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We prove sharp large time Gaussian estimates for heat kernels of elliptic and Schrodinger operators, including Schrodinger operators with magnetic fields. Our estimates are then used to prove that for general (magnetic) Schrodinger operators A = - Σ d k=1 (∂ ∂x k -ib k ) 2 + V, we have the L ∞ -estimate (for large t): where s(A):= inf σ(A) is the spectral bound of A. The same estimate holds for elliptic and Schrodinger operators on general domains.

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