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Journal of the London Mathematical Society 2002 65(3):757-768; doi:10.1112/S0024610702003125
© 2002 by London Mathematical Society
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© The London Mathematical Society

A Topological Criterion for the Existence of Half-Bound States

Gilles Carron

Département de Mathématiques, UMR 6629, Université de Nantes 2 rue de la Houssinière, F-44322 Nantes Cedex 03, France, gilles.carron{at}math.univ-nantes.fr

Received 1 October 1999. Revision received 10 July 2001.

The following theorem is proved: if (M4n+1,g) is a complete Riemannian manifold and {Sigma} sub 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 L2 but not in L2) with the signature of {Sigma}.


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