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.

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, let $0 < \epsilon \ll 1$ and take any Hermitian operator $P$. Then we can construct an example Hamiltonian $$ H = -\sum_{(i,j)} \sigma^{(z)}_i \sigma^{(z)}_j + \epsilon P t $$ on a 1D spin chain. The all-up state and the all-down are ground-states for $t = 0$; but 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 $$\Pi [H(T), \dot H(T)] = 0$$ — 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)?