[Paper Review] On eigenvalues of Lamé operator

Kouichi Takemura|ArXiv.org|Sep 15, 2004

Mathematical functions and polynomials11 references3 citations

TL;DR

This paper derives integral representations of monodromy for the Lamé equation using hyperelliptic and elliptic integrals, enabling hyperelliptic-to-elliptic reduction formulae and advancing the spectral theory of finite-gap potentials. A key result is the analytical confirmation of eigenvalue convergence radii via monodromy branching, reconciling perturbative and monodromy-based methods for n=1.

ABSTRACT

We introduce two integral representations of monodromy on Lamé equation. By applying them, we obtain results on hyperelliptic-to-elliptic reduction integral formulae, finite-gap potential and eigenvalues of Lamé operator.

Motivation & Objective

• To establish integral representations of monodromy for the Lamé equation using hyperelliptic and Hermite-Krichever Ansatz-based expressions.
• To derive hyperelliptic-to-elliptic reduction integral formulae by comparing two monodromy expressions.
• To clarify the relationship between boundary conditions (periodic, square-integrable) and finite-gap potential structure.
• To investigate the analytic continuation of eigenvalues in the modular parameter τ for n=1, particularly their convergence radii.
• To confirm consistency between perturbative expansions and monodromy-based analytic continuation of eigenvalues.

Proposed method

• Introduces two integral representations of monodromy: one via hyperelliptic integrals and another via Hermite-Krichever Ansatz.
• Applies the monodromy expressions to derive hyperelliptic-to-elliptic reduction formulae by equating the two forms.
• Uses the monodromy condition E = -℘(t₀) and -ζ(t₀) + 2η₁t₀ = mπi to parametrize eigenvalues analytically.
• Performs analytic continuation of eigenvalues Eₘ(p) along cycles 𝒞ₐ around branching points in the complex p-plane.
• Computes branching behavior by tracking monodromy transformations of eigenvalue labels (e.g., E₀ ↔ E₂) around singularities.
• Compares convergence radii from power series expansions (Table 1) with radii inferred from the closest branching point (|p| ≈ 0.743869).

Experimental results

Research questions

• RQ1 How can monodromy of the Lamé equation be represented via both hyperelliptic and elliptic integrals?
• RQ2 What integral formulae emerge from comparing these two monodromy representations?
• RQ3 How do boundary conditions (singly/doubly periodic, square-integrable) relate to finite-gap potential structure?
• RQ4 What is the convergence radius of eigenvalue expansions in p = exp(πiτ) for the n=1 Lamé operator?
• RQ5 How does analytic continuation along cycles around branching points confirm or contradict perturbative convergence radii?

Key findings

• The closest branching point to the origin in the p-plane is at p ≈ 0.258666 + 0.697448i, with |p| ≈ 0.743869.
• Eigenvalues E₀(p) and E₂(p) are connected by monodromy around this branching point, with E₀ ↔ E₂ and E₄ ↔ E₄ under analytic continuation.
• The convergence radius inferred from monodromy (≈0.743869) closely matches the perturbative estimate of ≈0.749 for E₀(p) and E₂(p).
• The agreement between perturbative and monodromy-based convergence radii confirms the consistency of both methods for eigenvalue expansions.
• The branching behavior confirms that E₀(p) and E₂(p) are analytically connected, explaining their similar convergence radii.
• For n=1, the eigenvalues Eₘ(p) are real-analytic in p² ∈ (−1,1), and their power series expansions converge within |p| < 1, though singularities limit the actual radius.