Factorization of Second-order strictly hyperbolic operators with non-smooth coefficients and microlocal diagonalization

TL;DR

This paper develops a microlocal diagonalization technique for second-order strictly hyperbolic operators with non-smooth coefficients, using semiclassical Colombeau equations to factorize and analyze their structure at infinity.

Contribution

It introduces a novel factorization and diagonalization method for non-smooth hyperbolic operators via semiclassical Colombeau equations, advancing microlocal analysis techniques.

Findings

Operator factorization into first-order constituents

Microlocal approximation at infinite points

Coupling effect computation

Abstract

We study strictly hyperbolic partial differential operators of second order with non-smooth coefficients. After modelling them as semiclassical Colombeau equations of log-type we provide a factorization procedure on some time-space-frequency domain. As a result the operator is written as a product of two semiclassical first-order constituents of log-type which approximates the modelled operator microlocally at infinite points. We then present a diagonalization method so that microlocally at infinity the governing equation is equal to a coupled system of two semiclassical first-order strictly hyperbolic pseudodifferential equations. Furthermore we compute the coupling effect. We close with some remarks on the results and future directions.