A note on the eigenfunction in a relativistic closed Toda chain

doi:10.3969/j.issn.0253-2778.2018.05.001

Abstract

Abstract: Recently, the eigenfunction for an N-particle relativistic closed Toda chain has been calculated by association with gauge theory. The results involve only the behavior of the eigenfunction of the energy level but do not include nonphysical energy. A conjecture was thus proposed to determine the eigenfunction of the nonphysical energy and a numerical calculation was conducted to prove that it is impossible to get an entire eigenfunction of the nonphysical energy. Because of the pole cancellation condition, the relation between the open and closed Bogomol’nyi-Prasad-Sommerfield (BPS) invariants could be derived.

Key words: eigenfunction, relativistic toda chain, BPS invariant, singularity

CLC Number:

WEN Xiaocheng. A note on the eigenfunction in a relativistic closed Toda chain[J]. Journal of University of Science and Technology of China, 2018, 48(5): 347-356.

References

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附录

这里对双正弦函数和与之相关的函数的性质做一个简单的介绍.双正弦函数的理论可以追溯到Barnes的论文.与量子二重对数函数之间紧密的联系是由Faddeev和Kashaev通过与量子反向散射方法给出的.双正弦函数的基本的性质已经在文献[13-15]中列了出来.

为了简单起见，引入符号ω=(ω1,ω2)，其中要求ω1,ω2>0.双正弦函数S2(x|ω)可以通过一个环路积分来定义

S2(x|ω)=exp∫Csinh(x-ω1+ω22)t2sinhω1 t2sinhω2 t2ln(-t)dt2πit(A1)

定义成立的范围为

0<Re x<ω1+ω2(A2)

在这个环路积分中，积分路径是从∞+0i到0，然后绕过原点之后再到∞-0i.与此同时还有一个与(A1)等价的积分表示由如下式子给出：

S2(x|ω)=exp(πi2B2,2(x|ω→))·exp∫R+0iexp(xt)(exp(ω1 t)-1)(exp(ω2 t)-1)dtt(A3)

式中，

B2,2(x|ω)=x2ω1ω2-ω1+ω2ω1ω2 x+ω21+ω22+3ω1ω26ω1ω2(A4)

从式（A4）中, 可以注意到有这样的性质：B2,2(ω1+ω2-x|ω)=B2,2(x|ω). 同样的，还可以从式（A1）中得到另一个等价的积分表示：

S2(x|ω)=exp∫∞0sinh(z-ω1+ω22)t2sinhω1 t2sinhω2 t2 -2x-ω1-ω2ω1ω2 tdtt(A5)

上述的两种积分表示在数值计算都非常难以计算，只能做解析证明时使用.所以如果希望能够做数值求解，就需要去将双正弦函数展开为一个收敛的数列求和.通过留数定理来计算式（A5），就可以得到双正弦函数的序列展开，适用范围是Im z>0或Im z<0.

S2(x|ω)=

expπi2B2,2(x|ω→) + ∑∞n=11nexp(2πinxw1)exp(2πinw1w2)-1+exp(2πinxw1)exp(2πinw2w1)-1， Im(x)>0；

exp-πi2B2,2(x|ω→) + ∑∞n=11nexp(-2πinxw1)exp(-2πinw1w2)-1+exp(-2πinxw1)exp(-2πinw2w1)-1， Im(x)<0
（A6）


通过对积分路径C做一个合适的变换，可以将双正弦函数（A1）的定义域解析延拓至除了ω1ω2∈R-之外ω1,ω2取任意复数.同样的，在Imω1ω2>0区域内也可以得到一个乘积形式的展开：

S2(x|ω)=expπi2B2,2(x|ω→)∏∞m=0(1-q2mexp(2πixω2))∏∞m=1(1-q-2mexp(2πixω1))=

exp(-πi2B2,2(x|ω→))∏∞m=0(1-q-2mexp( -2πixω1))∏∞m=1(1-q2mexp( -2πixω2))(A7)

式中,

q=exp(πiω1ω2), q=exp(πiω2ω1)(A8)

由theta函数θ1(x|ω1ω2)的模变换法则可以得到式（A6）与（A7）是等价的.

双正弦函数S2(x|ω)的零点和极点是其非常重要的性质，他们的位置分别是
poles: x=n1ω1+n2ω2, n1,n2≥1;
zeros: x=n1ω1+n2ω2, n1,n2≤0(A9)

并且在零点附近，函数满足如下的关系：

limx→0 S2(x|ω)=2πxω1ω2(A10)

()
()

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