The following theorem is proved: if $(M^{4n+1},g)$ is a complete Riemannian manifold and $\Sigma\subset M$ is an oriented hypersurface partitioning $M$ and with non-zero signature, then the spectrum of the Hodge–deRham Laplacian is $[0,\infty[$ . This result is obtained by a new Callias-type index. This new formula links half-bound harmonic forms (that is, nearly $L^2$ but not in $L^2$ ) with the signature of $\Sigma$ .