# Essential self-adjointness of perturbed quadharmonic operators on Riemannian manifolds with an application to the separation problem

## TL;DR

This paper establishes conditions under which perturbed quadharmonic operators on Riemannian manifolds are essentially self-adjoint and applies these results to demonstrate the separation property in $L^2$ for functions.

## Contribution

The paper provides new sufficient conditions for the essential self-adjointness of perturbed quadharmonic operators on manifolds with bounded geometry.

## Key findings

- Established essential self-adjointness criteria for $\Delta^4 + V$ operators.
- Proved the separation property in $L^2$ for functions under these operators.
- Extended the theory to operators on sections of Hermitian vector bundles.

## Abstract

We consider perturbed quadharmonic operators, $\Delta^4 + V$, acting on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential $V$ satisfying a bound from below by a non-positive function depending on the distance from a point. Under a bounded geometry assumption on the Hermitian vector bundle and the underlying Riemannian manifold, we give a sufficient condition for the essential self-adjointness of such operators. We then apply this to prove the separation property in $L^2$ when the perturbed operator acts on functions.