Real eigenvalues are determined by the recursion of eigenstates

TL;DR

This paper proposes a novel perspective showing that real eigenvalues in quantum systems can arise from recursive conditions of eigenstates, offering an alternative method to identify real spectra in non-Hermitian systems.

Contribution

It introduces a new approach based on recursive eigenstate conditions to determine real eigenvalues, independent of PT symmetry considerations.

Findings

Real eigenvalues can emerge from recursive eigenstate conditions.

This method offers an alternative to PT symmetry for obtaining real spectra.

The approach supports probability conservation in non-Hermitian systems.

Abstract

Quantum physics is generally concerned with real eigenvalues due to the unitarity of time evolution. With the introduction of $\mathcal{PT}$ symmetry, a widely accepted consensus is that, even if the Hamiltonian of the system is not Hermitian, the eigenvalues can still be pure real under specific symmetry. Hence, great enthusiasm has been devoted to exploring the eigenvalue problem of non-Hermitian systems. In this work, from a distinct perspective, we demonstrate that real eigenvalues can also emerge under the appropriate recursive condition of eigenstates. Consequently, our findings provide another path to extract the real energy spectrum of non-Hermitian systems, which guarantees the conservation of probability and stimulates future experimental observations.