Eta invariant

Remarks

The regularity at $s = 0$ was proved by Atiyah, Patodi and Singer, with the general case due to Gilkey. Some authors call $\xi_A = \tfrac12(\eta_A(0) + \dim \ker A)$ the reduced eta invariant.

Example: on $S^1 = \R / 2\pi\Z$ let $A = -i \, d/d\theta + a$ with $0 < a < 1$. The eigenfunctions are $e^{ik\theta}$, $k \in \Z$, with eigenvalues $k + a$. So $\eta_A(s) = \sum_{k \ge 0} (k + a)^{-s} - \sum_{k \ge 1} (k - a)^{-s} = \zeta(s, a) - \zeta(s, 1 - a)$, with $\zeta(s, a)$ the Hurwitz zeta function. Since $\zeta(0, a) = \tfrac12 - a$, the eta invariant is $\eta_A(0) = 1 - 2a$. For $a = 0$ the nonzero spectrum $\Z \setminus \{0\}$ is symmetric and $\eta_A(0) = 0$.

Non-example: $B = d/d\theta$ on $S^1$ is elliptic but not self-adjoint; its eigenvalues $ik$ are imaginary, so $\operatorname{sign}(\lambda)$ has no meaning and $B$ has no eta invariant.