Title: Sparse Gabor representations of metaplectic operators: controlled exponential decay and Schrödinger confinement Show Chinese title

Title: 稀疏Gabor表示的元射影算子：受控指数衰减和Schrödinger局域化

Abstract: Motivated by the phase space analysis of Schr\"odinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave packets. Quantitative bounds for the matrix coefficients arising in the Gabor wave packet decomposition of such operators are established, revealing precise exponential decay rates together with subtler dispersive and spreading phenomena. To this aim, we present several novel results concerning the time-frequency analysis of functions with controlled Gelfand-Shilov regularity, which are of independent interest. As a byproduct, we generalize Vemuri's Gaussian confinement results for the solutions of the quantum harmonic oscillator in two respects, namely by encompassing general exponential decay rates as well as arbitrary quadratic Schr\"odinger propagators. In particular, we extensively discuss some prominent models such as the harmonic oscillator, the free particle in a constant magnetic field and fractional Fourier transforms. Show Chinese abstract

Abstract: 受Schrödinger演化算子相空间分析的启发，本文研究了如何通过指数局部化的Gabor小波包，对相应的辛流沿其进行近似对角化。 建立了在这些算子的Gabor小波包分解中出现的矩阵系数的定量界，揭示了精确的指数衰减率以及更细微的色散和扩展现象。 为此，我们提出了几个关于具有可控Gelfand-Shilov正则性的函数的时间频率分析的新结果，这些结果具有独立的兴趣。 作为副产品，我们将Vemuri关于量子谐振子解的高斯约束结果推广到两个方面，即包括一般的指数衰减率以及任意二次Schrödinger传播器。 特别是，我们详细讨论了一些显著的模型，如谐振子、常磁场中的自由粒子以及分数傅里叶变换。