# Wave equation for operators with discrete spectrum and irregular propagation speed

## TL;DR

This paper studies the well-posedness of wave equations with operators having discrete spectra under various irregular propagation speeds, introducing 'very weak solutions' for distributional coefficients and demonstrating their uniqueness.

## Contribution

It introduces the concept of 'very weak solutions' for wave equations with distributional coefficients and proves their uniqueness, extending classical solution frameworks to irregular propagation speeds.

## Key findings

- Existence of unique very weak solutions for wave equations with distributional coefficients.
- Extension of classical solutions to irregular propagation speeds, including distributional cases.
- Application to various operators like harmonic oscillator, Landau Hamiltonian, and operators on manifolds.

## Abstract

Given a Hilbert space, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with discrete non-negative spectrum acting on it. We consider the cases when the time-dependent propagation speed is regular, H\"older, and distributional. We also consider cases when it it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of "very weak solutions" to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic oscillator and the Landau Hamiltonian on $\mathbb R^n$, uniformly elliptic operators of different orders on domains, H\"ormander's sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.