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Register a new userThe Wigner caustic on shell and singularities of odd functions
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We study the Wigner caustic on shell of a Lagrangian submanifold L of affine symplectic space. We present the physical motivation for studying singularities of the Wigner caustic on shell and present its mathematical definition in terms of a generating family. Because such a generating family is an odd deformation of an odd function, we study simple singularities in the category of odd functions and their odd versal deformations, applying these results to classify the singularities of the Wigner caustic on shell, interpreting these singularities in terms of the local geometry of L.
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We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostants coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables.
We study the essential singularities of geometric zeta functions $zeta_{mathcal L}$, associated with bounded fractal strings $mathcal L$. For any three prescribed real numbers $D_{infty}$, $D_1$ and $D$ in $[0,1]$, such that $D_{infty}<D_1le D$, we construct a bounded fractal string $mathcal L$ such that $D_{rm par}(zeta_{mathcal L})=D_{infty}$, $D_{rm mer}(zeta_{mathcal L})=D_1$ and $D(zeta_{mathcal L})=D$. Here, $D(zeta_{mathcal L})$ is the abscissa of absolute convergence of $zeta_{mathcal L}$, $D_{rm mer}(zeta_{mathcal L})$ is the abscissa of meromorphic continuation of $zeta_{mathcal L}$, while $D_{rm par}(zeta_{mathcal L})$ is the infimum of all positive real numbers $alpha$ such that $zeta_{mathcal L}$ is holomorphic in the open right half-plane ${{rm Re}, s>alpha}$, except for possible isolated singularities in this half-plane. Defining $mathcal L$ as the disjoint union of a sequence of suitable generalized Cantor strings, we show that the set of accumulation points of the set $S_{infty}$ of essential singularities of $zeta_{mathcal L}$, contained in the open right half-plane ${{rm Re}, s>D_{infty}}$, coincides with the vertical line ${{rm Re}, s=D_{infty}}$. We extend this construction to the case of distance zeta functions $zeta_A$ of compact sets $A$ in $mathbb{R}^N$, for any positive integer $N$.
Using the formalism of quantizers and dequantizers, we show that the characters of irreducible unitary representations of finite and compact groups provide kernels for star products of complex-valued functions of the group elements. Examples of permutation groups of two and three elements, as well as the SU(2) group, are considered. The k-deformed star products of functions on finite and compact groups are presented. The explicit form of the quantizers and dequantizers, and the duality symmetry of the considered star products are discussed.
We prove that there exists just one pair of complex four-dimensional Lie algebras such that a well-defined contraction among them is not equivalent to a generalized IW-contraction (or to a one-parametric subgroup degeneration in conventional algebraic terms). Over the field of real numbers, this pair of algebras is split into two pairs with the same contracted algebra. The example we constructed demonstrates that even in the dimension four generalized IW-contractions are not sufficient for realizing all possible contractions, and this is the lowest dimension in which generalized IW-contractions are not universal. Moreover, this is also the first example of nonexistence of generalized IW-contraction for the case when the contracted algebra is not characteristically nilpotent and, therefore, admits nontrivial diagonal derivations. The lower bound (equal to three) of nonnegative integer parameter exponents which are sufficient to realize all generalized IW-contractions of four-dimensional Lie algebras is also found.
In the first part of this paper I shall discuss the round-about way of how the integrable chiral Potts model was discovered about 30 years ago. As there should be more higher-genus models to be discovered, this might be of interest. In the second part I shall discuss some quantum group aspects, especially issues of odd versus even $N$ related to the Serre relations conjecture in our quantum loop subalgebra paper of 5 years ago and how we can make good use of coproducts, also borrowing ideas of Drinfeld, Jimbo, Deguchi, Fabricius, McCoy and Nishino.
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