Three Focused Study Routes

These routes turn the chapter map into concrete assignments. Each session has a limited core reading, one or two tasks, and a completion check. Follow links to the named sections; stop before the next section unless the table explicitly asks for it. The rest of each source page remains available for deeper study.

Allow 60–90 minutes for a core session, with a separate session for a numerical laboratory if installation or debugging is unfamiliar. These are provisional planning estimates. They have not yet been measured in graduate-reader trials. Do not use a stopwatch as an assessment of ability: record a missing prerequisite and revisit the relevant explanation.

Route A: analytic connection methods

Starting knowledge: second-order ODEs, elementary linear algebra, power series, and complex powers on a chosen logarithm branch. Destination: compute a normalized connection coefficient and distinguish numerical convergence from a truncated analytic approximation.

Session	Core reading	Limited assignment	Completion check
A1. One endpoint to another	Gauss starter, through “Interpret the endpoint behavior”	Exercises 1 and 2; optional integral derivation can wait	Explain why a bounded solution need not be analytic and how rescaling a basis changes its coefficient
A2. Why the coefficient is constant	Companion system, “Existence and normalized fundamental matrices,” and “Abel–Liouville identities”	Differentiate the relation between two fundamental matrices to prove the connection matrix is constant; derive Abel’s identity	Separate a generally varying Wronskian from its constant connection-coefficient ratio
A3. When the local basis changes	Indicial recurrence, “Nonresonant roots,” “Integer-separated roots and the obstruction,” and “Repeated roots”	Work the one-parameter obstruction example; solve $z^2y''+zy'=0$ on a fixed branch	An integer exponent difference signals a possible logarithm; the recurrence determines whether it occurs
A4. One coefficient selects a spectrum	Zero-to-one matrix and Jacobi spectrum	Trace the selected boundary condition to the reciprocal-gamma factor; change one endpoint basis by a nonzero constant	Identify which zeros are preserved by a normalization change and which boundary conditions are essential
A5. Reproduce and challenge the computation	Connection micro-lab	Run the Gauss checker; run the Heun command below once resolved and once underresolved	Explain why a determinant test alone can miss an inaccurate connection matrix

The Heun comparison keeps its conformal-block approximation at order two:

python3 general-heun-connection-check.py --t 0.04 --terms 120 --dps 50 --no-table --verify
python3 general-heun-connection-check.py --t 0.04 --terms 24 --dps 50 --no-table --verify

The first command should pass; the second should fail its matrix-drift check. Their finite-order block approximation error is a separate quantity. Download the Heun checker and follow the environment instructions in the computational appendix.

Final transfer task: suppose an equation has the same local exponents as another but a different accessory parameter. Explain why its connection matrix is not fixed by that local information. A complete answer identifies the changed equation and a global continuation or matching calculation that would distinguish the two problems.

Optional continuation: the remaining direct recurrence methods in Chapter 4, then the isomonodromy branch. Complete a direct example before replacing it with a modern correspondence.

Route B: exact WKB and quantum spectra

Starting knowledge: the Schrödinger equation, elementary asymptotic expansions, contour integration, and the boundary Wronskian from Route A or the Weber starter. Destination: distinguish a formal period, an exact boundary condition, and numerical evidence for a spectral value.

Session	Core reading	Limited assignment	Completion check
B1. Two turning points	Weber starter, through “An exact calculation for the oscillator”	Exercises 1 and 2	Derive the half-integer shift from the two phases, then identify the separate exact verification
B2. Build the formal solution	Triangular recursion and Amplitude from phase	For $\hbar^2y''=Q(x)y$ with $Q$ independent of $\hbar$, derive the first two logarithmic-derivative coefficients and the leading amplitude	State which quantities are formal and where the expansion breaks down
B3. Follow the correct cycle	Relative classes and “A lifted branch arc closes with a factor of two”	Recover the oscillator closed action from the half-cycle integral; reverse the cycle orientation	Explain the factor of two and which signs change together
B4. Impose the actual boundary condition	Admissible coefficient lines, “Transport both admissible lines into one frame,” and Two Airy shears	Multiply the displayed two Airy matrices and propagation matrix; complete Weber Exercise 3	Obtain the vanishing coefficient and explain why a half-line boundary condition selects a different spectrum
B5. Prepare the quartic equation	Quartic energy scaling, “Six chamber charges collapse to two orbit masses,” and Folded system	Derive $E=X^{4/3}$; identify the two unknown functions, convolution kernels, driving terms, and endpoint limits	Explain which quantities the nonlinear integral equations determine before any quantization condition is imposed
B6. Recover the spectral condition	Median and complex periods and “The exact condition separates even and odd levels”	Follow the displayed maps from pseudoenergies to $\mathcal A,\mathcal B$, then to the even and odd spectral conditions	Explain why a complex shift needs its stated strip or continuation prescription, and why the boundary condition is additional data
B7. Reproduce the quartic levels	Real-line solver and Direct diagonalization	Download the linked program and run python3 pure-quartic-tba-spectrum.py; compare quadrature, tail, basis-size, and basis-frequency refinements	Distinguish numerical agreement from an error enclosure and state why ODE/IM and its derived TBA are not automatically independent checks

Checkpoint for B2: on a chosen sheet, if $S=\hbar y'/y=S_0+\hbar S_1+\cdots$, then $S_0^2=Q$ and $S_1=-Q'/(4Q)$. Integrating $S_1$ supplies the $Q^{-1/4}$ amplitude. At a zero of $Q$, this local expansion fails and a turning-point construction is needed.

Scope of B4: for the matrix calculation, take the nonzero cycle multiplier $v$ as given and use $L(s)=\left(\begin{smallmatrix}1&0\\s&1\end{smallmatrix}\right)$. Deriving the Borel sum that defines $v$ belongs to the later summability sections. The assignment establishes the connection-matrix factorization in the displayed two-turning-point setting.

Preparation for B5–B7: these are three sessions beyond the Weber example. They introduce the supplied quartic integral equations and test their spectral output. Work with the model-specific charge reduction as a stated input; deriving it requires the separate wall-crossing chapter. If nonlinear integral equations are new, first read the three kinds of integral-equation unknowns. Here “pseudoenergy” names an unknown function of $\theta$; it is not a Schrödinger eigenvalue. The eigenvalues emerge only after B6’s condition and the scaling from B5. Allow an additional session for the convolution and principal-value integration if those are unfamiliar.

Final transfer task: perturb the oscillator to $V(x)=x^2+g x^4$ with $g>0$. Formulate the leading action condition and an independent numerical comparison. Do not assert that the harmonic-oscillator equality remains exact. A complete proposal specifies the endpoints, parameter value, normalization, and independent grid/interval refinements before comparing digits.

Optional continuation: summability, wall crossing, and resurgent transseries in Chapter 9, followed by the quantum-period dictionaries in Chapters 10–11. Those results need their own analytic hypotheses.

Route C: black holes and physical response

Starting knowledge: separation of a linear wave equation, elementary complex frequencies, Frobenius behavior, and connection Wronskians. If basis normalization is unfamiliar, complete A1–A2 first. Sessions C4–C5 additionally use the variational principle for a scalar field and asymptotically AdS geometry. If these are new, reserve a preparation session to derive the boundary term of the displayed scalar action in the renormalized-momentum section and identify the two boundary powers before interpreting their coefficients as source and response. Destination: translate a named physical boundary problem into a spectral condition or a response function with the correct normalization.

Session	Core reading	Limited assignment	Completion check
C1. Fix the physical endpoint solutions	Future event horizon and “Flat infinity is a sectorial Jost problem”	With time dependence $\ee^{-\ii\omega t}$, write the ingoing horizon and outgoing infinity factors; reverse the time convention consistently	Explain why a damped complex-frequency outgoing mode may grow along the real radial direction
C2. Translate endpoints into a spectral zero	Weighted Wronskian and “Normalization changes values, not genuine zero curves”	Rescale both endpoint bases by nonzero analytic functions; determine what happens to zeros and residues	Distinguish a preserved spectral divisor from a changed normalized amplitude
C3. Reproduce one named mode	Scalar Schwarzschild recurrence, “The minimal tail selects the outgoing line,” and “A Schwarzschild mode survives two independent tests”	Reproduce the scalar fundamental mode using both downloadable methods; record the length/time convention and refinement spread	Identify the scalar mode correctly and do not substitute the separate gravitational recurrence benchmark
C4. Turn a quotient into an observable	Renormalized momentum, “The ODE quotient becomes a retarded correlator,” and “Local counterterms do not change nonlocal spectral data”	Identify source, response, action factor, and local term in the displayed formula	Explain why the ODE quotient alone does not fix the complete physical correlator
C5. Change the boundary theory	Standard, alternate, and mixed theories and Rotating BTZ example	Write the zero condition for one allowed mixed boundary condition; check whether a numerator cancels a proposed pole	Keep physical boundary conditions, normalization, and pole multiplicity in the same calculation

Checkpoint for C1: with the stated time convention, the ingoing horizon factor is $\ee^{-\ii\omega r_*}$, and the outgoing flat-infinity factor is $\ee^{+\ii\omega r_*}$. If $\operatorname{Im}\omega<0$, temporal decay accompanies radial growth of the latter on the positive real $r_*$ axis. A real-axis boundedness test would impose the wrong quasinormal boundary condition.

Final transfer task: a paper gives the same mode frequencies in two representations and claims its retarded correlators are identical. State what additional evidence is required. A complete answer checks the source/response basis, action normalization, analytic nonzero factors, contact terms, and numerator behavior. Equal spectral zeros alone do not determine a normalized Green function.

Optional continuation: select the isomonodromic, conformal-block, SW/NS, or exact-WKB formulation only after checking that its dictionary applies to the named equation and physical endpoints.

Record what you can now do

Keep one page of results for the route: your equation and conventions, one derivation, one reproduced numerical record where applicable, and the final transfer answer. If a checkpoint fails, revisit that specific section before adding another framework. Worked solutions are useful feedback; being able to solve a changed problem is a separate learning goal.