Exact WKB analysis for PT -symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture

Abstract

We consider exact Wentzel-Kramers-Brillouin analysis to a PT symmetric quantum mechanics defined by the potential, V(x)=ω2x2+gx2(ix)ϵ=2 with ω∈R≥0, g∈R>0. We in particular aim to verify a conjecture proposed by Ai-Bender-Sarkar (ABS), that pertains to a relation between D-dimensional PT-symmetric theories and analytic continuation (AC) of Hermitian theories concerning the energy spectrum or Euclidean partition function. For the purpose, we construct energy quantization conditions by exact Wentzel-Kramers-Brillouin analysis and write down their transseries solution by solving the conditions. By performing alien calculus to the energy solutions, we verify validity of the ABS conjecture and seek a possibility of its alternative form by Borel resummation theory if it is violated. Our results claim that the validity of the ABS conjecture drastically changes depending on whether ω>0 or ω=0: If ω>0, then the ABS conjecture is violated when exceeding the semiclassical level of the first nonperturbative order, but its alternative form is constructable by Borel resummation theory. The PT and the AC energies are related to each other by a one-parameter Stokes automorphism, and a median resummed form, which corresponds to a formal exact solution, of the AC energy (resp. PT energy) is directly obtained by acting Borel resummation to a transseries solution of the PT energy (resp. AC energy). If ω=0, then, with respect to the inverse energy level-expansion, not only perturbative/nonperturbative structures of the PT and the AC energies but also their perturbative parts do not match with each other. These energies are independent solutions, and no alternative form of the ABS conjecture can be reformulated by Borel resummation theory. Published by the American Physical Society 2024