54 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
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
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
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
The Density of States and the Spectral Shift Density of Random Schroedinger Operators
In this article we continue our analysis of Schroedinger operators with a
random potential using scattering theory. In particular the theory of Krein's
spectral shift function leads to an alternative construction of the density of
states in arbitrary dimensions. For arbitrary dimension we show existence of
the spectral shift density, which is defined as the bulk limit of the spectral
shift function per unit interaction volume. This density equals the difference
of the density of states for the free and the interaction theory. This extends
the results previously obtained by the authors in one dimension. Also we
consider the case where the interaction is concentrated near a hyperplane.Comment: 1 figur
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
Well-Posedness and Symmetries of Strongly Coupled Network Equations
We consider a diffusion process on the edges of a finite network and allow
for feedback effects between different, possibly non-adjacent edges. This
generalizes the setting that is common in the literature, where the only
considered interactions take place at the boundary, i. e., in the nodes of the
network. We discuss well-posedness of the associated initial value problem as
well as contractivity and positivity properties of its solutions. Finally, we
discuss qualitative properties that can be formulated in terms of invariance of
linear subspaces of the state space, i. e., of symmetries of the associated
physical system. Applications to a neurobiological model as well as to a system
of linear Schroedinger equations on a quantum graph are discussed.Comment: 25 pages. Corrected typos and minor change
Scattering Theory Approach to Random Schroedinger Operators in One Dimension
Methods from scattering theory are introduced to analyze random Schroedinger
operators in one dimension by applying a volume cutoff to the potential. The
key ingredient is the Lifshitz-Krein spectral shift function, which is related
to the scattering phase by the theorem of Birman and Krein. The spectral shift
density is defined as the "thermodynamic limit" of the spectral shift function
per unit length of the interaction region. This density is shown to be equal to
the difference of the densities of states for the free and the interacting
Hamiltonians. Based on this construction, we give a new proof of the Thouless
formula. We provide a prescription how to obtain the Lyapunov exponent from the
scattering matrix, which suggest a way how to extend this notion to the higher
dimensional case. This prescription also allows a characterization of those
energies which have vanishing Lyapunov exponent.Comment: 1 figur
Integrable Models From Twisted Half Loop Algebras
This paper is devoted to the construction of new integrable quantum
mechanical models based on certain subalgebras of the half loop algebra of
gl(N). Various results about these subalgebras are proven by presenting them in
the notation of the St Petersburg school. These results are then used to
demonstrate the integrability, and find the symmetries, of two types of
physical system: twisted Gaudin magnets, and Calogero-type models of particles
on several half-lines meeting at a point.Comment: 22 pages, 1 figure, Introduction improved, References adde
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