Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators

We study exact Wentzel-Kramers-Brillouin analysis (EWKB) for a $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics (QM) defined by the potential that $V_{\mathcal{P}\mathcal{T}}(x)={\omega }^2{x}^2+g{x}^{2K}(ix{)}^{\epsilon }$ with $\omega \in {\mathbb{R}}_{\ge 0}$, $g\in {\mathbb{R}}_{>0}$ and $K,\epsilon \in \mathbb{N}$ to clarify its perturbative/non-perturbative structure. In our analysis, we mainly consider the massless cases, i.e., $\omega =0$, and derive the exact quantization conditions (QCs) for arbitrary $(K,\epsilon )$ including all perturbative/non-perturbative corrections. From the exact QCs, we clarify full transseries structure of the energy spectra with respect to the inverse energy level expansion, and then formulate the Gutzwiller trace formula, the spectral summation form, and the Euclidean path-integral. For the massive cases, i.e., $\omega >0$, we show the fact that, by requiring existence of solution of the exact QCs, the path of analytic continuation in EWKB is uniquely determined for a given $N=2K+\epsilon$, and in consequence the exact QCs, the energy spectra, and the three formulas are all perturbative. Similarities to Hermitian QMs and resurgence are also discussed as additional remarks.