2,429 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
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
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
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 single-mode quantum transport in serial-structure geometric scatterers
We study transport in quantum systems consisting of a finite array of N
identical single-channel scatterers. A general expression of the S matrix in
terms of the individual-element data obtained recently for potential scattering
is rederived in this wider context. It shows in particular how the band
spectrum of the infinite periodic system arises in the limit . We
illustrate the result on two kinds of examples. The first are serial graphs
obtained by chaining loops or T-junctions. A detailed discussion is presented
for a finite-periodic "comb"; we show how the resonance poles can be computed
within the Krein formula approach. Another example concerns geometric
scatterers where the individual element consists of a surface with a pair of
leads; we show that apart of the resonances coming from the decoupled-surface
eigenvalues such scatterers exhibit the high-energy behavior typical for the
delta' interaction for the physically interesting couplings.Comment: 36 pages, a LaTeX source file with 2 TeX drawings, 3 ps and 3 jpeg
figures attache
Non-Weyl asymptotics for quantum graphs with general coupling conditions
Inspired by a recent result of Davies and Pushnitski, we study resonance
asymptotics of quantum graphs with general coupling conditions at the vertices.
We derive a criterion for the asymptotics to be of a non-Weyl character. We
show that for balanced vertices with permutation-invariant couplings the
asymptotics is non-Weyl only in case of Kirchhoff or anti-Kirchhoff conditions,
while for graphs without permutation numerous examples of non-Weyl behaviour
can be constructed. Furthermore, we present an insight helping to understand
what makes the Kirchhoff/anti-Kirchhoff coupling particular from the resonance
point of view. Finally, we demonstrate a generalization to quantum graphs with
nonequal edge weights.Comment: minor changes, to appear in Pierre Duclos memorial issue of J. Phys.
A: Math. Theo
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