"Quantization, coherent states and complex orthogonal polynomials: applications to physics and signal analysis"

我的研究在过去的六年一直是一个持续以及正在进行的计划的延伸，涉及几个研究生和以下同事：J。P. Antoine（Louvain-la-Neuve），F. Bagarello（巴勒莫），M. Engli·s（布拉格），J.- Gazeau（巴黎），这项工作是我们几十年来对相干态和平方可积群表示及其在量化、小波分析和信号处理中的应用的研究的结果。大多数早期的结果的工作已出版的两本专著，而一些最近的工作已被描述在两个评论文章，其中之一是在数学物理百科全书。这项工作已应用于信号分析和图像处理，在量子光学和原子物理学和数学方面，在谐波分析和时间频率和小波分析领域。我在未来几年的工作预计将继续沿着上述路线，以及进入一些新的方向。这些新的工作领域之一将是在广义的相干态希尔伯特空间更一般的希尔伯特模块。这一点，除了其内在的数学价值，在相关的次正规算子的研究中，也将发现应用于非对易量子力学的问题，更一般地说，非对易空间的理论。我们也希望在这个背景下研究量子计算中的纠缠问题。第二个新的工作方向将在该地区的正交多项式相关的相干态和第三贝叶斯对偶在经典统计和相干态之间的连接。众所周知的电子在恒定磁场中的问题有一个特别有趣的代数结构。在过去的几年里，我们已经对此进行了详细的研究，现在正在将这个系统推广到包括赝玻色子。我的研究触及一些最活跃的数学物理领域。

My research over the past six years has been a continuation as well as extension of an ongoing programme, involving several graduate students and the following coworkers: J.-P. Antoine (Louvain-la-Neuve), F. Bagarello (Palermo), M. Engli·s (Prague), J.-P. Gazeau (Paris), The work has been an outgrowth of our decades long research on coherent states and square-integrable group representations and their use in quantization, wavelet analysis and signal processing. Most of the earlier results of the work have been published in two monographs while some of the more recent work has been described in two review articles, one of them being in the Encyclopedia of Mathematical Physics. This work has had applications to signal analysis and image processing, in quantum optics and atomic physics and on the mathematical side, in the field of harmonic analysis and time-frequency and wavelet analysis. My work over the next few years is expected to continue along the above lines, as well as moving into a number of new directions. One of these new areas of work will be in a generalization of coherent states on Hilbert spaces to more general Hilbert modules. This, apart from its intrinsic mathematical value, in the study of related subnormal operators, would also find applications to problems in non-commuting quantum mechanics and more generally to the theory of non-commutative spaces. We also expect to study the problem of entanglement in quantum computing within this setting. A second new direction of work will be in the area of orthogonal polynomials related to coherent states and a third to the connection between Baysian duality in classical statistics and coherent states. The well known problem of the electron in a constant magnetic field has a particularly interesting algebraic structure. We have studied this in some detail in the past couple of years and are now in the process of generalizing the system to include pseudo-bosons. My research touches on some of the most actively worked upon areas of mathematical physics.