  We establish a bijection between the self-adjoint extensions of the Laplace
operator on a bounded regular domain and the unitary operators on the boundary.
Each unitary encodes a specific relation between the boundary value of the
function and its normal derivative. This bijection sets up a characterization
of all physically admissible dynamics of a nonrelativistic quantum particle
confined in a cavity. More- over, this correspondence is discussed also at the
level of quadratic forms. Finally, the connection between this parametrization
of the extensions and the classical one, in terms of boundary self-adjoint
operators on closed subspaces, is shown.
