Generalised Stirling operators, with Applications

广义斯特林算子及其应用

• 批准号: 2602423
• 金额: --
• 依托单位: Swansea University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2602423
• 关键词: Generalised; Stirling; operators; Applications

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.

在他的开创性的说明“组合方面的玻色子代数”，Katriel证明，斯特林数的第二种出现在代表权力的粒子密度运营商在玻色子代数作为一个总和威克有序条款。这个结果导致了许多重要的推广斯特林数，当一个取代玻色子代数与另一个代数。这种推广的一个例子是一对满足q-对易关系的创造和湮灭算子的情况。然而，所有可用的斯特林数的推广仅在单模情况下进行，即，当所考虑的代数是由一对创建和湮灭算子生成时。在最近的一篇论文的导师的学生与合作者（出现在杂志的功能分析），空间对应的组合理论的斯特林号码的发展。在这个理论中，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，这意味着用一个配置空间来代替自然数的集合。在这个框架中，空间对应的斯特林数是斯特林运营商的对称多元函数的空间。证明了斯特林算符自然地作为Wick序项之和出现在无限维玻色子代数中粒子密度算符的幂表示中。本课题的主要目的是：（i）对几类重要的无限维代数的生成和湮灭算子，导出相应的广义斯特林算子，（ii）研究广义斯特林算子的组合性质;（iii）考虑广义斯特林数在随机测度理论中的应用，该项目将首先研究由作用于无限维多项式空间的下列算子生成的代数：与变量相乘的运算符（被视为创建运算符）和差分导数（被视为湮灭运算符）。预计相应的颗粒密度，即，在一个点上的创建和湮灭算子的涂抹乘积将引导我们到伽马随机测度和负二项点过程。另一类要考虑的代数是那些用q-微分算子代替差分算子的代数。在这种情况下，产生和湮灭算子满足无限维q-对易关系。虽然相应的广义斯特林数是很好地理解在一个模式的情况下，q-变形的斯特林运营商将提出一个重要的和高度非平凡的研究对象。