Zero-temperature gap integral equals $\operatorname{arsinh}(\hbar\omega_D/\Delta)$

Proved

BCSTheory.gapIntegralZeroTemp_eq_arsinh

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

bcs-theorysuperconductivity

For every Debye energy $\hbar\omega_D > 0$ and every $\Delta > 0$ ,

\int_0^{\hbar\omega_D} \frac{d\eta}{\sqrt{\eta^2+\Delta^2}} = \operatorname{arsinh}\frac{\hbar\omega_D}{\Delta}.

Preamble

import Mathlib
import Definitions.Def_BCSTheory_GapEquation
open Filter Topology

Formal statement

namespace BCSTheory

theorem gapIntegralZeroTemp_eq_arsinh (hbarOmegaD Δ : ℝ) (hω : 0 < hbarOmegaD)
    (hΔ : 0 < Δ) :
    gapIntegralZeroTemp hbarOmegaD Δ = Real.arsinh (hbarOmegaD / Δ) := by sorry

end BCSTheory

Source

Wikipedia (de), Artikel „BCS-Theorie“ (PDF-Ausdruck vom 05.10.2026), Abschnitt „Erfolge der BCS-Theorie“, Unterabschnitt „Existenz einer Energielücke“, erste Formel (T = 0).

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; non-blind

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statements, with full knowledge of the source text and of the intended meaning. It was not produced by a blind, independent auditor and must not be mistaken for independent testimony; an independent read-back is still advisable before launch.

For all real numbers $\hbar\omega_D$ and $\Delta$ with $\hbar\omega_D>0$ and $\Delta>0$ ,

\int_0^{\hbar\omega_D}\frac{d\eta}{\sqrt{\eta^2+\Delta^2}} = \operatorname{arsinh}\Bigl(\frac{\hbar\omega_D}{\Delta}\Bigr),

where the left-hand side is the zero-temperature gap integral $I_0(\hbar\omega_D,\Delta)$ of the definition file and $\operatorname{arsinh}$ is the real inverse hyperbolic sine. There are no other hypotheses; both positivity conditions exclude the degenerate cases $\Delta=0$ and $\hbar\omega_D\le0$ .

Human review

Endorsed by Shuze Chen · Oct 10, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 10, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Zero-temperature gap integral equals arsinh⁡(ℏωD/Δ)\operatorname{arsinh}(\hbar\omega_D/\Delta)arsinh(ℏωD​/Δ)

Zero-temperature gap integral equals $\operatorname{arsinh}(\hbar\omega_D/\Delta)$