##
**L\({}^ 2\)-index theorems on certain complete manifolds.**
*(English)*
Zbl 0722.58043

Let M be a Riemannian manifold, E,F - Hermitian vector bundles over M and D: \(C^{\infty}(E)\to C^{\infty}(F)\) a first order elliptic differential operator. The author considers D which need not be Fredholm, but have a finite \(L^ 2\)-index in the sense that ker \(D\cap L^ 2(E)\) and ker D\({}'\cap L^ 2(F)\) are finite dimensional \((D'\) is the formal adjoint to D):
\[
L^ 2-ind D\overset \circ =\dim \ker D\cap L^ 2(E)-\dim \ker D'\cap L^ 2(F).
\]
It is supposed that M has finitely many ends and every end is a warped product with a warping function of at most linear growth. In this setting the author shows that D has a particularly simple normal form and constructs a weight function g such that gDg is unitarily equivalent to a regular singular operator in the sense of the author and R. T. Seeley [Am. J. Math. 110, No.4, 659- 714 (1988; Zbl 0664.58035)]. This allows to obtain an index formula for such operators, which contains interior terms, involving the geometry of the whole manifold, the spectral invariants of the cross-section such as the \(\eta\)-invariant, and the global contributions which can be expressed in terms of the solutions of an ordinary differential equation on \({\mathbb{R}}_+\).

Reviewer: V.B.Marenich (Novosibirsk)

### MSC:

58J22 | Exotic index theories on manifolds |

58J20 | Index theory and related fixed-point theorems on manifolds |

47A53 | (Semi-) Fredholm operators; index theories |