The Cosmological Constant as an Eigenvalue of a Sturm-Liouville Problem and its Renormalization

TL;DR

This paper explores the cosmological constant as an eigenvalue in a Sturm-Liouville problem, using a variational approach with Gaussian wave functionals, one-loop approximation, and zeta function regularization to address divergences and renormalization.

Contribution

It introduces a novel approach linking the cosmological constant to eigenvalues of a Sturm-Liouville problem using variational methods and regularization techniques.

Findings

Eigenvalue formulation of the cosmological constant.

Application of zeta function regularization in curved spacetime.

Renormalization group equations for the cosmological constant.

Abstract

We discuss the case of massive gravitons and their relation with the cosmological constant, considered as an eigenvalue of a Sturm-Liouville problem. A variational approach with Gaussian trial wave functionals is used as a method to study such a problem. We approximate the equation to one loop in a Schwarzschild background and a zeta function regularization is involved to handle with divergences. The regularization is closely related to the subtraction procedure appearing in the computation of Casimir energy in a curved background. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation.