Brownian motion,Lp properties of Schrödinger operators and the localization of binding

We use Brownian motion ideas to study Schrödinger operators $\text{H =}\text{built?1}\text{2}\text{Δ + V}\text{on}\text{L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{v}}\text{)}$. In particular: (a) We prove that lim_t→∞t^?1In ∥ e^?tH ∥_p,p is p-independent for a very large class of V's where ∥ A ∥_p,p = norm of A as an operator from L^pto L^p. (b) For v ? 3 and $\text{V ? L}{}^{\mathrm{<mtext>v</mtext><mtext>2\; ?\; ?</mtext>}}\text{∩ L}{}^{\mathrm{<mtext>v</mtext><mtext>2\; +\; ?</mtext>}}$, we show that sup ∥ e^?tH ∥_∞,∞ < ∞ if and only if H has no negative eigenvalues or zero energy resonances. (c) We relate the “localization of binding” recently noted by Sigal to Brownian hitting probabilities.