## Prof. Dr. Matthias Lesch

### Project leader

Rheinische Friedrich-Wilhelms-Universität Bonn

E-mail: lesch(at)math.uni-bonn.de

Telephone: +49 228 73 7641

Homepage: http://www.math.uni-bonn.de/people/lesch…

## Project

**23**Spectral geometry, index theory and geometric flows on singular spaces

## Publications within SPP2026

We consider a generalized Dirac operator on a compact stratified space with an iterated cone-edge metric. Assuming a spectral Witt condition, we prove its essential self-adjointness and identify its domain and the domain of its square with weighted edge Sobolev spaces. This sharpens previous results where the minimal domain is shown only to be a subset of an intersection of weighted edge Sobolev spaces. Our argument does not rely on microlocal techniques and is very explicit. The novelty of our approach is the use of an abstract functional analytic notion of interpolation scales. Our results hold for the Gauss-Bonnet and spin Dirac operators satisfying a spectral Witt condition.

Journal | JOURNAL OF SPECTRAL THEORY |

Volume | Volume 8, Issue 4, 2018, pp. 1295–1348 |

Link to preprint version | |

Link to published version |

**Related project(s):****23**Spectral geometry, index theory and geometric flows on singular spaces

Let (M,g) be a compact smoothly stratified pseudomanifold with an iterated cone-edge metric satisfying a spectral Witt condition. Under these assumptions the Hodge-Laplacian Δ is essentially self-adjoint. We establish the asymptotic expansion for the resolvent trace of Δ. Our method proceeds by induction on the depth and applies in principle to a larger class of second-order differential operators of regular-singular type, e.g., Dirac Laplacians. Our arguments are functional analytic, do not rely on microlocal techniques and are very explicit. The results of this paper provide a basis for studying index theory and spectral invariants in the setting of smoothly stratified spaces and in particular allow for the definition of zeta-determinants and analytic torsion in this general setup.

**Related project(s):****23**Spectral geometry, index theory and geometric flows on singular spaces