1,865 research outputs found
Bound states in point-interaction star-graphs
We discuss the discrete spectrum of the Hamiltonian describing a
two-dimensional quantum particle interacting with an infinite family of point
interactions. We suppose that the latter are arranged into a star-shaped graph
with N arms and a fixed spacing between the interaction sites. We prove that
the essential spectrum of this system is the same as that of the infinite
straight "polymer", but in addition there are isolated eigenvalues unless N=2
and the graph is a straight line. We also show that the system has many
strongly bound states if at least one of the angles between the star arms is
small enough. Examples of eigenfunctions and eigenvalues are computed
numerically.Comment: 17 pages, LaTeX 2e with 9 eps figure
Magnetic layers with periodic point perturbations
We study spectral properties of a spinless quantum particle confined to an
infinite planar layer with hard walls which interacts with a periodic lattice
of point perturbations and a homogeneous magnetic field perpendicular to the
layer. It is supposed that the lattice cell contains a finite number of
impurities and the flux through the cell is rational. Using the Landau-Zak
transformation, we convert the problem into investigation of the corresponding
fiber operators which is performed by means of Krein's formula. This yields an
explicit description of the spectral bands which may be absolutely continuous
or degenerate, depending on the parameters of the model.Comment: LaTeX 2e, 30 pages; with minor revisions, to appear in Rep. Math.
Phy
Quantum phases of supersymmetric lattice models
We review recent results on lattice models for spin-less fermions with strong
repulsive interactions. A judicious tuning of kinetic and interaction terms
leads to a model possessing supersymmetry. In the 1D case, this model displays
critical behavior described by superconformal field theory. On 2D lattices we
generically find superfrustration, characterized by an extensive ground state
entropy. For certain 2D lattices analytical results on the ground state
structure reveal yet another quantum phase, which we tentatively call
'supertopological'.Comment: 5 pages, 1 figure, 1 table, contribution to the proceedings of the
XVI International Congress on Mathematical Physics (2009) in Prague, Czeck
Republi
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
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
Quantum mechanics of layers with a finite number of point perturbations
We study spectral and scattering properties of a spinless quantum particle
confined to an infinite planar layer with hard walls containing a finite number
of point perturbations. A solvable character of the model follows from the
explicit form of the Hamiltonian resolvent obtained by means of Krein's
formula. We prove the existence of bound states, demonstrate their properties,
and find the on-shell scattering operator. Furthermore, we analyze the
situation when the system is put into a homogeneous magnetic field
perpendicular to the layer; in that case the point interactions generate
eigenvalues of a finite multiplicity in the gaps of the free Hamiltonian
essential spectrum.Comment: LateX 2e, 48 pages, with 3 ps and 3 eps 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
On the discrete spectrum of Robin Laplacians in conical domains
We discuss several geometric conditions guaranteeing the finiteness or the
infiniteness of the discrete spectrum for Robin Laplacians on conical domains.Comment: 12 page
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