A new type of connected component created by singular potentials

Prof. Augusto Ponce
Université Catholique de Louvain

Abstract: The Schrödinger operator −∆ + V in R^N has been extensively studied for potentials V in L^∞ e L^p with any exponent p > N/2. In a joint work with L. Orsina, we have investigated what happens to potentials V:Ω → [0, +∞] that are merely Borel-measurable in an open subset Ω⊂R^N and discovered that, correctly interpreted, the strong maximum principle is still available no matter how singular the potential is. The heart of the matter consists in splitting Ω as a combination of Sobolev-connected components that exclude the set of points where the Green's function is not available. Surprisingly, such a decomposition propagates to any function in W^1,2(Ω)∩L¹(Ω;V dx).