3 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]
Band spectra of rectangular graph superlattices
We consider rectangular graph superlattices of sides l1, l2 with the
wavefunction coupling at the junctions either of the delta type, when they are
continuous and the sum of their derivatives is proportional to the common value
at the junction with a coupling constant alpha, or the "delta-prime-S" type
with the roles of functions and derivatives reversed; the latter corresponds to
the situations where the junctions are realized by complicated geometric
scatterers. We show that the band spectra have a hidden fractal structure with
respect to the ratio theta := l1/l2. If the latter is an irrational badly
approximable by rationals, delta lattices have no gaps in the weak-coupling
case. We show that there is a quantization for the asymptotic critical values
of alpha at which new gap series open, and explain it in terms of
number-theoretic properties of theta. We also show how the irregularity is
manifested in terms of Fermi-surface dependence on energy, and possible
localization properties under influence of an external electric field.
KEYWORDS: Schroedinger operators, graphs, band spectra, fractals,
quasiperiodic systems, number-theoretic properties, contact interactions, delta
coupling, delta-prime coupling.Comment: 16 pages, LaTe