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Wigner Representation of Schrödinger Propagators

Elena Cordero, Gianluca Giacchi, Luigi Rodino

TL;DR

This work develops a Wigner-based microlocal framework for Fourier integral operators to analyze Schrödinger propagators generated by quadratic Hamiltonians with bounded pseudodifferential perturbations. By introducing the FIO(S) class and establishing a Wiener algebra structure, the authors obtain a global, caustic-robust description of propagators via Wigner kernels: for $H=a(x,D)+\sigma(x,D)$ and $t\in\mathbb{R}$, the propagator $e^{itH}$ belongs to $FIO(S_t)$ with a kernel $k(t,z,w)=h_t(z,S_t w)$. The paper also provides explicit Wigner-kernel formulas for FIOs of type I, analyzes their boundedness and compositional properties, and demonstrates the approach through canonical Schrödinger examples such as the free particle and magnetic potentials. The resulting framework offers a robust, globally valid representation of Schrödinger dynamics, including at caustics, and opens avenues for numerical computation and further generalizations to tame-phase FIOs. Overall, the work extends Wigner analysis to a broad class of FIOs, establishing both structural (algebraic) and practical (propagator) results with potential impact on quantum mechanics and PDE microlocal analysis.

Abstract

We perform a Wigner analysis of Fourier integral operators (FIOs), whose main examples are Schrödinger propagators arising from quadratic Hamiltonians with bounded perturbations. The perturbation is given by a pseudodifferential operator $σ(x,D)$ with symbol in the Hörmander class $S^0_{0,0}(\mathbb{R}^{2d})$. We compute and study the Wigner kernel of these operators. They are special instances of a more general class of FIOs named $FIO(S)$, with $S$ the symplectic matrix representing the classical symplectic map. We shall show the algebra and the Wiener's property of this class. The algebra will be the fundamental tool to represent the Wigner kernel of the Schrödinger propagator for every $t\in\mathbb{R}^d$, also in the caustic points. This outcome underlines the validity of the Wigner analysis for the study of Schrödinger equations.

Wigner Representation of Schrödinger Propagators

TL;DR

This work develops a Wigner-based microlocal framework for Fourier integral operators to analyze Schrödinger propagators generated by quadratic Hamiltonians with bounded pseudodifferential perturbations. By introducing the FIO(S) class and establishing a Wiener algebra structure, the authors obtain a global, caustic-robust description of propagators via Wigner kernels: for and , the propagator belongs to with a kernel . The paper also provides explicit Wigner-kernel formulas for FIOs of type I, analyzes their boundedness and compositional properties, and demonstrates the approach through canonical Schrödinger examples such as the free particle and magnetic potentials. The resulting framework offers a robust, globally valid representation of Schrödinger dynamics, including at caustics, and opens avenues for numerical computation and further generalizations to tame-phase FIOs. Overall, the work extends Wigner analysis to a broad class of FIOs, establishing both structural (algebraic) and practical (propagator) results with potential impact on quantum mechanics and PDE microlocal analysis.

Abstract

We perform a Wigner analysis of Fourier integral operators (FIOs), whose main examples are Schrödinger propagators arising from quadratic Hamiltonians with bounded perturbations. The perturbation is given by a pseudodifferential operator with symbol in the Hörmander class . We compute and study the Wigner kernel of these operators. They are special instances of a more general class of FIOs named , with the symplectic matrix representing the classical symplectic map. We shall show the algebra and the Wiener's property of this class. The algebra will be the fundamental tool to represent the Wigner kernel of the Schrödinger propagator for every , also in the caustic points. This outcome underlines the validity of the Wigner analysis for the study of Schrödinger equations.
Paper Structure (13 sections, 23 theorems, 189 equations, 1 figure)

This paper contains 13 sections, 23 theorems, 189 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The symplectic group $Sp(d,\mathbb{R})$ and the metaplectic operators Gos11
  4. Metaplectic Wigner distributions
  5. Pseudodifferential operators
  6. Tame phase functions and related canonical transformations
  7. Properties of the Wigner Kernel
  8. Properties of FIO($S$)
  9. FIOs of type I
  10. Properties of the Wigner kernel
  11. Application to Schrödinger equations
  12. Examples
  13. Comments and additional references

Key Result

Theorem 1.3

(i) An operator $T\in FIO(S)$ is bounded on $L^2(\mathbb{R}^d)$. (ii) If $T_i\in FIO(S_i)$, $i=1,2$, then $T_1 T_2\in FIO(S_1 S_2)$. (iii) If $T\in FIO(S)$ is invertible on $L^2(\mathbb{R}^d)$, then $T^{-1}\in FIO(S^{-1})$.

Figures (1)

  • Figure 1: Wigner analysis explained with an example.

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Properties of the class FIO($S$)
  • Theorem 1.4: Properties of the Wigner kernel
  • Proposition 2.1
  • Definition 2.2
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 38 more