AI 中文总结
该研究探究由各向同性奥恩斯坦-乌伦贝克算子生成的薛定谔演化，通过对应关系揭示其与自由薛定谔结构的关联，得到时空型限制定理、酉群全局描述及相关估计。
AI 中文摘要
我们研究由各向同性奥恩斯坦-乌伦贝克算子 \mathscr L=\Delta-\langle x,\nabla\rangle 在 $L^2(\mathbb R^d,d\gamma)$ 中生成的薛定谔演化，其中 $d\gamma$ 是不变高斯测度。结合自由薛定谔方程与谐振子之间的经典透镜对应关系以及高斯共轭，我们识别出隐藏在奥恩斯坦-乌伦贝克演化中的自由薛定谔结构。主要结果是一个时空型的动态限制定理：经时空变量的显式变换后，奥恩斯坦-乌伦贝克延拓算子的伴随成为限制在经典薛定谔抛物面上的普通时空傅里叶变换，在奥恩斯坦-乌伦贝克侧对应于显式依赖时间的高斯时空测度，由此产生的限制几何既非由 \mathscr L 的符号所暗示，也非由奥恩斯坦-乌伦贝克方程的标度对称所选取，而是从自由薛定谔表示中动态涌现。该对应关系还识别出齐次与非齐次奥恩斯坦-乌伦贝克 Strichartz 估计中的加权高斯空间，并将质量临界非线性薛定谔方程转化为 $i\partial_tu+\mathscr Lu = \mu e^{-|x|^2/d}|u|^{4/d}u$。我们还得到了酉奥恩斯坦-乌伦贝克群的全局描述，包括其在焦散外的振荡表示、焦散时刻的精确值，以及相关的尖锐加权色散估计和 Hardy 型不确定原理。
英文摘要
We study the Schrödinger evolution generated by the isotropic Ornstein--Uhlenbeck operator \[ \mathscr L=Δ-\langle x,\nabla\rangle \] in $L^2(\mathbb R^d,dγ)$, where $dγ$ is the invariant Gaussian measure. Combining the classical lens correspondence between the free Schrödinger equation and the harmonic oscillator with Gaussian conjugation, we identify the free Schrödinger structure hidden in the Ornstein--Uhlenbeck evolution. The main consequence is a dynamic restriction theorem of spacetime type. After an explicit transformation of spacetime variables, the adjoint Ornstein--Uhlenbeck extension operator becomes an ordinary spacetime Fourier transform restricted to the classical Schrödinger paraboloid. On the Ornstein--Uhlenbeck side this corresponds to an explicitly time-dependent Gaussian spacetime measure. Thus a restriction geometry which is neither suggested by the symbol of $\mathscr L$ nor selected by a scaling symmetry of the Ornstein--Uhlenbeck equation emerges dynamically from the free Schrödinger representation. The same correspondence identifies the weighted Gaussian spaces in the homogeneous and inhomogeneous Ornstein--Uhlenbeck Strichartz estimates and transports the mass-critical nonlinear Schrödinger equation to \[ i\partial_tu+\mathscr Lu = μe^{-|x|^2/d}|u|^{4/d}u. \] We also obtain a global description of the unitary Ornstein--Uhlenbeck group, including its oscillatory representation away from the caustics and its exact values at the caustic times, together with the associated sharp weighted dispersive estimate and Hardy-type uncertainty principles.
