 2 (R 2n ) which consist of the eigenfunctions of the generalized Hermite operator and of the generalized Landau operator. Applications to an L 2 estimate for the solutions of initial value problems for the heat equations governed by L λ respectivelyÃ, in terms of L p norm, 1 ≤ p ≤ ∞ of the initial data are given. Mathematics Subject Classification (2010). Primary 47G10, 47G30; Secondary 35S10.Following Wong's point of view (see [6], by Wong), we give a formula for the one-parameter strongly continuous semigroup e −tL λ , t > 0, generated by the generalized Hermite operator L λ , for a fixed λ ∈ R \ {0}, in terms of the Weyl transforms. Then we use it to obtain an L 2 estimate for the solution of the initial value problem for the heat equation governed by L λ , in terms of the L p norm, 1 ≤ p ≤ ∞, of the initial data.Similar results have also been derived for the generalized Landau operatorà which was firstly introduced by M.A. de Gosson (see [3] by de Gosson) who has studied its spectral properties.This work was completed with the support of our T E X-pert.

On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces

“…To compare with other results in the literature, we observe that also in [31,32], and in [6] the authors use Wigner distributions and pseudodifferential operators as tools for their main results. In the latter paper the author gives a formula for the oneparameter strongly continuous semigroup e −t H β in terms of the Weyl transforms of a L 2 -orthonormal basis made of generalized Hermite eigenfunctions.…”

“…Well-posedness of the heat equation has been studied by many authors, see e.g. [10,21] and the many contributions by Wong, for instance [31,32], see also [6]. In particular, heat equations associated to fractional Hermite operators were recently studied in [3], for related results see also [5].…”