  Based on empirical evidence, quantum systems appear to be strictly linear and
gauge invariant. This work uses concise mathematics to show that quantum
eigenvalue equations on a one dimensional ring can either be gauge invariant or
have a linear boundary condition, but not both. Further analysis shows that
non-linear boundaries for the ring restore gauge invariance but lead
unexpectedly to eigenfunctions with a continuous eigenvalue spectrum, a
discreet subset of which forms a Hilbert space with energy bands. This Hilbert
space maintains the principle of superposition of eigenfunctions despite the
nonlinearity. The momentum operator remains Hermitian. If physical reality
requires gauge invariance, it would appear that quantum mechanics should
incorporate these nonlinear boundary conditions.
