On Dunkl Schrödinger semigroups with Green bounded potentials

2022/04/07 by Jacek Dziubański, Agnieszka Hejna, Dziubański, Jacek +1
Mathematics · #33C52 #35J10 #35K08 #39A70 #43A32 #44A20 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2204.03443

openalex publication_date 2022/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

On \mathbb R^N equipped with a normalized root system R, a multiplicity function k(α) > 0, and the associated measure dw(\mathbf x)=∏_{α∈ R}|⟨ \mathbf x,α⟩|^k(α) d\mathbf x, we consider a Dunkl Schrödinger operator L=-Δ_k+V, where Δ_k is the Dunkl Laplace operator and V∈ L¹_{\rm loc} (dw) is a non-negative potential. Let h_t(\mathbf x,\mathbf y) and k^\V\_t(\mathbf x,\mathbf y) denote the Dunkl heat kernel and the integral kernel of the semigroup generated by -L respectively. We prove that k^\V\_t(\mathbf x,\mathbf y) satisfies the following heat kernel lower bounds: there are constants C, c>0 such that h_ct(\mathbf x,\mathbf y)≤ C k^\V\_t(\mathbf x,\mathbf y) if and only if sup_{\mathbf x∈\mathbb R^N} ∫₀^∞ ∫_{\mathbb R^N} V(\mathbf y)w(B(\mathbf x,√(t)))^-1e^{-‖\mathbf x-\mathbf y‖2/t} dw(\mathbf y) dt

Related

• Upper and lower bounds for Dunkl heat kernel
• Integral kernels of Schrödinger semigroups with nonnegative locally bounded potentials
• Semigroup and Riesz transform for the Dunkl- Schrödinger operators
• Two-sided heat kernel estimates for Schrödinger operators with unbounded potentials
• On Schroedinger type operators with unbounded coefficients: Generation and heat kernel estimates