论文标题
通过符号表示形式和对Schrödinger方程的应用来表征调制空间
Characterization of modulation spaces by symplectic representations and applications to Schrödinger equations
论文作者
论文摘要
在过去的二十年中，H。G。Feichtinger于1983年引入的调制空间已成功地用于信号分析的研究，PDE，Pseudodidiffertential Operators，Quantum Mechanics，由数百个贡献。在2011年，De Gosson M. De Gosson表明，可以用Wigner分布代替了定义调制空间的工具的短时傅立叶变换（STFT）。这个想法进一步推广到[9]中的$τ$ -WIGNER表示。在本文中，时间频率表示为通过元容器的符号矩阵的图像。这种新的观点强调，时间频率分析的主角是跨度运算符和symplectic矩阵$ \ mathcal {a} \ in sp（2d，\ mathbb {r}）$。我们在$ \ Mathcal {a} $上找到条件，相关的符号时间频率表示$ W_ \ MATHCAL {A} $可以替换STFT并为加权调制空间提供等效的规范。特别是，我们研究了协变量矩阵$ \ MATHCAL {a} $的情况，即，它们相应的$ W_ \ Mathcal {a} $是Cohen类的成员。 最后，我们表明符合时间频率表示$ W_ \ MATHCAL {A} $可以有效地用于Schrödinger方程的研究中。这种新方法可能在量子力学和PDE方面有进一步的应用。
In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to $τ$-Wigner representations in [9]. In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices $\mathcal{A} \in Sp(2d,\mathbb{R})$. We find conditions on $\mathcal{A}$ for which the related symplectic time-frequency representation $W_\mathcal{A}$ can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices $\mathcal{A}$, i.e., their corresponding $W_\mathcal{A}$ are members of the Cohen class. Finally, we show that symplectic time-frequency representations $W_\mathcal{A}$ can be efficiently employed in the study of Schrödinger equations. This new approach may have further applications in quantum mechanics and PDE's.
