# The Schr\"odinger equation for general non-hermitian quantum system

## TL;DR

This paper introduces a new time-dependent Schr"odinger equation for non-Hermitian quantum systems, revealing unique phenomena like geometric phase generation and Lieb-Robinson bound violation, with implications for anyonic models.

## Contribution

The authors derive a symmetric TDSE for non-Hermitian systems and demonstrate novel effects such as geometric phase manipulation and instantaneous action, linking these to anyonic models.

## Key findings

- Exchanges of quasi-particles generate geometric phases.
- Non-Hermitian systems can violate Lieb-Robinson bounds.
- Non-Hermitian single-particle models share features with anyonic models.

## Abstract

We derive a new time-dependent Schr\"odinger equation(TDSE) for quantum models with non-hermitian Hamiltonian. Within our theory, the TDSE is symmetric in the two Hilbert spaces spanned by the left and the right eigenstates, respectively. The physical quantities are also identical in these two spaces. Based on this TDSE, we show that exchanging two quasi-particles in a non-hermitian model can generate arbitrary geometric phase. The system can also violate the Lieb-Robinson bound in non-relativistic quantum mechanics so that an action in one place will immediately cause a change in the distance. We show that the above two surprising behaviors can also appear in anyonic model, which makes us propose that the non-hermitian single particle model may possess many common features with anyonic model.