Fourth-order operators on manifolds with a boundary

Recent work in the literature has studied fourth-order elliptic operators o n manifolds with a boundary. This paper proves that, in the case of the squ ared Laplace operator, the boundary conditions which require that the eigen functions and their normal derivative should vanish at the boundary lead to self-adjointness of the boundary-value problem. On studying, for simplicit y, the squared Laplace operator in one dimension, on a closed interval of t he 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 ei genfunctions at the ends of the interval. For the first four choices of bou ndary conditions, the resulting 1-loop divergence is evaluated for a real s calar field on the portion of flat Euclidean 4-space bounded by a 3-sphere, or by two concentric 3-spheres.