A Riemannian manifold is $L^p$-positivity preserving if any $L^p$ distributional solution of $(-\Delta + 1)u \ge 0$ is necessarily non-negative. In his seminal works on the spectral properties of Schrödinger operators with locally $L^p$ potentials on the Euclidean spaces, T. Kato proved the self-adjointness of such operators by combining his celebrated inequality with the $L^p$-positivity preservation property of $R^n$. While trying to extend Kato's investigations and techniques to Schrödinger operators on Riemannian manifolds, in 2002 M. Braverman, O. Milatovic and M. Shubin conjectured that any complete manifold is $L^2$-positivity preserving.

In this talk, we will first review the different approaches proposed to face this conjecture and the partial results obtained so far. Then we will present a new strategy based on regularity properties of subsolutions of elliptic PDEs, which permits to settle the problem. All the techniques will be mostly presented in the Euclidean setting, so as to avoid geometric technicalities as much as possible. It is a joint work with Stefano Pigola.

Meeting ID: 881 1662 2980 Passcode: 002621