Asymptotic solvability of an imaginary cubic oscillator with spikes

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.