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

我过去六年的研究一直是一个正在进行的项目的延续和扩展，涉及几名研究生和以下同事：安托万（卢万-拉-纽夫），F. Bagarello（巴勒莫），M. Engli·s（布拉格），j . p .。Gazeau (Paris)，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成果。这项工作的大部分早期成果已发表在两本专著中，而一些较新的工作已在两篇评论文章中进行了描述，其中一篇发表在《数学物理百科全书》中。这项工作在信号分析和图像处理、量子光学和原子物理以及数学方面，在谐波分析、时频和小波分析领域都有应用。在接下来的几年里，我的工作预计将沿着上述路线继续下去，并进入一些新的方向。其中一个新的研究领域是将希尔伯特空间上的相干态推广到更一般的希尔伯特模。这除了其固有的数学价值外，在相关的次正规算子的研究中，也可以应用于非交换量子力学的问题，更广泛地应用于非交换空间的理论。我们也期望在这种情况下研究量子计算中的纠缠问题。第二个新的工作方向将是与相干态相关的正交多项式领域，第三个方向是经典统计学中的贝叶斯对偶性与相干态之间的联系。众所周知的恒定磁场中的电子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一些详细的研究，现在正在将这个系统推广到包括伪玻色子。我的研究涉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.