100 research outputs found
A range description for the planar circular Radon transform
The transform considered in the paper integrates a function supported in the
unit disk on the plane over all circles centered at the boundary of this disk.
Such circular Radon transform arises in several contemporary imaging
techniques, as well as in other applications. As it is common for transforms of
Radon type, its range has infinite co-dimension in standard function spaces.
Range descriptions for such transforms are known to be very important for
computed tomography, for instance when dealing with incomplete data, error
correction, and other issues. A complete range description for the circular
Radon transform is obtained. Range conditions include the recently found set of
moment type conditions, which happens to be incomplete, as well as the rest of
conditions that have less standard form. In order to explain the procedure
better, a similar (non-standard) treatment of the range conditions is described
first for the usual Radon transform on the plane.Comment: submitted for publicatio
On Parseval frames of exponentially decaying composite Wannier functions
Let be a periodic self-adjoint linear elliptic operator in with
coefficients periodic with respect to a lattice \G, e.g. Schr\"{o}dinger
operator with periodic magnetic and
electric potentials , or a Maxwell operator in a periodic medium. Let also be a finite part of
its spectrum separated by gaps from the rest of the spectrum. We address here
the question of existence of a finite set of exponentially decaying Wannier
functions such that their \G-shifts w_{j,\g}(x)=w_j(x-\g) for
\g\in\G span the whole spectral subspace corresponding to . It was shown
by D.~Thouless in 1984 that a topological obstruction sometimes exists to
finding exponentially decaying w_{j,\g} that form an orthonormal (or any)
basis of the spectral subspace. This obstruction has the form of non-triviality
of certain finite dimensional (with the dimension equal to the number of
spectral bands in ) analytic vector bundle (Bloch bundle). It was shown in
2009 by one of the authors that it is always possible to find a finite number
of exponentially decaying Wannier functions such that their
\G-shifts form a tight (Parseval) frame in the spectral subspace. This
appears to be the best one can do when the topological obstruction is present.
Here we significantly improve the estimate on the number of extra Wannier
functions needed, showing that in physical dimensions the number can be
chosen equal to , i.e. only one extra family of Wannier functions is
required. This is the lowest number possible in the presence of the topological
obstacle. The result for dimension four is also stated (without a proof), in
which case functions are needed.
The main result of the paper was announced without a proof in Bull. AMS, July
2016.Comment: Submitted. arXiv admin note: text overlap with arXiv:0807.134
Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths
We study the dependence of the quantum graph Hamiltonian, its resolvent, and
its spectrum on the vertex conditions and graph edge lengths. In particular,
several results on the interlacing (bracketing) of the spectra of graphs with
different vertex conditions are obtained and their applications are discussed.Comment: 19 pages, 1 figur
Integral representations and Liouville theorems for solutions of periodic elliptic equations
The paper contains integral representations for certain classes of
exponentially growing solutions of second order periodic elliptic equations.
These representations are the analogs of those previously obtained by S. Agmon,
S. Helgason, and other authors for solutions of the Helmholtz equation. When
one restricts the class of solutions further, requiring their growth to be
polynomial, one arrives to Liouville type theorems, which describe the
structure and dimension of the spaces of such solutions. The Liouville type
theorems previously proved by M. Avellaneda and F.-H. Lin, and J. Moser and M.
Struwe for periodic second order elliptic equations in divergence form are
significantly extended. Relations of these theorems with the analytic structure
of the Fermi and Bloch surfaces are explained.Comment: 48 page
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