# Eigenvalue estimates without Bakry-Emery-Ricci bounds

## TL;DR

This paper derives lower bounds for the eigenvalues of a Laplace-Beltrami operator with an $L^
Infty$-drift, without requiring self-adjointness or regularity assumptions, extending spectral estimates beyond Bakry-Emery-Ricci bounds.

## Contribution

It provides new eigenvalue estimates for Laplace-Beltrami operators with drift, independent of Bakry-Emery-Ricci bounds and regularity conditions.

## Key findings

- Lower bounds for eigenvalues with $L^
Infty$-drift
- Results hold without self-adjointness assumptions
- Spectral bounds depend only on geometry and drift magnitude

## Abstract

We establish a lower bound for the real eigenvalues of a Laplace-Beltrami operator with an $L^\infty$-drift term. We make no assumptions that the operator is self-adjoint or that the drift has any additional regularity. In the case where the operator is self-adjoint, this establishes a lower bound on the spectrum without assuming a lower bound for the Bakry-Emery Ricci tensor. Put colloquially, this result states that no matter which way the wind blows, heat will diffuse at a definite rate depending only on the geometry of the underlying space and the maximal wind speed.