It's folklore dating back to von Neumann and Wigner that [time-dependent Hamiltonian systems tend not to have level crossings of their energy eigenvalues](http://en.wikipedia.org/wiki/Avoided_crossing). However, we can of course consider smoothly varying Hamiltonians which have been engineered to have level crossings. These don't even have to be complicated: for instance, any Hamiltonian of the form
$$ H = -\sum_{(i,j)} \sigma^{(z)}_i \sigma^{(z)}_j + \epsilon P t $$
on a 1D spin chain, for $0 < \epsilon \ll 1$ and any Hermitian operator $P$, is an example: the all-up state and the all-down are ground-states for $t = 0$, though for $t \ne 0$ such symmetry is typically broken, so that for $|t| \ll 1$ we expect to have eigenstates close to the all-up and all-down states but with distinct eigenvalues.

After some investigation, I've come to suspect the following:

> **Conjecture.** If $H$ has a level crossing between energy levels $E_0, E_1$ at some time $T$, and $\Pi$ is the projection onto the span of the $E_0$- and $E_1$-eigenstates then there are well-defined (continuously varying) eigenvectors through the level crossing only if $\Pi [H(T), \dot H(T)] = 0$ &mdash; that is, if the change in the Hamiltonian at the level crossing is only a change in the values of the two crossing eigenvalues, for a common pair of eigenvectors. Specifically: if this equality does not hold, then any (unitary) time-dependent change-of-basis operator from the standard basis to the energy eigenbasis of $H$ at time $t$ which is continuous for some neighborhood $t \in (T,T+\epsilon]$ will oscillate infinitely rapidly as $t \to T$.

 * Is this true generally? (If not, can you point to a counterexample?)

 * Is this a known result, and is there a reference that I can refer to where this question is treated clearly, and more-or-less formally for bounded operators (e.g. on finite-dimensional systems)?