168 research outputs found
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
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
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
Distribution theory for Schr\"odinger's integral equation
Much of the literature on point interactions in quantum mechanics has focused
on the differential form of Schr\"odinger's equation. This paper, in contrast,
investigates the integral form of Schr\"odinger's equation. While both forms
are known to be equivalent for smooth potentials, this is not true for
distributional potentials. Here, we assume that the potential is given by a
distribution defined on the space of discontinuous test functions.
First, by using Schr\"odinger's integral equation, we confirm a seminal
result by Kurasov, which was originally obtained in the context of
Schr\"odinger's differential equation. This hints at a possible deeper
connection between both forms of the equation. We also sketch a generalisation
of Kurasov's result to hypersurfaces.
Second, we derive a new closed-form solution to Schr\"odinger's integral
equation with a delta prime potential. This potential has attracted
considerable attention, including some controversy. Interestingly, the derived
propagator satisfies boundary conditions that were previously derived using
Schr\"odinger's differential equation.
Third, we derive boundary conditions for `super-singular' potentials given by
higher-order derivatives of the delta potential. These boundary conditions
cannot be incorporated into the normal framework of self-adjoint extensions. We
show that the boundary conditions depend on the energy of the solution, and
that probability is conserved.
This paper thereby confirms several seminal results and derives some new
ones. In sum, it shows that Schr\"odinger's integral equation is viable tool
for studying singular interactions in quantum mechanics.Comment: 23 page
On the discrete spectrum of spin-orbit Hamiltonians with singular interactions
We give a variational proof of the existence of infinitely many bound states
below the continuous spectrum for spin-orbit Hamiltonians (including the Rashba
and Dresselhaus cases) perturbed by measure potentials thus extending the
results of J.Bruening, V.Geyler, K.Pankrashkin: J. Phys. A 40 (2007)
F113--F117.Comment: 10 pages; to appear in Russian Journal of Mathematical Physics
(memorial volume in honor of Vladimir Geyler). Results improved in this
versio
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
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
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
Schrödinger operators with δ and δ′-potentials supported on hypersurfaces
Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity
A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators
Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator
are studied under the assumption that the weight function has one turning
point. An abstract approach to the problem is given via a functional model for
indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues
are obtained. Also, operators with finite singular critical points are
considered.Comment: 38 pages, Proposition 2.2 and its proof corrected, Remarks 2.5, 3.4,
and 3.12 extended, details added in subsections 2.3 and 4.2, section 6
rearranged, typos corrected, references adde
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