11 research outputs found
Point interactions in a strip
We study the behavior of a quantum particle confined to a hard--wall strip of
a constant width in which there is a finite number of point
perturbations. Constructing the resolvent of the corresponding Hamiltonian by
means of Krein's formula, we analyze its spectral and scattering properties.
The bound state--problem is analogous to that of point interactions in the
plane: since a two--dimensional point interaction is never repulsive, there are
discrete eigenvalues, , the lowest of which is
nondegenerate. On the other hand, due to the presence of the boundary the point
interactions give rise to infinite series of resonances; if the coupling is
weak they approach the thresholds of higher transverse modes. We derive also
spectral and scattering properties for point perturbations in several related
models: a cylindrical surface, both of a finite and infinite heigth, threaded
by a magnetic flux, and a straight strip which supports a potential independent
of the transverse coordinate. As for strips with an infinite number of point
perturbations, we restrict ourselves to the situation when the latter are
arranged periodically; we show that in distinction to the case of a
point--perturbation array in the plane, the spectrum may exhibit any finite
number of gaps. Finally, we study numerically conductance fluctuations in case
of random point perturbations.Comment: a LaTeX file, 38 pages, to appear in Ann. Phys.; 12 figures available
at request from [email protected]
Bound states and scattering in quantum waveguides coupled laterally through a boundary window
We consider a pair of parallel straight quantum waveguides coupled laterally
through a window of a width in the common boundary. We show that such
a system has at least one bound state for any . We find the
corresponding eigenvalues and eigenfunctions numerically using the
mode--matching method, and discuss their behavior in several situations. We
also discuss the scattering problem in this setup, in particular, the turbulent
behavior of the probability flow associated with resonances. The level and
phase--shift spacing statistics shows that in distinction to closed
pseudo--integrable billiards, the present system is essentially non--chaotic.
Finally, we illustrate time evolution of wave packets in the present model.Comment: LaTeX text file with 12 ps figure
Leaky quantum graphs: approximations by point interaction Hamiltonians
We prove an approximation result showing how operators of the type in , where is a graph,
can be modeled in the strong resolvent sense by point-interaction Hamiltonians
with an appropriate arrangement of the potentials. The result is
illustrated on finding the spectral properties in cases when is a ring
or a star. Furthermore, we use this method to indicate that scattering on an
infinite curve which is locally close to a loop shape or has multiple
bends may exhibit resonances due to quantum tunneling or repeated reflections.Comment: LaTeX 2e, 31 pages with 18 postscript figure
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
Spectra of soft ring graphs
We discuss of a ring-shaped soft quantum wire modeled by interaction
supported by the ring of a generally nonconstant coupling strength. We derive
condition which determines the discrete spectrum of such systems, and analyze
the dependence of eigenvalues and eigenfunctions on the coupling and ring
geometry. In particular, we illustrate that a random component in the coupling
leads to a localization. The discrete spectrum is investigated also in the
situation when the ring is placed into a homogeneous magnetic field or threaded
by an Aharonov-Bohm flux and the system exhibits persistent currents.Comment: LaTeX 2e, 17 pages, with 10 ps figure
Bound states in open coupled asymmetrical waveguides and quantum wires
The behavior of bound states in asymmetric cross, T and L shaped
configurations is considered. Because of the symmetries of the wavefunctions,
the analysis can be reduced to the case of an electron localized at the
intersection of two orthogonal crossed wires of different width. Numerical
calculations show that the fundamental mode of this system remains bound for
the widths that we have been able to study directly; moreover, the
extrapolation of the results obtained for finite widths suggests that this
state remains bound even when the width of one arm becomes infinitesimal. We
provide a qualitative argument which explains this behavior and that can be
generalized to the lowest energy states in each symmetry class. In the case of
odd-odd states of the cross we find that the lowest mode is bounded when the
width of the two arms is the same and stays bound up to a critical value of the
ratio between the widths; in the case of the even-odd states we find that the
lowest mode is unbound up to a critical value of the ratio between the widths.
Our qualitative arguments suggest that the bound state survives as the width of
the vertical arm becomes infinitesimal.Comment: 11 pages, 19 figures, 3 table