Painlevé IV and degenerate Gaussian unitary ensembles
Painlevé IV and degenerate Gaussian unitary ensembles
复制标题
DOI：
10.1088/0305-4470/39/40/007
复制
发表时间：
2006-06
期刊：
影响因子：
--
通讯作者：
Yang Chen;M. Feigin
中科院分区：
文献类型：
--
作者：
Yang Chen;M. Feigin
考虑特征值具有规定多重性的高斯酉系综，得到特征值的联合概率密度。在最简单的情况下，只有一个多重特征值t，这导致正交多项式的Hermite权值被一个因子扰动，该因子在t处具有多个零。我们通过一对阶梯算子证明，对角递推系数满足任何实多重性的特定painlevel IV方程。如果多重性是偶的，则用广义埃尔米特多项式表示，以t为自变量。
We consider those Gaussian unitary ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painleve IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable.