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

Abstract

We discuss the exact quantization of general one-dimensional potentials in view of the exact-WKB formalism. Building on our previous work, we perform analytic continuations across different sectors via the complexification to the spectral (energy) parameter $u$ and identify continuous and discontinuous transitions of the exact spectrum for generic potentials. When the transition is discontinuous, it is characterized by the Stokes phenomena, inducing different exact (median) quantization conditions, thereby distinct trans-series structures valid in different sectors. We analyze two illustrative examples, namely asymmetric triple-well (ATW) and tilted double-well (TDW), and verify the general qualitative analysis by deriving exact (median) quantization conditions in each sector. Moreover, by obtaining the trans-series solutions for each system, we identify bion/bounce configurations and show that the trans-series of ATW is organized in accordance with the cluster expansion of the bion gas and there should exist a previously neglected complex saddle in the TDW system. These identifications further strengthen the link between path integral and exact-WKB formalisms, while also demonstrating the predictive power of the latter. In parallel, for the P-NP relations of genus-1 systems, we derive transformation rules between any perturbative and non-perturbative pair of WKB-cycles. Our results show that the entire resurgence data of a genus-1 system transforms only by the change of classical parameters, i.e. frequencies and bion/bounce actions, and the perturbative energy series. This also reveals the underlying reasons of the previously found $S$-duality transformations.

arXiv comment64 pages, 18 figures, v2. published version

References in corpus (81)

• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Instantons and recursion relations in N=2 Susy gauge theory
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• The ODE/IM Correspondence
• Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions
• Quasi-exactly solvable quartic potential
• Spectral networks
• Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions
• Abelian duality, confinement, and chiral symmetry breaking in QCD(adj)
• Uniform WKB, Multi-instantons, and Resurgent Trans-Series
• Continuity, Deconfinement, and (Super) Yang-Mills Theory
• Generating Non-perturbative Physics from Perturbation Theory
• QCD-like Theories on R_3\times S_1: a Smooth Journey from Small to Large r(S_1) with Double-Trace Deformations
• Resurgence theory, ghost-instantons, and analytic continuation of path integrals
• Multi-Instantons and Exact Results II: Specific Cases, Higher-Order Effects, and Numerical Calculations
• Matone's Relation in the Presence of Gravitational Couplings
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• Conformality or confinement: (IR)relevance of topological excitations
• Multi-Instantons and Exact Results III: Unified Description of the Resonances of Even and Odd Anharmonic Oscillators
• Complexified path integrals, exact saddles and supersymmetry
• Neutral bions in the $\mathbb{C}{P}^{N-1}$ model
• TBA equations and resurgent Quantum Mechanics
• Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems
• Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence
• Adjoint QCD on ${\mathbb{R}}^3\times S^1$ with twisted fermionic boundary conditions
• Deconstructing zero: resurgence, supersymmetry and complex saddles
• Instantons in Quantum Mechanics and Resurgent Expansions
• Non-perturbative Contributions from Complexified Solutions in $\mathbb{C}{P}^{N-1}$ Models
• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• Resumming the string perturbation series
• Microscopic Structure of Magnetic Bions
• Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB
• Exact WKB analysis of the vacuum pair production by time-dependent electric fields
• Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes
• Bion non-perturbative contributions versus infrared renormalons in two-dimensional $\mathbb{C}{P}^{N-1}$ models
• Exact Resurgent Trans-series and Multi-Bion Contributions to All Orders
• Holomorphic Anomaly and Quantum Mechanics
• RG-Whitham dynamics and complex Hamiltonian systems
• On exact-WKB analysis, resurgent structure, and quantization conditions
• Exact WKB and abelianization for the $T_3$ equation
• Remark on the Dunne-Unsal relation in exact semi-classics
• Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$
• Quantization condition from exact WKB for difference equations
• Exact WKB Analysis of N = 2 Gauge Theories
• Non-BPS exact solutions and their relation to bions in $\mathbb{C}{P}^{N-1}$ models
• Resurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics
• Critical Points at Infinity, Non-Gaussian Saddles, and Bions
• TBA Equations and Quantization Conditions
• Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries
• Phase structure of the twisted $SU(3)/U(1{)}^2$ flag sigma model on $\mathbb{R}\times S^1$
• TBA equations for the Schrödinger equation with a regular singularity
• Neutral bions in the CP^{N-1} model for resurgence
• Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization
• Confinement-Deconfinement Crossover in the Lattice $\mathbb{C}{P}^{N-1}$ Model
• Exact conditions for Quasi-normal modes of extremal M5-branes and Exact WKB analysis
• Exact WKB analysis and TBA equations for the Mathieu equation
• Lattice $\mathbb{C}{P}^{N-1}$ model with ${\mathbb{Z}}_N$ twisted boundary condition: bions, adiabatic continuity and pseudo-entropy
• More on Homological Supersymmetric Quantum Mechanics
• Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves
• Exact WKB analysis for $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture
• The Exact WKB analysis and the Stokes phenomena of the Unruh effect and Hawking radiation
• Resurgence and Lefschetz thimble in 3d N=2 supersymmetric Chern-Simons matter theories
• $1/\epsilon$ problem in resurgence
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics
• Exact WKB Analysis of ${\mathbb{CP}}^1$ Holomorphic Blocks
• Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators
• Lattice study on the twisted $\mathbb{C}{P}^{N-1}$ models on $\mathbb{R}\times S^1$
• Cluster expansion and resurgence in Polyakov model
• Resurgence and semiclassical expansion in two-dimensional large-$N$ sigma models
• All-order Resurgence from Complexified Path Integral in a Quantum Mechanical System with Integrability
• Unified genus-1 potential and a parametric perturbative/nonperturbative relation
• Unified exact WKB framework for resonance -- Zel'dovich/complex-scaling regularization and rigged Hilbert space
• Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB Analysis
• Bions and Instantons in Triple-well and Multi-well Potentials
• Resurgence of deformed genus-1 curves: A novel perturbative/nonperturbative relation
• Exact WKB in all sectors I: Potentials with degenerate saddles
• Thermodynamic Bethe ansatz and wall crossing for deformed supersymmetric quantum mechanics
• Exact WKB method for radial Schrödinger equation
• Exact-WKB Analysis of Two-level Floquet Systems