Schr\"odinger operators and unique continuation. Towards an optimal result

TL;DR

This paper proves a unique continuation property for solutions of certain differential inequalities involving Schrödinger operators with a broad class of potentials, advancing understanding of when solutions are uniquely determined by their behavior.

Contribution

It establishes unique continuation for solutions of | abla^2 u| leq |V u| with potentials in L^{d/2, inf}_{loc} and solutions in a space containing all eigenfunctions, extending previous results.

Findings

Unique continuation holds for a wide class of potentials including L^{d/2, inf}_{loc}

Solutions in a specific function space exhibit quasianalyticity

Advances understanding of Schrödinger operators and solution uniqueness

Abstract

In this article we prove the property of unique continuation (also known for C^\infty functions as quasianalyticity) for solutions of the differential inequality |\Delta u| \leq |Vu| for V from a wide class of potentials (including L^{d/2,\infty}_{\loc}(R^d) class) and u in a space of solutions Y_V containing all eigenfunctions of the corresponding self-adjoint Schr\"odinger operator. Motivating question: is it true that for potentials V, for which self-adjoint Schr\"odinger operator is well defined, the property of unique continuation holds?