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Advanced ODE Guide
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ArXiv
Go to
ArXiv
arXiv search
hep-th new
hep-ph new
gr-qc new
math-ph new
INSPIRE
Advanced ODE Guide
Overview
How to use the guide
Further directions
Mathematical Methods
NIST Digital Library of Mathematical Functions
EqWorld
Cambridge Mathematical Methods
MIT Advanced Calculus for Engineers
Stone–Goldbart: Mathematics for Physics
Nearing: Mathematical Tools for Physics
Molinari: Mathematical Methods for Physics
Calculus and Differential Equations Review
Visual & Computational
3Blue1Brown
Visual Complex Analysis
MIT Mathlets
VisualPDE
Fundamentals of Numerical Computation
MIT Computational Science and Engineering
Exploring ODEs
Physics
AdS/CFT Duality
Black Hole
Quantum Field Theory
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Guides
Conventions & maps
HeunG (General)
HeunC (Confluent)
HeunD (Double Confluent)
HeunB (Biconfluent)
HeunT (Triconfluent)
Advanced Methods for Linear ODEs
Overview
How to use this guide
First connection problem
First spectral problem
Focused study routes
Run the laboratories
Notation and status
Mathematical toolkit
1. Complex linear ODEs
Overview and diagnostics
Scalar equations and systems
Liouville form and opers
Regular singularities
Irregular singularities
Sectorial normalization
Confluence and singular limits
Case files and problems
2. Global connection data
Continuation and monodromy
Wild monodromy
Connections and Wronskians
Riemann–Hilbert and characters
Spectral theory interlude
Boundary functions
Hypergeometric benchmark
Global-data problems
3. Hypergeometric and Heun laboratories
Canonical local models
The general Heun equation
Accessory parameters and four-point geometry
The confluence hierarchy
Canonical local and asymptotic bases
Transformations
Parameter crosswalks
Polynomial and QES sectors
4. Wronskians and recurrences
Wronskian matching
Frobenius and Jaffé series
Dominant and minimal solutions
Perron–Kreuser and fractions
Large-order asymptotics
Truncation and Hill determinants
Green functions and QNMs
Verified numerics
5. Isomonodromic deformation
Connections and monodromy moduli
Lax compatibility and Schlesinger
Hamiltonians and accessory parameters
Jimbo–Miwa–Ueno tau functions
Painlevé Lax systems and Heun reductions
Confluence to Painlevé V–II
Inverse monodromy and spectra
Series, Fredholm forms, and numerics
6. 2D CFT and linear ODEs
Virasoro representations and blocks
Degenerate fields and null vectors
BPZ equations as second-order ODEs
Degenerate fusion and braiding
Heavy–light classical limit
Classical blocks, opers, and accessories
Irregular states and confluent BPZ equations
Normalization conventions: a worked example
7. Conformal-block connection formulae
Two regimes: c = 1 versus c → ∞
c = 1 tau expansions
Classical blocks and accessories
General Heun connection matrices
Irregular blocks and confluent-Heun connections
From blocks to full coefficients
Frobenius versus Kac loci
Recurrence tests and mathematical status
8. Formal WKB and quantum periods
Normal form and Riccati equation
Recursion and even–odd split
Spectral curves and sheets
WKB differentials and homology
All-orders quantum periods
Picard–Fuchs methods
Airy, Weber, and Mathieu
Problems: cycles and covariance
9. Exact WKB and resurgence
Gevrey and Borel sums
Resurgence and jumps
Quadratic differentials and graphs
Canonical solutions and local connections
Voros symbols and wall crossing
Boundary conditions and quantization
Transseries and large order
Complex turning points and resonances
Borel–Padé spectral tests
10. Seiberg–Witten theory and the NS limit
4d N=2 background
Curves, cycles, special geometry
Omega background and Nekrasov function
NS limit and twisted superpotential
Quantum curves and ordering
WKB and quantum SW periods
Matone and accessory relations
Bethe vacua and completion
11. AGT and the unified dictionary
AGT operational content
Regular punctures and HeunG
Decoupling and irregular punctures
Degenerate insertions and defects
Exponents, masses, and momenta
Internal momentum and accessory
Monodromy, SW, and WKB cycles
Complete connection example
12. Spectral determinants and ODE/IM
Entire spectral determinants
Stokes multipliers and rotated problems
Symanzik rotations and prototypes
Baxter Q-functions and Bethe roots
A₁ TQ, Wronskian, and Y-system
Wronskians and exact quantization
Polynomial and exponential potentials
Analyticity, scope, and failure modes
13. TBA and exact spectra
From TQ and Y-systems to TBA
Strips, kernels, and contours
Exact-WKB/GMN Riemann–Hilbert equations
When the two constructions coincide
Excited states and wall crossing
Numerical iteration and error
Exact WKB and NS periods
Pure quartic spectrum
14. Quantum-mechanical spectra
Oscillator calibration
Floquet and Mathieu bands
Double wells and transseries
Modified Mathieu and DCHE
Razavy hyperbolic double well
Quartic versus hyperbolic wells
QES versus generic exact WKB
Three controlled capstones
15. Black holes and holography
Separation and Heun classes
Horizon and boundary bases
QNM Wronskians and recurrences
Isomonodromy and conformal blocks
SW/NS, exact WKB, and ODE/IM
Retarded holographic correlators
Distinct spectral singularities
Named-equation capstones
Method selection and open problems
Further directions
Reference
A. Notation and normalization
B. Complex analysis and asymptotics
C. Computational laboratory
D. Exercises, hints, and solutions
E. Bibliography and source map
Version and corrections
ODE general
Decoupling coupled linear ODEs
Transforming ODEs to Schrödinger-like equations
Spectral Curves (2nd‑order ODEs)
Riemann–Hilbert Problems and Painlevé VI
SO(3) vs Virasoro
Seiberg–Witten theory: an introduction
Series solutions near an irregular singular point
Hypergeometric functions
Local solutions
Connection formula
Heun functions
Local solutions
Double confluent Heun function (HeunD)
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Jie Ren (
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