Integral kernels of Schrödinger semigroups with nonnegative locally bounded potentials
Streszczenie
We give upper and lower estimates of the heat kernels for Schrödinger operators $H=-\Delta +V$ with nonnegative and locally bounded potentials $V$ in $\mathbb{R} ^d$, $d \geq 1$. We observe a factorization: the contribution of the potential is described separately for each spatial variable, and the interplay between the spatial variables is seen only through the Gaussian kernel – optimal in the lower bound and nearly optimal in the upper bound. In some regimes we observe the exponential decay in time with the rate corresponding to the bottom of the spectrum of $H$. The upper estimate is more local; it applies to general potentials, including confining ones (i.e. $V(x) \to \infty $ as $|x| \to \infty $) and decaying ones (i.e. $V(x) \to 0$ as $|x| \to \infty $), even if they are nonradial, and their mixtures. The lower bound is specialized to the confining case, and the contribution of the potential is described in terms of its radial upper profile. Our results take the sharpest form for confining potentials that are comparable to radial monotone profiles with sufficiently regular growth – in this case they lead to two-sided qualitatively sharp estimates. In particular, we describe the large-time behaviour of nonintrinsically ultracontractive Schrödinger semigroups – this has been a long-standing open problem. Our methods combine probabilistic techniques with analytic ideas.
