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Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary --- or, in general, projective unitary --- representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.
It is shown that planar quantum dynamics can be related to 3-body quantum dynamics in the space of relative motion with a special class of potentials. As an important special case the $O(d)$ symmetry reduction from $d$ degrees of freedom to one degre
In the first part of the paper, we classify linear integrable (multi-dimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $mathbb C$, enjoying natural symmetries and the property that the restriction of their solutions to th
The notion of monodromy was introduced by J. J. Duistermaat as the first obstruction to the existence of global action coordinates in integrable Hamiltonian systems. This invariant was extensively studied since then and was shown to be non-trivial in
We generalize the concept of coherent states, traditionally defined as special families of vectors on Hilbert spaces, to Hilbert modules. We show that Hilbert modules over $C^*$-algebras are the natural settings for a generalization of coherent state
We consider the generic quadratic first integral (QFI) of the form $I=K_{ab}(t,q)dot{q}^{a}dot{q}^{b}+K_{a}(t,q)dot{q}^{a}+K(t,q)$ and require the condition $dI/dt=0$. The latter results in a system of partial differential equations which involve the