# Projective Hilbert space structures at exceptional points

## TL;DR

This paper investigates the geometric and algebraic structures of projective Hilbert spaces near exceptional points in non-Hermitian systems, revealing how to resolve normalization issues and analyze phase behavior.

## Contribution

It introduces a projective extension of Hilbert space to resolve normalization contradictions at EPs and analyzes phase and PT-symmetry aspects in this framework.

## Key findings

- Normalization conditions are unified via projective extension.
- Phase rigidity effectively measures distance to EPs.
- Geometric phase analysis reveals phase jump behavior.

## Abstract

A non-Hermitian complex symmetric 2x2 matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behavior in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase jump behavior are analyzed and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT-symmetrically extended Quantum Mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated.