  Remarkably we find that for a ring with linear boundary conditions such that
the eigenvector and its derivative are continuous, there does not seem to be a
way for the well-known de Broglie relation to be gauge invariant. Certain
nonlinear boundary conditions assure gauge invariance, and lead to
eigenfunctions with a discontinuous but differentiable phase and a continuous
spectrum. A discrete subset of this spectrum forms a Hilbert space, while
another subset is excluded by the nonlinear boundaries. We conclude that
discontinuous momentum eigenfunctions are tenable, and that it is possible that
quantum mechanics can have nonlinear boundary conditions in some circumstances.
