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A Topological Criterion for the Existence of Half‐Bound States
Author(s) -
Carron Gilles
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003125
Subject(s) - hypersurface , signature (topology) , mathematics , manifold (fluid mechanics) , zero (linguistics) , atiyah–singer index theorem , upper and lower bounds , pure mathematics , spectrum (functional analysis) , laplace operator , harmonic , topology (electrical circuits) , combinatorics , mathematical analysis , physics , geometry , quantum mechanics , mechanical engineering , linguistics , philosophy , engineering
The following theorem is proved: if ( M 4 n +1 ,g) is a complete Riemannian manifold and Σ ⊂ M is an oriented hypersurface partitioning M and with non‐zero signature, then the spectrum of the Hodge–deRham Laplacian is [0,∞]. 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 Σ.