On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates

Abstract

We consider the Schrödinger type operator A = (1+|x| α )∆−|x| β , for α ∈ [0, 2] and β ≥ 0. We prove that, for any p ∈ (1, ∞), the minimal realization of operator A in L p (R N ) generates a strongly continuous analytic semigroup (Tp(t)) t≥0 .For α ∈ [0, 2) and β ≥ 2, we then prove some upper estimates for the heat kernel k associated to the semigroup (Tp(t)) t≥0 . As a consequence we obtain an estimate for large |x| of the eigenfunctions of A. Finally, we extend such estimates to a class of divergence type elliptic operators.2000 Mathematics Subject Classification. 47D07; 35J10, 35K05, 35K10.

On a polynomial scalar perturbation of a Schrödinger system in L-spaces

“…Furthermore, we obtain sufficient conditions for analyticity of the semigroup and compactness of the resolvent. Motivated by the kernel estimates existing in literature for scalar Schrödinger operators, see for instance [16,22,23,15,13,9,10], we prove Gaussian and other kernel estimates of the obtained semigroups. As a consequence we study in the symmetric case the asymptotic distribution of the eigenvalues.…”

Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces

“…Let us begin with the generation results for suitable realizations L p of the operator L in L p (R N ), 1 < p < ∞. Such results have been proved in [6,9,11]. More specifically, the case α ≤ 2 has been investigated in [6] for 1 < α ≤ 2 and in [9] for α ≤ 1, where the authors proved the following result.…”

Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms

“…In [11] (resp. [3]) it is proved that the realization Ap of A in L p (R N ) for 1 < p < ∞ with domain Dp(A) = {u ∈ W 2,p (R N ) | (1+|x| α )|D 2 u|, (1+|x| α ) 1/2 ∇u, |x| β u ∈ L p (R N )} generates a strongly continuous and analytic semigroup Tp(·) for α ∈ [0, 2] and β > 0 (resp.…”

“…[2], [12]). In [11] estimates of the kernel k(t, x, y) for α ∈ [0, 2) and β > 2 were obtained. Our contribution in this paper is to show similar upper bounds for the case α ≥ 2 and β > α − 2.…”