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Qualitative analysis of integrable Hamiltonian systems.
Qualitative analysis of highly excited vibrational, rotational-vibrational states of molecules.
Monodromy in quantum and classical physics.
Qualitative analysis is based on the study of symmetry of the problem, and the topology of the phase space (or the rotational energy of the surface) for integrable Hamiltonian systems (or molecules, respectively). Thus, a relatively small data set on the problem allows one to make some qualitative conclusions for the whole class of molecules, in particular, on energy levels clusterization.
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| For Manakov top problem with one concrete choice of parameters the classical energy-momentum diagram is represented in Fig. 1 together with the joint spectrum of two commuting quantum operators. The parameters were chosen so that to produce the most symmetric form of the image of EM map and to have a sufficient number of quantum states to see the characteristic pattern in each qualitatively different part of the
diagram.
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| The analysis of the possible propagation of the quantum cell through the joint quantum spectrum
of two commuting observables for the Manakov top model was studied in Zhilinskii B.I., Sinitsyn E.A.
paper "Qualitative Analysis of the Classical and Quantum Manakov Top" for the first
time. The presentation of the material here is done on completely heuristic physical ground.
Nevertheless, the authors hope that our result about quantum monodromy will find more serious
description in classical as well as in quantum mechanics. We believe that further analysis will
lead to the formulation of new important qualitative features of classical and quantum problems
and allow to make further important steps in formulating general qualitative theory of highly
excited quantum systems which is the ultimate goal of the authors.
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