11 research outputs found
Quantum Graphs II: Some spectral properties of quantum and combinatorial graphs
The paper deals with some spectral properties of (mostly infinite) quantum
and combinatorial graphs. Quantum graphs have been intensively studied lately
due to their numerous applications to mesoscopic physics, nanotechnology,
optics, and other areas.
A Schnol type theorem is proven that allows one to detect that a point
belongs to the spectrum when a generalized eigenfunction with an subexponential
growth integral estimate is available. A theorem on spectral gap opening for
``decorated'' quantum graphs is established (its analog is known for the
combinatorial case). It is also shown that if a periodic combinatorial or
quantum graph has a point spectrum, it is generated by compactly supported
eigenfunctions (``scars'').Comment: 4 eps figures, LATEX file, 21 pages Revised form: a cut-and-paste
blooper fixe
On the skeleton method and an application to a quantum scissor
In the spectral analysis of few one dimensional quantum particles interacting
through delta potentials it is well known that one can recast the problem into
the spectral analysis of an integral operator (the skeleton) living on the
submanifold which supports the delta interactions. We shall present several
tools which allow direct insight into the spectral structure of this skeleton.
We shall illustrate the method on a model of a two dimensional quantum particle
interacting with two infinitely long straight wires which cross one another at
a certain angle : the quantum scissor.Comment: Submitte
On occurrence of spectral edges for periodic operators inside the Brillouin zone
The article discusses the following frequently arising question on the
spectral structure of periodic operators of mathematical physics (e.g.,
Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can
obtain the correct spectrum by using the values of the quasimomentum running
over the boundary of the (reduced) Brillouin zone only, rather than the whole
zone? Or, do the edges of the spectrum occur necessarily at the set of
``corner'' high symmetry points? This is known to be true in 1D, while no
apparent reasons exist for this to be happening in higher dimensions. In many
practical cases, though, this appears to be correct, which sometimes leads to
the claims that this is always true. There seems to be no definite answer in
the literature, and one encounters different opinions about this problem in the
community.
In this paper, starting with simple discrete graph operators, we construct a
variety of convincing multiply-periodic examples showing that the spectral
edges might occur deeply inside the Brillouin zone. On the other hand, it is
also shown that in a ``generic'' case, the situation of spectral edges
appearing at high symmetry points is stable under small perturbations. This
explains to some degree why in many (maybe even most) practical cases the
statement still holds.Comment: 25 pages, 10 EPS figures. Typos corrected and a reference added in
the new versio
Covering graphs, magnetic spectral gaps and applications to polymers and nanoribbons
In this article, we analyze the spectrum of discrete magnetic Laplacians
(DML) on an infinite covering graph with (Abelian) lattice group and periodic magnetic potential
. We give sufficient conditions for the existence of
spectral gaps in the spectrum of the DML and study how these depend on
. The magnetic potential may be interpreted as a control
parameter for the spectral bands and gaps. We apply these results to describe
the spectral band/gap structure of polymers (polyacetylene) and of nanoribbons
in the presence of a constant magnetic field.Comment: 17 pages; 6 figure
Numerical Analysis of Linear and Nonlinear Schrodinger Equations on Quantum Graphs
A wide variety of problems in quantum mechanics can be modeled with Schrodinger-type equations on quantum graphs. More specifically, graphs are useful for simplifying models of physical systems that feature nano-scaled branching structures. Since analytical solutions can only be found for a few trivial cases, it is necessary to consider how to accurately solve this type of problem numerically. While a wide variety of tools exist to solve these types of partial differential equations on lines, this is not well studied in the case of graphs - one critical difference between the two cases being that graphs require more complicated boundary conditions. This paper utilizes a new approach to include the boundary conditions in the discretized operator that preserves high levels of accuracy. Thus, a proper time evolution scheme must work in conjunction with a spatial operator that has incorporated boundary conditions and preserve the accuracy of our spatial component, and this is accomplished by implementing methods from differential algebraic equations. We study the numerical computation of linear and nonlinear states for Schrodinger equations on graphs, as well as numerically compute their linear stability properties. The latter result has implications for the future study of relative periodic orbits on graphs. All numerical components are being adapted into a MATLAB software package called QGLAB jointly with Roy Goodman on Github.Doctor of Philosoph