Fourth-Order Operators on Manifolds with Boundary

TL;DR

This paper investigates boundary conditions for fourth-order elliptic operators on manifolds with boundary, establishing conditions for self-adjointness and analyzing quantum field divergences under these conditions.

Contribution

It characterizes boundary conditions ensuring self-adjointness of the squared Laplace operator and evaluates quantum divergences for these conditions in simplified models.

Findings

Boundary conditions for self-adjointness are identified for the squared Laplace operator.

One-loop divergences are computed for various boundary conditions in a flat Euclidean space model.

Alternative boundary conditions beyond Dirichlet are shown to preserve self-adjointness and influence quantum divergences.

Abstract

Recent work in the literature has studied fourth-order elliptic operators on manifolds with boundary. This paper proves that, in the case of the squared Laplace operator, the boundary conditions which require that the eigenfunctions and their normal derivative should vanish at the boundary lead to self-adjointness of the boundary-value problem. On studying, for simplicity, the squared Laplace operator in one dimension, on a closed interval of the real line, alternative conditions which also ensure self-adjointness set to zero at the boundary the eigenfunctions and their second derivatives, or their first and third derivatives, or their second and third derivatives, or require periodicity, i.e. a linear relation among the values of the eigenfunctions at the ends of the interval. For the first four choices of boundary conditions, the resulting one-loop divergence is evaluated for a real scalar field on the portion of flat Euclidean four-space bounded by a three-sphere, or by two concentric three-spheres.