Global-Data Problems: Monodromy, Basis Changes, and Resonance

The chapter has spent seven pages separating objects that are often denoted by the same letter: monodromy matrices, Stokes factors, connection coefficients, boundary Wronskians, and several determinant constructions. This capstone asks whether those distinctions survive an actual calculation. The problems progress from convention checks to resonant limits, operator domains, and a reproducible hypergeometric computation.

Complete solutions are provided for the closed problems. The computational lab gives pseudocode and acceptance criteria rather than software-specific output, and the final research problem gives milestones rather than pretending that every model has a universal closed-form answer.

The convention card

Use the chapter conventions throughout:

$$\begin{aligned} \Phi^\gamma &= \Phi M_\gamma,\\ \Phi_\beta &= \Phi_\alpha C_{\alpha\beta},\\ \widetilde C_{\alpha\beta} &= H_\alpha^{-1}C_{\alpha\beta}H_\beta,\\ C_{\alpha\gamma} &= C_{\alpha\beta}C_{\beta\gamma}. \end{aligned}$$

The path product $\gamma_1\gamma_2$ traverses $\gamma_2$ first and then $\gamma_1$, so

$$M_{\gamma_1\gamma_2} = M_{\gamma_1}M_{\gamma_2}.$$

For a scalar equation in monic form,

$$y''+p(z)y'+q(z)y=0,$$

the book’s Wronskian and Abel identity are

$$\Wr[f,g]=fg'-f'g, \qquad \Wr'=-p\Wr.$$

Every answer involving a connection coefficient should name its ordered bases, direction, branches, and continuation path. Every spectral answer should additionally name the Hilbert space or resonance sheet and the operator domain or asymptotic boundary condition.

Problem	Level	Principal audit
1	Core	Requested datum and missing hypotheses
2	Core	Based-loop order and conjugacy
3	Intermediate	Four different transformations
4	Intermediate	Connection cocycle and boundary entry
5	Intermediate	A commuting Riemann–Hilbert reconstruction
6	Advanced	Wild factorization and residual gauge
7	Advanced	Coalescing bases and Jordan monodromy
8	Advanced	Hurwitz moves and marked trace data
9	Advanced	Robin boundary function and Fredholm ratio
10	Advanced	Resonant Legendre quantization
11	Computational lab	Exact and near-resonant numerical audits
12	Research	A complete global-data dossier

Monodromy words and frame dependence

1. Name the datum before calculating

For each request below, identify the primary mathematical object and at least one datum or hypothesis that must be added before the request is well posed.

1. Continue a normalized germ once around a puncture.
2. Compare unit-leading Frobenius bases at two singular points.
3. Measure the jump between two lateral asymptotic sums at an irregular singularity.
4. Find the values of a parameter for which a left-selected solution also satisfies a right endpoint condition.
5. Evaluate $\det(I-K(\lambda))$.
6. Evaluate $\det_\zeta T$.

Solution

The objects and their minimum ledgers are:

Request	Primary object	Data or hypothesis still required
Continued germ	A based monodromy matrix, or its representation class	Base point, based homotopy class, ordered frame, loop orientation, and path-composition convention
Two local bases	A path-labelled connection matrix	Exact basis normalizations, matrix direction, branches, and continuation path
Lateral asymptotic jump	A Stokes factor inside wild monodromy data	Formal normal form, sector labels, lateral summation directions, and orientation of the crossing
Two-end condition	A selected connection entry or boundary Wronskian	Hilbert space and operator domain for eigenvalues, or time convention, asymptotic condition, and sheet for resonances
$\det(I-K)$	A Fredholm determinant, or possibly a regularized determinant	A trace-class hypothesis for the ordinary determinant; if only $K\in\mathcal S_p$, the appropriate $\det_p$ must be declared
$\det_\zeta T$	A zeta-regularized operator determinant	A closed operator with suitable discrete spectral behavior, a spectral cut when needed, a summability half-plane, and regular continuation to the evaluation point

The first four objects arise directly from solution spaces and analytic continuation. The last two are operator constructions with separate existence theorems. A boundary Wronskian is not promoted to a Fredholm or zeta determinant merely because its zeros are spectral.

2. A based-loop product in an explicit frame

Let

$$M_0 = \begin{pmatrix} 1&a\\ 0&1 \end{pmatrix}, \qquad M_1 = \begin{pmatrix} 1&0\\ b&1 \end{pmatrix}, \qquad a,b\neq0.$$

The distinguished loops obey $\gamma_0\gamma_1\gamma_\infty=1$ in the book’s convention.

1. Compute $M_\infty$.
2. Verify the full product and its determinant.
3. Change the common base frame by $H=\operatorname{diag}(h,h^{-1})$. Compute the new $M_0$ and $M_1$, and identify a quantity built from $a,b$ that survives.

Solution

Because the representation respects the declared path product,

$$M_\infty = (M_0M_1)^{-1}.$$

Now

$$M_0M_1 = \begin{pmatrix} 1+ab&a\\ b&1 \end{pmatrix},$$

whose determinant is one. Therefore

$$M_\infty = \begin{pmatrix} 1&-a\\ -b&1+ab \end{pmatrix}.$$

Direct multiplication gives $M_0M_1M_\infty=I$. Each matrix has determinant one, so the determinant relation is also satisfied.

The frame change acts by simultaneous conjugation:

$$\widetilde M_j = H^{-1}M_jH.$$

Hence

$$\widetilde M_0 = \begin{pmatrix} 1&h^{-2}a\\ 0&1 \end{pmatrix}, \qquad \widetilde M_1 = \begin{pmatrix} 1&0\\ h^2b&1 \end{pmatrix}.$$

The separate off-diagonal entries depend on the frame, but their product $ab$ does not. Equivalently,

$$\operatorname{tr}M_\infty=2+ab$$

is a simultaneous-conjugacy invariant. The loop relation itself is also preserved:

$$\widetilde M_0\widetilde M_1\widetilde M_\infty = H^{-1} \bigl( M_0M_1M_\infty \bigr) H =I.$$

3. Four transformations that should not be conflated

For a flat frame $\Phi$, a connection matrix $C_{\alpha\beta}$, and a boundary function $E(\lambda)$ built from selected solution lines, compare the following operations:

1. a constant change of the common global frame;
2. independent changes of normalized local bases;
3. a single-valued invertible left gauge $G(z)$;
4. a multivalued scalar gauge $g(z)$ with $g^\gamma=\chi(\gamma)g$.

State how monodromy and connection data transform, and which spectral information in $E$ is invariant under a nowhere-zero analytic renormalization.

Solution

A constant global frame change $\widetilde\Phi=\Phi H$ gives

$$\widetilde M_\gamma = H^{-1}M_\gamma H.$$

Thus traces, determinants, Jordan form, and all simultaneous-conjugacy invariants survive, while individual entries generally do not.

Independent local changes

$$\widetilde\Phi_\alpha=\Phi_\alpha H_\alpha, \qquad \widetilde\Phi_\beta=\Phi_\beta H_\beta$$

give the two-sided law

$$\widetilde C_{\alpha\beta} = H_\alpha^{-1}C_{\alpha\beta}H_\beta.$$

This is not usually a conjugation because the two endpoint normalizations are independent. If $H_\alpha$ and $H_\beta$ preserve the selected boundary lines and rescale their generators by nowhere-zero analytic factors, then the associated scalar boundary function changes by a nowhere-zero analytic factor. Its zero set and zero multiplicities are unchanged.

For a single-valued invertible left gauge,

$$\widetilde\Phi=G\Phi,$$

analytic continuation gives

$$\widetilde\Phi^\gamma = G\Phi^\gamma = \widetilde\Phi M_\gamma.$$

The numerical right monodromy matrices are unchanged when the same solution frame is retained. Consistently gauged local frames also retain the same connection matrices.

A multivalued scalar gauge instead yields

$$\widetilde M_\gamma = \chi(\gamma)M_\gamma.$$

It twists the $GL(n)$ representation by a character. Projective monodromy is unchanged, but traces and determinants of a chosen lift generally are not. These four operations coincide only in special circumstances and should never be inferred from the visual similarity of their formulas.

Connection groupoids and reconstruction

4. A cocycle, a path change, and one boundary zero

Suppose three normalized frames satisfy

$$\Phi_0=\Phi_1A, \qquad \Phi_1=\Phi_\infty B.$$

1. Find $C_{\infty0}$ in $\Phi_0=\Phi_\infty C_{\infty0}$.

2. Let $\widetilde\Phi_j=\Phi_jH_j$. Verify the transformed cocycle.

3. If a new comparison path replaces the endpoint frames by $\Phi_\infty D_\infty$ and $\Phi_0D_0$, find the new $C_{\infty0}$.

4. Write

   $$C_{\infty0} = \begin{pmatrix} p&q\\ r&s \end{pmatrix}.$$

   If the first solution at zero is required to lie in the first selected solution line at infinity, identify the boundary equation. Explain why diagonal, nowhere-zero analytic renormalizations preserve it.

Solution

Composition gives

$$\Phi_0 = \Phi_\infty BA,$$

so

$$C_{\infty0}=BA.$$

The two factors transform as

$$\widetilde A = H_1^{-1}AH_0, \qquad \widetilde B = H_\infty^{-1}BH_1.$$

Their product is

$$\widetilde B\widetilde A = H_\infty^{-1}BAH_0 = \widetilde C_{\infty0},$$

and the middle normalization $H_1$ cancels exactly as a cocycle requires.

For the changed endpoint branches, solve

$$\Phi_0D_0 = \Phi_\infty D_\infty C_{\infty0}^{\mathrm{new}}.$$

This yields

$$C_{\infty0}^{\mathrm{new}} = D_\infty^{-1}C_{\infty0}D_0.$$

Finally, the first column says

$$f_0=p\,h_1+r\,h_2.$$

Requiring $f_0$ to lie in the line spanned by $h_1$ is therefore the boundary equation

$$r(\lambda)=0.$$

Diagonal rescalings multiply $r$ by nonzero analytic factors from the source and target normalizations, so its zeros and their orders survive. A general nondiagonal basis change can mix the selected line with its complement and describes a different boundary condition.

Reconstruction, wild data, and marked representations

5. A commuting Riemann–Hilbert reconstruction

Let

$$A_0= \begin{pmatrix} \alpha&0\\ 0&-\alpha \end{pmatrix}, \qquad A_1= \begin{pmatrix} \beta&0\\ 0&-\beta \end{pmatrix},$$

and consider the prescribed local monodromies

$$M_0=\ee^{2\pi\ii A_0}, \qquad M_1=\ee^{2\pi\ii A_1}.$$

1. Construct a Fuchsian system on $\mathbb P^1\setminus\{0,1,\infty\}$ with these monodromies.
2. Determine its residue and local monodromy at infinity.
3. Show that the same monodromy representation does not determine unique residue logarithms.
4. Explain geometrically what the ambiguity changes.

Solution

Because $A_0$ and $A_1$ commute, chosen branches of the matrix powers give

$$Y(z)=z^{A_0}(z-1)^{A_1}.$$

It is a fundamental matrix of

$$\frac{\dd Y}{\dd z} = \left( \frac{A_0}{z} + \frac{A_1}{z-1} \right)Y.$$

A positive loop about zero adds $2\pi\ii$ to $\Log z$ and gives $M_0=\ee^{2\pi\ii A_0}$; the analogous loop about one gives $M_1=\ee^{2\pi\ii A_1}$.

In the local coordinate $w=1/z$, the residue at infinity is

$$A_\infty=-(A_0+A_1).$$

Therefore

$$\begin{aligned} M_\infty &= \ee^{2\pi\ii A_\infty}\\ &= \left( M_0M_1 \right)^{-1}, \end{aligned}$$

and the based product $M_0M_1M_\infty=I$ holds. Here commutativity removes the ordering difficulty; it does not remove the need to state the distinguished loop system.

Now choose traceless integral diagonal matrices

$$K_j= \begin{pmatrix} m_j&0\\ 0&-m_j \end{pmatrix}, \qquad m_j\in\mathbb Z.$$

Then

$$\ee^{2\pi\ii(A_j+K_j)} = \ee^{2\pi\ii A_j}.$$

The single-valued meromorphic gauge

$$Y\longmapsto z^{K_0}(z-1)^{K_1}Y$$

shifts the finite residues by $K_0$ and $K_1$ while leaving the local system on the punctured sphere unchanged. Thus the representation determines the exponents only modulo integers. Choosing particular logarithms amounts to choosing how the flat bundle is extended across the punctures—a logarithmic lattice, such as a Deligne extension after a preferred strip for the real parts has been fixed.

This example is deliberately abelian. In a noncommuting reconstruction, existence of a logarithmic connection with a prescribed pole structure is an additional Riemann–Hilbert problem; one cannot obtain it by choosing three unrelated matrix logarithms.

6. Wild factors hidden by ordinary monodromy

At a rank-two irregular singularity, assume that the formal exponential type has two distinct, labelled one-dimensional blocks. Block swaps are not allowed: residual formal gauges must preserve this exponential grading. Take

$$M_{\mathrm f} = \begin{pmatrix} q&0\\ 0&q^{-1} \end{pmatrix}, \qquad S_0=I+sE_{12}, \qquad S_1=I+tE_{21},$$

with $q\neq0$. In the orientation and fixed-frame convention of the wild-monodromy page, the actual local monodromy is

$$M_{\mathrm{loc}} = M_{\mathrm f}S_1^{-1}S_0^{-1}.$$

1. Compute $M_{\mathrm{loc}}$, its determinant, and its trace.
2. Determine how $s$ and $t$ transform under the residual formal-frame change $H=\operatorname{diag}(h,h^{-1})$.
3. Which combination of the two Stokes multipliers can be recovered from $\operatorname{tr}M_{\mathrm{loc}}$ when $q$ is known?
4. Explain precisely what ordinary monodromy forgets.

Solution

The inverse Stokes factors give

$$S_1^{-1}S_0^{-1} = \begin{pmatrix} 1&-s\\ -t&1+st \end{pmatrix},$$

so

$$M_{\mathrm{loc}} = \begin{pmatrix} q&-qs\\ -q^{-1}t&q^{-1}(1+st) \end{pmatrix}.$$

Consequently,

$$\det M_{\mathrm{loc}}=1, \qquad \operatorname{tr}M_{\mathrm{loc}} = q+q^{-1}(1+st).$$

Because $H$ preserves the exponential grading—and commutes with $M_{\mathrm f}$—it is a residual normalization of the formal eigenlines. Conjugation gives

$$\begin{aligned} H^{-1}S_0H &= I+h^{-2}sE_{12},\\ H^{-1}S_1H &= I+h^2tE_{21}. \end{aligned}$$

Thus $s$ and $t$ are framed quantities, whereas $st$ is invariant under this centralizer action. If $q$ and the trace are known, then

$$st = q\left( \operatorname{tr}M_{\mathrm{loc}} -q-q^{-1} \right).$$

Even this invariant does not reconstruct the direction-labelled factorization. For example, if $st=0$ and $q^2\neq1$, the cases

$$(s,t)=(0,0), \qquad (s,0), \qquad (0,t)$$

all have ordinary monodromy conjugate to $\operatorname{diag}(q,q^{-1})$, provided the displayed nonzero multiplier is allowed. The three wild data sets differ: one has no jump, and the other two place a jump in different Stokes groups. Ordinary conjugacy retains only the product matrix; wild monodromy retains the formal monodromy, ordered singular directions, sector normalizations, and Stokes factors.

Resonant limits and marked monodromy

7. Coalescing Euler modes and Jordan monodromy

For $\varepsilon\neq0$, consider

$$x^2y''+(1-\varepsilon)xy'=0$$

on the slit plane with a chosen $\Log x$. A nonresonant basis is

$$f_\varepsilon(x)=1, \qquad g_\varepsilon(x)=x^\varepsilon.$$

1. Explain why this basis becomes defective as $\varepsilon\to0$.
2. Construct a basis with a finite, nonzero Wronskian limit.
3. Transform the monodromy into that basis and take the limit.
4. Check the answer directly from the limiting differential equation.

Solution

The Wronskian of the raw pair is

$$\Wr[f_\varepsilon,g_\varepsilon] = \varepsilon x^{\varepsilon-1},$$

which tends to zero because both columns tend to $1$. The singularity is in the chosen basis, not in the two-dimensional solution space.

Set

$$u_\varepsilon=f_\varepsilon, \qquad v_\varepsilon = \frac{ g_\varepsilon-f_\varepsilon }{\varepsilon}.$$

Then

$$\Wr[u_\varepsilon,v_\varepsilon] = x^{\varepsilon-1} \longrightarrow x^{-1},$$

and $v_\varepsilon\to\Log x$. If

$$\Phi_\varepsilon = \left( f_\varepsilon,g_\varepsilon \right), \qquad \Psi_\varepsilon = \left( u_\varepsilon,v_\varepsilon \right),$$

then

$$\Phi_\varepsilon = \Psi_\varepsilon S_\varepsilon, \qquad S_\varepsilon = \begin{pmatrix} 1&1\\ 0&\varepsilon \end{pmatrix}.$$

In the raw basis, positive continuation around zero is

$$D_\varepsilon = \operatorname{diag} \left( 1,\ee^{2\pi\ii\varepsilon} \right).$$

The right monodromy in the convergent basis is therefore

$$\begin{aligned} \widehat M_\varepsilon &= S_\varepsilon D_\varepsilon S_\varepsilon^{-1}\\ &= \begin{pmatrix} 1& \dfrac{ \ee^{2\pi\ii\varepsilon}-1 }{\varepsilon}\\ 0&\ee^{2\pi\ii\varepsilon} \end{pmatrix}. \end{aligned}$$

Taking the limit gives the nontrivial Jordan matrix

$$\widehat M_0 = \begin{pmatrix} 1&2\pi\ii\\ 0&1 \end{pmatrix}.$$

At $\varepsilon=0$, the equation is $x^2y''+xy'=0$ and the limiting basis is $(1,\Log x)$. Since a positive loop sends $\Log x$ to $\Log x+2\pi\ii$, the direct continuation gives the same matrix. Taking the limit of the diagonal eigenvalues alone would have produced the identity and lost the logarithmic extension data.

8. A Hurwitz move on marked character data

Let a four-puncture monodromy tuple satisfy $ABCD=I$, with

$$A= \begin{pmatrix} 1&a\\ 0&1 \end{pmatrix}, \quad B= \begin{pmatrix} 1&0\\ b&1 \end{pmatrix}, \quad C= \begin{pmatrix} r&0\\ 0&r^{-1} \end{pmatrix},$$

where $abr(r-r^{-1})\neq0$, and set $D=(ABC)^{-1}$. Perform the Hurwitz move

$$(A,B,C,D) \longmapsto \left( B,B^{-1}AB,C,D \right).$$

1. Verify that the product constraint survives and that the first two local conjugacy classes are exchanged.
2. Compute the three pair traces $\operatorname{tr}(AB)$, $\operatorname{tr}(BC)$, and $\operatorname{tr}(AC)$ before the move.
3. Compute the corresponding traces after the move.
4. Why is this a change of marking rather than an ordinary simultaneous conjugation?

Solution

Write

$$A'=B, \qquad B'=B^{-1}AB.$$

Then

$$A'B'=AB,$$

so $A'B'CD=ABCD=I$. Moreover, $A'$ is the old $B$, while $B'$ is conjugate to the old $A$; hence their local conjugacy classes are exchanged.

Before the move,

$$\begin{aligned} \operatorname{tr}(AB) &= 2+ab,\\ \operatorname{tr}(BC) &= r+r^{-1},\\ \operatorname{tr}(AC) &= r+r^{-1}. \end{aligned}$$

Direct calculation gives

$$B^{-1}AB = \begin{pmatrix} 1+ab&a\\ -ab^2&1-ab \end{pmatrix}.$$

Therefore the new pair traces are

$$\begin{aligned} \operatorname{tr}(A'B') &= 2+ab,\\ \operatorname{tr}(A'C) &= r+r^{-1},\\ \operatorname{tr}(B'C) &= r+r^{-1} + ab(r-r^{-1}). \end{aligned}$$

The last value differs from both old traces under the stated assumptions. A simultaneous conjugation preserves every trace of every labelled word, so it cannot produce this change. The Hurwitz move instead changes the distinguished generators of the punctured-sphere fundamental group. It is a braid or mapping-class action on the marked character variety: local conjugacy classes and the total product remain compatible, but the trace coordinates attached to labelled loop words transform.

Boundary functions and quantization

9. A Robin boundary function and a Fredholm ratio

Fix $L>0$ and $h\in\mathbb R$. Let

$$T_h=-\frac{\dd^2}{\dd x^2}$$

in $L^2(0,L)$, with

$$\mathcal D(T_h) = \left\{ y\in H^2(0,L): y(0)=0,\ y'(L)+hy(L)=0 \right\}.$$

Let $k^2=\lambda$ and use the entire-in-$\lambda$ left-normalized solution

$$u(x,\lambda) = \frac{\sin(kx)}{k}, \qquad u(0,\lambda)=0, \qquad u'(0,\lambda)=1.$$

1. Construct an entire boundary function and identify its zeros.
2. Prove directly that its eigenvalue zeros are simple.
3. When is its normalized ratio an ordinary Fredholm determinant?
4. What fails at the exceptional value $h=-1/L$?

Solution

Applying the right boundary functional gives

$$\begin{aligned} F_h(\lambda) &= u'(L,\lambda)+hu(L,\lambda)\\ &= \cos(kL) + h\frac{\sin(kL)}{k}. \end{aligned}$$

Both apparent functions of $\sqrt\lambda$ have power series in $\lambda$, so $F_h$ is entire. Its zeros are exactly the eigenvalues of the specified self-adjoint domain: the left-normalized solution then also satisfies the Robin condition.

Differentiate

$$-u''=\lambda u$$

with respect to $\lambda$, writing $\dot u=\partial_\lambda u$. The Wronskian identity is

$$\frac{\dd}{\dd x} \Wr[u,\dot u] = -u^2.$$

At an eigenvalue $\lambda_n$, the left normalization gives zero Wronskian at $x=0$, while the Robin condition gives

$$\Wr[u,\dot u](L) = u(L,\lambda_n)F_h'(\lambda_n).$$

Hence

$$u(L,\lambda_n)F_h'(\lambda_n) = -\int_0^L u(x,\lambda_n)^2\,\dd x.$$

The endpoint value $u(L,\lambda_n)$ cannot vanish, since the Robin condition would then force both Cauchy data at $L$ to vanish. The integral is positive, so $F_h'(\lambda_n)\neq0$.

At zero,

$$F_h(0)=1+hL.$$

If $h\neq-1/L$, then $T_h$ is invertible. Its inverse is trace class, because the eigenvalues of $T_h$ grow quadratically. Since $F_h$ has order $1/2$ as an entire function of $\lambda$, its normalized genus-zero factorization is

$$\frac{F_h(\lambda)}{F_h(0)} = \prod_n \left( 1-\frac{\lambda}{\lambda_n} \right).$$

The same convergent product defines the Fredholm determinant, so

$$\frac{F_h(\lambda)}{F_h(0)} = \det\nolimits_{\mathrm F} \left( I-\lambda T_h^{-1} \right).$$

This equality uses the declared domain, trace-class inverse, growth, and normalization; it does not follow from the zero set alone.

For $h=-1/L$, the function $u(x,0)=x$ satisfies both boundary conditions. Thus zero is an eigenvalue, $F_h(0)=0$, and neither the displayed quotient nor $T_h^{-1}$ exists. The unnormalized boundary function remains valid. One may factor out the zero mode or normalize at a shifted spectral parameter, but that is a different determinant statement. An absolute zeta determinant would require its own continuation, normalization, and, for a non-self-adjoint problem, spectral-cut data.

10. Resonant Legendre quantization

Let

$$\tau y = -\frac{\dd}{\dd x} \left[ (1-x^2)y' \right]$$

and begin with the maximal domain

$$\begin{aligned} \mathcal D_{\max} = \bigl\{ y:\;& y,\ (1-x^2)y'\in AC_{\mathrm{loc}}(-1,1),\\ & \tau y\in L^2(-1,1) \bigr\}. \end{aligned}$$

Consider the endpoint-regular self-adjoint restriction of $\tau$ defined by

$$-\frac{\dd}{\dd x} \left[ (1-x^2)y' \right] = \lambda y$$

in $L^2(-1,1)$, with the separated conditions

$$\lim_{x\to-1^+} (1-x^2)y'(x)=0, \qquad \lim_{x\to1^-} (1-x^2)y'(x)=0.$$

Write $\lambda=\nu(\nu+1)$ and choose the solution regular at $x=1$,

$$P_\nu(x) = {}_2F_1 \left( \begin{matrix} -\nu,\ \nu+1\\ 1 \end{matrix} ; \frac{1-x}{2} \right).$$

1. Show that the hypergeometric equation is zero-balanced at $x=-1$.
2. Compute the coefficient of the logarithmic branch there.
3. Derive the spectrum selected by regularity at both endpoints.
4. Explain why “the equation is resonant” is not itself a quantization condition.

Solution

Set

$$z=\frac{1-x}{2}, \qquad t=1-z=\frac{1+x}{2}.$$

The hypergeometric parameters are

$$a=-\nu, \qquad b=\nu+1, \qquad c=1=a+b.$$

Thus the exponent difference at $z=1$, corresponding to $x=-1$, is zero for every $\nu$. For $\nu\notin\mathbb Z$, the zero-balanced continuation formula has leading term

$${}_2F_1(a,b;a+b;z) = \frac{ -\Log(1-z)+O(1) }{ \Gamma(a)\Gamma(b) }$$

because $\Gamma(a+b)=\Gamma(1)=1$. Euler’s reflection formula gives

$$\Gamma(-\nu)\Gamma(\nu+1) = -\frac{\pi}{\sin(\pi\nu)}.$$

Therefore, as $x\to-1^+$,

$$P_\nu(x) = \frac{\sin(\pi\nu)}{\pi} \Log\left( \frac{1+x}{2} \right) + O(1).$$

The logarithmic coefficient $\sin(\pi\nu)/\pi$ extends to integer $\nu$ by the parameter limit, even though the two gamma factors in the intermediate formula then have poles. The endpoint-regular domain removes the logarithmic branch, so

$$\sin(\pi\nu)=0.$$

Using the symmetry $\nu\leftrightarrow-\nu-1$, choose $\nu=n\in\mathbb Z_{\ge0}$. The eigenvalues and eigenfunctions are

$$\lambda_n=n(n+1), \qquad y_n(x)=P_n(x).$$

The singular point was resonant for every value of $\nu$; most of those values produce a logarithm. Quantization occurs only when the parameter-dependent logarithmic connection coefficient vanishes. At integer $\nu$, the hypergeometric series terminates and the limiting solution is a Legendre polynomial. This is the distinction between a local exponent collision and a global two-end boundary condition.

Computational synthesis

11. Reproduce a hypergeometric global-data audit

Build one high-precision computation with

$$a=\frac13, \qquad b=\frac25, \qquad c=\frac76, \qquad \delta=\frac{13}{30}.$$

Use the unit-leading bases and upper/lower phase conventions from the hypergeometric benchmark. Your computation must do all of the following:

1. form $C_{10}$ and the four path-labelled matrices $C_{\infty0}^{(s)}$, $C_{\infty1}^{(s)}$ for $s=\pm1$ from gamma quotients;
2. verify the connection cocycle and the three exact determinant targets;
3. assemble the lower- and upper-stem based monodromy products;
4. repeat with $c=a+b+\varepsilon$ and compare the raw $z=1$ basis with its logarithmic recombination as $\varepsilon\to0$.

Solution and acceptance criteria

The exact determinant targets are

$$\begin{aligned} \det C_{10} &= \frac5{13},\\ \det C_{\infty0}^{(+)} &= \frac52\ee^{17\pi\ii/30},\\ \det C_{\infty1}^{(+)} &= -\frac{13}{2}\ee^{-13\pi\ii/30}. \end{aligned}$$

The upper matrices must obey

$$C_{\infty0}^{(+)} = C_{\infty1}^{(+)}C_{10}.$$

With the lower stem, transport the three diagonal local matrices into the zero frame and check

$$M_0^{(0)}M_1^{(0)}M_\infty^{(0)}=I.$$

With the upper stem and its corresponding infinity matrix, check instead

$$M_1^{(0)}M_0^{(0)}M_\infty^{(+)}=I.$$

Testing the wrong word against the right matrices is expected to fail.

A language-neutral implementation can follow this outline:

set working precision P and guard digits G
define gamma quotients A_f, A_g, B_f, B_g
construct C10 and Cinf0(s), Cinf1(s) for s = -1, +1
record every logarithm branch and every lateral phase

for each claimed identity:
    compute a scale-aware matrix residual
    repeat at P, 2P, and 4P digits
    compare determinants separately with their exact targets

construct D0, D1, Dinf
transport them with the declared connection matrices
test the lower word and the upper word independently

for epsilon in 10^(-2), 10^(-4), 10^(-6), ...:
    set c = a + b + epsilon
    form the raw C10(epsilon)
    set S = [[1, 1], [0, epsilon]]
    form Chat(epsilon) = S * C10(epsilon)
    record condition numbers of C10 and Chat
    compare Chat with its gamma–digamma limit

The numerical report passes only if:

• relative determinant errors and scaled matrix residuals decrease as the working precision increases;
• the upper cocycle and both correctly ordered monodromy words converge to zero residual;
• the raw near-resonant condition number grows on the scale $1/|\varepsilon|$, while the transformed matrix remains bounded and converges when $a+b\neq1$;
• an independent check—Wronskian evaluation, direct ODE integration along the declared paths, or overlapping local series—agrees with the matrix calculation;
• the report records software and version, precision, branch conventions, path geometry, matrix norm, and all parameter values.

A table of residual versus precision is evidence; a single string of machine digits is not. Near resonance, evaluate the recombined expressions directly or with sufficient guard digits, since subtracting two $O(1/\varepsilon)$ terms at ordinary precision recreates the instability that the basis change was meant to remove.

Research extension

12. Build a complete global-data dossier

Choose one nontrivial model: a Bessel, Airy, Whittaker, Heun, or compactly-supported Schrödinger equation is suitable. Formulate one global question—connection, scattering, bound-state, resonance, or quasinormal-mode—and answer it as completely as the model permits.

Your dossier must separate the following layers:

1. differential equation, parameter domain, and singularity types;
2. normalized local or asymptotic bases, including branches and sectors;
3. base point, path system, and ordinary or wild monodromy data;
4. the connection entry or boundary Wronskian that answers the question;
5. Hilbert space and operator domain, or time convention, outgoing condition, and resonance sheet;
6. the status of any Fredholm, zeta, or canonical-product determinant;
7. at least two independent analytic or numerical checks;
8. a ledger marking each statement as proved, numerically supported, or conjectural.

Guidance and known milestones

A successful dossier is reproducible before it is ambitious. Begin with a one-page convention ledger. Normalize every basis by a leading coefficient or Cauchy datum; draw or describe every cut and path; and reserve different symbols for local monodromy, Stokes factors, connection matrices, and spectral functions.

Then meet these milestones:

1. Derive local exponents or formal exponential parts directly from the equation.
2. Compute at least one connection quantity in two independent ways, such as a Wronskian ratio and numerical continuation.
3. Transport every local monodromy matrix to one base frame before testing the global product.
4. If an irregular point is present, give the ordered Stokes factorization and its residual centralizer action.
5. Derive the spectral condition from the stated domain or asymptotics, rather than importing a familiar spectrum from another realization.
6. State separately whether the resulting scalar function is merely a boundary function, a canonical product, a Fredholm determinant, or a zeta determinant. Cite the comparison theorem for every asserted equality.
7. Include machine-readable parameters and a precision-convergence table. One check should be structurally different from the formula being tested.

For a Bessel model, the integer-order logarithmic limit is a natural resonance test. Airy supplies a clean Stokes audit. Whittaker combines regular and irregular singularities. Heun exposes accessory-parameter dependence, while a compactly-supported Schrödinger model makes the distinction between physical and nonphysical sheets unavoidable. A negative result—such as proving that the available data do not determine a claimed determinant—is a valid research conclusion when the missing hypothesis is identified precisely.

References

• NIST Digital Library of Mathematical Functions, §15.10: Hypergeometric Differential Equation fixes standard solutions and connection formulae for the Gauss equation; §15.8 records the transformations and limiting cases used in the resonant problems.
• NIST Digital Library of Mathematical Functions, Chapter 14: Legendre and Related Functions gives the Legendre normalizations and endpoint behavior used in Problem 10.
• A. A. Bolibrukh, “The Riemann–Hilbert problem”, Russian Mathematical Surveys 45 (1990), 1–58, explains why prescribed monodromy and a prescribed Fuchsian realization are distinct questions.
• P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics 163, Springer, 1970, develops the regular-singular correspondence and logarithmic extensions.
• K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991, treats marked monodromy, braid actions, and hypergeometric connection data.
• W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover, 1987, develops formal solutions, sectors, Stokes matrices, and irregular monodromy.
• B. Simon, Trace Ideals and Their Applications, second edition, American Mathematical Society, 2005, gives the operator-theoretic hypotheses for Fredholm determinants.
• A. Zettl, Sturm–Liouville Theory, American Mathematical Society, 2005, treats self-adjoint domains, endpoint classification, and separated boundary conditions.