= Solution
Define a transformed <wavefunction> by
$$
\boxed{\widetilde\psi(x)=e^{-iax/\hbar}\psi(x).}
$$
Indeed, writing $\psi=e^{iax/\hbar}\widetilde\psi$ gives
$$
(p-a)\psi=e^{iax/\hbar}p\widetilde\psi,
\qquad H_a\psi=e^{iax/\hbar}\frac{p^2}{2m}\widetilde\psi.
$$
This is the electromagnetic <gauge transformation> generated by $\chi(x)=-A_xx$: $A_x'=A_x+\partial_x\chi=0$ and $\psi'=e^{iq\chi/\hbar}\psi$. The transformed <Hamiltonian operator> is consequently $p^2/(2m)$, but the original <periodic boundary condition> becomes the <twisted boundary condition>
$$
\boxed{\widetilde\psi(x+L)=e^{-iaL/\hbar}\widetilde\psi(x).}
$$
For the domain of the <self-adjoint> kinetic <Hamiltonian operator>, the derivative has the same twist, $\widetilde\psi'(x+L)=e^{-iaL/\hbar}\widetilde\psi'(x)$. Thus the <magnetic flux> has moved into the <boundary condition>. The transformation is a single-valued gauge on periodic representatives only when $aL/\hbar$ is an integer multiple of $2\pi$; otherwise it is understood on the cut interval with the displayed identification of its endpoints.
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