arXiv 0801.48182008-01cond-mat.mes-hallcond-mat.stat-mechhep-th

随机Aharonov-Bohm涡与某些精确积分族：II部分

Random Aharonov-Bohm vortices and some exact families of integrals: Part II

Stefan Mashkevich, Stéphane Ouvry

备注：: 9 pages, LaTeX 2e. A few sentences rephrased more exactly, misprints corrected
发表信息：: J. Stat. Mech. (2008) P03018
DOI：: 10.1088/1742-5468/2008/03/P03018

摘要：

在微扰理论的第六阶，杂质密度的第二阶随机磁性杂质问题缩小到具有最大杂质线交叉的单个费曼图的评估。该图可以重写为普通积分和修正贝塞尔函数$K_ν$和$I_ν$的乘积的嵌套二重积分的和，其中$ν=0,1$。该总和，反过来，被证明是一个线性组合的合理系数的$(2^5-1)ζ(5)$，$\int_0^{\infty} u K_0(u)^6 du$和$\int_0^{\infty} u^3 K_0(u)^6 du$。与低阶情况不同，这两个积分不是具有欧拉和有理系数的线性组合，尽管它们与$ζ(5)$组合出现。另一方面，任何积分$\int_0^{\infty} u^{n+1} K_0(u)^p (uK_1(u))^q du$的重量$p+q=6$和一个甚至$n$是一个线性组合，上述两个积分和1的有理系数，可以很容易地推广到任何重量$p+q=k$的结果。在$n$中建立了这种积分的矩阵递推关系。初始条件是渐近行为由转移矩阵的最小特征值决定。

英文摘要：

At 6th order in perturbation theory, the random magnetic impurity problem at second order in impurity density narrows down to the evaluation of a single Feynman diagram with maximal impurity line crossing. This diagram can be rewritten as a sum of ordinary integrals and nested double integrals of products of the modified Bessel functions $K_ν$ and $I_ν$, with $ν=0,1$. That sum, in turn, is shown to be a linear combination with rational coefficients of $(2^5-1)ζ(5)$, $\int_0^{\infty} u K_0(u)^6 du$ and $\int_0^{\infty} u^3 K_0(u)^6 du$. Unlike what happens at lower orders, these two integrals are not linear combinations with rational coefficients of Euler sums, even though they appear in combination with $ζ(5)$. On the other hand, any integral $\int_0^{\infty} u^{n+1} K_0(u)^p (uK_1(u))^q du$ with weight $p+q=6$ and an even $n$ is shown to be a linear combination with rational coefficients of the above two integrals and 1, a result that can be easily generalized to any weight $p+q=k$. A matrix recurrence relation in $n$ is built for such integrals. The initial conditions are such that the asymptotic behavior is determined by the smallest eigenvalue of the transition matrix.

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英文摘要：

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