APS η-Invariant: Spectral Asymmetry

Atiyah-Patodi-Singer: measuring the asymmetry of the spectrum of a self-adjoint operator

Mode

η(s) = Σ sign(λ)|λ|⁻ˢ
η(0) = #(λ>0) − #(λ<0)

The η-invariant measures spectral asymmetry: how many more positive eigenvalues than negative ones a self-adjoint operator has (after ζ-regularization at s=0).

Operator Parameters

Asymmetry parameter a:

a = 0.0

Number of eigenvalues N:

N = 10

η(0) = 0.00

η(0) ∈ ℝ (not necessarily integer). The fractional part ξ = (η(0)+dim ker D)/2 appears in the APS index formula.

APS η-Invariant — Spectral Asymmetry

Mode

Operator Parameters