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At least 523 records · Page 29Linked to original sources
arXiv2013

Scattering theory with finite-gap backgrounds: Transformation operators and characteristic properties of scattering data

We develop direct and inverse scattering theory for Jacobi operators (doubly infinite second order difference operators) with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give necessary and sufficient conditions for the scattering data in the case of perturbations with finite second (or higher) moment.

math.SP↗
arXiv2012

Distinguished three-qubit 'magicity' via automorphisms of the split Cayley hexagon

Disregarding the identity, the remaining 63 elements of the generalized three-qubit Pauli group are found to contain 12096 distinct copies of Mermin's magic pentagram. Remarkably, 12096 is also the number of automorphisms of the smallest split Cayley hexagon. We give a few solid arguments showing that this may not be a mere coincidence. These arguments are mainly tied to the structure of certain types of geometric hyperplanes of the hexagon. It is further demonstrated that also an (18_{2}, 12_{3})-type of magic configurations, recently proposed by Waegell and Aravind (J. Phys. A: Math. Theor. 45 (2012) 405301), seems to be intricately linked with automorphisms of the hexagon. Finally, the entanglement properties exhibited by edges of both pentagrams and these particular Waegell-Aravind configurations are addressed.

math-ph↗
arXiv2011

Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution

We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the a limiting horizontal mean counting-measure of the zeroes exists almost surely, and that it is non-random if and only if the spectral measure is continuous (or degenerate). In this case, the mean zero-counting measure is computed in terms of the spectral measure. We compare the behavior with Gaussian analytic function with symmetry around the real axis. These results extend a work by Norbert Wiener.

math.PR↗
arXiv2008

Discrete Accidental Symmetry for a Particle in a Constant Magnetic Field on a Torus

A classical particle in a constant magnetic field undergoes cyclotron motion on a circular orbit. At the quantum level, the fact that all classical orbits are closed gives rise to degeneracies in the spectrum. It is well-known that the spectrum of a charged particle in a constant magnetic field consists of infinitely degenerate Landau levels. Just as for the $1/r$ and $r^2$ potentials, one thus expects some hidden accidental symmetry, in this case with infinite-dimensional representations. Indeed, the position of the center of the cyclotron circle plays the role of a Runge-Lenz vector. After identifying the corresponding accidental symmetry algebra, we re-analyze the system in a finite periodic volume. Interestingly, similar to the quantum mechanical breaking of CP invariance due to the $θ$-vacuum angle in non-Abelian gauge theories, quantum effects due to two self-adjoint extension parameters $θ_x$ and $θ_y$ explicitly break the continuous translation invariance of the classical theory. This reduces the symmetry to a discrete magnetic translation group and leads to finite degeneracy. Similar to a particle moving on a cone, a particle in a constant magnetic field shows a very peculiar realization of accidental symmetry in quantum mechanics.

quant-ph↗
arXiv2012

On a family of (1+1)-dimensional scalar field theory models: kinks, stability, one-loop mass shifts

In this paper we construct a one-parametric family of (1+1)-dimensional one-component scalar field theory models supporting kinks. Inspired by the sine-Gordon and $ϕ^4$ models, we look at all possible extensions such that the kink second-order fluctuation operators are Schrödinger differential operators with Pöschl-Teller potential wells. In this situation, the associated spectral problem is solvable and therefore we shall succeed in analyzing the kink stability completely and in computing the one-loop quantum correction to the kink mass exactly. When the parameter is a natural number, the family becomes the hierarchy for which the potential wells are reflectionless, the two first levels of the hierarchy being the sine-Gordon and $ϕ^4$ models.

hep-th↗
arXiv2012

Non-intrusive Low-Rank Separated Approximation of High-Dimensional Stochastic Models

This work proposes a sampling-based (non-intrusive) approach within the context of low-rank separated representations to tackle the issue of curse-of-dimensionality associated with the solution of models, e.g., PDEs/ODEs, with high-dimensional random inputs. Under some conditions discussed in details, the number of random realizations of the solution, required for a successful approximation, grows linearly with respect to the number of random inputs. The construction of the separated representation is achieved via a regularized alternating least-squares regression, together with an error indicator to estimate model parameters. The computational complexity of such a construction is quadratic in the number of random inputs. The performance of the method is investigated through its application to three numerical examples including two ODE problems with high-dimensional random inputs.

math-ph↗
arXiv2013

Velocity Polytopes of Periodic Graphs and a No-Go Theorem for Digital Physics

A periodic graph in dimension $d$ is a directed graph with a free action of $\Z^d$ with only finitely many orbits. It can conveniently be represented in terms of an associated finite graph with weights in $\Z^d$, corresponding to a $\Z^d$-bundle with connection. Here we use the weight sums along cycles in this associated graph to construct a certain polytope in $\R^d$, which we regard as a geometrical invariant associated to the periodic graph. It is the unit ball of a norm on $\R^d$ describing the large-scale geometry of the graph. It has a physical interpretation as the set of attainable velocities of a particle on the graph which can hop along one edge per timestep. Since a polytope necessarily has distinguished directions, there is no periodic graph for which this velocity set is isotropic. In the context of classical physics, this can be viewed as a no-go theorem for the emergence of an isotropic space from a discrete structure.

math-ph↗
arXiv2012

Cardy-Frobenius extension of algebra of cut-and-join operators

Motivated by the algebraic open-closed string models, we introduce and discuss an infinite-dimensional counterpart of the open-closed Hurwitz theory describing branching coverings generated both by the compact oriented surfaces and by the foam surfaces. We manifestly construct the corresponding infinite-dimensional equipped Cardy-Frobenius algebra, with the closed and open sectors are represented by conjugation classes of permutations and the pairs of permutations, i.e. by the algebra of Young diagrams and bipartite graphes respectively.

hep-th↗
arXiv2013

A physical approach to the classification of indecomposable Virasoro representations from the blob algebra

In the context of Conformal Field Theory (CFT), many results can be obtained from the representation theory of the Virasoro algebra. While the interest in Logarithmic CFTs has been growing recently, the Virasoro representations corresponding to these quantum field theories remain dauntingly complicated, thus hindering our understanding of various critical phenomena. We extend in this paper the construction of Read and Saleur (2007), and uncover a deep relationship between the Virasoro algebra and a finite-dimensional algebra characterizing the properties of two-dimensional statistical models, the so-called blob algebra (a proper extension of the Temperley--Lieb algebra). This allows us to explore vast classes of Virasoro representations (projective, tilting, generalized staggered modules, etc.), and to conjecture a classification of all possible indecomposable Virasoro modules (with, in particular, L_0 Jordan cells of arbitrary rank) that may appear in a consistent physical Logarithmic CFT where Virasoro is the maximal local chiral algebra. As by-products, we solve and analyze algebraically quantum-group symmetric XXZ spin chains and sl(2|1) supersymmetric spin chains with extra spins at the boundary, together with the "mirror" spin chain introduced by Martin and Woodcock (2004).

hep-th↗
arXiv2013

Attractive and Repulsive Casimir Vacuum Energy with General Boundary Conditions

The infrared behavior of quantum field theories confined in bounded domains is strongly dependent on the shape and structure of space boundaries. The most significant physical effect arises in the behaviour of the vacuum energy. The Casimir energy can be attractive or repulsive depending on the nature of the boundary. We calculate the vacuum energy for a massless scalar field confined between two homogeneous parallel plates with the most general type of boundary conditions depending on four parameters. The analysis provides a powerful method to identify which boundary conditions generate attractive or repulsive Casimir forces between the plates. In the interface between both regimes we find a very interesting family of boundary conditions which do not induce any type of Casimir force. We also show that the attractive regime holds far beyond identical boundary conditions for the two plates required by the Kenneth-Klich theorem and that the strongest attractive Casimir force appears for periodic boundary conditions whereas the strongest repulsive Casimir force corresponds to anti-periodic boundary conditions. Most of the analysed boundary conditions are new and some of them can be physically implemented with metamaterials.

hep-th↗
arXiv2013

Some properties of generalized Fisher information in the context of nonextensive thermostatistics

We present two extended forms of Fisher information that fit well in the context of nonextensive thermostatistics. We show that there exists an interplay between these generalized Fisher information, the generalized $q$-Gaussian distributions and the $q$-entropies. The minimum of the generalized Fisher information among distributions with a fixed moment, or with a fixed $q$-entropy is attained, in both cases, by a generalized $q$-Gaussian distribution. This complements the fact that the $q$-Gaussians maximize the $q$-entropies subject to a moment constraint, and yields new variational characterizations of the generalized $q$-Gaussians. We show that the generalized Fisher information naturally pop up in the expression of the time derivative of the $q$-entropies, for distributions satisfying a certain nonlinear heat equation. This result includes as a particular case the classical de Bruijn identity. Then we study further properties of the generalized Fisher information and of their minimization. We show that, though non additive, the generalized Fisher information of a combined system is upper bounded. In the case of mixing, we show that the generalized Fisher information is convex for $q\geq1.$ Finally, we show that the minimization of the generalized Fisher information subject to moment constraints satisfies a Legendre structure analog to the Legendre structure of thermodynamics.

math-ph↗
arXiv2013

Polynomial Supersymmetry for Matrix Hamiltonians

We study intertwining relations for matrix one-dimensional, in general, non-Hermitian Hamiltonians by matrix differential operators of arbitrary order. It is established that for any matrix intertwining operator Q_N^- of minimal order N there is a matrix operator Q_{N'}^+ of different, in general, order N' that intertwines the same Hamiltonians as Q_N^- in the opposite direction and such that the products Q_{N'}^+Q_N^- and Q_N^-Q_{N'}^+ are identical polynomials of the corresponding Hamiltonians. The related polynomial algebra of supersymmetry is constructed. The problems of minimization and of reducibility of a matrix intertwining operator are considered and the criteria of minimizability and of reducibility are presented. It is shown that there are absolutely irreducible matrix intertwining operators, in contrast to the scalar case.

quant-ph↗
arXiv2012

Markovian Dynamics on Complex Reaction Networks

Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multi-agent networks. A common approach to modeling such networks is by a master equation that governs the dynamic evolution of the joint probability mass function of the underling population process and naturally leads to Markovian dynamics for such process. Due however to the nonlinear nature of most reactions, the computation and analysis of the resulting stochastic population dynamics is a difficult task. This review article provides a coherent and comprehensive coverage of recently developed approaches and methods to tackle this problem. After reviewing a general framework for modeling Markovian reaction networks and giving specific examples, the authors present numerical and computational techniques capable of evaluating or approximating the solution of the master equation, discuss a recently developed approach for studying the stationary behavior of Markovian reaction networks using a potential energy landscape perspective, and provide an introduction to the emerging theory of thermodynamic analysis of such networks. Three representative problems of opinion formation, transcription regulation, and neural network dynamics are used as illustrative examples.

math-ph↗
arXiv2013

Convexity and quasi-uniformizability of closed preordered spaces

In many applications it is important to establish if a given topological preordered space has a topology and a preorder which can be recovered from the set of continuous isotone functions. Under antisymmetry this property, also known as quasi-uniformizability, allows one to compactify the topological space and to extend its order dynamics. In this work we study locally compact $σ$-compact spaces endowed with a closed preorder. They are known to be normally preordered, and it is proved here that if they are locally convex, then they are convex, in the sense that the upper and lower topologies generate the topology. As a consequence, under local convexity they are quasi-uniformizable. The problem of establishing local convexity under antisymmetry is studied. It is proved that local convexity holds provided the convex hull of any compact set is compact. Furthermore, it is proved that local convexity holds whenever the preorder is compactly generated, a case which includes most examples of interest, including preorders determined by cone structures over differentiable manifolds. The work ends with some results on the problem of quasi-pseudo-metrizability. As an application, it is shown that every stably causal spacetime is quasi-uniformizable and every globally hyperbolic spacetime is strictly quasi-pseudo-metrizable.

math.GN↗
arXiv2012

Norm resolvent convergence of singularly scaled Schrödinger operators and δ'-potentials

For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as εgoes to 0, of a family S_εof one-dimensional Schrödinger operators on the line of the form S_ε:= -D^2 + ε^{-2} V(x/ε). Under certain conditions the family of potentials converges in the sense of distributions to the first derivative of the Dirac delta-function, and then the limit of S_εmight be considered as a "physically motivated" interpretation of the one-dimensional Schrödinger operator with potential δ'.

math.SP↗
arXiv2023

Macroscopic limit of a Fokker-Planck model of swarming rigid bodies

We consider self-propelled rigid-bodies interacting through local body-attitude alignment modelled by stochastic differential equations. We derive a hydrodynamic model of this system at large spatio-temporal scales and particle numbers in any dimension $n \geq 3$. This goal was already achieved in dimension $n=3$, or in any dimension $n \geq 3$ for a different system involving jump processes. However, the present work corresponds to huge conceptual and technical gaps compared with earlier ones. The key difficulty is to determine an auxiliary but essential object, the generalized collision invariant. We achieve this aim by using the geometrical structure of the rotation group, namely, its maximal torus, Cartan subalgebra and Weyl group as well as other concepts of representation theory and Weyl's integration formula. The resulting hydrodynamic model appears as a hyperbolic system whose coefficients depend on the generalized collision invariant.

math-ph↗
arXiv2013

NIFTY - Numerical Information Field Theory - a versatile Python library for signal inference

NIFTY, "Numerical Information Field Theory", is a software package designed to enable the development of signal inference algorithms that operate regardless of the underlying spatial grid and its resolution. Its object-oriented framework is written in Python, although it accesses libraries written in Cython, C++, and C for efficiency. NIFTY offers a toolkit that abstracts discretized representations of continuous spaces, fields in these spaces, and operators acting on fields into classes. Thereby, the correct normalization of operations on fields is taken care of automatically without concerning the user. This allows for an abstract formulation and programming of inference algorithms, including those derived within information field theory. Thus, NIFTY permits its user to rapidly prototype algorithms in 1D, and then apply the developed code in higher-dimensional settings of real world problems. The set of spaces on which NIFTY operates comprises point sets, n-dimensional regular grids, spherical spaces, their harmonic counterparts, and product spaces constructed as combinations of those. The functionality and diversity of the package is demonstrated by a Wiener filter code example that successfully runs without modification regardless of the space on which the inference problem is defined.

astro-ph.IM↗