Exact WKB analysis of inverted triple-well: resonance, PT-symmetry breaking, and resurgence

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Comprehensive Research Summary: Exact WKB Analysis of Non-Hermitian Quantum Mechanics in an Inverted Triple-Well Potential

This research employs the Exact WKB (EWKB) formalism as a powerful, unifying framework to study spectral problems and resurgence theory within one-dimensional non-Hermitian quantum mechanics, specifically focusing on an inverted triple-well (ITW) potential with parity symmetry. The study rigorously derives the exact quantization conditions (QCs) for three distinct physical scenarios defined by different Siegert boundary conditions: PT-symmetric systems, resonance systems, and anti-resonance systems.

Core Methodological Framework

The central technique involves deriving the exact QCs for these three cases on an equal footing. By identifying the relevant resurgent structures and associated cancellations within these non-Hermitian setups, the authors obtain the median-summed trans-series contributions to the physical spectra. This approach leverages Alien calculus applied to median QCs, which reveals a universal minimal trans-series that participates in resurgence cancellations across all studied systems and parameter regimes. Furthermore, explicit links are established between these results and the semi-classical path integral formalism, elucidating the roles of bounce and bion configurations. The analysis utilizes the Stokes automorphism to extract exact median-summed non-perturbative spectra from the median QCs.

Key Findings by System Type

The analysis yields distinct, yet interconnected, results for each quantum problem:

1. PT-Symmetric Systems

• Symmetry and Breaking: The study predicts PT-symmetry breaking, which is subsequently verified numerically.

• Exceptional Point Analysis: Using the median quantization conditions, the authors establish an exact equation for the exceptional point, which emerges as a remarkably simple algebraic relation between the bounce and bion actions: B2 = 2B14 (1 + B1).

• Phase Determination: The nature of the spectrum is determined by comparing bounce and bion actions:

• Unbroken PT-Symmetric Phase: Characterized by B2(1 + B1) > 2B1.

• Broken Phase: Occurs when 2B1 > B2(1 + B1).

• Non-Perturbative Corrections: The median-summed non-perturbative correction to the spectrum vanishes exactly at the exceptional point. In the unbroken phase, this correction is given by a real and unambiguous expression: S 0 delta_UBr_PT epsilon-K 2S 0hP(1)0i + K 2 1K 2S 0[PB2I] + K 2 2S 0hP(2)0P(21i − 4K 3/8)]. In the broken phase, the imaginary part of the spectrum is non-zero and determined by terms involving K 1 and K 2, confirming non-perturbative PT-symmetry breaking.

2. Resonance and Anti-Resonance Systems

• Time Reversal Invariance: The exact quantization conditions (QCs) for these systems are not invariant under time reversal. Instead, the QCs for resonance (R) and anti-resonance (AR) systems are related by complex conjugation, meaning they map onto each other as time-reversed partners. This reflects their intrinsically irreversible and non-equilibrium nature.

• Spectrum Nature: The exact median-summed spectra for these cases cannot be real; they are necessarily complex.

• Minimal Trans-Series: Despite the spectral differences, both resonance and anti-resonance systems share the same minimal trans-series, indicating a deep underlying universality in their non-perturbative structure. The exact non-perturbative part is found as J(plus or minus) NP = plus or minus i 1 - i epsilon p/p B2/1 + B1/p B2.

Unified Insights and Formal Connections

The analysis successfully bridges several advanced theoretical domains:

• Unification: The EWKB formalism provides a powerful, unified perspective on the quantization of non-Hermitian theories by treating PT-symmetric, resonance, and anti-resonance cases on an equal footing.

• Resurgence Structure: The identification of resurgent structures and cancellations via Alien calculus demonstrates that all systems participate in the same universal minimal trans-series structure for resurgence cancellations.

• Semi-Classical Verification: Asymptotic limits (expanding around perturbative states using J = J P + J NP) confirm the semi-classical analysis, showing that the median-summed non-perturbative contributions recover results from previous studies involving bounce and bion configurations.

• Exceptional Points: Exact exceptional points are identified as fugacities of semi-classical bounce and bion configurations. The competition between these classical actions is highlighted as the physical mechanism driving PT-symmetry breaking.

In conclusion, this paper provides a unified, general perspective on the quantization of non-Hermitian theories by establishing exact relationships between different quantum regimes (PT-symmetric, resonance, anti-resonance) through the lens of median quantization conditions and resurgence theory. It precisely maps out the conditions for spectral reality versus complexity and rigorously defines the non-perturbative corrections across these diverse physical settings.

1. Exact Quantization of Non-Hermitian Systems: The improved system can determine whether a quantum system is in an unbroken or broken PT-symmetric phase by calculating DiscPT, which is defined as "DiscPT 0, if K2 > K1, (5.11) and DiscPT = 0 gives the exact condition for the exceptional point of the PT-symmetric system, which we analyze now."

2. Non-Perturbative Spectrum Calculation: The system can calculate non-perturbative corrections to the spectrum by using the median-summed non-perturbative contributions to the spectrum as described in equation (4.31), which is a real and unambiguous expression that confirms that the "K1 < K2 limit falls into the unbroken PT-symmetric phase."

3. Resurgence Structure Mapping: The improved system can uncover the universal minimal trans-series by using Alien calculus to relate the Borel summations and non-perturbative configurations, allowing it to identify where the remaining non-perturbative corrections cancel exactly.

4. Transition Point Determination: The system can determine the exact transition point between phases by solving for the exceptional point condition, which is given by ΠB2 = Π2 B1 4 (1 + ΠB1) and analyzing the condition "Unbroken phase: ΠB2 (1 + ΠB1) > Π 2 B1, (5.25) Broken phase: Π2 B1 > ΠB2 (1 + ΠB1), (5.26)."

5. Spectral Properties of Resonance Systems: The system can distinguish between resonance and anti-resonance cases by observing that the median-summed trans-series solutions for resonance and antiresonance cases are complex conjugate of each other: S0 h δ(±) R i = CS0 h δ(±) AR i, (4.54).

6. Unambiguous Spectrum Extraction: The system can extract the final single-valued spectrum by using the median summation to remove ambiguities, as shown in equation (4.29), which leads to a real and single-valued expression as ∓iS± h P(1)0 i ∼ (+1) (1 + O(ħ)), (4.29).

We study non-Hermitian quantum mechanics of an inverted triple-well potential within the exact WKB framework. For a single classical potential, different Siegert boundary conditions define three distinct quantum problems: the PT-symmetric, resonance, and anti-resonance systems. For each case, we derive the exact quantization condition and construct the associated trans-series solution. By identifying the resurgent structures and cancellations in these non-Hermitian setups, we obtain the median-summed series, clarifying when the spectra are real or complex in accordance with the physical properties of each system. Establishing explicit links to the semi-classical path integral formalism, we elucidate the roles of bounce and bion configurations in these non-Hermitian systems. This analysis predicts PT-symmetry breaking, which we also verify numerically. Using the median quantization conditions, we prove the existence of this symmetry breaking and establish an exact equation for the exceptional point, which emerges as a remarkably simple algebraic relation between the bounce and bion actions. We further show that the median-summed non-perturbative correction to the spectrum vanishes at the exceptional point, while the resurgent structure survives through a universal minimal trans-series. For the resonance and anti-resonance systems, we find that the exact median-summed spectra are related by complex conjugation, representing time reversal in this setting, are necessarily complex, and do not exhibit an exceptional point. Although their spectra differ significantly from the PT-symmetric case, they share the same minimal trans-series. By maintaining explicit links with the path integral saddles and the formal theory of resurgence, our analysis provides a unified and general perspective on the quantization of non-Hermitian theories.