PT-Rotations, PT-Spherical Harmonics and the PT-Hydrogen Atom

TL;DR

This paper introduces non-Hermitian operators satisfying SO(3) and SO(4) Lie algebras, generating rotations in a modified space, and applies them to analyze a PT-Hydrogen atom with complex terms, revealing conserved quantities.

Contribution

It constructs non-Hermitian rotation generators and spherical harmonics, and applies these to solve a PT-Hydrogen atom with complex potential terms, identifying conserved quantities.

Findings

Non-Hermitian operators generate rotations in a modified momentum space.

PT-spherical harmonics are PT-orthonormal.

Conserved non-Hermitian Runge-Lenz vector in PT-Hydrogen atom.

Abstract

We have constructed a set of non-Hermitian operators that satisfy the commutation relations of the SO(3)-Lie algebra. It is shown that this operators generate rotations in the configuration space and not in the momentum space but in a modified non-Hermitian momentum space. This generators are related with a new type of spherical harmonics that result to be PT-orthonormal. Additionally, we have shown that this operators represent conserved quantities for a non-Hermitian Hamiltonian with an additional complex term. As a particular case, the solutions of the corresponding PT-Hydrogen atom that includes a complex term are obtained, and it is found that a non-Hermitian Runge-Lenz vector is a conserved quantity. In this way, we obtain a set of non-Hermitian operators that satisfy the SO(4)-Lie algebra.