725 research outputs found
Spectral convergence of non-compact quasi-one-dimensional spaces
We consider a family of non-compact manifolds X_\eps (``graph-like
manifolds'') approaching a metric graph and establish convergence results
of the related natural operators, namely the (Neumann) Laplacian \laplacian
{X_\eps} and the generalised Neumann (Kirchhoff) Laplacian \laplacian {X_0}
on the metric graph. In particular, we show the norm convergence of the
resolvents, spectral projections and eigenfunctions. As a consequence, the
essential and the discrete spectrum converge as well. Neither the manifolds nor
the metric graph need to be compact, we only need some natural uniformity
assumptions. We provide examples of manifolds having spectral gaps in the
essential spectrum, discrete eigenvalues in the gaps or even manifolds
approaching a fractal spectrum. The convergence results will be given in a
completely abstract setting dealing with operators acting in different spaces,
applicable also in other geometric situations.Comment: some references added, still 36 pages, 4 figure
Band width estimates via the Dirac operator
Let be a closed connected spin manifold such that its spinor Dirac
operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian
metric on with scalar curvature bounded below by , the distance between the boundary components of is at most
, where with being a universal constant. This verifies a conjecture of Gromov
for such manifolds. In particular, our result applies to all high-dimensional
closed simply connected manifolds which do not admit a metric of positive
scalar curvature. We also establish a quadratic decay estimate for the scalar
curvature of complete metrics on manifolds, such as ,
which contain as a codimension two submanifold in a suitable way.
Furthermore, we introduce the "-width" of a closed manifold and
deduce that infinite -width is an obstruction to positive scalar
curvature.Comment: 24 pages, 2 figures; v2: minor additions and improvements; v3: minor
corrections and slightly improved estimates. To appear in J. Differential
Geo
Quantitative spectral perturbation theory for compact operators on a Hilbert space
We introduce compactness classes of Hilbert space operators by grouping
together all operators for which the associated singular values decay at a
certain speed and establish upper bounds for the norm of the resolvent of
operators belonging to a particular compactness class. As a consequence we
obtain explicitly computable upper bounds for the Hausdorff distance of the
spectra of two operators belonging to the same compactness class in terms of
the distance of the two operators in operator norm.Comment: 26 page
SchrĂśdinger operators with δ and δâ˛-potentials supported on hypersurfaces
Self-adjoint SchrĂśdinger operators with δ and δâ˛-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the BirmanâSchwinger principle and a variant of Kreinâs formula are shown. Furthermore, Schattenâvon Neumann type estimates for the differences of the powers of the resolvents of the SchrĂśdinger operators with δ and δâ˛-potentials, and the SchrĂśdinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed SchrĂśdinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity
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