论文标题
某些Bethe Ansatz可溶度模型的差异方程方法的方法
Method of difference-differential equations for some Bethe ansatz solvable models
论文作者
论文摘要
在对一维bethe ansatz可溶度模型的研究中，经常出现在有限间隔内具有差异内核的第二种弗雷德姆积分方程。该方程通常不接受封闭形式的解决方案，因此其分析非常复杂。在这里，我们研究了这样的方程式的家庭，专注于它们的时刻。我们以差异方程式的形式找到了矩之间的确切关系。后者的结果大大推进了分析，从而使人们能够从最低的明确知识中确定所有时刻。作为应用，首先，我们研究Lieb-Liniger模型中的准分布分布的力矩，并找到明确的分析结果。后一刻决定了几个基本数量，例如$ n $ body局部相关功能。我们证明了文献中三体局部相关函数的不同表达式之间的等效性，并根据quasimomentum分布的力矩找到了四体局部相关函数的确切结果。我们最终在弱和强相互作用的制度中以渐近系列的形式找到了三体和四体相关函数的分析结果。接下来，我们研究杨高丁物模型描述的两个分量的bose气体中镁（二极体）激发的低能光谱的确切形式。我们发现其显式形式，取决于Lieb-Liniger模型的准分布的矩。然后，我们解决了一个看似无关的圆形电容电容的问题，并表达了参数形式的电容的确切结果。在最有趣的短板分离的情况下，参数形式具有单个对数项。这应该与具有复杂的对数结构的明确结果形成鲜明对比。
In studies of one-dimensional Bethe ansatz solvable models, a Fredholm integral equation of the second kind with a difference kernel on a finite interval often appears. This equation does not generally admit a closed-form solution and hence its analysis is quite complicated. Here we study a family of such equations concentrating on their moments. We find exact relations between the moments in the form of difference-differential equations. The latter results significantly advance the analysis, enabling one to practically determine all the moments from the explicit knowledge of the lowest one. As applications, first we study the moments of the quasimomentum distribution in the Lieb-Liniger model and find explicit analytical results. The latter moments determine several basic quantities, e.g., the $N$-body local correlation functions. We prove the equivalence between different expressions found in the literature for the three-body local correlation functions and find an exact result for the four-body local correlation function in terms of the moments of the quasimomentum distributions. We eventually find the analytical results for the three- and four-body correlation functions in the form of asymptotic series in the regimes of weak and strong interactions. Next, we study the exact form of the low-energy spectrum of a magnon (a polaron) excitation in the two-component Bose gas described by the Yang-Gaudin model. We find its explicit form, which depends on the moments of the quasimomentum distributions of the Lieb-Liniger model. Then, we address a seemingly unrelated problem of capacitance of a circular capacitor and express the exact result for the capacitance in the parametric form. In the most interesting case of short plate separations, the parametric form has a single logarithmic term. This should be contrasted with the explicit result that has a complicated structure of logarithms.
