= Solution
The <Hamiltonian operator> annihilates every vector orthogonal to both of its defining states:
$$
H_G|\phi\rangle=\Delta\bigl[|\psi\rangle\langle\psi|\phi\rangle+|w\rangle\langle w|\phi\rangle\bigr]=0.
$$
Every positive power also annihilates that vector. The exponential series therefore gives
$$
\boxed{e^{-iH_Gt}|\phi\rangle=|\phi\rangle.}
$$
It is a stationary zero-energy state under this <unitary time evolution>. The displayed evolution uses $\hbar=1$, as in the question. For $N=1$ the two independent defining vectors already span the whole qubit space, so no nonzero vector satisfies both orthogonality conditions; for larger $N$ their orthogonal complement has dimension $2^N-2$.
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