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Geometric operators on singular domains

Many phenomena in physics can be formulated by boundary value problems for linear,  elliptic partial differential equations. The prototype of a mixed boundary value problem for the Laplacian is the Poisson problem on a polygon with partly Dirichlet and partly Neumann boundary conditions. A very large number of papers were published about boundary value   problems on singular domains, among which the papers of Kondratiev  and Mazya-Plamenevskij  have played a pioneering role. We want to study such problems for a wider set of singularities and in higher-dimensional contexts from a geometric point of view. The basic idea is to use a conformal blow-up to transfer the problem from singular spaces on weighted regularity scales to noncompact spaces with uniform regularity scales.

## Publications

We prove regularity estimates for the eigenfunctions of Schrödinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual N-body operators are covered by our result; in that case, the weight is in terms of the (euclidean) distance to the collision planes. The technique of proof is based on blow-ups and Lie manifolds. More precisely, we first blow-up the spheres at infinity of the collision planes to obtain the Georgescu-Vasy compactification and then we blow-up the collision planes. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher order operators and matrices of scalar operators.

 Journal to appear in Letters Math. Phys. Link to preprint version

Related project(s):
35Geometric operators on singular domains

Let (N,h) be a time- and space-oriented Lorentzian spin manifold, and let M be a compact spacelike hypersurface of N with induced Riemannian metric g and second fundamental form K. If (N,h) satisfies the dominant energy condition in a strict sense, then the Dirac-Witten operator of M⊂ N is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial data pairs on $M$ satisfying the dominant energy condition in a strict sense. The central tool will be a Lorentzian analogue of Hitchin's α-invariant. In case that the dominant energy condition only holds in a weak sense, the Dirac-Witten operator may be non-invertible, and we will study the kernel of this operator in this case.

We will show that the kernel may only be non-trivial if π1(M) is virtually solvable of derived length at most 2. This allows to extend the index theoretical methods to spaces of initial data pairs, satisfying the dominant energy condition in the weak sense.

We will show further that a spinor φ is in the kernel of the Dirac--Witten operator on (M,g,K) if and only if (M,g,K,φ) admits an extension to a Lorentzian manifold (N,h) with parallel spinor ψ such that M is a Cauchy hypersurface of (N,h), such that g and K are the induced metric and second fundamental form of M, respectively, and φ is the restriction of ψ to M.

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Related project(s):
35Geometric operators on singular domains

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## Team Members

Prof. Dr. Bernd Ammann
Universität Regensburg
ammann(at)berndammann.de

Albert-Ludwigs-Universität Freiburg im Breisgau