8,432 research outputs found
Spectral triples and manifolds with boundary
We investigate manifolds with boundary in noncommutative geometry. Spectral
triples associated to a symmetric differential operator and a local boundary
condition are constructed. For a classical Dirac operator with a chiral
boundary condition, we show that there is no tadpole.Comment: 18 pages To appear in J. Funct. Ana
Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems
For a strongly elliptic second-order operator on a bounded domain
it has been known for many years how to interpret
the general closed -realizations of as representing boundary
conditions (generally nonlocal), when the domain and coefficients are smooth.
The purpose of the present paper is to extend this representation to nonsmooth
domains and coefficients, including the case of H\"older
-smoothness, in such a way that pseudodifferential
methods are still available for resolvent constructions and ellipticity
considerations. We show how it can be done for domains with
-smoothness and operators with -coefficients, for
suitable and . In particular, Kre\u\i{}n-type resolvent
formulas are established in such nonsmooth cases. Some unbounded domains are
allowed.Comment: 62 page
Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains
We consider nonlinear diffusive evolution equations posed on bounded space
domains, governed by fractional Laplace-type operators, and involving porous
medium type nonlinearities. We establish existence and uniqueness results in a
suitable class of solutions using the theory of maximal monotone operators on
dual spaces. Then we describe the long-time asymptotics in terms of
separate-variables solutions of the friendly giant type. As a by-product, we
obtain an existence and uniqueness result for semilinear elliptic non local
equations with sub-linear nonlinearities. The Appendix contains a review of the
theory of fractional Sobolev spaces and of the interpolation theory that are
used in the rest of the paper.Comment: Keywords: Fractional Laplace operators, Porous Medium diffusion,
Existence and uniqueness theory, Asymptotic behaviour, Fractional Sobolev
Space
Decompositions of Hilbert Spaces, Stability Analysis and Convergence Probabilities for Discrete-Time Quantum Dynamical Semigroups
We investigate convergence properties of discrete-time semigroup quantum
dynamics, including asymptotic stability, probability and speed of convergence
to pure states and subspaces. These properties are of interest in both the
analysis of uncontrolled evolutions and the engineering of controlled dynamics
for quantum information processing. Our results include two Hilbert space
decompositions that allow for deciding the stability of the subspace of
interest and for estimating of the speed of convergence, as well as a formula
to obtain the limit probability distribution for a set of orthogonal invariant
subspaces.Comment: 14 pages, no figures, to appear in Journal of Physics A, 201
Self-adjoint extensions and unitary operators on the boundary
We establish a bijection between the self-adjoint extensions of the Laplace
operator on a bounded regular domain and the unitary operators on the boundary.
Each unitary encodes a specific relation between the boundary value of the
function and its normal derivative. This bijection sets up a characterization
of all physically admissible dynamics of a nonrelativistic quantum particle
confined in a cavity. More- over, this correspondence is discussed also at the
level of quadratic forms. Finally, the connection between this parametrization
of the extensions and the classical one, in terms of boundary self-adjoint
operators on closed subspaces, is shown.Comment: 16 page
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