Sharp Gaussian bounds and 𝐿^{𝑝}-growth of semigroups associated with elliptic and Schrödinger operators

Ouhabaz, El Maati

doi:10.1090/S0002-9939-06-08430-9

Sharp Gaussian bounds and $L^p$-growth of semigroups associated with elliptic and Schrödinger operators
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Proc. Amer. Math. Soc. 134 (2006), 3567-3575 Request permission

Abstract:

We prove sharp large time Gaussian estimates for heat kernels of elliptic and Schrödinger operators, including Schrödinger operators with magnetic fields. Our estimates are then used to prove that for general (magnetic) Schrödinger operators $A=-\sum _{k = 1}^d (\tfrac {\partial }{\partial x_k}-i b_k)^2 + V$, we have the $L^\infty$-estimate (for large $t$): \[ \Vert e^{-tA} \Vert _{{\mathcal L}(L^\infty (\mathbb {R}^d))} \le C e^{-s(A)t} ( t\ln t)^{d/4}\] where $s(A) := \inf \sigma (A)$ is the spectral bound of $A.$ The same estimate holds for elliptic and Schrödinger operators on general domains.

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