Level crossings in complex two-dimensional potentials

摘要

Two-dimensional $mathcal{PT}$-symmetric quantum-mechanical systems with the complex cubic potential e?‘‰12 = e?‘￥2 + e?‘|2 + e?‘–e?‘”e?‘￥e?‘|2 and the complex H??nona€“Heiles potential e?‘‰HH = e?‘￥2 + e?‘|2 + e?‘–e?‘”(e?‘￥e?‘|2 a?’ e?‘￥3/3) are investigated. Using numerical and perturbative methods, energy spectra are obtained to high levels. Although both potentials respect the $mathcal{PT}$ symmetry, the complex energy eigenvalues appear when level crossing happens between same parity eigenstates.