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

The Hamiltonian H = p(2)+x(4)+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\ --> infinity and E --> infinity, at the points of coalescence the ratio a = \A\E-3/4 approaches a constant whose numerical value is a(crit) = 1.1838363072914. Conventional WKB theory determines the high-lying energy levels but cannot be used to calculate a(crit). This critical value is predicted exactly by complex WKB theory.