Alessandro Carbotti: “A Brunn-Minkowski inequality for Schrödinger operators with Kato class potentials”

In this talk we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schrödinger type operator H ∶= − div(A ∇) + V, where V is convex and Kato decomposable, using the trace class property of the generated semigroup. As a consequence, using the ultracontractivity of the
semigroup we obtain the log-concavity of the ground state which is also strong log-concave under additional assumptions on Ω and V.