双曲Landau能级上的Husmi Q函数
Husmi Q-functions attached to hyperbolic Landau levels
摘要：
我们研究了一个相空间概率分布，它是密度算符关于一组相干态$\vert\widetildeκ_{z,B,R,m}\rangle$的Husimi $Q$ -函数，相干态附着在$m$双曲朗道能级上，用半径为$R$的开放圆盘上的点$z$标记，其中$B>0$与磁场强度成正比.对于表示Fock态$\left\vert j\right\rangle$上的投影的密度算符，我们得到了$Q_{j}$分布，并讨论了它的一些基本性质，如它的特征函数和它的主要统计参数。我们实现了相同的程序的热密度算符（混合态）的保序振子，我们建立了一个下界的相关的量子势。我们恢复的欧几里德设置（平坦的情况下）的参数$R$走向无穷大的结果，通过呼吁涉及正交多项式和特殊功能的渐近公式。作为工具，我们建立了特殊Kampé de Fériet函数$\digamma _{2:0:0}^{1:2:2}$的求和公式.
英文摘要：
We are concerned with a phase-space probability distribution which is known as Husimi $Q$-function of a density operator with respect to a set of coherent states $\vert\widetildeκ_{z,B,R,m}\rangle$ attached to an $m$th hyperbolic Landau level and labeled by points $z$ of an open disk of radius $R$, where $B>0$ is proportional to a magnetic field strength. For a density operator representing a projector on a Fock state $\left\vert j\right\rangle$ we obtain the $Q_{j}$ distribution and discuss some of its basic properties such as its characteristic function and its main statistical parameters. We achieve the same program for the thermal density operator (mixed states) of the isotonic oscillator for which we establish a lower bound for the associated thermodynamical potential. We recover most of the results of the Euclidean setting (flat case) as the parameter $R$ goes to infinity by making appeal to asymptotic formulas involving orthogonal polynomials and special functions. As a tool, we establish a summation formula for the special Kampé de Fériet function $\digamma _{2:0:0}^{1:2:2}$.
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