1,692 research outputs found
Dirichlet to Neumann Maps for Infinite Quantum Graphs
The Dirichlet problem and Dirichlet to Neumann map are analyzed for elliptic
equations on a large collection of infinite quantum graphs. For a dense set of
continuous functions on the graph boundary, the Dirichlet to Neumann map has
values in the Radon measures on the graph boundary
Quantum graphs with singular two-particle interactions
We construct quantum models of two particles on a compact metric graph with
singular two-particle interactions. The Hamiltonians are self-adjoint
realisations of Laplacians acting on functions defined on pairs of edges in
such a way that the interaction is provided by boundary conditions. In order to
find such Hamiltonians closed and semi-bounded quadratic forms are constructed,
from which the associated self-adjoint operators are extracted. We provide a
general characterisation of such operators and, furthermore, produce certain
classes of examples. We then consider identical particles and project to the
bosonic and fermionic subspaces. Finally, we show that the operators possess
purely discrete spectra and that the eigenvalues are distributed following an
appropriate Weyl asymptotic law
Some remarks on the Krein--von Neumann extension of different Laplacians
We discuss the Krein--von Neumann extensions of three Laplacian-type
operators -- on discrete graphs, quantum graphs, and domains. In passing we
present a class of one-dimensional elliptic operators such that for any infinitely many elements of the class have -dimensional null
space.Comment: 13 page
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