Ali Hafoud, Allal Ghanmi
We derive a novel integral representations of Jacobi polynomials in terms of the Gauss hypergeometric function. Such representation is then used to give the explicit integral representation for the Heat kernel on the quantized Riemann sphere.
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U(n) Yang-Mills theory on the fuzzy sphere S^2_N is quantized using random matrix methods. The gauge theory is formulated as a matrix model for a single Hermitian matrix subject to a constraint, and a potential with two degenerate minima. This allows to reduce the path integral over the gauge fields to an integral over eigenvalues, which can be evaluated for large N. The partition function of U(n) Yang-Mills theory on the classical sphere is recovered in the large N limit, as a sum over instanton contributions. The monopole solutions are found explicitly.
Global Solution of a Functional Hamilton-Jacobi Equation associated with a Hard Sphere Gas
本文针对硬球气体低密度极限下得到的泛函Hamilton-Jacobi方程，通过分析耦合Boltzmann方程组构造全局解，并证明其在适当假设下长时间收敛到非平凡稳态解。
Jacobi fields of classical solutions of a Hamiltonian mechanical system are quantized in the framework of vertical-extended Hamiltonian formalism. Quantum Jacobi fields characterize quantum transitions between classical solutions.
The paper collates a complete list of nodeless regular almost-everywhere holomorphic (AEH) solutions for a subset of rational canonical Sturm-Liouville equations (RCSLEs) exactly quantized on a finite interval by classical Jacobi polynomials. The subset was constrained by the requirement that the appropriate Liouville transformation results in the Schrodinger equation on the line. The common remarkable feature of the selected nodeless solutions co-existent with the discrete energy spectrum is that they can be used as seed functions for multi-step 'canonical Liouville-Darboux transformations' (…
We consider a hierarchy of Poisson structures defined on rational functions on the Riemann sphere. This hierarchy is originated in the theory of the integrable Camassa-Holm equation associated with the Krein's string spectral problem. Previously the proof of Jacobi identity was obtained by reducing the bracket to canonical Darboux coordinates. The main result of this note is a direct proof of the Jacobi identity. It turns out that the direct proof of the Jacobi identity is far from trivial. We also give an example of another hierarchy of Poisson brackets and construct Darboux coordinates for i…
We investigate the geometric interpretation of quantized Nambu-Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin-Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras, as well as the approach based on harmonic analysi…
The classical Maxwell--Born--Infeld field equations coupled with a Hamilton--Jacobi law of point charge motion are partially quantized by coupling the Hamilton-Jacobi phase function with an amplitude function, which combines with the phase function into a single complex wave function satisfying a relativistic Klein--Gordon equation self-consistently coupled to the evolution equations for the electromagnetic fields with generic point source (explicitly worked out for one particle; options for many particles briefly discussed). Radiation-free stationary states exist. The hydrogen spectrum with i…
Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi-Moser system, we disclose a novel integrable system on the sphere $S^n$, namely the "dual Moser" system. The latter falls, along with the Jacobi-Moser and Neumann-Uhlenbeck systems, into the category of (locally) Stäckel systems. Moreover, it is proved that quantum integrability of both Neumann-Uhlenbeck and dual Moser systems is insured by means of the conformally equivariant quantization procedure.
The dynamical algebra of the generic superintegrable model on the two-sphere
研究二维球面上一般二次超可积模型的动力学代数结构，通过嵌入su(1,1)⊗3获得物理表示，并利用该表示推导出模型的精确解，波函数以双变量Jacobi多项式表示。
We consider the generic quantum superintegrable system on the $d$-sphere with potential $V(y)=\sum_{k=1}^{d+1}\frac{b_k}{y_k^2}$, where $b_k$ are parameters. Appropriately normalized, the symmetry operators for the Hamiltonian define a representation of the Kohno-Drinfeld Lie algebra on the space of polynomials orthogonal with respect to the Dirichlet distribution. The Gaudin subalgebras generated by Jucys-Murphy elements are diagonalized by families of Jacobi polynomials in $d$ variables on the simplex. We define a set of generators for the symmetry algebra and we prove that their action on t…
Quantized Bertrand models on two-dimensional spaces of constant curvature
本文研究常曲率二维空间（球面、罗氏空间）上的Bertrand系统，采用基于Laplace-Beltrami算子的量子化技术处理经典模型，分析其量子行为。
While dealing with the constant-strength magnetic Laplacian on the annulus, we complete J. Peetre's work. In particular, the eigenspaces associated with its discrete spectrum are true-polyanalytic spaces with respect to the invariant Cauchy-Riemann operator, and we write down explicit formulas for their reproducing kernels. The latter are expressed by means of the fourth Jacobi theta function and of its logarithmic derivatives when the magnetic field strength is an integer. Under this quantization condition, we also derive the transformation rule satisfied by the reproducing kernel under the a…
The goal of the present paper is to provide a detailed study of irreducible representations of the algebra generated by the symmetries of the generic quantum superintegrable system on the $d$-sphere. Appropriately normalized, the symmetry operators preserve the space of polynomials. Under mild conditions on the free parameters, maximal abelian subalgebras of the symmetry algebra, generated by Jucys-Murphy elements, have unique common eigenfunctions consisting of families of Jacobi polynomials in $d$ variables. We describe the action of the symmetries on the basis of Jacobi polynomials in terms…
The paper presents the unified technique for constructing SUSY ladders of rational Liouville potentials (RLPs) starting from the so-called "Gauss-reference" (GRef) potentials exactly quantized on the line via classical Jacobi, classical (generalized) Laguerre, or Romanovski-Routh polynomials with energy-dependent indexes. Each RLP is obtained by means of the Liouville transformation (LT) of the appropriate rational canonical Sturm-Liouville equation (RCSLE) with second-order poles. The presented analysis takes advantage of the generic factorization of canonical Sturm-Liouville equations (CSLEs…
Using the theory of quantized equivariant vector bundles over compact coadjoint orbits we determine the Chern characters of all noncommutative line bundles over the fuzzy sphere with regard to its derivation based differential calculus. The associated Chern numbers (topological charges) arise to be non-integer, in the commutative limit the well known integer Chern numbers of the complex line bundles over the 2-sphere are recovered.
A trajectory isomorphism between the two Newtonian fixed center problem in the sphere and two associated planar two fixed center problems is constructed by performing two simultaneous gnomonic projections in $S^2$. This isomorphism converts the original quadratures into elliptic integrals and allows the bifurcation diagram of the spherical problem to be analyzed in terms of the corresponding ones of the planar systems. The dynamics along the orbits in the different regimes for the problem in $S^2$ is expressed in terms of Jacobi elliptic functions.
The $9j$ symbols of $\mathfrak{su}(1,1)$ are studied within the framework of the generic superintegrable system on the 3-sphere. The canonical bases corresponding to the binary coupling schemes of four $\mathfrak{su}(1,1)$ representations are constructed explicitly in terms of Jacobi polynomials and are seen to correspond to the separation of variables in different cylindrical coordinate systems. A triple integral expression for the $9j$ coefficients exhibiting their symmetries is derived. A double integral formula is obtained by extending the model to the complex three-sphere and taking the c…
The quantized magnetic flux $Φ=-4πN\sk, N=0,\pm1,...$ of non-topological vortices in the non-relativistic Chern-Simons theory is related to the topological degree of the $S^2\to S^2$ mapping defined by lifting the problem to the Riemann spheres. Regular solutions with finite degree only arise for rational functions, whose topological degree, $N$, is the commun number of their zeros and poles on the Riemann sphere, also called their algebraic order.
We extend the construction of 2D superintegrable Hamiltonians with separation of variables in spherical coordinates using combinations of shift, ladder, and supercharge operators to models involving rational extensions of the two-parameter Lissajous systems on the sphere. These new families of superintegrable systems with integrals of arbitrary order are connected with Jacobi exceptional orthogonal polynomials (EOP) of type I (or II) and supersymmetric quantum mechanics (SUSYQM). Moreover, we present an algebraic derivation of the degenerate energy spectrum for the one- and two-parameter Lissa…
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