Asymptotic solvability of an imaginary cubic oscillator with spikes

TL;DR

This paper investigates the asymptotic behavior of eigenstates in a PT symmetric potential with spikes, revealing that in the large angular momentum limit, the eigenstates resemble shifted harmonic oscillators, aiding numerical analysis.

Contribution

It introduces an approximate Hermitization method for a non-Hermitian PT symmetric oscillator in the large angular momentum limit, valid where the spectrum remains real.

Findings

Eigenstates match shifted harmonic oscillators at high angular momentum

Approximate Hermitization is numerically efficient

Discontinuity occurs at PT regularization vanishing point

Abstract

For the PT symmetric potential of Dorey, Dunning and Tateo we show that in the large angular momentum (i.e., strongly spiked) limit the low-lying eigenstates of this popular non-Hermitian problem coincide with the shifted Hermitian harmonic oscillators calculated at the zero angular momentum. This type of an approximate Hermitization is valid in all the domain where the spectrum of energies remains real. It proves very efficient numerically. The construction is asymmetric with respect to the sign of the subdominant square-root spike, and exhibits a discontinuity at the point where the PT symmetric regularization vanishes.