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

The HamiltonianH = p2+x4+iAx, whereA 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 ofA 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. PACS numbers: 0230M, 1110K, 1110L, 1130E In this Letter we examine the Hamiltonian H = p + x + iAx, (1) where A is a real parameter. This Hamiltonian is an additive complex deformation of the conventional Hermitian HamiltonianH = p2 +x4, which represents the pure quartic oscillator. This Hamiltonian is a special case of a slightly more general Hamiltonian previously examined by Delabaere and Pham [1]5. This Hamiltonian is of interest because, while the HamiltonianH is complex for allA = 0, its entire spectrum is discrete, real, and positive for |A| < 3.169 (see figure 1). The reality of the spectrum is apparently due to the PT invariance of the Hamiltonian. However, not all 3 Web site: http://www.phy.bris.ac.uk/staff/berry mv.html 4 Permanent address: Gazi Universitesi, Fen Edebiyat Fakultesi, Fizik Bolumu, 06500 Teknikokullar-Ankara, Turkey. 5 Very recently, an oscillator problem like that in (1) except with x4 replaced by x3 was studied by Delabaere and Trinh [2]. 0305-4470/01/060031+06$30.00 © 2001 IOP Publishing Ltd Printed in the UK L31 L32 Letter to the Editor 0 10 20 -10 -20 0 10 20 30 40