Spectral Curves for Second‑Order Linear ODEs

0. What is a “spectral curve”?

For second‑order scalar ODEs there are three complementary notions of spectral curve:

1. Classical/WKB (oper) spectral curve.
   Put the ODE in normal (Schrödinger) form $\psi''+T(z)\,\psi=0$. The leading WKB relation $S_0^2=T$ yields the algebraic curve

   $$\boxed{\,\Sigma_{\mathrm{cl}}:\ y^2 = T(z)\,}$$

   — a 2‑sheeted cover of the base Riemann surface. This is the “classical limit” (quantum curve viewpoint).

2. Floquet (periodic) spectral curve.
   For $-y''+V(x)\,y=\lambda y$ with $V$ periodic, the one‑period monodromy $M(\lambda)$ has eigenvalues $\mu^{\pm1}$ and discriminant $\Delta(\lambda)=\mathrm{tr}\,M(\lambda)$. The dispersion relation

   $$\boxed{\,\Sigma_{\mathrm{Floquet}}:\ \mu+\mu^{-1}=\Delta(\lambda)\,}$$

   encodes bands/gaps; for finite‑gap potentials this is hyperelliptic.

3. Monodromy/character‑variety curve (Fuchsian problems).
   For a Fuchsian ODE on $\mathbb P^1\!\setminus\!\{z_i\}$, the monodromies $M_i\in\mathrm{SL}_2(\mathbb C)$ satisfy algebraic trace relations (Fricke–Klein/Jimbo–Fricke). Fixing local conjugacy classes cuts out algebraic curves parametrizing accessory data.

In what follows, (1) is the backbone; (2)–(3) appear where illuminating.

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1. From a second‑order ODE to the classical spectral curve

1.1 Normal form and the projective connection

Start from

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

with $p,q$ meromorphic. Gauge away the first‑derivative term via

$$y(z)=\exp\!\Big(-\tfrac12\!\int^z\! p(\zeta)\,d\zeta\Big)\,\psi(z),$$

to obtain the normal form

$$\psi''(z)+T(z)\,\psi(z)=0,\qquad T=q-\tfrac12 p'-\tfrac14 p^2.$$

Here $T(z)\,dz^2$ is a meromorphic quadratic differential. Define the classical spectral curve

$$\boxed{\,\Sigma_{\mathrm{cl}}:\ y^2=T(z)\,}.$$

1.2 Local exponents, accessory parameters, and $T$

At a regular singular point $z=z_i$ with local exponent difference $\theta_i$ (i.e. solutions behave like $(z-z_i)^{\rho_\pm}$ with $\rho_+-\rho_-=\theta_i$), one has

$$T(z)=\frac{\Delta_i}{(z-z_i)^2}+\frac{c_i}{z-z_i}+\text{holomorphic},\qquad \Delta_i=\frac{1-\theta_i^2}{4}.$$

The coefficients $c_i$ are accessory parameters; for $n$ singular points on $\mathbb P^1$ there are $n-3$ independent $c_i$’s controlling the global monodromy.

1.3 Branch points and genus (Fuchsian case on $\mathbb P^1$)

For a second‑order Fuchsian ODE with $n$ regular singular points on the sphere:

• $T$ has only double poles at those $n$ points and $T(z)=\mathcal O(z^{-2})$ at $\infty$;
• generically, the finite zeros of $T$ are simple and they alone produce branching.

A divisor count yields $B=2(n-2)$ branch points. Hence the 2‑sheeted cover has genus

$$\boxed{\,g(\Sigma_{\mathrm{cl}})=\frac{B}{2}-1=n-3\,}.$$

So: hypergeometric ($n=3$) $\Rightarrow g=0$; Heun ($n=4$) $\Rightarrow g=1$; in general, $g=n-3$.

Remark. On a base of genus $g_0$, Riemann–Hurwitz gives $g(\Sigma)=2g_0-1+\tfrac{B}{2}$.

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2. Hypergeometric class: explicit $T$ and $\Sigma_{\mathrm{cl}}$

Place singularities at $z=0,1,\infty$ and write exponent differences $(\theta_0,\theta_1,\theta_\infty)$. In normal form,

$$T_{\mathrm{hyp}}(z)= \frac{1-\theta_0^2}{4\,z^2} +\frac{1-\theta_1^2}{4\,(z-1)^2} +\frac{\theta_0^2+\theta_1^2-\theta_\infty^2-1}{4\,z(z-1)}.$$

Clearing denominators with $Y=y\,z(z-1)$ gives

$$\Sigma_{\mathrm{hyp}}:\quad Y^2=P_2(z)\,,$$

where $P_2$ is a quadratic polynomial whose two simple zeros are the branch points. Thus $\Sigma_{\mathrm{cl}}$ is a sphere ($g=0$).

• Rigid monodromy. There is no accessory parameter; specifying $(\theta_0,\theta_1,\theta_\infty)$ fixes $T$ globally.
• Algebraic special cases. When $P_2$ has a double root (discriminant $0$), the curve degenerates and the monodromy becomes finite (Schwarz list).

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3. Heun class: genus‑one spectral curve and the accessory parameter

Put singularities at $z=0,1,a,\infty$ (with $a\neq0,1$) and exponents $(\theta_0,\theta_1,\theta_a,\theta_\infty)$. In the standard Heun form

$$y''+\Big(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\varepsilon}{z-a}\Big)y' +\frac{\alpha\beta\,z-q}{z(z-1)(z-a)}\,y=0,\qquad \gamma+\delta+\varepsilon=\alpha+\beta+1,$$

the normal‑form potential is

$$T_{\mathrm{Heun}}(z)= \sum_{s\in\{0,1,a\}}\frac{1-\theta_s^2}{4\,(z-s)^2} +\sum_{s\in\{0,1,a\}}\frac{c_s}{z-s},\qquad \sum_s c_s=0.$$

Here the single accessory parameter $q$ enters linearly in the $c_s$. Clearing denominators with $Y=y\,z(z-1)(z-a)$,

$$\Sigma_{\mathrm{Heun}}:\quad Y^2=P_4(z)\,,$$

with a quartic $P_4$ having four simple zeros: a genus‑one (elliptic) curve.

• Moduli. For fixed $(\theta_0,\theta_1,\theta_a,\theta_\infty,a)$, varying $q$ moves the branch points of $P_4$ and changes the complex structure of $\Sigma_{\mathrm{Heun}}$.
• Isomonodromy. Varying $a$ at fixed monodromy gives $q(a)$ governed by Painlevé VI; geometrically, the elliptic curve varies isomonodromically.

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4. Confluence: from regular to irregular singularities

Merging regular singularities produces irregular ones; the classical curve remains $y^2=T(z)$ but its asymptotics and branching at $\infty$ change.

• Confluent hypergeometric (Kummer). One regular singularity at $0$ and a rank‑1 irregular at $\infty$; $T$ acquires higher‑order terms at $\infty$.
• Confluent Heun variants. (Confluent, doubly confluent, biconfluent, triconfluent.) Each step raises the rank at $\infty$ and modifies the branching (often introducing branching “at infinity” in the compactified picture).

In periodic/elliptic settings (e.g. Lamé/Mathieu), the Floquet spectral curve $\mu+\mu^{-1}=\Delta(\lambda)$ is the natural object for the band/gap problem; for finite‑gap cases it is hyperelliptic of genus equal to the gap number.

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5. How the three spectral curves fit together

• Classical curve $y^2=T(z)$ is intrinsic to the scalar ODE in normal form; it controls WKB periods $\oint y\,dz$, branch points, and genus ($g=n-3$ on $\mathbb P^1$). It is the spectral curve of a rank‑2 Hitchin system with quadratic differential $\phi_2=T\,dz^2$ (quantization $y\mapsto \hbar\partial_z$ returns the ODE: “quantum curve”).

• Floquet curve is tailored to periodic spectral problems; it encodes the spectrum via the discriminant and is algebraic in finite‑gap cases.

• Character‑variety curves organize monodromy: for $n=4$ they are elliptic; the accessory parameter is a coordinate on that curve. Isomonodromic deformations (PVI) move the Heun branch points without changing monodromy.

They are different “projections” of the same global analytic structure: solutions live on branched covers of the base, with monodromy/Stokes/Floquet data algebraic on suitable ambient curves.

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6. Worked recipes

6.1 From an ODE to $\Sigma_{\mathrm{cl}}$

1. Normalize. Compute $T=q-\tfrac12 p'-\tfrac14 p^2$.
2. Local exponents. At each regular singular $z_i$, solve the indicial equation; set $\Delta_i=(1-\theta_i^2)/4$.
3. Assemble $T$. On $\mathbb P^1$, $$T(z)=\sum_{i=1}^{n-1}\frac{\Delta_i}{(z-z_i)^2} +\sum_{i=1}^{n-1}\frac{c_i}{z-z_i},\qquad \sum c_i=0,$$ with $n-3$ independent accessory parameters among the $c_i$.
4. Define the curve. $\Sigma_{\mathrm{cl}}:\ y^2=T(z)$. Clear denominators to write $Y^2=P_{2(n-2)}(z)$ with $Y=y\prod (z-z_i)$.
5. Genus/branch points. Generically $g=n-3$; collisions of zeros of $T$ lower $g$ and signal special monodromy (algebraic solutions/apparent singularities).

6.2 Hypergeometric (explicit)

With $(\theta_0,\theta_1,\theta_\infty)$,

$$T(z)=\frac{1-\theta_0^2}{4z^2} +\frac{1-\theta_1^2}{4(z-1)^2} +\frac{\theta_0^2+\theta_1^2-\theta_\infty^2-1}{4z(z-1)}.$$

Set $Y=y\,z(z-1)$ to get $Y^2=P_2(z)$ (genus 0).

6.3 Heun (explicit structure)

With singularities at $0,1,a,\infty$ and exponents $(\theta_0,\theta_1,\theta_a,\theta_\infty)$,

$$T(z)=\sum_{s\in\{0,1,a\}}\frac{1-\theta_s^2}{4(z-s)^2} +\sum_{s\in\{0,1,a\}}\frac{c_s(q)}{z-s},\qquad \sum c_s=0.$$

Set $Y=y\,z(z-1)(z-a)$ to get $Y^2=P_4(z)$ (genus 1). Varying $q$ moves the branch points; varying $a$ at fixed monodromy gives $q(a)$ via PVI.

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7. Minimal glossary

• Regular/irregular singularity. Frobenius power‑law/log growth vs. Stokes/exponential growth.
• Exponent difference $\theta$. If local exponents are $\rho_\pm$, then $\theta=\rho_+-\rho_-$ and $\Delta=(1-\theta^2)/4$ in $T$.
• Accessory parameter. Coefficients of simple poles in $T$; determine global monodromy for $n\ge4$.
• Oper/quantum curve. Quantizing $y^2=T(z)$ by $y\mapsto \hbar\partial_z$ yields the original ODE.
• Character variety. Algebraic variety of monodromy representations; elliptic for four punctures.
• Finite‑gap. Periodic Schrödinger potentials with algebraic Floquet curves.

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8. Further reading (entry points)

• Classical Fuchsian ODEs and monodromy: standard texts on complex ODEs and the Riemann–Hilbert problem.
• Hypergeometric/Heun families: monographs on Heun’s equation and its confluent forms.
• Hitchin systems and opers: reviews on spectral curves and quadratic differentials in integrable systems.