伪微扰展开法;非多项式、截断库仑势和库仑加对数势

Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.

纯相动力学伊辛模型的一些新结果

Some New Results on the Kinetic Ising Model in a Pure Phase

我们考虑一类广义Glauber动力学在$\bbZ^d$中相对于标准伊辛模型是可逆的，其外磁场为零，逆温度$\gb$严格大于2维的临界值$\gb_c$或所谓的$d \geq 3$维的“板阈值”$\hat \b_c$.我们首先证明，在一个大的立方体的侧$N$加上边界条件的逆谱间隙是，除了对数校正，大于$N$在$d=2$，而对数Sobolev常数，而不是大于$N^2$在任何维度。这样的结果大大改善了所有以前现有的界限，并同意在一个框架中获得的一个类似的计算的基础上平均曲率运动的一维玩具模型。该证明基于H. T.几年前，Yau明确地构建了一个微妙的测试函数，该函数迫使负相的大液滴进入正相。一般$d\ge 2$的相关界限，然后通过仔细使用最近的$\bbL^1$ -方法的武尔夫建设。最后，我们证明，在$d=2$的概率，两个独立的初始配置，分布根据无限体积加相位和发展下的任何耦合，同意在原点的时间$t$是有界的从下面的拉伸指数$\exp(-\sqrt{t})$，再次除了对数校正。这样的结果应该被看作是严格证明的第一步，正如Fisher和Huse几年前所指出的那样，在$d=2$中，原点自旋的平衡时间自相关以拉伸指数衰减。

Construction of potential functions associated with a given energy spectrum

Using a formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters, we present a method that gives the class of potential functions for exactly solvable problems corresponding to a given energy spectrum. In this work, we study the class of problems with a mix of continuous and discrete energy spectrum that are associated with the continuous dual Hahn polynomial. These include the one-dimensional logarithmic potential and the three-dimensional Coulomb plus linear potential.