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We introduce a $Q$-operator $\mathcal{Q}_z$ for the hyperbolic Calogero--Moser system as a one-parameter family of explicit integral operators. We establish the standard properties of a $Q$-operator, i.e.~invariance of Hamiltonians,…
We consider the canonical map from the Calogero-Moser space to symmetric powers of the affine line, sending conjugacy classes of pairs of n by n matrices to their eigenvalues. We show that the character of a natural C^*-action on the…
We extend our previous analysis of the classical integrable models of Calogero in several respects. Firstly we provide the algebraic resaons of their quantum integrability.Secondly we show why these systems allow their initial value problem…
A complete set of supertraces on the algebras of observables of the rational Calogero models with harmonic interaction based on the classical root systems of B_N, C_N and D_N types is found. These results extend the results known for the…
In this paper we construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is generated…
We continue the study of Calogero-Moser spaces associated with dihedral groups by investigating in more details the equal parameter case: we obtain explicit equations, some informations about the Poisson bracket, the structure of the Lie…
The hidden-variable question is whether or not various properties --- randomness or correlation, for example --- that are observed in the outcomes of an experiment can be explained via introduction of extra (hidden) variables which are…
Use of certain non-commuting variables is considered in first-order differential equations. Superspace variables are discussed within the setting of first-order ordinary differential equations and n-ary algebras. Results on quadratic…
We seek to clarify some of the physical aspects of the Ruijsenaars-Schneider models. This important class of models was presented as a relativistic generalisation of the Calogero-Moser models but, as we shall argue, this description is…
In this article we consider two particular examples of general construction proposed in arXiv:2012.15529. We consider the integrable extensions of the classical elliptic Calogero-Moser model of N particles with spin and the integrable…
We reconsider a model of two relativistic particles interacting via a multiplicative potential, as an example of a simple dynamical system with sectors, or branches, with different dynamics and degrees of freedom.The presence or absence of…
We derive reciprocal integral relations between phases and amplitude moduli for a class of wave functions that are cyclically varying in time. The relations imply that changes of a certain kind (e.g. not arising from the dynamic phase)…
In this work we discuss connections between a one-dimensional system of $N$ particles interacting with a repulsive inverse square potential and confined in a harmonic potential (Calogero-Moser model) and the log-gas model which appears in…
We prove a canonical Kuenneth decomposition of the relative motive with rational coefficients of a smooth commutative group scheme over a noetherian finite dimensional base. This paper is a follow-up of "On the motive of a commutative…
There are some theoretical arguments about possible variations of fundamental constants with cosmic time. We review the fact that all conversion factors depend on these quantities and consider how their variations may affect transformations…
The SL(2,R) invariant Hamiltonian systems are discussed within the frame- work of the orbit method. It is shown that both dynamics and symmetry trans- formations are globally well-defined on phase space. The flexibility in the choice of…
We quantize the spin Calogero-Moser model in the $R$-matrix formalism. The quantum $R$-matrix of the model is dynamical. This $R$-matrix has already appeared in Gervais-Neveu's quantization of Toda field theory and in Felder's quantization…
Integrable deformations of the hyperbolic and trigonometric ${\mathrm{BC}}_n$ Sutherland models were recently derived via Hamiltonian reduction of certain free systems on the Heisenberg doubles of ${\mathrm{SU}}(n,n)$ and…
The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by…
The article deals with the subject of solving the problem of canonical-map synthesis for Hamiltonian systems. For this purpose, the controlling-function method has been developed that allows appropriate changes of the variables in terms of…