532 research outputs found
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
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
On Cayley Identity for Self-Adjoint Operators in Hilbert Spaces
We prove an analogue to the Cayley identity for an arbitrary self-adjoint
operator in a Hilbert space. We also provide two new ways to characterize
vectors belonging to the singular spectral subspace in terms of the analytic
properties of the resolvent of the operator, computed on these vectors. The
latter are analogous to those used routinely in the scattering theory for the
absolutely continuous subspace
Scattering theory for a class of non-selfadjoint extensions of symmetric operators
This work deals with the functional model for a class of extensions of
symmetric operators and its applications to the theory of wave scattering. In
terms of Boris Pavlov's spectral form of this model, we find explicit formulae
for the action of the unitary group of exponentials corresponding to almost
solvable extensions of a given closed symmetric operator with equal deficiency
indices. On the basis of these formulae, we are able to construct wave
operators and derive a new representation for the scattering matrix for pairs
of such extensions in both self-adjoint and non-self-adjoint situations.Comment: 32 pages; This is the continuation of arXiv:1703.06220 (and formerly
contained in v1); this version is as accepted by the journal (Operator
Theory: Advances and Applications
Phase transition in a periodic tubular structure
We consider an ε-periodic (ε→0) tubular structure, modelled as a magnetic Laplacian on a metric graph, which is periodic along a single axis. We show that the corresponding Hamiltonian admits norm-resolvent convergence to an ODE on R which is fourth order at a discrete set of values of the magnetic potential (\emph{critical points}) and second-order generically. In a vicinity of critical points we establish a mixed-order asymptotics. The rate of convergence is also estimated. This represents a physically viable model of a phase transition as the strength of the (constant) magnetic field increases
Mathematical Heritage of Sergey Naboko: Functional Models of Non-Self-Adjoint Operators
This is an overview of mathematical heritage of Sergey Naboko in the area of
functional models of non-self-adjoint operators. It covers the works by Sergey
in model construction, the analysis of absolutely continuous and singular
spectra and the construction of the scattering theory in model terms.Comment: 14 pages. arXiv admin note: text overlap with arXiv:2204.0119
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