The BMS conjecture

TL;DR

This paper discusses the BMS conjecture, an open problem in differential geometry and mathematical physics, relating positivity of solutions to Schrödinger operators and their self-adjointness on complete Riemannian manifolds.

Contribution

The paper clarifies the BMS conjecture and explores its connection to essential self-adjointness of covariant Schrödinger operators, highlighting its unresolved status for over 14 years.

Findings

Explains the BMS conjecture and its significance.

Connects the conjecture to self-adjointness problems.

Highlights the open status of the conjecture for over 14 years.

Abstract

I explain an open conjecture by Braverman/Milatovic/Shubin (BMS) on the positivity of square integrable solutions $f$ of $(-\Delta+1)f\geq 0$ on a geodescially complete Riemannian manifold, and its connection to essential self-adjointness problems of covariant Schr\"odinger operators. The latter conjecture has remained open for more than 14 years now.