In a quantum system with constant parameters and dynamics in bounded
region the Schrodinger equation gives rise to a discrete
spectrum of eigenvalues, the allowed energy levels.
In systems with periodic variation of parameters in time,
and in discrete time systems, instead of the energy spectrum
people introduce the ** spectrum of quasienergies**.
Let the evolution of state vector over a period
t
is governed by a unitary operator

For maps, t
is one step of discrete time. As the evolution operator for quantum map on torus
is represented by a matrix *N*x*N*, the spectrum must contain
*N* eigenvalues
l* _{s}*=exp(

The trace of the matrix Tr(**U**) is equal to the sum
of all the eigenvalues, and for the matrix **U**^{k}
it is a sum of their *k*-th powers:

The "trace-последовательность" *S _{k}* has the period

It is seen that the coefficient at the *r*-th term of the
expansion

is a number of degeneracy for the eigenvalue
w* ^{r}* in the spectrum of the
evolution operator.
As follows, to obtain the spectrum of the evolution operator
it is sufficient to take the trace sequence and produce the discrete Fourier
transform. The very straightforward way is to compute subsequent powers of the
matrix

The next figure presents comparison of the
dependencies of quasienergies on the quantum parameter *N*
for the maps of hyperbolic (a), parabolic (b), and elliptic (c) type.

In the distribution of the levels of quasienergy of maps with regular dynamics one can distinguish ordered structures, but in the Arnold cat map ("chaos") such structures are obviously not visible.

The next interesting question: Can we see any
difference in the structure of eigenvectors in the quantum
maps with regular and chaotic dynamics
of classic analogs? Let us take some probe vector
|y_{0}> and, acting with the
evolution operator **U** step by step construct the sequence
|y* _{k}*>=

As "coefficients" in this expression we have the (non-normalized) eigenvectors of the evolution operator. A vector

corresponds to the eigenvalue
l* _{s}*=exp(2p

The figure below shows spatial distributions and
Husimi distributions for several eigenvectors of the quantum Arnold cat map at
*N*=131. For comparison the next two pictures
show examples of eigenvectors for the parabolic and elliptic maps
(regular dynamics in classic analogs).

Arnold's cat map

Parabolic map

Elliptic map

of theoretical nonlinear

dynamics

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