Generalised Stirling operators, with Applications

在他开创性的注记《玻色子代数的组合方面》中，Ktriel证明了第二类Stirling数出现在玻色子代数中粒子密度算符的幂的表示中，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，这一结果导致了许多重要的Stirling数的推广。这种推广的一个例子是满足Q对易关系的一对创建和湮灭算子的情况。然而，所有可用的Stirling数的推广都只在单模的情况下进行，即当所考虑的代数是由一对创建和湮灭算子生成的时候。在这位学生和合著者的导师最近发表的一篇论文中(发表在《泛函分析杂志》上)，Stirling数的组合理论在空间上得到了相应的发展。在这一理论中，自然数(解释为人口的大小)被空间分布的人口所取代。在数学上，这意味着一个人用一个配置空间取代了自然数的集合。在这个框架中，Stirling数的空间对应是作用在对称多元函数空间上的Stirling算子。证明了在无限维玻色子代数中，Stirling算符以Wick有序项之和的形式自然地出现在粒子密度算符的幂表示中。本项目的主要目标如下：(I)对于几类重要的无限维代数的生成和湮灭算子，导出相应的广义Stirling算子；(Ii)研究广义Stirling算子的组合性质；(Iii)考虑广义Stirling数在随机测度论中的应用，以及其他方面。本项目将从研究作用于无限维多项式空间的下列算子生成的代数开始：与变量相乘的算子(被视为创建算子)；以及差分导数(被视为湮灭算子)。预计相应的粒子密度，即在某一点上产生和湮灭算符的涂抹乘积，将把我们引向伽马随机测量和负二项式点过程。另一类要考虑的代数是那些用q-微分算子代替差算子的代数。在这种情况下，产生和湮灭算符满足无限维Q-对易关系。当相应的广义Stirling数在单模情况下被很好地理解时，q-变形的Stirling算符将是一个重要的和非常重要的研究对象。

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.