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

We consider exact Wentzel-Kramers-Brillouin analysis to a $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics defined by the potential, $V(x)={\omega }^2{x}^2+g{x}^2(ix{)}^{ϵ=2}$ 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 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 $\omega >0$ or $\omega =0$: If $\omega >0$, 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 $\mathcal{P}\mathcal{T}$ 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/nonperturbative 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.

Figure 1

Schematic figure of our modified ABS conjecture for $\omega >0$. Borel resummation ${\mathcal{S}}_{0_{\pm }}$ and a one-parameter Stokes automorphism ${\mathfrak{S}}_0^{\nu \in \mathbb{R}}$ are defined around $\arg (g)=0$. Formal transseries of the $\mathcal{P}\mathcal{T}$ and the AC energies (resp. QCs), denoted by $E_{\mathcal{P}\mathcal{T}/\mathrm{AC}}$ (resp. ${\mathfrak{D}}_{\mathcal{P}\mathcal{T}/\mathrm{AC}}$), are connected to each other by the one-parameter Stokes automorphism with $\nu =\pm 1/2$. The superscript, $\arg (\lambda )=\pm \pi$, corresponds to $\lambda =-g+i{0}_{\pm }$ in Eq. (4). Acting the median resummation ${\mathcal{S}}_{\mathrm{med}}$ to the formal transseries gives their median resummed forms, ${\widehat{E}}_{\mathcal{P}\mathcal{T}/\mathrm{AC}}$ (resp. ${\widehat{\mathfrak{D}}}_{\mathcal{P}\mathcal{T}/\mathrm{AC}}$), that reproduce the original transseries by taking $\hslash \to 0_{+}$. By introducing another one-parameter Stokes automorphism acting to Borel resummed forms, ${\widehat{\mathfrak{S}}}^{\nu \in \mathbb{R}}$, the similar relations among the Borel resummed forms of the $\mathcal{P}\mathcal{T}$ and the AC energies (resp. QCs) hold formally. Borel resummation ${\mathcal{S}}_{0_{\pm }}$ makes a direct connection from $E_{\mathcal{P}\mathcal{T}/\mathrm{AC}}$ (resp. ${\mathfrak{D}}_{\mathcal{P}\mathcal{T}/\mathrm{AC}}$) to ${\widehat{E}}_{\mathrm{AC}/\mathcal{P}\mathcal{T}}$ (resp. ${\widehat{\mathfrak{D}}}_{\mathrm{AC}/\mathcal{P}\mathcal{T}}$).

Figure 2

Example of Stokes graph, a double-well potential. Turning points are denoted by blue dots, $a_{1,\dots ,4}$. The black solid and red wave lines mean Stokes lines and branch-cuts, respectively. The label, $+$ ($\mathrm{resp}\mathrm{.}-$), denotes an asymptotic domain where $\mathrm{Re}[{\int }_{a_j}^{x}d{x}^{\prime }{S}_{\mathrm{od},-1}(x^{\prime })]\to +\infty$ ($\mathrm{resp}\mathrm{.}-\infty$). We perform analytic continuation along the blue line slightly below the real axis from $x=-\infty$ to $x=+\infty$. In these cycles, only the $B$-cycle is nonperturbative, and ${\mathrm{C}}_{\mathrm{NP},\theta =0}=\{B\}$. The other cycles, $A$ and $C$, intersect with $B$ once, so that action of the Stokes automorphism to them is nontrivial and related to $B$. In this case, the intersection number is given by $⟨A,B⟩=-⟨C,B⟩=-1$.

Figure 3

Example of degeneracy of two Stokes lines. When $\arg (\hslash )=0$, two Stokes lines emerging from turning points (blue dots) on the one side go into the other side and degenerate. This degeneracy can be resolved by adding an infinitesimal complex phase to $\hslash$ as $\arg (\hslash )=0_{\pm }$.

Figure 4

Airy-type Stokes graph defined by a simple turning point, $a$ (blue dot). The colored arrows crossing the Stokes lines and the branch-cut are expressed by $M_{+}$ (green), $M_{-}$ (red), and $T$ (blue) in Eq. (66), respectively.

Figure 5

Stokes graph of the Hermitian potential with $\omega >0$. We take the analytic continuation along the path denoted the blue line to obtain the monodromy matrix.

Figure 6

Stokes graph of the Hermitian potential with a quadratic term. Varying $E_0\to 0_{+}$, the two simple turning points, $a_1$ and $a_4$, collide to each other and then become a double turning point (green dot) at $E_0=0$, where $E_0$ is the classical part of the energy.

Figure 7

Stokes graphs for $\arg (\lambda )=0,{\theta }_{*}=\frac{3\pi E|\lambda |}{4{\omega }^4}+O(E^{3/2})$, and $+\pi$. The similar phenomena occur at $-{\theta }_{*}$ and $-\pi$ by varying $\arg (\lambda )$ from 0 to $-\pi$.

Figure 8

Stokes graph of the negative coupling potential with a quadratic term. The paths for the analytic continuation are denoted by colored lines, ${\gamma }_{3\to 1}$ (red), ${\gamma }_{2\to 4}$ (green), and ${\gamma }_{3\to 4}$ (blue).

Figure 9

Stokes graph with $E_0=0$ corresponding to Fig. 8, where $E_0$ is the classical part of the energy. By taking $E_0\to 0_{+}$, two simple turning points, $a_2$ and $a_3$, collide to each other and then become a double turning point (green dot) at $E_0=0$.

Figure 10

Stokes graph of the negative coupling potential without a quadratic term. The paths for analytic continuation are denoted by colored lines, ${\gamma }_{3\to 1}$ (red), ${\gamma }_{2\to 4}$ (green), and ${\gamma }_{3\to 4}$ (blue).

Figure 11

Truncated $\mathcal{P}\mathcal{T}$ energy spectrum for first five energy levels, $k=0,\dots ,4$, evaluated by Eq. (208) with $g{\hslash }^4=1$. $n_{\max }$ in the horizontal axis denotes the truncation order of the truncated $\mathcal{P}\mathcal{T}$ energy solution. The black dashed lines are exact values from Ref. [8].

Figure 12

Stokes graphs for $\arg [{\tilde{Q}}_0{\kappa }^2]=0,\mp \frac{\pi }{2}$. The QCs in Eqs. (200) and (201) are constructed under the assumption that $\arg [{\tilde{Q}}_0{\kappa }^2]=0$ shown in (a). The solution of ${\mathfrak{D}}_{\mathrm{AC}}^{\arg (\tilde{\eta })=0}=0$ violates the assumption and reproduces either (b) or (c), where another Stokes phenomenon occurs, depending on the paths of analytic continuation, ${\gamma }_{3\to 1}$ or ${\gamma }_{2\to 4}$.

Figure 13

Stokes graph for the degenerate Weber equation. The green dot is a double turning point. The black solid and red wave lines denote Stokes lines and a branch-cut, respectively.