Lemniscate Operator H_L

The Laplacian H_L = −d²/dx² on L = S¹ ∨ S¹ with Kirchhoff boundary conditions at the crossing point. Standard self-adjoint operator on a metric graph. Compact resolvent, discrete spectrum.

Lemniscate Operator H_L

Summary

The Laplacian H_L = −d²/dx² on L = S¹ ∨ S¹ with Kirchhoff boundary conditions at the crossing point. Standard self-adjoint operator on a metric graph. Compact resolvent, discrete spectrum.

Statement

\label{def:lemniscate-operator}
The \textbf{lemniscate operator} $H_L : \mathrm{Dom}(H_L) \subset L^2(L) \to L^2(L)$ is given by
\[
H_L f = -\frac{d^2 f}{dx^2}
\]
on each edge $e_B$ and $e_C$, with domain
\[
\mathrm{Dom}(H_L) = \left\{ f \in H^2(L) : \begin{array}{l}
f_B(\omega) = f_C(\omega) \quad \text{(continuity)} \\[0.5ex]
f'_B(\omega) + f'_C(\omega) = 0 \quad \text{(Kirchhoff)}
\end{array} \right\}.
\]
The Kirchhoff condition is the \textbf{current conservation} law: the sum of outward normal derivatives at $\omega$ vanishes.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 64
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch23-the-lemniscate-operator.tex lines 77-91

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Doors.LemniscateOperator
• Name: lemniscate_eigenvalue

Dependencies

• Canonical: I.D18, III.D11

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.