135 research outputs found
The Generalized Star Product and the Factorization of Scattering Matrices on Graphs
In this article we continue our analysis of Schr\"odinger operators on
arbitrary graphs given as certain Laplace operators. In the present paper we
give the proof of the composition rule for the scattering matrices. This
composition rule gives the scattering matrix of a graph as a generalized star
product of the scattering matrices corresponding to its subgraphs. We perform a
detailed analysis of the generalized star product for arbitrary unitary
matrices. The relation to the theory of transfer matrices is also discussed
Global Bounds for the Lyapunov Exponent and the Integrated Density of States of Random Schr\"odinger Operators in One Dimension
In this article we prove an upper bound for the Lyapunov exponent
and a two-sided bound for the integrated density of states at an
arbitrary energy of random Schr\"odinger operators in one dimension.
These Schr\"odinger operators are given by potentials of identical shape
centered at every lattice site but with non-overlapping supports and with
randomly varying coupling constants. Both types of bounds only involve
scattering data for the single-site potential. They show in particular that
both and decay at infinity at least like
. As an example we consider the random Kronig-Penney model.Comment: 9 page
Boundary condition at the junction
The quantum graph plays the role of a solvable model for a two-dimensional
network. Here fitting parameters of the quantum graph for modelling the
junction is discussed, using previous results of the second author.Comment: Replaces unpublished draft on related researc
Sharpening the norm bound in the subspace perturbation theory
Let A be a self-adjoint operator on a Hilbert space H. Assume that {\sigma}
is an isolated component of the spectrum of A, i.e. dist({\sigma},{\Sigma})=d>0
where {\Sigma}=spec(A)\{\sigma}. Suppose that V is a bounded self-adjoint
operator on H such that ||V||<d/2 and let L=A+V. Denote by P the spectral
projection of A associated with the spectral set {\sigma} and let Q be the
spectral projection of L corresponding to the closed ||V||-neighborhood of
{\sigma}. We prove a bound of the form arcsin(||P-Q||)\leq M(||V||/d), M:
[0,1/2)-->R^+, that is essentially stronger than the previously known estimates
for ||P-Q||. In particular, the bound obtained ensures that ||P-Q||<1 and,
thus, that the spectral subspaces Ran(P) and Ran(Q) are in the acute-angle case
whenever ||V||<cd with c=0.454169... (the precise expression for c is also
given). Our proof of the above results is based on using the triangle
inequality for the maximal angle between subspaces and on employing the a
priori generic \sin2\theta estimate for the variation of a spectral subspace.
As an example, the boundedly perturbed quantum harmonic oscillator is
discussed.Comment: Some typos fixed; minor changes in the text; a new reference adde
Физико-химическая характеристика и кривые разгонки газового конденсата Мыльджинского месторождения
The industrial application of laser welding implies a reliable and efficient production process. In order to facilitate this, modeling and simulation is used to reveal the crucial points of the welding process. The welding process is described by transport phenomena for mass, momentum and energy. The three involved phases (solid, liquid, gaseous) interact over the free moving phase boundaries. The present boundary layer characters allow to reduce the dimension of these Free Boundary Problems. However, the boundary layer character of the melt flow is lost close to the stagnation point. This can be understood as perturbation of the asymptotic solution in the boundary layer. This perturbation is identified to be singular. The description of the flow near the stagnation point leads to a modified Hiemenz problem which can be solved analytically. The resulting welding model reproduces the time-dependent spatially 3d-distributed welding process. Technical relevant prediction of seam width and depth are compared to experimental results. The reproduction of thermal monitoring signals from the interaction zone between laser and material will be approached
The Berry-Keating operator on L^2(\rz_>, x) and on compact quantum graphs with general self-adjoint realizations
The Berry-Keating operator H_{\mathrm{BK}}:=
-\ui\hbar(x\frac{
\phantom{x}}{
x}+{1/2}) [M. V. Berry and J. P. Keating,
SIAM Rev. 41 (1999) 236] governing the Schr\"odinger dynamics is discussed in
the Hilbert space L^2(\rz_>,
x) and on compact quantum graphs. It is
proved that the spectrum of defined on L^2(\rz_>,
x) is
purely continuous and thus this quantization of cannot yield
the hypothetical Hilbert-Polya operator possessing as eigenvalues the
nontrivial zeros of the Riemann zeta function. A complete classification of all
self-adjoint extensions of acting on compact quantum graphs
is given together with the corresponding secular equation in form of a
determinant whose zeros determine the discrete spectrum of .
In addition, an exact trace formula and the Weyl asymptotics of the eigenvalue
counting function are derived. Furthermore, we introduce the "squared"
Berry-Keating operator which is a special case of the
Black-Scholes operator used in financial theory of option pricing. Again, all
self-adjoint extensions, the corresponding secular equation, the trace formula
and the Weyl asymptotics are derived for on compact quantum
graphs. While the spectra of both and on
any compact quantum graph are discrete, their Weyl asymptotics demonstrate that
neither nor can yield as eigenvalues the
nontrivial Riemann zeros. Some simple examples are worked out in detail.Comment: 33p
Lagrangian structures in time-periodic vortical flows
The Lagrangian trajectories of fluid particles are experimentally studied in an oscillating four-vortex velocity field. The oscillations occur due to a loss of stability of a steady flow and result in a regular reclosure of streamlines between the vortices of the same sign. The Eulerian velocity field is visualized by tracer displacements over a short time period. The obtained data on tracer motions during a number of oscillation periods show that the Lagrangian trajectories form quasi-regular structures. The destruction of these structures is determined by two characteristic time scales: the tracers are redistributed sufficiently fast between the vortices of the same sign and much more slowly transported into the vortices of opposite sign. The observed behavior of the Lagrangian trajectories is quantitatively reproduced in a new numerical experiment with two-dimensional model of the velocity field with a small number of spatial harmonics. A qualitative interpretation of phenomena observed on the basis of the theory of adiabatic chaos in the Hamiltonian systems is given. <br><br> The Lagrangian trajectories are numerically simulated under varying flow parameters. It is shown that the spatial-temporal characteristics of the Lagrangian structures depend on the properties of temporal change in the streamlines topology and on the adiabatic parameter corresponding to the flow. The condition for the occurrence of traps (the regions where the Lagrangian particles reside for a long time) is obtained
Zeta functions of quantum graphs
In this article we construct zeta functions of quantum graphs using a contour
integral technique based on the argument principle. We start by considering the
special case of the star graph with Neumann matching conditions at the center
of the star. We then extend the technique to allow any matching conditions at
the center for which the Laplace operator is self-adjoint and finally obtain an
expression for the zeta function of any graph with general vertex matching
conditions. In the process it is convenient to work with new forms for the
secular equation of a quantum graph that extend the well known secular equation
of the Neumann star graph. In the second half of the article we apply the zeta
function to obtain new results for the spectral determinant, vacuum energy and
heat kernel coefficients of quantum graphs. These have all been topics of
current research in their own right and in each case this unified approach
significantly expands results in the literature.Comment: 32 pages, typos corrected, references adde
Kirchhoff's Rule for Quantum Wires
In this article we formulate and discuss one particle quantum scattering
theory on an arbitrary finite graph with open ends and where we define the
Hamiltonian to be (minus) the Laplace operator with general boundary conditions
at the vertices. This results in a scattering theory with channels. The
corresponding on-shell S-matrix formed by the reflection and transmission
amplitudes for incoming plane waves of energy is explicitly given in
terms of the boundary conditions and the lengths of the internal lines. It is
shown to be unitary, which may be viewed as the quantum version of Kirchhoff's
law. We exhibit covariance and symmetry properties. It is symmetric if the
boundary conditions are real. Also there is a duality transformation on the set
of boundary conditions and the lengths of the internal lines such that the low
energy behaviour of one theory gives the high energy behaviour of the
transformed theory. Finally we provide a composition rule by which the on-shell
S-matrix of a graph is factorizable in terms of the S-matrices of its
subgraphs. All proofs only use known facts from the theory of self-adjoint
extensions, standard linear algebra, complex function theory and elementary
arguments from the theory of Hermitean symplectic forms.Comment: 40 page
Grassmann-Gaussian integrals and generalized star products
In quantum scattering on networks there is a non-linear composition rule for
on-shell scattering matrices which serves as a replacement for the
multiplicative rule of transfer matrices valid in other physical contexts. In
this article, we show how this composition rule is obtained using Berezin
integration theory with Grassmann variables.Comment: 14 pages, 2 figures. In memory of Al.B. Zamolodichiko
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