# Non-self-adjointness of the Klein-Gordon operator on a globally hyperbolic and geodesically complete manifold. An example

## TL;DR

This paper presents an example of a globally hyperbolic, geodesically complete Lorentzian manifold where the Klein-Gordon operator is not essentially self-adjoint, challenging assumptions about operator self-adjointness in such spacetimes.

## Contribution

It provides a specific example demonstrating that the Klein-Gordon operator can fail to be essentially self-adjoint even on well-behaved, globally hyperbolic, and geodesically complete manifolds.

## Key findings

- The Klein-Gordon operator is not essentially self-adjoint on the constructed manifold.
- The example challenges common expectations about operator self-adjointness in physically relevant spacetimes.
- The manifold is both globally hyperbolic and geodesically complete, yet exhibits non-self-adjointness of the Klein-Gordon operator.

## Abstract

We describe a Lorentzian manifold that is globally hyperbolic and geodesically complete, but such that the (minimally coupled) Klein-Gordon operator with the standard domain is not essentially self-adjoint.