Generalised Stirling operators, with Applications

在他的开创性笔记“玻色子代数的组合方面”中，Katriel证明了第二类斯特林数出现在玻色子代数中粒子密度算子的幂表示中，作为Wick有序项的和。当人们用另一种代数代替玻色子代数时，这个结果导致了许多重要的斯特林数推广。这种推广的一个例子是一对满足q对易关系的产生和湮灭算子。然而，所有可用的斯特林数的推广都只在单模情况下进行，即当所考虑的代数是由单个对产生和湮灭算子生成时。在该学生的导师和合作者最近发表的一篇论文中（发表在《功能分析杂志》上），发展了斯特林数组合理论的空间对应物。在这个理论中，自然数（解释为种群的大小）被空间分布的种群所取代。在数学上，这意味着用一个位形空间来代替自然数的集合。在这个框架中，斯特林数的空间对应物是作用于对称多元函数空间的斯特林算子。证明了在无限维玻色子代数中，斯特林算符自然地以威克序项和的形式出现在粒子密度算符的幂表示中。本课题的主要目的如下：(1)对于几类重要的无限维产生算子和湮灭算子，导出了相应的广义Stirling算子；（ii）研究广义Stirling算子的组合性质；（iii）考虑广义斯特林数在随机测度理论及其以外的应用。该项目将从研究以下运算符在无限维多项式空间上产生的代数开始：与变量相乘的运算符（被视为创建运算符）；差分导数（作为湮灭算符）我们期望相应的粒子密度，即在一个点上的产生算子和湮灭算子的涂抹积，将导致我们得到伽马随机测量和负二项点过程。另一类要考虑的代数是用q微分算子代替差分算子的代数。在这种情况下，产生和湮灭算符满足无穷维q对易关系。当相应的广义斯特林数在单模情况下被很好地理解时，q-变形斯特林算子将提供一个重要的和高度不平凡的研究对象。

In his seminal note "Combinatorial aspects of boson algebra", Katriel proved that Stirling numbers of the second kind appear in the representation of powers of the particle density operator in the boson algebra as a sum of Wick ordered terms. This result led to numerous important generalisations of Stirling numbers, when one replaces the boson algebra with another algebra. An example of such a generalisation is the case of a pair of creation and annihilation operators satisfying the q-commutation relation. However, all the available generalisations of Stirling numbers have only be carried out in the one-mode case, i.e., when the algebra under consideration is generated by a single pair of creation and annihilation operators. In a recent paper by the supervisor of the student with co-authors (to appear in Journal of Functional Analysis) , a spatial counterpart of the combinatorial theory of Stirling numbers was developed. In this theory, the natural numbers (interpreted as the size of a population) are replaced by a population distributed in space. Mathematically, this means that one replaces the set of natural numbers by a configuration space. In this framework, the spatial counterpart of Stirling numbers are Stirling operators acting on spaces of symmetric multivariate functions. It was proved that Stirling operators naturally appear in the representation of powers of the particle density operators in the infinite-dimensional boson algebra as a sum of Wick-ordered terms. The main objective of the present project are as follows:(i) for several important classes of infinite-dimensional algebras of creation and annihilation operators, derive the corresponding generalised Stirling operators;(ii) study combinatorial properties of the generalised Stirling operators;(iii) consider applications of the generalised Stirling numbers in the theory of random measures, and beyond.The project will start with a study of the algebra generated by the following operators acting on the space of infinite-dimensional polynomials: the operators of multiplication by a variable (treated as creation operators); and the difference derivatives (treated as annihilation operators). It is expected that the corresponding particle density, i.e., the smeared product of the creation and annihilation operators at a point, will lead us to the gamma random measure and the negative binomial point processes. Another class of the algebras to be considered are those where the difference operators are replaced by the operators of q-differentiation. In that case, the creation and annihilation operators satisfy the infinite-dimensional q-commutation relations. While the corresponding generalised Stirling numbers are well understood in the one-mode case, the q-deformed Stirling operators will present an important and highly non-trivial object of studies.