775 research outputs found
On simultaneous diagonalization via congruence of real symmetric matrices
Simultaneous diagonalization via congruence (SDC) for more than two symmetric
matrices has been a long standing problem. So far, the best attempt either
relies on the existence of a semidefinite matrix pencil or casts on the complex
field. The problem now is resolved without any assumption. We first propose
necessary and sufficient conditions for SDC in case that at least one of the
matrices is nonsingular. Otherwise, we show that the singular matrices can be
decomposed into diagonal blocks such that the SDC of given matrices becomes
equivalently the SDC of the sub-matrices. Most importantly, the sub-matrices
now contain at least one nonsingular matrix. Applications to simplify some
difficult optimization problems with the presence of SDC are mentioned
(2+1)-Dimensional Quantum Gravity as the Continuum Limit of Causal Dynamical Triangulations
We perform a non-perturbative sum over geometries in a (2+1)-dimensional
quantum gravity model given in terms of Causal Dynamical Triangulations.
Inspired by the concept of triangulations of product type introduced
previously, we impose an additional notion of order on the discrete, causal
geometries. This simplifies the combinatorial problem of counting geometries
just enough to enable us to calculate the transfer matrix between boundary
states labelled by the area of the spatial universe, as well as the
corresponding quantum Hamiltonian of the continuum theory. This is the first
time in dimension larger than two that a Hamiltonian has been derived from such
a model by mainly analytical means, and opens the way for a better
understanding of scaling and renormalization issues.Comment: 38 pages, 13 figure
Quantum Einstein Gravity
We give a pedagogical introduction to the basic ideas and concepts of the
Asymptotic Safety program in Quantum Einstein Gravity. Using the continuum
approach based upon the effective average action, we summarize the state of the
art of the field with a particular focus on the evidence supporting the
existence of the non-trivial renormalization group fixed point at the heart of
the construction. As an application, the multifractal structure of the emerging
space-times is discussed in detail. In particular, we compare the continuum
prediction for their spectral dimension with Monte Carlo data from the Causal
Dynamical Triangulation approach.Comment: 87 pages, 13 figures, review article prepared for the New Journal of
Physics focus issue on Quantum Einstein Gravit
Dynamical control of correlated states in a square quantum dot
In the limit of low particle density, electrons confined to a quantum dot
form strongly correlated states termed Wigner molecules, in which the Coulomb
interaction causes the electrons to become highly localized in space. By using
an effective model of Hubbard-type to describe these states, we investigate how
an oscillatory electric field can drive the dynamics of a two-electron Wigner
molecule held in a square quantum dot. We find that, for certain combinations
of frequency and strength of the applied field, the tunneling between various
charge configurations can be strongly quenched, and we relate this phenomenon
to the presence of anti-crossings in the Floquet quasi-energy spectrum. We
further obtain simple analytic expressions for the location of these
anti-crossings, which allows the effective parameters for a given quantum dot
to be directly measured in experiment, and suggests the exciting possibility of
using ac-fields to control the time evolution of entangled states in mesoscopic
devices.Comment: Replaced with version to be published in Phys. Rev.
From General Relativity to Quantum Gravity
In general relativity (GR), spacetime geometry is no longer just a background
arena but a physical and dynamical entity with its own degrees of freedom. We
present an overview of approaches to quantum gravity in which this central
feature of GR is at the forefront. However, the short distance dynamics in the
quantum theory are quite different from those of GR and classical spacetimes
and gravitons emerge only in a suitable limit. Our emphasis is on communicating
the key strategies, the main results and open issues. In the spirit of this
volume, we focus on a few avenues that have led to the most significant
advances over the past 2-3 decades.Comment: To appear in \emph{General Relativity and Gravitation: A Centennial
Survey}, commissioned by the International Society for General Relativity and
Gravitation and to be published by Cambridge University Press. Abhay Ashtekar
served as the `coordinating author' and combined the three contribution
Exceptional Gegenbauer polynomials via isospectral deformation
In this paper, we show how to construct exceptional
orthogonal polynomials (XOP) using isospectral
deformations of classical orthogonal polynomials. The
construction is based on confluent Darboux transformations,
where repeated factorizations at the same
eigenvalue are allowed. These factorizations allow us
to construct Sturm–Liouville problems with polynomial
eigenfunctions that have an arbitrary number of realvalued
parameters. We illustrate this new construction
by exhibiting the class of deformed Gegenbauer polynomials,
which are XOP families that are isospectral
deformations of classical Gegenbauer polynomials.Spanish MINECO through Juan de la Cierva fellowship FJC2019-039681-I, Spanish State Research Agency through BCAM Severo Ochoa excellence accreditation SEV-2017-0718, Basque Government through the BERC Programme 2022-2025, projects PGC2018-096504-B-C33 and RTI2018-100754-B-I00 from FEDER/Ministerio de Ciencia e Innovacion-Agencia Estatal de Investigacion, the European Union under the 2014-2020 ERDF Operational Programme, and the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia (project FEDER-UCA18-108393
Exceptional Gegenbauer polynomials via isospectral deformation
In this paper, we show how to construct exceptional orthogonal polynomials (XOP) using isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, where repeated factorizations at the same eigenvalue are allowed. These factorizations allow us to construct Sturm-Liouville problems with polynomial eigenfunctions that have an arbitrary number of realvalued parameters. We illustrate this new construction by exhibiting the class of deformed Gegenbauer polynomials, which are XOP families that are isospectral deformations of classical Gegenbauer polynomials
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