The Heat Kernel and Green Function of the Generalized Hermite Operator, and the Abstract Cauchy Problem for the Abstract Hermite Operator

Abstract

Following Wong’s point of view (see [14], by Wong) we give a formula for the heat kernel of the generalized Hermite operator Lλ on \(\mathbb{R}^2\) , \(\lambda \, \epsilon \, \mathbb{R}\, \backslash \,\{0\}\) .This formula is derived by means of pseudo-differential operators of the Weyl type, i.e., Weyl transforms, Fourier-Wigner transforms and Wigner transforms of generalized Hermite functions, which are the eigenfunctions of the generalized Hermite operators and form an orthonormal basis of \(L^2(\mathbb{R}^2)\) (see [2], by Catană). By means of the heat kernel, we give a formula for the Green function of Lλ , \(\lambda \, \epsilon \, \mathbb{R}\, \backslash \,\{0\}\) . Using the Green function and the heat kernel we give some applications concerning the global hypoellipticity of Lλ in the sense of Schwartz distributions, the ultracontractivity and the hypercontractivity of the strongly continuous one-parameter semigroup \(e{^{-tL}}^{\lambda}\), t > 0, \(\lambda \, \epsilon \, \mathbb{R}\, \backslash \,\{0\}\) . We also give a formula for the one-parameter strongly continuous semigroup e-tA generated by the abstract Hermite operator A. The formula is derived by means of the abstract Weyl operators, the abstract Fourier-Wigner operator and the abstract Wigner operators.