  Quantization procedure of the Gardner-Zakharov-Faddeev and Magri brackets by
means of the fermionic representation for the KdV field is considered. It is
shown that in both cases the corresponding Hamiltonians are given as sums of
two well defined operators. Each of them is bilinear and diagonal with respect
to either fermion, or boson (current) creation-annihilation operators. As a
result the quantization procedure needs no any space cut-off and can be
performed on the whole axis. Existence of the solitonic states in the Hilbert
space as well as quantization of soliton parameters are shown to result from
this approach. As a by-product it is also demonstrated that the dispersionless
KdV is uniquely and explicitly solvable in the quantum case.

Exactly Solvable and Integrable Systems;High Energy Physics - Theory