9,027 research outputs found
On the critical exponent in an isoperimetric inequality for chords
The problem of maximizing the norms of chords connecting points on a
closed curve separated by arclength arises in electrostatic and
quantum--mechanical problems. It is known that among all closed curves of fixed
length, the unique maximizing shape is the circle for , but this
is not the case for sufficiently large values of . Here we determine the
critical value of above which the circle is not a local maximizer
finding, in particular, that . This corrects a claim
made in \cite{EHL}.Comment: LaTeX, 12 pages, with 1 eps figur
Approximations by graphs and emergence of global structures
We study approximations of billiard systems by lattice graphs. It is
demonstrated that under natural assumptions the graph wavefunctions approximate
solutions of the Schroedinger equation with energy rescaled by the billiard
dimension. As an example, we analyze a Sinai billiard with attached leads. The
results illustrate emergence of global structures in large quantum graphs and
offer interesting comparisons with patterns observed in complex networks of a
different nature.Comment: 6 pages, RevTeX with 5 ps figure
Approximation of a general singular vertex coupling in quantum graphs
The longstanding open problem of approximating all singular vertex couplings
in a quantum graph is solved. We present a construction in which the edges are
decoupled; an each pair of their endpoints is joined by an edge carrying a
potential and a vector potential coupled to the "loose" edges by a
coupling. It is shown that if the lengths of the connecting edges
shrink to zero and the potentials are properly scaled, the limit can yield any
prescribed singular vertex coupling, and moreover, that such an approximation
converges in the norm-resolvent sense.Comment: LaTeX Elsevier format, 36 pages, 1 PDF figur
Inequalities for means of chords, with application to isoperimetric problems
We consider a pair of isoperimetric problems arising in physics. The first
concerns a Schr\"odinger operator in with an attractive
interaction supported on a closed curve , formally given by
; we ask which curve of a given length
maximizes the ground state energy. In the second problem we have a loop-shaped
thread in , homogeneously charged but not conducting,
and we ask about the (renormalized) potential-energy minimizer. Both problems
reduce to purely geometric questions about inequalities for mean values of
chords of . We prove an isoperimetric theorem for -means of chords
of curves when , which implies in particular that the global extrema
for the physical problems are always attained when is a circle. The
article finishes with a discussion of the --means of chords when .Comment: LaTeX2e, 11 page
A Klein Gordon Particle Captured by Embedded Curves
In the present work, a Klein Gordon particle with singular interactions
supported on embedded curves on Riemannian manifolds is discussed from a more
direct and physical perspective, via the heat kernel approach. It is shown that
the renormalized problem is well-defined, and the ground state energy is unique
and finite. The renormalization group invariance of the model is discussed, and
it is observed that the model is asymptotically free.Comment: Published version, 13 pages, no figures. arXiv admin note:
substantial text overlap with arXiv:1202.356
An isoperimetric problem for point interactions
We consider Hamiltonian with point interactions in all
with the same coupling constant, placed at vertices of an equilateral polygon
\PP_N. It is shown that the ground state energy is locally maximized by a
regular polygon. The question whether the maximum is global is reduced to an
interesting geometric problem.Comment: LaTeX 2e, 10 page
Scattering by local deformations of a straight leaky wire
We consider a model of a leaky quantum wire with the Hamiltonian in , where is a compact
deformation of a straight line. The existence of wave operators is proven and
the S-matrix is found for the negative part of the spectrum. Moreover, we
conjecture that the scattering at negative energies becomes asymptotically
purely one-dimensional, being determined by the local geometry in the leading
order, if is a smooth curve and .Comment: Latex2e, 15 page
A remark on helical waveguides
Motivated by a proposal to create an optical helix-shaped waveguides for cold
atoms and molecules, we discuss local perturbations which can create bound
states in such a setting. This is known about a local slowdown of the twist; we
show that a similar effect can result from a local tube protrusion or a change
of the helix radius in correlation with its pitch angle.Comment: LaTeX, 12 page
An isoperimetric problem for leaky loops and related mean-chord inequalities
We consider a class of Hamiltonians in with attractive
interaction supported by piecewise smooth loops of a fixed
length , formally given by with .
It is shown that the ground state of this operator is locally maximized by a
circular . We also conjecture that this property holds globally and
show that the problem is related to an interesting family of geometric
inequalities concerning mean values of chords of .Comment: LaTeX, 16 page
Schroedinger operators with singular interactions: a model of tunneling resonances
We discuss a generalized Schr\"odinger operator in , with an attractive singular interaction supported by a
-dimensional hyperplane and a finite family of points. It can be
regarded as a model of a leaky quantum wire and a family of quantum dots if
, or surface waves in presence of a finite number of impurities if .
We analyze the discrete spectrum, and furthermore, we show that the resonance
problem in this setting can be explicitly solved; by Birman-Schwinger method it
is cast into a form similar to the Friedrichs model.Comment: LaTeX2e, 34 page
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