  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.
