Richardson方程的“概率”方法
"Probabilistic" approach to Richardson equations
摘要：
众所周知，Richardson方程的解可以表示为平面上经典自由电荷“能量”的驻点。我们建议考虑电荷系统在相互作用常数给定的有效温度下占据组态空间中某些状态的“概率”，该温度在热力学极限下趋于零。值得注意的是，“概率”的表达式与劳克林波函数的平方有相似之处。接下来，我们引入“配分函数“，由此可以确定初始量子力学系统的基态能量。“配分函数“是由一个多维积分给出的，它类似于出现在共形场论和随机-矩阵模型中的Selberg积分。作为这种方法的第一个应用，我们考虑一个系统的能量状态的密度恒定在任意填充的能量区间，其中潜在的行为。在这种情况下，“配分函数“使用范德蒙矩阵的性质相当容易地评估。因此，我们的方法产生了相当简单和短的方式来找到基态能量，这是由一个单一的表达式描述所有从稀到密对制度。它还提供了对库珀配对态物理学的额外见解。
英文摘要：
It is known that solutions of Richardson equations can be represented as stationary points of the "energy" of classical free charges on the plane. We suggest to consider "probabilities" of the system of charges to occupy certain states in the configurational space at the effective temperature given by the interaction constant, which goes to zero in the thermodynamical limit. It is quite remarkable that the expression of "probability" has similarities with the square of Laughlin wave function. Next, we introduce the "partition function", from which the ground state energy of the initial quantum-mechanical system can be determined. The "partition function" is given by a multidimensional integral, which is similar to Selberg integrals appearing in conformal field theory and random-matrix models. As a first application of this approach, we consider a system with the constant density of energy states at arbitrary filling of the energy interval, where potential acts. In this case, the "partition function" is rather easily evaluated using properties of the Vandermonde matrix. Our approach thus yields quite simple and short way to find the ground state energy, which is shown to be described by a single expression all over from the dilute to the dense regime of pairs. It also provides additional insights into the physics of Cooper-paired states.
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