I am reading an [article](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.74.115406) about [Bloch-Floquet state](http://www.google.com/search?as_q=floquet+bloch+state). My questions is in *Part II.B* and *Appendix A* of this paper, I will describe them below.

The original Schordinger equation we consider is: 

$$i\hbar\frac{\partial}{\partial t} \tilde{\Psi}(r,t)=\tilde{H}(t)\tilde{\Psi}(r,t)$$

where: 

$$\tilde{H}(t)=\frac{1}{2m_e}(\frac{\hbar}{i}\nabla+\frac{e\vec{A}(t)}{c})^2+V_c(r) $$

with $A(t)$ periodic in time and $V_c(r)$ periodic in space. 

According to Floquet theorem, this time-periodic Hamiltonian has the  wave function in the form:

$$\tilde{\Psi}(r,t)=e^{-i\tilde{\epsilon}(k)t/\hbar}e^{ik\cdot r}\tilde{\phi}_{\tilde{\epsilon},k}(r,t)$$
with $\tilde{\phi}_{\tilde{\epsilon},k}(r,t)$ periodic in space and time, we call $\tilde{\epsilon}(k)$ the Bloch-Floquet quasienergy.

The author did a following transform to avoid dealing with the square term of $A(t)$:

$$\tilde{\Psi}(r,t)=\exp(-\frac{ie^2}{2m_e\hbar c^2}\int^t dt'A^2(t'))\Psi(r,t)$$

Substitute this to the original Schordinger equation we arrive at:

$$i\hbar\frac{\partial}{\partial t} \Psi(r,t)=H(t)\Psi(r,t)$$

where:

$$H(t)=H_0+\frac{e}{m_ec}\vec{A}(t)\cdot \frac{\hbar}{i}\nabla$$
$H_0$ is the field free Hamiltonian.
Finally, the author claimed that the physics quantities are invariant under such a transformation and give an example of the invariance  of the current density(I can see the identity).

My question is:

 - Is this really true for all observables, how is this related to gauge invariance?

 - After the transformation, the author is using this new Schordinger equation to calculate the Bloch-Floquet quasienergy, is these qusienergies same as the ones obtained using the original Schordinger equation?

 - If I calculate this quantity $\int d\vec{r}\Psi^*H\Psi$ and $\int d\vec{r}\tilde{\Psi}^*\tilde{H}\tilde{\Psi}$ , what's the physical meaning of them? One can easily see that they are not equal. Also what's the difference and relation between this quantity and
the floquet energy?

Please help if you have answer(s), also any comments are welcomed.