\(L^ 2\)-index for certain Dirac-Schrödinger operators
Let \(M\) be a complete noncompact Riemannian manifold of odd dimension and \(D\) a generalized Dirac operator acting on the compactly supported sections of a Clifford bundle \(E\) with a Hermitean structure. The authors introduce a skew adjoint potential \(V=-V':C_ 0^ \infty(E)\to C_ 0^ \infty(E)\) and a one parameter family of ``Dirac-Schrödinger operators'' \(D_ \lambda:=D+\lambda V\), \(\lambda\in\mathbb{R}\). They give sufficient conditions that the \(L^ 2\)-index of \(D_ \lambda\) is finite. To this end they assume that there exists an open subset \(U\) of \(M\) such that \(M_ 1:=M\backslash U\) is a compact manifold with boundary and the operators \(D_ \lambda\) have a special nice representation over \(U\). Under some additional assumptions on \(D\) and \(V\) they deduce from results of the first author [ibid. 257-309 (1992; see the review below)] an \(L^ 2\)-index formula for \(D_ \lambda\) when \(\lambda\) is sufficiently large. This formula only contains the \(\eta\)-invariant of a certain boundary term of the given representation. The abstract index theorem is applied to the case that \(M\) is a spin manifold and \(U=(0,\infty)\times\partial M_ 1\) with metric \(g=dy^ 2+g_ y\), where \(g_ y\) is a smooth family of metrics on \(\partial M_ 1\), \(y\geq 0\). \(D\) is the Dirac operator acting on \(C^ \infty_ 0(S\otimes E)\), where \(S\) is the spinor bundle of \(M\) and \(E\) a Hermitean vector bundle over \(M\). The potential \(V\) is defined as \(\sqrt{-1}I\otimes\Phi\) with \(\Phi\in C^ \infty(\text{End} E)\) and \(\Phi=\Phi^*\). The resulting index formula is related to results of \textit{N. Anghel} [Commun. Math. Phys. 128, No. 1, 77-97 (1990; Zbl 0697.58048)] and \textit{C. Callias} [Commun. Math. Phys. 62, 213-234 (1978; Zbl 0416.58024)].
scientific article; zbMATH DE number 78541
scientific article