# New families of non-parity-time-symmetric complex potentials with all-real spectra

## TL;DR

This paper introduces new non-parity-time-symmetric complex potentials with all-real spectra, derived via supersymmetry and pseudo-Hermiticity methods, featuring explicit analytical forms and tunable spectral properties.

## Contribution

It presents novel methods to construct complex potentials with real spectra without relying on discrete eigenmodes, expanding the class of potentials with controllable spectral characteristics.

## Key findings

- Derived explicit analytical expressions for complex potentials with real spectra.
- Identified conditions under which eigenvalues are real or form conjugate pairs.
- Demonstrated phase transition phenomena with complex eigenvalues emerging.

## Abstract

New families of non-parity-time-symmetric complex potentials with all-real spectra are derived by the supersymmetry method and the pseudo-Hermiticity method. With the supersymmetry method, we find families of non-parity-time-symmetric complex partner potentials which share the same spectrum as base potentials with known real spectra, such as the (complex) Wadati potentials. Different from previous supersymmetry derivations of potentials with real spectra, our derivation does not utilize discrete eigenmodes of base potentials. As a result, our partner potentials feature explicit analytical expressions which contain free functions. With the pseudo-Hermiticity method, we derive a new class of non-parity-time-symmetric complex potentials with free functions and constants, whose eigenvalues appear as conjugate pairs. This eigenvalue symmetry forces the spectrum to be all-real for a wide range of choices of these functions and constants in the potential. Tuning these free functions and constants, phase transition can also be induced, where conjugate pairs of complex eigenvalues emerge in the spectrum.