Complex WKB analysis of energy-level degeneracies of non-Hermitian Hamiltonians

The Hamiltonian H = p2 + x4 + iAx, where A is a real parameter, is investigated. The spectrum of H is discrete and entirely real and positive for |A|<3.169. As |A| increases past this point, adjacent pairs of energy levels coalesce and then become complex, starting with the lowest-lying energy levels. For large energies, the values of A at which this merging occurs scale as the three-quarters power of the energy. That is, as |A|→∞ and E→∞, at the points of coalescence the ratio a = |A|E-3/4 approaches a constant whose numerical value is acrit = 1.1838363072914···. Conventional WKB theory determines the high-lying energy levels but cannot be used to calculate acrit. This critical value is predicted exactly by complex WKB theory.