The spherical sector of the Calogero model as a reduced matrix model

We investigate the matrix-model origin of the spherical sector of the rational Calogero model and its constants of motion. We develop a diagrammatic technique which allows us to find explicit expressions of the constants of motion and calculate their Poisson brackets. In this way we obtain all functionally independent constants of motion to any given order in the momenta. Our technique is related to the valence-bond basis for singlet states.     

The tetrahexahedric Calogero model

We consider the spherical reduction of the rational Calogero model (of type A _n-1, without the center of mass) as a maximally superintegrable quantum system. It describes a particle on the (n = 2)-sphere in a very special potential. A detailed analysis is provided of the simplest non-separable case, n = 4, whose potential blows up at the edges of a spherical tetrahexahedron, tesselating the two-sphere into 24 identical right isosceles spherical triangles in which the particle is trapped. We construct a complete set of independent conserved charges and of Hamiltonian intertwiners and elucidate their algebra. The key structure is the ring of polynomials in Dunkl-deformed angular momenta, in particular the subspaces invariant and antiinvariant under all Weyl reflections, respectively.     

Hidden symmetries of integrable conformal mechanical systems

We split the generic conformal mechanical system into a “radial” and an “angular” part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We reduce the analysis of the constants of motion of the full system to the study of certain differential equations on this orbit. For integrable mechanical systems, the conformal invariance renders them superintegrable, yielding an additional series of conserved quantities originally found by Wojciechowski in the rational Calogero model. Finally, we show that, starting from any N=4 supersymmetric “angular” Hamiltonian system one may construct a new system with full N=4 superconformal D(1,2;α) symmetry.     

Multicenter Higgs oscillators and Calogero model

We show that the spherical part of N-particle Calogero model describes, after exclusion of the center of mass, the motion of the particle on (N − 2)-dimensional sphere interacting with N(N − 1)/2 force centers with Higgs oscillator potential. In the case of four-particle system these force centers are located at the vertexes of cuboctahedron. The geometry of the five-particle case is also investigated.     

Some comments on the rational solutions of the Zakharov-Schabat equations

We describe a family of the rational solutions of the Zakharov—Schabat equations. This family is characterized by extremely simple superposition principle, following directly from the Darboux-invariance of the Zakharov-Schabat equations proved in the works [1, 4]. Particularly we present an infinite sequence of polynomials P _n (x, y, t, t ₄, ..., t _m), m≤n, so that the formula $$u = 2\partial _x^2 Log\left( {\sum\limits_{i = 1}^N {c_i P_i } } \right)$$ where c _i are the arbitrary constants, represents some class of solutions of the Kadomtcev—Petviashvily equation. The paramters t ₄, ..., t _K represent the explicit action of the commuting flows, related with the Zakharov—Schabat operators of the higher order, on the solutions of the K—P equation, and can be fixed independently in each P _i. The polynomials P _n are closely related with the second Waring formular well known in algebra. This relation imposes some specific constraints on the motion of the N particle Moser—Calogero system generated by P _n.     

Contribution of the multipole polarization to the elastic constants in fluorite structure crystals

Numerical calculations are presented for the contribution of the multipole polarization to the elastic constants in fluorite structure crystals. The multipole polarizability is calculated by the self-consistent field treatment of the local density approximation and the spherical solid model. The elastic constants are significantly affected by the multipole polarization of the ions. The contributions of the multipole polarization of the ions to the elastic constant C₁₁, C₁₂, and C₄₄ are about 25%, 40% and 20% of the experimental values, respectively. The calculated values of the deviation from the Cauchy relation are in good agreement with the experimental values.     

Action-angle variables for dihedral systems on the circle

A nonrelativistic particle on a circle and subject to a cos−2(kφ) potential is related to the two-dimensional (dihedral) Coxeter system I₂(k), for k∈N. For such ‘dihedral systems’ we construct the action-angle variables and establish a local equivalence with a free particle on the circle. We perform the quantization of these systems in the action-angle variables and discuss the supersymmetric extension of this procedure. By allowing radial motion one obtains related two-dimensional systems, including A₂, BC₂ and G₂ three-particle rational Calogero models on R, which we also analyze.     

Dunkl operator,integrability, and pairwise scattering in rational Calogero model

The integrability of the Calogero model can be expressed as zero curvature condition using Dunkl operators. The corresponding flat connections are non-local gauge transformations, which map the Calogero wave functions to symmetrized wave functions of the set of N free particles, i.e. it relates the corresponding scattering matrices to each other. The integrability of the Calogero model implies that any k-particle scattering is reduced to successive pairwise scatterings. The consistency condition of this requirement is expressed by the analog of the Yang–Baxter relation.     

Generalized Lorentzian triangulations and the Calogero Hamiltonian

We introduce and solve a generalized model of (1+1)D Lorentzian triangulations in which a certain subclass of outgrowths is allowed, the occurrence of these being governed by a coupling constant β. Combining transfer matrix-, saddle point- and path integral-techniques we show that for β<1 it is possible to take a continuum limit in which the model is described by a 1D quantum Calogero Hamiltonian. The coupling constant β survives the continuum limit and appears as a parameter of the Calogero potential.     

The effect of singular potentials on the harmonic oscillator

We address the problem of a quantum particle moving under interactions presenting singularities. The self-adjoint extension approach is used to guarantee that the Hamiltonian is self-adjoint and to fix the choice of boundary conditions. We specifically look at the harmonic oscillator added of either a δ-function potential or a Coulomb potential (which is singular at the origin). The results are applied to Landau levels in the presence of a topological defect, the Calogero model and to the quantum motion on the noncommutative plane.     

Quantum completely integrable systems connected with semi-simple Lie algebras

The quantum version of the dynamical systems whose integrability is related to the root systems of semi-simple Lie algebras are considered. It is proved that the operators _k introduced by Calogero et al. are integrals of motion and that they commute. The explicit form of another class of integrals of motion is given for systems with few degrees of freedom.     

The rotational dependence of diagonal Coriolis coupling

The diagonal Coriolis coupling in degenerate vibrational states of symmetric- or spherical-top molecules has a rotational dependence of degree three in the components of the total angular momentum vector. A general formula is obtained for the coefficients of these terms and is specialized to the cases of symmetric tops (particularly C_3v molecules), where the coefficients are Maes' constants η_tJ and η_tK, and of cubic spherical tops, where the coefficients are Hecht's constants F_t (t ∈ E), F_uS and F_uT (u ∈ F_1,2). It is shown that these coefficients obey certain sum rules in which many of the contributions cancel. The values of the sums can be expressed in terms of the ordinary quartic centrifugal constants D_J, D_JK, and D_K for a symmetric top, or D and D_T for a cubic spherical top. The sum rules for methane are approximately satisfied by the experimental values of the constants.     

Percolation and variable-range hopping in random systems with spherical site symmetry

We show that the numerical work of Seager and Pike^1,2 suggests that the critical volume fraction (CVF) is a constant for sites of spherical symmetry in n dimensions, with $\text{CVF}\text{?nπ}{}^{\mathrm{1?n}}$ for small n. The average number of bonds per site, $\text{B}\text{?}{}_{\mathrm{c}}$, is calculated for a random distribution of site radii, and shown to agree with the Monte Carlo calculation. Analysis of a model having spherical site symmetry in (position-energy) space yields percolation constants C₂ = 2.1, C₃ = 2.6. This calculation indicates that there is an anomaly in some estimated values for the AHL percolation model. The physical significance of our model and its possible use in hard-core problems is discussed.     

Paraxial imaging electron optics and its spatial-temporal aberrations for a bi-electrode concentric spherical system with electrostatic focusing

The paraxial solutions play an important role in studying electron optical imaging system and its spatial-temporal aberrations, as was discussed in previous paper [1], but investigation of a bi-electrode concentric spherical system with electrostatic focusing directly from paraxial electron ray equation and paraxial electron motion equation has not been done before. In this paper, we shall use the paraxial equations to study the spatial-temporal trajectories and their aberrations for a bi-electrode concentric spherical system with electrostatic focusing.In the present paper, start from the paraxial ray equation and paraxial motion equation, the paraxial spatial-temporal trajectory of moving electron emitted from the photocathode has been solved for a bi-electrode concentric spherical system with electrostatic focusing. The paraxial static and dynamic electron optics, as well as the paraxial spatial-temporal aberrations in this system are then discussed, the general regularity of imaging in paraxial optical system has been explored. The paraxial spatial aberrations, as well as the paraxial temporal aberrations with different orders, have been defined and deduced, that are classified by the order of (?_z/?_ac)^1/2 and (?_T/?_ac)^1/2. Thus we get same conclusions about paraxial spatial and temporal aberrations as we have given in the previous paper and it completely shows that the paraxial spatial-temporal aberrations can be investigated directly from the paraxial ray equation and paraxial motion equation.     

Two-dimensional matrix Calogero model of the <Emphasis Type="Italic">BC</Emphasis><Subscript>2</Subscript> type

The new matrix Calogero model of the BC ₂ type is constructed. The central idea of the construction is exact solvability of the model. The text was submitted by the authors in English.     

Lax form of the quantum mechanical eigenvalue problem

The quantum mechanical eigenvalue problem with the hamiltonian H = H₀ + tV is written as a set of dynamical equations for the eigenvalues x_n(t) and the matrix elements V_nm(t) regarding the parameter t as time. By appropriate changes of variables it can be expressed as a pair of matrix equations with the Lax form, hence we are able to write all the possible constants of the motion explicitly. Implications of these constants to the statistical properties of levels are discussed.     

Pseudopotential in metals and alloys

In this communication, the pseudopotential investigation of, the various properties of non-transition metals and alloys, is discussed. Various one parametric model pseudopotentials, derived from well known spherical functions s_l(x), are employed in the calculations. Many recurrence relations of the s_l(x) function have been described. The effects of exchange and correlation on conduction electrons are also considered separately by using different dielectric screenings in various properties. The ion-ion interaction, force constants, phonon spectrum, temperature coefficient of Knight shift and electronic transport coefficients of certain metals and alloys are evaluated. The results are compared with available experimental values. Generally good agreement is achieved. The screening charge density of certain metals in low and high density region are also determined.     

A Class of Calogero Type Reductions of Free Motion on a Simple Lie Group

The reductions of the free geodesic motion on a non-compact simple Lie group G based on the G ₊ × G ₊ symmetry given by left- and right-multiplications for a maximal compact subgroup are investigated. At generic values of the momentum map this leads to (new) spin Calogero type models. At some special values the ‘spin’ degrees of freedom are absent and we obtain the standard BC _n Sutherland model with three independent coupling constants from SU(n + 1,n) and from SU(n,n). This generalization of the Olshanetsky-Perelomov derivation of the BC _n model with two independent coupling constants from the geodesics on G/G ₊ with G = SU(n + 1,n) relies on fixing the right-handed momentum to a non-zero character of G ₊. The reductions considered permit further generalizations and work at the quantized level, too, for non-compact as well as for compact G.