28 research outputs found
Trace formulae for Schrodinger operators on metric graphs with applications to recovering matching conditions
The paper is a continuation of the study started in \cite{Yorzh1}.
Schrodinger operators on finite compact metric graphs are considered under the
assumption that the matching conditions at the graph vertices are of
type. Either an infinite series of trace formulae (provided that edge
potentials are infinitely smooth) or a finite number of such formulae (in the
cases of and edge potentials) are obtained which link together two
different quantum graphs under the assumption that their spectra coincide.
Applications are given to the problem of recovering matching conditions for a
quantum graph based on its spectrum.Comment: arXiv admin note: substantial text overlap with arXiv:1403.761
Trace formulae for graph Laplacians with applications to recovering matching conditions
Graph Laplacians on finite compact metric graphs are considered under the
assumption that the matching conditions at the graph vertices are of either
or type. In either case, an infinite series of trace
formulae which link together two different graph Laplacians provided that their
spectra coincide is derived. Applications are given to the problem of
reconstructing matching conditions for a graph Laplacian based on its spectrum
Time-dispersive behavior as a feature of critical-contrast media
Motivated by the urgent need to attribute a rigorous mathematical meaning to
the term "metamaterial", we propose a novel approach to the homogenisation of
critical-contrast composites. This is based on the asymptotic analysis of the
Dirichlet-to-Neumann map on the interface between different components ("stiff"
and "soft") of the medium, which leads to an asymptotic approximation of
eigenmodes. This allows us to see that the presence of the soft component makes
the stiff one behave as a class of time-dispersive media. By an inversion of
this argument, we also offer a recipe for the construction of such media with
prescribed dispersive properties from periodic composites.Comment: 24 pages, 6 fugure
Asymptotic analysis of operator families and applications to resonant media
We give an overview of operator-theoretic tools that have recently proved
useful in the analysis of boundary-value and transmission problems for
second-order partial differential equations, with a view to addressing, in
particular, the asymptotic behaviour of resolvents of physically motivated
parameter-dependent operator families. We demonstrate the links of this rich
area, on the one hand, to functional frameworks developed by S. N. Naboko and
his students, and on the other hand, to concrete applications of current
interest in the physics and engineering communities.Comment: 60 pages, 2 figures; a survey of recent results in the area, see also
arXiv:2010.13318, arXiv:1808.03961, arXiv:1703.06220, arXiv:1510.0336
Unified approach to critical-contrast homogenisation with explicit links to time-dispersive media
A novel approach to critical-contrast homogenisation is proposed.
Norm-resolvent asymptotics are explicitly constructed. An essential feature of
our approach is that it relates homogenisation limits to a class of
time-dispersive media.Comment: 37 pages, 3 figures; as accepted by Transactions of the Moscow
Mathematical Society; continues 1803.09372 and supplements 1808.0396