As known, for the one-dimensional quantum motion the wave function
in position representation
*x*)*p*)
are linked as the forward and backward Fourier transform:

where *h* is the Planck constant.
Let now the functions y(*x*) and
j(*p*)
have periods
*L _{x}* and

Hence, in the momentum representation
the wave function is fully determined by *N*
complex coefficients j_{k},
the factors at
d-peaks.

Analogous consideration may be performed in the inverse order
to conclude that the function y
(*x*) is a "comb" of
d-functions on the *x*-axis
at points *x=hm/L _{p}*, that is with a step
D

Thus,

1) the Hilbert space of states for
our system is the *N*-dimensional complex vector space.

2) Non-contradictory quantization is possible only
under restriction on the parameters of the system
expressed by relation *L _{x}L_{p}=hN*.

As known, one of the basic statements in the quantum mechanics
is a requirement that at
*h*®0 the correspondence
with classic mechanics should take place.
In the dynamics on torus, we cannot treat *h*
as a continuous variable, but we can consider the passage to the limit
on a discrete set of values allowed by the formula
*L _{x}L_{p}=hN*.
In this sense, the classic limit corresponds to

Speaking on the position representation,
we mean now simply an *N*-dimensional vector
y_{m},
and in the momentum representation a vector
j_{k}, respectively.
It is easy to check that these two sets of numbers
y_{m}
and j_{k}
are linked by the discrete Fourier transformation:

Here we use the notion
a=exp(2p*i*/*N*),
which will be used below.
For operators of the forward and backward Fourier transformation
we adopt the symbols **F** and **F**^{+}.
They are represented by matrices of size
*N*x*N*, with elements
*F _{mn}*=a

If the periods *L _{x}* and

In the commonly used Dirac notion,
the column vector is called the *ket-vector*
|*a*>, where *a* is a symbol marking the given state.
A conjugate row vector is called
*bra-vector* <*b*|,
and their scalar product ("bracket")
is <*b*|*a*>.
In our case these are the *N*-component vectors:

|*a*>={*a*_{0}, *a*_{1},...*a _{N}*-1}

of theoretical nonlinear

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