Exact WKB analysis for $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture

Abstract

We consider exact WKB analysis to a $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics defined by the potential, $V(x)={\omega }^2{x}^2+g{x}^2(ix{)}^{\epsilon =2}$ with $\omega \in {\mathbb{R}}_{\ge 0}$, $g\in {\mathbb{R}}_{>0}$. We in particular aim to verify a conjecture proposed by Ai-Bender-Sarkar (ABS), that pertains to a relation between $D$-dimensional $\mathcal{P}\mathcal{T}$-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 WKB 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 $\omega >0$ or $\omega =0$: If $\omega >0$, then the ABS conjecture is violated when exceeding the semi-classical level of the first non-perturbative order, but its alternative form is constructable by Borel resummation theory. The $\mathcal{P}\mathcal{T}$ 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. $\mathcal{P}\mathcal{T}$ energy) is directly obtained by acting Borel resummation to a transseries solution of the $\mathcal{P}\mathcal{T}$ energy (resp. AC energy). If $\omega =0$, then, with respect to the inverse energy level-expansion, not only perturbative/non-perturbative structures of the $\mathcal{P}\mathcal{T}$ 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.

arXiv comment40 pages, 13 figures, v2: typo and minor corrections, v3: typo and grammar corrections, references added, another example added in Sec. 3.1, structure slightly modified, v4: edit format is changed, minor revision, Sec VI A is modified, the conclusion is unchanged, accepted in PRD

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• Exact WKB in all sectors II: Potentials with non-degenerate saddles