623 research outputs found
On perturbations of Dirac operators with variable magnetic field of constant direction
We carry out the spectral analysis of matrix valued perturbations of
3-dimensional Dirac operators with variable magnetic field of constant
direction. Under suitable assumptions on the magnetic field and on the
pertubations, we obtain a limiting absorption principle, we prove the absence
of singular continuous spectrum in certain intervals and state properties of
the point spectrum. Various situations, for example when the magnetic field is
constant, periodic or diverging at infinity, are covered. The importance of an
internal-type operator (a 2-dimensional Dirac operator) is also revealed in our
study. The proofs rely on commutator methods.Comment: 12 page
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Model risk â daring to open up the black box
With the increasing use of complex quantitative models in applications throughout the financial world, model risk has become a major concern. Such risk is generated by the potential inaccuracy and inappropriate use of models in business applications, which can lead to substantial financial losses and reputational damage. In this paper we deal with the management and measurement of model risk.
First, a model risk framework is developed, adapting concepts such as risk appetite, monitoring, and mitigation to the particular case of model risk. The usefulness of such a framework for preventing losses associated with model risk is demonstrated through case studies. Second, we investigate the ways in which different ways of using and perceiving models within an organisation both lead to different model risks. We identify four distinct model cultures and argue that in conditions of deep model uncertainty, each of those cultures makes a valuable contribution to model risk governance. Thus the space of legitimate challenges to models is expanded, such that, in addition to a technical critique, operational and commercial concerns are also addressed. Third, we discuss through the examples of proxy modelling, longevity risk and investment advice, common methods and challenges for quantifying model risk. Difficulties arise in mapping model errors to actual financial impact. In the case of irreducible model uncertainty, it is necessary to employ a variety of measurement approaches, based on statistical inference, fitting multiple models, and stress and scenario analysis
Quantum Simulation of Spin Chains Coupled to Bosonic Modes with Superconducting Circuits
We propose the implementation of a digital quantum simulation of spin chains
coupled to bosonic field modes in superconducting circuits. Gates with high
fidelities allows one to simulate a variety of Ising magnetic pairing
interactions with transverse field, Tavis-Cummings interaction between spins
and a bosonic mode, and a spin model with three-body terms. We analyze the
feasibility of the implementation in realistic circuit quantum electrodynamics
setups, where the interactions are either realized via capacitive couplings or
mediated by microwave resonators.Comment: Chapter in R. S. Anderssen et al. (eds.), Mathematics for Industry 11
(Springer Japan, 2015
Essential spectra of difference operators on \sZ^n-periodic graphs
Let (\cX, \rho) be a discrete metric space. We suppose that the group
\sZ^n acts freely on and that the number of orbits of with respect to
this action is finite. Then we call a \sZ^n-periodic discrete metric
space. We examine the Fredholm property and essential spectra of band-dominated
operators on where is a \sZ^n-periodic discrete metric space.
Our approach is based on the theory of band-dominated operators on \sZ^n and
their limit operators.
In case is the set of vertices of a combinatorial graph, the graph
structure defines a Schr\"{o}dinger operator on in a natural way. We
illustrate our approach by determining the essential spectra of Schr\"{o}dinger
operators with slowly oscillating potential both on zig-zag and on hexagonal
graphs, the latter being related to nano-structures
Measuring multipartite entanglement via dynamic susceptibilities
Entanglement plays a central role in our understanding of quantum many body
physics, and is fundamental in characterising quantum phases and quantum phase
transitions. Developing protocols to detect and quantify entanglement of
many-particle quantum states is thus a key challenge for present experiments.
Here, we show that the quantum Fisher information, representing a witness for
genuinely multipartite entanglement, becomes measurable for thermal ensembles
via the dynamic susceptibility, i.e., with resources readily available in
present cold atomic gas and condensed-matter experiments. This moreover
establishes a fundamental connection between multipartite entanglement and
many-body correlations contained in response functions, with profound
implications close to quantum phase transitions. There, the quantum Fisher
information becomes universal, allowing us to identify strongly entangled phase
transitions with a divergent multipartiteness of entanglement. We illustrate
our framework using paradigmatic quantum Ising models, and point out potential
signatures in optical-lattice experiments.Comment: 5+5 pages, 3+2 figure
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