127 research outputs found
Deformation Quantization: Twenty Years After
We first review the historical developments, both in physics and in
mathematics, that preceded (and in some sense provided the background of)
deformation quantization. Then we describe the birth of the latter theory and
its evolution in the past twenty years, insisting on the main conceptual
developments and keeping here as much as possible on the physical side. For the
physical part the accent is put on its relations to, and relevance for,
"conventional" physics. For the mathematical part we concentrate on the
questions of existence and equivalence, including most recent developments for
general Poisson manifolds; we touch also noncommutative geometry and index
theorems, and relations with group theory, including quantum groups. An
extensive (though very incomplete) bibliography is appended and includes
background mathematical literature.Comment: 39 pages; to be published with AIP Press in Proceedings of the 1998
Lodz conference "Particles, Fields and Gravitation". LaTeX (compatibility
mode) with aipproc styl
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods
Challenges in Optimal Control of Nonlinear PDE-Systems
The workshop focussed on various aspects of optimal control problems for systems of nonlinear partial differential equations. In particular, discussions around keynote presentations in the areas of optimal control of nonlinear/non-smooth systems, optimal control of systems involving nonlocal operators, shape and topology optimization, feedback control and stabilization, sparse control, and associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, also aspects of control of fluid structure interaction problems as well as problems arising in the optimal control of quantum systems were considered
Recommended from our members
Isospectral algorithms, Toeplitz matrices and orthogonal polynomials
An isospectral algorithm is one which manipulates a matrix without changing its spectrum. In this thesis we study three interrelated examples of isospectral algorithms, all pertaining to Toeplitz matrices in some fashion, and one directly involving orthogonal polynomials.
The first set of algorithms we study come from discretising a continuous isospectral flow designed to converge to a symmetric Toeplitz matrix with prescribed eigenvalues. We analyse constrained, isospectral gradient flow approaches and an isospectral flow studied by Chu in 1993. The second set of algorithms compute the spectral measure of a Jacobi operator, which is the weight function for the associated orthogonal polynomials and can include a singular part. The connection coefficients matrix, which converts between different bases of orthogonal polynomials, is shown to be a useful new tool in the spectral theory of Jacobi operators. When the Jacobi operator is a finite rank perturbation of Toeplitz, here called pert-Toeplitz, the connection coefficients matrix produces an explicit, computable formula for the spectral measure. Generalisation to trace class perturbations is also considered.
The third algorithm is the infinite dimensional QL algorithm. In contrast to the finite dimensional case in which the QL and QR algorithms are equivalent, we find that the QL factorisations do not always exist, but that it is possible, at least in the case of pert-Toeplitz Jacobi operators, to implement shifts to generate rapid convergence of the top left entry to an eigenvalue. A fascinating novelty here is that the infinite dimensional matrices are computed in their entirety and stored in tailor made data structures.
Lastly, the connection coefficients matrix and the orthogonal transformations computed in the QL iterations can be combined to transform these pert-Toeplitz Jacobi operators isospectrally to a canonical form. This allows us to implement a functional calculus for pert-Toeplitz Jacobi operators.UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1
Theory of robust quantum many-body scars in long-range interacting systems
Quantum many-body scars (QMBS) are exceptional energy eigenstates of quantum
many-body systems associated with violations of thermalization for special
non-equilibrium initial states. Their various systematic constructions require
fine-tuning of local Hamiltonian parameters. In this work we demonstrate that
the setting of long-range interacting quantum spin systems generically hosts
robust QMBS. We analyze spectral properties upon raising the power-law decay
exponent of spin-spin interactions from the solvable
permutationally-symmetric limit . First, we numerically establish
that despite spectral signatures of chaos appear for infinitesimal ,
the towers of energy eigenstates with large collective spin are
smoothly deformed as is increased, and exhibit characteristic QMBS
features. To elucidate the nature and fate of these states in larger systems,
we introduce an analytical approach based on mapping the spin Hamiltonian onto
a relativistic quantum rotor non-linearly coupled to an extensive set of
bosonic modes. We exactly solve for the eigenstates of this interacting
impurity model, and show their self-consistent localization in large-spin
sectors of the original Hamiltonian for . Our theory unveils the
stability mechanism of such QMBS for arbitrary system size and predicts
instances of its breakdown e.g. near dynamical critical points or in presence
of semiclassical chaos, which we verify numerically in long-range quantum Ising
chains. As a byproduct, we find a predictive criterion for presence or absence
of heating under periodic driving for , beyond existing
Floquet-prethermalization theorems. Broader perspectives of this work range
from independent applications of the technical toolbox developed here to
informing experimental routes to metrologically useful multipartite
entanglement.Comment: 25+13 pages, 15+3 figure
- …