2,866 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
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
Interlaced dense point and absolutely continuous spectra for Hamiltonians with concentric-shell singular interactions
We analyze the spectrum of the generalized Schrodinger operator in
, with a general local, rotationally invariant singular
interaction supported by an infinite family of concentric, equidistantly spaced
spheres. It is shown that the essential spectrum consists of interlaced
segments of the dense point and absolutely continuous character, and that the
relation of their lengths at high energies depends on the choice of the
interaction parameters; generically the p.p. component is asymptotically
dominant. We also show that for there is an infinite family of
eigenvalues below the lowest band.Comment: LaTeX, 18 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
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