Integral Representations for the Class of Generalized Metaplectic Operators

TL;DR

This paper derives explicit integral formulas for generalized metaplectic operators, a class of Fourier integral operators related to Schrödinger equations, enabling detailed analysis of solutions including at caustic points.

Contribution

It provides new integral representations for generalized metaplectic operators using symplectic matrix properties and time-frequency analysis, extending classical formulas to broader contexts.

Findings

Explicit integral formulas for generalized metaplectic operators.

Integral representation for Schrödinger equation solutions at all times.

Application to solutions with bounded perturbations, including caustic points.

Abstract

This article gives explicit integral formulas for the so-called generalized metaplectic operators, i.e. Fourier integral operators (FIOs) of Schr\"odinger type, having a symplectic matrix as canonical transformation. These integrals are over specific linear subspaces of R^d, related to the d x d upper left-hand side submatrix of the underlying 2d x 2d symplectic matrix. The arguments use the integral representations for the classical metaplectic operators obtained by Morsche and Oonincx in a previous paper, algebraic properties of symplectic matrices and time-frequency tools. As an application, we give a specific integral representation for solutions to the Cauchy problem of Schr\"odinger equations with bounded perturbations for every instant time t in R, even in the so-called caustic points.