Spectral_function
The spectral function for the normalized state \psi is defined as (sometimes modulo a factor 2*pi):
where H is the Hamiltonian and the overline denotes averaging over disorder configurations. By construction, it is normalized to unity:
and its average value is:
The spectral function can be computed by Fourier transform of the autocorrelation function:
and:
\psi(t) is computed using the Temporal propagation method. For the standard spectral function, \psi(0) is a plane wave, but the program allows any initial state.
One has to specify the required energy resolution (which will give the total propagation time 2\pi/energy_resolution) and the total energy range: because of the performed FFT, the spectrum is folded in an interval Delta E = 2\pi/dt, where dt is the elementary time step for propagation. Thus, the program uses dt=2\pi/energy_range. The spectrum is output in the energy range chosen by the user.
Note that the CPU time spent in the Fourier transform time->energy is usually negligible.
The spectral function can be computed in the interacting case, simply as the Fourier transform of C(t), but there are various possibilities. One may, in the definition of the spectral function, incorporate or not the nonlinear interaction. That is define the function C(t) by:
where the nonlinear evolution operator U(t) is generated by the nonlinear Hamitonian:
\alpha is here an arbitrary real number.
For \alpha=0, one is back to the original Schroedinger definition.
For \alpha=1, the nonlinear evolution is the one of the Gross-Pitaevksii equation.
These definitions have a drwaback. While the spectral function is properly normalized, the average energy:
is not preserved during the temporal evolution.
Another solution is to choose \alpha=1/2, which ensures energy conservation..
Which choice is the best is not clear|-(.