687 research outputs found

### Schr\"odinger operator on homogeneous metric trees: spectrum in gaps

The paper studies the spectral properties of the Schr\"odinger operator
$A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying
real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the
free Laplacian $A_0 = -\Delta$ has a band-gap structure with a single
eigenvalue of infinite multiplicity in the middle of each finite gap. The
perturbation $gV$ gives rise to extra eigenvalues in the gaps. These
eigenvalues are monotone functions of $g$ if the potential $V$ has a fixed
sign. Assuming that the latter condition is satisfied and that $V$ is
symmetric, i.e. depends on the distance to the root of the tree, we carry out a
detailed asymptotic analysis of the counting function of the discrete
eigenvalues in the limit $g\to\infty$. Depending on the sign and decay of $V$,
this asymptotics is either of the Weyl type or is completely determined by the
behaviour of $V$ at infinity.Comment: AMS LaTex file, 47 page

### Exact Casimir Interaction Between Semitransparent Spheres and Cylinders

A multiple scattering formulation is used to calculate the force, arising
from fluctuating scalar fields, between distinct bodies described by
$\delta$-function potentials, so-called semitransparent bodies. (In the limit
of strong coupling, a semitransparent boundary becomes a Dirichlet one.) We
obtain expressions for the Casimir energies between disjoint parallel
semitransparent cylinders and between disjoint semitransparent spheres. In the
limit of weak coupling, we derive power series expansions for the energy, which
can be exactly summed, so that explicit, very simple, closed-form expressions
are obtained in both cases. The proximity force theorem holds when the objects
are almost touching, but is subject to large corrections as the bodies are
moved further apart.Comment: 5 pages, 4 eps figures; expanded discussion of previous work and
additional references added, minor typos correcte

### On the continuous spectral component of the Floquet operator for a periodically kicked quantum system

By a straightforward generalisation, we extend the work of Combescure from
rank-1 to rank-N perturbations. The requirement for the Floquet operator to be
pure point is established and compared to that in Combescure. The result
matches that in McCaw. The method here is an alternative to that work. We show
that if the condition for the Floquet operator to be pure point is relaxed,
then in the case of the delta-kicked Harmonic oscillator, a singularly
continuous component of the Floquet operator spectrum exists. We also provide
an in depth discussion of the conjecture presented in Combescure of the case
where the unperturbed Hamiltonian is more general. We link the physics
conjecture directly to a number-theoretic conjecture of Vinogradov and show
that a solution of Vinogradov's conjecture solves the physics conjecture. The
result is extended to the rank-N case. The relationship between our work and
the work of Bourget on the physics conjecture is discussed.Comment: 25 pages, published in Journal of Mathematical Physic

### On the spectrum of the periodic Dirac operator

The absolute continuity of the spectrum for the periodic Dirac operator $\hat D=\sum_{j=1}^n(-i\frac {\partial}{\partial x_j}-A_j)\hat \alpha_j + \hat
V^{(0)}+\hat V^{(1)}, x\in R^n, n\geq 3,$ is proved given that either $A\in
C(R^n;R^n)\cap H^q_{loc}(R^n;R^n)$, 2q > n-2, or the Fourier series of the
vector potential $A:R^n\to R^n$ is absolutely convergent. Here, $\hat
V^{(s)}=(\hat V^{(s)})^*$ are continuous matrix functions and \hat V^{(s)}\hat
\alpha_j=(-1}^s\hat \alpha_j\hat V^{(s)} for all anticommuting Hermitian
matrices $\hat \alpha_j$, $\hat \alpha_j^2=\hat I$, s=0,1.Comment: 17 page

### Cosmic Strings Stabilized by Fermion Fluctuations

We provide a thorough exposition of recent results on the quantum
stabilization of cosmic strings. Stabilization occurs through the coupling to a
heavy fermion doublet in a reduced version of the standard model. The study
combines the vacuum polarization energy of fermion zero-point fluctuations and
the binding energy of occupied energy levels, which are of the same order in a
semi-classical expansion. Populating these bound states assigns a charge to the
string. Strings carrying fermion charge become stable if the Higgs and gauge
fields are coupled to a fermion that is less than twice as heavy as the top
quark. The vacuum remains stable in the model, because neutral strings are not
energetically favored. These findings suggest that extraordinarily large
fermion masses or unrealistic couplings are not required to bind a cosmic
string in the standard model.Comment: Based on talk by HW at QFEXT 11 (Benasque, Spain), 15p, uses
ws-ijmpcs.cls (incl

### Planar waveguide with "twisted" boundary conditions: discrete spectrum

We consider a planar waveguide with combined Dirichlet and Neumann conditions
imposed in a "twisted" way. We study the discrete spectrum and describe it
dependence on the configuration of the boundary conditions. In particular, we
show that in certain cases the model can have discrete eigenvalues emerging
from the threshold of the essential spectrum. We give a criterium for their
existence and construct them as convergent holomorphic series

### Smilansky's model of irreversible quantum graphs, II: the point spectrum

In the model suggested by Smilansky one studies an operator describing the
interaction between a quantum graph and a system of K one-dimensional
oscillators attached at different points of the graph. This paper is a
continuation of our investigation of the case K>1. For the sake of simplicity
we consider K=2, but our argument applies to the general situation. In this
second paper we apply the variational approach to the study of the point
spectrum.Comment: 18 page

### Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction

A general technique for the study of embedded quantum graphs with magnetic
fields and spin-orbit interaction is presented. The analysis is used to
understand the contribution of Rashba constant to the extreme localization
induced by magnetic field in the T3 shaped quantum graph. We show that this
effect is destroyed at generic values of the Rashba constant. On the other
hand, for certain combinations of the Rashba constant and the magnetic
parameters another series of infinitely degenerate eigenvalues appears.Comment: 25 pages, typos corrected, references extende

### A rigorous analysis of high order electromagnetic invisibility cloaks

There is currently a great deal of interest in the invisibility cloaks
recently proposed by Pendry et al. that are based in the transformation
approach. They obtained their results using first order transformations. In
recent papers Hendi et al. and Cai et al. considered invisibility cloaks with
high order transformations. In this paper we study high order electromagnetic
invisibility cloaks in transformation media obtained by high order
transformations from general anisotropic media. We consider the case where
there is a finite number of spherical cloaks located in different points in
space. We prove that for any incident plane wave, at any frequency, the
scattered wave is identically zero. We also consider the scattering of finite
energy wave packets. We prove that the scattering matrix is the identity, i.e.,
that for any incoming wave packet the outgoing wave packet is the same as the
incoming one. This proves that the invisibility cloaks can not be detected in
any scattering experiment with electromagnetic waves in high order
transformation media, and in particular in the first order transformation media
of Pendry et al. We also prove that the high order invisibility cloaks, as well
as the first order ones, cloak passive and active devices. The cloaked objects
completely decouple from the exterior. Actually, the cloaking outside is
independent of what is inside the cloaked objects. The electromagnetic waves
inside the cloaked objects can not leave the concealed regions and viceversa,
the electromagnetic waves outside the cloaked objects can not go inside the
concealed regions. As we prove our results for media that are obtained by
transformation from general anisotropic materials, we prove that it is possible
to cloak objects inside general crystals.Comment: The final version is now published in Journal of Physics A:
Mathematical and Theoretical, vol 41 (2008) 065207 (21 pp). Included in
IOP-Selec

### On absolute continuity of the spectrum of a periodic magnetic Schr\"odinger operator

We consider the Schr\"odinger operator in ${\mathbb R}^n$, $n\geq 3$, with
the electric potential $V$ and the magnetic potential $A$ being periodic
functions (with a common period lattice) and prove absolute continuity of the
spectrum of the operator in question under some conditions which, in
particular, are satisfied if $V\in L^{n/2}_{{\mathrm {loc}}}({\mathbb R}^n)$
and $A\in H^q_{{\mathrm {loc}}}({\mathbb R}^n;{\mathbb R}^n)$, $q>(n-1)/2$.Comment: 25 page

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