By inserting the results (2.6) and (2.7) for
and
respectively into the
Hamiltonian (2.15) and after integration one gets the following
result for the ground state energy
(2.17)
The first term in the above equation simply gives the mean field energy
(see eq.(1.14)) with the correction due to the presence of
the random external potential. This becomes clearer if we rewrite this
term in terms of the coupling constant
.
The beyond mean-field correction to the ground state energy is
given by
(2.18)
The correction due to disorder in the above result is proportional to