Disheveled Arnold’s cat and the problem of quantum–classic correspondence

Quantum Arnold’s cat map is studied for a case of perfect square inverse Planck’s constant, N = M2. The classic limit is analyzed on a subset of numbers N increasing as 4k. The quantum problem in this case allows exact reduction to the classic cat map defined on a discrete lattice of size M × M and supplemented by evolution of a phase variable. A link between the classic periodic orbits and spectrum of eigenvalues of the quantum evolution operator is outlined. For M growing as 2k genetic analysis is developed for periodic orbits, and they are classified by means of a tree-like graph. A phase shift, accumulated over a period of the orbits, evolves from level to level of the graph according to a certain rule, governed by non-periodic binary code. Representation of a localized Gaussian wave packet in a basis of eigenvectors of the evolution operator gives rise to a probability measure distributed on a unit circle, where the eigenvalues are located. This measure looks like spectrum of a finite-time sample of a stationary random process (periodogram): (1) majority of the eigenstates have intensities of comparable order of magnitude, (2) the spectral distribution is of locally random-like nature, i.e. statistical variance of the amplitudes has the same order as the amplitudes themselves. This combination of properties in very straightforward manner follows from chaotic nature of the classic map and is conjectured to be the most fundamental attribute of quantum chaos.