A random necklace model

We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables. The integrated density of states of this Laplace operator is shown to have discontinuities provided that the distribution of arc lengths of the loops has a nontrivial pure point part. Some numerical illustrations are also presented.     

Systems of Linear Dyson–Schwinger Equations

Mathematical Physics, Analysis and Geometry - In this study, we consider a quantum waveguide with random boundary conditions . Precisely we consider Laplace operator restricted to a two dimensional...     

A stochastic approach to Wilson loops

Wilson loops exp (i A (x) dx) are investigated in two-dimensional Euclidean space-time. The electromagnetic vector potential A is regarded as a generalized random field given by the stochastic partial differential equation A = F where is a first-order differential operator and F is white noise. We give a rigorous definition of Wilson loops and examine the properties of the N-loop Schwinger functions.     

An exactly soluble relaxation problem

The problem of a particle moving in a two-valued random potential occurred in a recent paper by Pomeau. The exact time-dependent solution is here obtained for a quadratic potential by two different methods. The first method treats the problem as a stochastic differential equation and leads to the characteristic function of the probability distribution of the particle coordinate. In the second method the equation for the joint probability density of particle and potential is solved, which leads to the temporal Laplace transform of the distribution. The spectral properties of the evolution operator are examined.     

Critical behaviour in deterministic motion in a random environment

A numerical study of deterministic motion in a random environment in two dimensions is performed. All trajectories are localised except at two isolated points in the parameter space. The distribution of trajectory lengths shows “critical” behaviour as those points are approached. Contact with an analytic estimate, based on analogy with gelation theory, is made in one region of the parameter space. The spatial extent of the trajectories scales with arc length as if they were ideal random walks, surprisingly.     

Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement

The Laplace distribution of random processes was observed in numerous situations that include glasses, colloidal suspensions, live cells, and firm growth. Its origin is not so trivial as in the case of Gaussian distribution, supported by the central limit theorem. Sums of Laplace distributed random variables are not Laplace distributed. We discovered a new mechanism leading to the Laplace distribution of observable values. This mechanism changes the contribution ratio between a jump and a continuous parts of random processes. Our concept uses properties of Bernstein functions and subordinators connected with them.     

Diffusion on the Scaling Limit of the Critical Percolation Cluster in the Diamond Hierarchical Lattice

We construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal. A Dirichlet form can be constructed on the limit set and we consider the properties of the associated Laplace operator and diffusion process. In particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on-diagonal heat kernel for the percolation clusters and for the underlying lattice. In this setting a number of features of the lattice are inherited by the critical cluster.     

On the symmetry of a linear second-order equation of nonparabolic type

Determining equations are obtained in covariant form for the coordinates of a differential symmetry operator of second order of a nonparabolic equation. The possibility of constructing in Riemannian space a Laplace operator having the complete symmetry of the space is discussed.     

Value Distribution of the Eigenfunctions and Spectral Determinants of Quantum Star Graphs

We compute the value distributions of the eigenfunctions and spectral determinant of the Schrödinger operator on families of star graphs. The values of the spectral determinant are shown to have a Cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related. This is in contrast to the spectral determinants of random matrices, for which the logarithm is known to satisfy a Gaussian limit distribution. The value distribution of the eigenfunctions also differs from the corresponding random matrix result. We argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of eba-type billiards.     

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.     

矢量微分算符▽的探讨

本文试图从微分几何的角度探讨矢量微分算符▽在各种正交坐标系的梯度、散度、旋度和拉普拉斯符中的不变形式;并探讨了运用▽算符时容易出现混乱的根源,提出了‘括弧矢量优先’的判断与解决方法.从而扫除了运用▽算符的若干障碍.     

Generalized optical theorems for the reconstruction of Green's function of an inhomogeneous elastic medium

This paper investigates the reconstruction of elastic Green's function from the cross-correlation of waves excited by random noise in the context of scattering theory. Using a general operator equation-the resolvent formula-Green's function reconstruction is established when the noise sources satisfy an equipartition condition. In an inhomogeneous medium, the operator formalism leads to generalized forms of optical theorem involving the off-shell T-matrix of elastic waves, which describes scattering in the near-field. The role of temporal absorption in the formulation of the theorem is discussed. Previously established symmetry and reciprocity relations involving the on-shell T-matrix are recovered in the usual far-field and infinitesimal absorption limits. The theory is applied to a point scattering model for elastic waves. The T-matrix of the point scatterer incorporating all recurrent scattering loops is obtained by a regularization procedure. The physical significance of the point scatterer is discussed. In particular this model satisfies the off-shell version of the generalized optical theorem. The link between equipartition and Green's function reconstruction in a scattering medium is discussed.     

On Odd Laplace Operators

We consider odd Laplace operators acting on densities of various weights on an odd Poisson (= Schouten) manifold M. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an 'orbit space' of volume forms. This includes earlier results for the odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on M is partitioned into orbits by the action of a natural groupoid whose arrows correspond to the solutions of the quantum Batalin–Vilkovisky equations. We compare this situation with that of Riemannian and even Poisson manifolds. In particular, we show that the square of an odd Laplace operator is a Poisson vector field defining an analog of Weinstein's 'modular class'.     

A Clifford analysis approach to superspace

A new framework for studying superspace is given, based on methods from Clifford analysis. This leads to the introduction of both orthogonal and symplectic Clifford algebra generators, allowing for an easy and canonical introduction of a super-Dirac operator, a super-Laplace operator and the like. This framework is then used to define a super-Hodge coderivative, which, together with the exterior derivative, factorizes the Laplace operator. Finally both the cohomology of the exterior derivative and the homology of the Hodge operator on the level of polynomial-valued super-differential forms are studied. This leads to some interesting graphical representations and provides a better insight in the definition of the Berezin-integral.     

Transparent boundary conditions for the Helmholtz equation in some ramified domains with a fractal boundary

The paper addresses a class of boundary value problems in some self-similar ramified domains, with the Laplace or Helmholtz equations. Much stress is placed on transparent boundary conditions which allow the solutions to be computed in subdomains. A self similar finite element method is proposed and tested. It can be used for numerically computing the spectrum of the Laplace operator with Neumann boundary conditions, as well as the eigenmodes. The eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out. Finally, a numerical method for the wave equation is addressed.     

Hyperbolic transverse patterns in nonlinear optical resonators

We consider a nonlinear optical system in general, and a broad aperture laser, in particular, in a resonator where the diffraction coefficients are of opposite signs along two transverse directions. The system is described by the hyperbolic Maxwell-Bloch equations, where the spatial coupling is provided by the D'Alambert operator rather than by the Laplace operator. We show that this system supports hyperbolic transverse patterns residing on hyperbolas in far-field domain, and consisting of stretched vortices in near-field domain.     

Free Energies and Fluctuations for the Unitary Brownian Motion

We show that the Laplace transforms of traces of words in independent unitary Brownian motions converge towards an analytic function on a non trivial disc. These results allow one to study the asymptotic behavior of Wilson loops under the unitary Yang–Mills measure on the plane with a potential. The limiting objects obtained are shown to be characterized by equations analogue to Schwinger–Dyson’s ones, named here after Makeenko and Migdal.     

First-order operators and boundary triples

The aim of the present paper is to introduce a first-order approach to the abstract concept of boundary triples for Laplace operators. Our main application is the Laplace operator on a manifold with boundary; a case in which the ordinary concept of boundary triples does not apply directly. In our first-order approach, we show that we can use the usual boundary operators in abstract Green’s formula as well. Another motivation for the first-order approach is to give an intrinsic definition of the Dirichlet-to-Neumann map and intrinsic norms on the corresponding boundary spaces. We also show how the first-order boundary triples can be used to define a usual boundary triple leading to a Dirac operator. In memoriam Vladimir A. Geyler (1943–2007)     

Integrable and Chaotic Systems Associated with Fractal Groups

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.     

Localization on Quantum Graphs with Random Edge Lengths

The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple ${\mathbb Z^d}$ -lattice with δ-type boundary conditions at the vertices, and we assume that the edge lengths are randomly independently identically distributed. Under the assumption that the coupling constant at the vertices does not vanish, we show that the operator exhibits the Anderson localization near the spectral edges situated outside a certain forbidden set.