212 research outputs found
Time delay is a common feature of quantum scattering theory
We prove that the existence of time delay defined in terms of sojourn times,
as well as its identity with Eisenbud-Wigner time delay, is a common feature of
two Hilbert space quantum scattering theory. All statements are
model-independent.Comment: 15 page
New expressions for the wave operators of Schroedinger operators in R^3
We prove new and explicit formulas for the wave operators of Schroedinger
operators in R^3. These formulas put into light the very special role played by
the generator of dilations and validate the topological approach of Levinson's
theorem introduced in a previous publication. Our results hold for general (not
spherically symmetric) potentials decaying fast enough at infinity, without any
assumption on the absence of eigenvalue or resonance at 0-energy.Comment: 11 page
A few results on Mourre theory in a two-Hilbert spaces setting
We introduce a natural framework for dealing with Mourre theory in an
abstract two-Hilbert spaces setting. In particular a Mourre estimate for a pair
of self-adjoint operators (H,A) is deduced from a similar estimate for a pair
of self-adjoint operators (H_0,A_0) acting in an auxiliary Hilbert space. A new
criterion for the completeness of the wave operators in a two-Hilbert spaces
setting is also presented.Comment: 13 page
Explicit formulas for the Schroedinger wave operators in R^2
In this note, we derive explicit formulas for the Schroedinger wave operators
in R^2 under the assumption that 0-energy is neither an eigenvalue nor a
resonance. These formulas justify the use of a recently introduced topological
approach of scattering theory to obtain index theorems.Comment: 6 page
Commutator methods for unitary operators
We present an improved version of commutator methods for unitary operators
under a weak regularity condition. Once applied to a unitary operator, the
method typically leads to the absence of singularly continuous spectrum and to
the local finiteness of point spectrum. Large families of locally smooth
operators are also exhibited. Half of the paper is dedicated to applications,
and a special emphasize is put on the study of cocycles over irrational
rotations. It is apparently the first time that commutator methods are applied
in the context of rotation algebras, for the study of their generators.Comment: 15 page
The method of the weakly conjugate operator: Extensions and applications to operators on graphs and groups
In this review we present some recent extensions of the method of the weakly
conjugate operator. We illustrate these developments through examples of
operators on graphs and groups.Comment: 11 page
Spectral analysis for adjacency operators on graphs
We put into evidence graphs with adjacency operator whose singular subspace
is prescribed by the kernel of an auxiliary operator. In particular, for a
family of graphs called admissible, the singular continuous spectrum is absent
and there is at most an eigenvalue located at the origin. Among other examples,
the one-dimensional XY model of solid-state physics is covered. The proofs rely
on commutators methods.Comment: 16 pages, 9 figure
Detecting communities using asymptotical Surprise
Nodes in real-world networks are repeatedly observed to form dense clusters,
often referred to as communities. Methods to detect these groups of nodes
usually maximize an objective function, which implicitly contains the
definition of a community. We here analyze a recently proposed measure called
surprise, which assesses the quality of the partition of a network into
communities. In its current form, the formulation of surprise is rather
difficult to analyze. We here therefore develop an accurate asymptotic
approximation. This allows for the development of an efficient algorithm for
optimizing surprise. Incidentally, this leads to a straightforward extension of
surprise to weighted graphs. Additionally, the approximation makes it possible
to analyze surprise more closely and compare it to other methods, especially
modularity. We show that surprise is (nearly) unaffected by the well known
resolution limit, a particular problem for modularity. However, surprise may
tend to overestimate the number of communities, whereas they may be
underestimated by modularity. In short, surprise works well in the limit of
many small communities, whereas modularity works better in the limit of few
large communities. In this sense, surprise is more discriminative than
modularity, and may find communities where modularity fails to discern any
structure
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