I would like to know sufficient conditions for a non-zero eigengap of a time-dependent Hamiltonian.
Suppose we have a time-dependent Hamiltonian $H(t)$ defined as follows: $$H(t) = (1-s(t))H_{init} + s(t)H_{final}$$ where $0 \leq s(t) \leq 1$ is monotonically increasing continuous function for $t \in [0,T]$. Suppose that we know and can prepare the ground state $|\psi_0\rangle$ of $H_{init}$. The final goal is to evolve $|\psi_0\rangle$ into the ground state of $H_{final}$. For this to work, the minimum eigengap of $H(t)$ must be non-zero for all $t \in [0,T]$.
I know that the necessary condition for non-zero eigengap is the non-commutativity of $H_{init}$ and $H_{prob}$. However, I don't know if this is a sufficient condition, i.e., the non-commutativity $\implies$ non-zero eigengap?