Lagrangian distributions and Fourier integral operators with quadratic phase functions and Shubin amplitudes

Shubin type Fourier integral operators and evolution equations

We study the Cauchy problem for an evolution equation of Schrödinger type. The Hamiltonian is the Weyl quantization of a real homogeneous quadratic form with a pseudodifferential perturbation of negative order from Shubin's class. We prove that the propagator is a Fourier integral operator of Shubin type of order zero. Using results for such operators and corresponding Lagrangian distributions, we study the propagator and the solution, and derive phase space estimates for them.

Shubin--Vakilian--Wolff 半密度不确定性原理

The Shubin--Vakilian--Wolff Uncertainty Principle at Half Density

该研究否定回答了 Shubin-Vakilian-Wolff 关于半密度以下不确定性原理的猜想，构造了 $1/2$ 密度薄集的反例，并给出了 $\kappa$ 小时临界密度趋于 $1/2$ 的正估计。

Null-controllability of evolution equations associated with fractional Shubin operators through quantitative Agmon estimates

We consider the anisotropic Shubin operators $(-Δ)^m + \vert x\vert^{2k}$ acting on the space $L^2(\mathbb R^n)$, with $k, m \geq1$ some positive integers. We provide sharp quantitative estimates in Gelfand-Shilov spaces for the eigenfunctions of these selfadjoint differential operators, that is, exponential decay estimates both for these functions and their Fourier transforms in $L^2(\mathbb R^n)$. The strategy implemented is based on the classical approach to obtain Agmon estimates in spectral theory. By using a Weyl law for the eigenvalues of the anisotropic Shubin operators, we also descri…

带复相位拉格朗日分布的主象征映射

The Principal Symbol Map for Lagrangian Distributions with Complex Phase

本文针对带复相位的拉格朗日分布，提出主象征映射的新构造方法，从而支持傅里叶积分算子干净复合后的主象征计算。

On traces of Fourier integral operators on submanifolds

Given a smooth embedding $i\colon X \hookrightarrow M$ of manifolds and a Fourier integral operator $Φ= Φ(Λ)$ on $M$ associated with a Lagrangian submanifold $Λ\subset T^*(X\times X) \setminus \{0\}$, we consider its trace $i^!(Λ)$ on the submanifold $X$, i.e. the composition $i^* Φi_*$, where $i^*$ and $i_*$ are the boundary and coboundary operators, respectively. We establish the conditions under which the trace $i^!(Φ)$ is also a Fourier integral operator and calculate its amplitude in canonical local coordinates.

Fourier integral operators with open umbrellas and seismic inversion for cusp caustics

In general the composition of Fourier integral operators (FIOs) need not be an FIO. Motivated by the problem of linearized seismic inversion in the presence of cusp caustics for the background sound speed, we consider FIOs whose canonical relations have certain two-sided cusp degeneracies, and show that the resulting compositions have wave-front relations in the union of the diagonal and an open umbrella, the simplest type of singular Lagrangian manifold.