For a general subcritical second-order elliptic operator $P$ in a domain $Omega subset mathbb{R}^n$ (or noncompact manifold), we construct Hardy-weight $W$ which is optimal in the following sense. The operator $P - lambda W$ is subcritical in $Omega$
for all $lambda < 1$, null-critical in $Omega$ for $lambda = 1$, and supercritical near any neighborhood of infinity in $Omega$ for any $lambda > 1$. Moreover, if $P$ is symmetric and $W>0$, then the spectrum and the essential spectrum of $W^{-1}P$ are equal to $[1,infty)$, and the corresponding Agmon metric is complete. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation $Pu=0$, the existence of which depends on the subcriticality of $P$ in $Omega$.
We study the adiabatic response of open systems governed by Lindblad evolutions. In such systems, there is an ambiguity in the assignment of observables to fluxes (rates) such as velocities and currents. For the appropriate notion of flux, the formul
as for the transport coefficients are simple and explicit and are governed by the parallel transport on the manifold of instantaneous stationary states. Among our results we show that the response coefficients of open systems, whose stationary states are projections, is given by the adiabatic curvature.
We study strong ratio limit properties of the quotients of the heat kernels of subcritical and critical operators which are defined on a noncompact Riemannian manifold.
