Exact quantization of a PT-symmetric (reversible) Li\'enard-type nonlinear oscillator

TL;DR

This paper presents an exact quantum and semiclassical analysis of a PT-symmetric nonlinear oscillator, revealing that its spectrum matches that of a linear harmonic oscillator despite different eigenfunctions.

Contribution

It introduces a novel exact quantization method for a PT-symmetric nonlinear oscillator using momentum space solutions and confirms the spectrum's equivalence to a harmonic oscillator.

Findings

Eigenvalues match harmonic oscillator spectrum

Eigenfunctions differ significantly from harmonic oscillator

Semiclassical and quantum results are consistent

Abstract

We carry out an exact quantization of a PT symmetric (reversible) Li\'{e}nard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard type and is invariant under a combined coordinate reflection and time reversal transformation. We use von Roos symmetric ordering procedure to write down the appropriate quantum Hamiltonian. While the quantum problem cannot be tackled in coordinate space, we show how the problem can be successfully solved in momentum space by solving the underlying Schr\"{o}dinger equation therein. We obtain explicitly the eigenvalues and eigenfunctions (in momentum space) and deduce the remarkable result that the spectrum agrees exactly with that of the linear harmonic oscillator, which is also confirmed by a semiclassical modified Bohr-Sommerfeld quantization rule, while the eigenfunctions are completely different.