Metaplectic 算子的 Pitt 不等式与对数型不确定性原理

中文摘要 AI 辅助

本文对 Metaplectic 算子的 Pitt 不等式给出了完整的刻画。Metaplectic 算子背后的辛几何区分了有效方向与奇异方向：在有效方向上，其作用类似于 Fourier 变换；在奇异方向上，浓度保持不变。为此，我们采用两种互补的视角，从而同时得到各向同性的 Pitt 不等式（模拟经典的 Pitt 定理）和适应 Metaplectic 群几何特征的各向异性不等式。作为副产品，我们得到了沿 $\ d$ 的子空间的 Fourier 变换的 Pitt 不等式，以及 Metaplectic 算子在齐次 Sobolev 空间上的新的定量时间相关有界性结果，并应用于由二次 Hamilton 量生成的 Schrödinger 演化。此外，我们推导了 Metaplectic 群的对数型和熵型不确定性原理。我们证明，当有效方向和奇异方向同时存在时，经典的熵型不确定性原理失效。在这种情况下，一个通常无界的修正熵项解释了浓度保持方向，并恢复了有意义的下界。对于二次 Schrödinger 演化，该修正遵循非色散方向的时间相关几何，并相应地调整不确定性估计。

英文摘要

In this work, we provide a complete characterisation of Pitt's inequality for metaplectic operators. The symplectic geometry underlying a metaplectic operator distinguishes effective directions, along which its action is Fourier-type, from singular directions, along which concentration is preserved. For this reason, we adopt two complementary perspectives, thereby obtaining both an isotropic Pitt's inequality, emulating the classical theorem of Pitt, and an anisotropic inequality that adapts to the geometric features of the metaplectic group. As a by-product, we obtain Pitt's inequality for Fourier transforms along subspaces of $\rd$ and new quantitative time-dependent boundedness results for metaplectic operators on homogeneous Sobolev spaces, with applications to Schrödinger evolutions generated by quadratic Hamiltonians. Additionally, we derive logarithmic and entropic uncertainty principles for the metaplectic group. We show that the classical entropic uncertainty principle fails when both the effective and singular directions are present. In this case, a generally unbounded corrective entropy term accounts for the concentration-preserving directions and restores a meaningful lower bound. For quadratic Schrödinger evolutions, this correction follows the time-dependent geometry of the nondispersive directions and adjusts the uncertainty estimate accordingly.

发表机构

• Università della Svizzera italiana（瑞士意大利语大学）
• University of Birmingham（伯明翰大学）

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