2,217 research outputs found

### Inequalities for means of chords, with application to isoperimetric problems

We consider a pair of isoperimetric problems arising in physics. The first
concerns a Schr\"odinger operator in $L^2(\mathbb{R}^2)$ with an attractive
interaction supported on a closed curve $\Gamma$, formally given by
$-\Delta-\alpha \delta(x-\Gamma)$; we ask which curve of a given length
maximizes the ground state energy. In the second problem we have a loop-shaped
thread $\Gamma$ in $\mathbb{R}^3$, homogeneously charged but not conducting,
and we ask about the (renormalized) potential-energy minimizer. Both problems
reduce to purely geometric questions about inequalities for mean values of
chords of $\Gamma$. We prove an isoperimetric theorem for $p$-means of chords
of curves when $p \leq 2$, which implies in particular that the global extrema
for the physical problems are always attained when $\Gamma$ is a circle. The
article finishes with a discussion of the $p$--means of chords when $p > 2$.Comment: LaTeX2e, 11 page

### An isoperimetric problem for point interactions

We consider Hamiltonian with $N$ point interactions in $\R^d, d=2,3,$ all
with the same coupling constant, placed at vertices of an equilateral polygon
\PP_N. It is shown that the ground state energy is locally maximized by a
regular polygon. The question whether the maximum is global is reduced to an
interesting geometric problem.Comment: LaTeX 2e, 10 page

### Scattering by local deformations of a straight leaky wire

We consider a model of a leaky quantum wire with the Hamiltonian $-\Delta
-\alpha \delta(x-\Gamma)$ in $L^2(\R^2)$, where $\Gamma$ 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 $\Gamma$ is a smooth curve and $\alpha \to\infty$.Comment: Latex2e, 15 page

### Schroedinger operators with singular interactions: a model of tunneling resonances

We discuss a generalized Schr\"odinger operator in $L^2(\mathbb{R}^d),
d=2,3$, with an attractive singular interaction supported by a
$(d-1)$-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
$d=2$, or surface waves in presence of a finite number of impurities if $d=3$.
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

### Resonances from perturbations of quantum graphs with rationally related edges

We discuss quantum graphs consisting of a compact part and semiinfinite
leads. Such a system may have embedded eigenvalues if some edge lengths in the
compact part are rationally related. If such a relation is perturbed these
eigenvalues may turn into resonances; we analyze this effect both generally and
in simple examples.Comment: LaTeX source file with 10 pdf figures, 24 pages; a replaced version
with minor improvements, to appear in J. Phys. A: Math. Theo

### 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

### A lower bound to the spectral threshold in curved tubes

We consider the Laplacian in curved tubes of arbitrary cross-section rotating
together with the Frenet frame along curves in Euclidean spaces of arbitrary
dimension, subject to Dirichlet boundary conditions on the cylindrical surface
and Neumann conditions at the ends of the tube. We prove that the spectral
threshold of the Laplacian is estimated from below by the lowest eigenvalue of
the Dirichlet Laplacian in a torus determined by the geometry of the tube.Comment: LaTeX, 13 pages; to appear in R. Soc. Lond. Proc. Ser. A Math. Phys.
Eng. Sc

### An isoperimetric problem for leaky loops and related mean-chord inequalities

We consider a class of Hamiltonians in $L^2(\R^2)$ with attractive
interaction supported by piecewise $C^2$ smooth loops $\Gamma$ of a fixed
length $L$, formally given by $-\Delta-\alpha\delta(x-\Gamma)$ with $\alpha>0$.
It is shown that the ground state of this operator is locally maximized by a
circular $\Gamma$. 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 $\Gamma$.Comment: LaTeX, 16 page

### Quantum waveguides with a lateral semitransparent barrier: spectral and scattering properties

We consider a quantum particle in a waveguide which consists of an infinite
straight Dirichlet strip divided by a thin semitransparent barrier on a line
parallel to the walls which is modeled by a $\delta$ potential. We show that if
the coupling strength of the latter is modified locally, i.e. it reaches the
same asymptotic value in both directions along the line, there is always a
bound state below the bottom of the essential spectrum provided the effective
coupling function is attractive in the mean. The eigenvalues and
eigenfunctions, as well as the scattering matrix for energies above the
threshold, are found numerically by the mode-matching technique. In particular,
we discuss the rate at which the ground-state energy emerges from the continuum
and properties of the nodal lines. Finally, we investigate a system with a
modified geometry: an infinite cylindrical surface threaded by a homogeneous
magnetic field parallel to the cylinder axis. The motion on the cylinder is
again constrained by a semitransparent barrier imposed on a ``seam'' parallel
to the axis.Comment: a LaTeX source file with 12 figures (11 of them eps); to appear in J.
Phys. A: Math. Gen. Figures 3, 5, 8, 9, 11 are given at 300 dpi; higher
resolution originals are available from the author

### Non-Weyl asymptotics for quantum graphs with general coupling conditions

Inspired by a recent result of Davies and Pushnitski, we study resonance
asymptotics of quantum graphs with general coupling conditions at the vertices.
We derive a criterion for the asymptotics to be of a non-Weyl character. We
show that for balanced vertices with permutation-invariant couplings the
asymptotics is non-Weyl only in case of Kirchhoff or anti-Kirchhoff conditions,
while for graphs without permutation numerous examples of non-Weyl behaviour
can be constructed. Furthermore, we present an insight helping to understand
what makes the Kirchhoff/anti-Kirchhoff coupling particular from the resonance
point of view. Finally, we demonstrate a generalization to quantum graphs with
nonequal edge weights.Comment: minor changes, to appear in Pierre Duclos memorial issue of J. Phys.
A: Math. Theo

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