Related papers: Laplace operators in gamma analysis
We consider a family of manifolds with a class of degenerating warped product metrics $g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$, with $M$ compact, $\rho$ homogeneous degree one, $a \le -1$ and $b > 0$. We study the…
We consider several harmonic analysis operators in the multi-dimensional context of the Dunkl Laplacian with the underlying group of reflections isomorphic to $\mathbb{Z}_2^n$ (also negative values of the multiplicity function are…
In this paper, an explicit expression is obtained for the conformally invariant higher spin Laplace operator $\mathcal{D}_{\lambda}$, which acts on functions taking values in an arbitrary (finite-dimensional) irreducible representation for…
Grade-$d$ measures on a $\sigma$-algebra $\mathcal{A}\subseteq 2^X$ over a set $X$ are generalizations of measures satisfying one of a hierarchy of weak additivity-type conditions initially introduced as interference operators in quantum…
In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we study a selfadjoint strongly elliptic operator $A_\varepsilon$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$. Here $g({\mathbf…
In this text we consider discrete Laplace operators defined on lattices based on finitely-ramified self-similar sets, and their continuous analogous defined on the self-similar sets themselves. We are interested in the spectral properties…
We consider a family of self-adjoint operators [H_\omega = - \Delta + \lambda V_\omega, \quad \omega \in \Omega = \bigtimes_{k \in \ZZ^d} \RR,] on the Hilbert space $\ell^2 (\ZZ^d)$ or $L^2 (\RR^d)$. Here $\Delta$ denotes the Laplace…
The purpose of this paper is to study the $L^p$ boundedness of operators of the form \[ f\mapsto \psi(x) \int f(\gamma_t(x))K(t)\: dt, \] where $\gamma_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in…
We consider discrete periodic operator on $\mathbb Z^d$ with respect to lattices $\Gamma\subset\mathbb Z^d$ of full rank. We describe the class of lattices $\Gamma$ for which the operator may have a spectral gap for arbitrarily small…
We construct invariant differential operators acting on sections of vector bundles of densities over a smooth manifold without using a Riemannian metric. The spectral invariants of such operators are invariant under both the diffeomorphisms…
In this short note, we show that the continuous intrinsic graph Laplace operator with Gaussian kernel on a compact Riemannian manifold without boundary uniquely determines both the Riemannian metric and the sampling density, provided the…
We investigate the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces. We show that the covariance operator of a weak second-order Radon probability measure on such a space is uniquely…
We have run numerical simulations of Euclidean lattice quantum gravity for metrics which are time-independent and spherically symmetric. The radial variable is discretized as $r=hL_{Planck}$, with $h=0,1,...,N$ and $N$ up to $10^5$. The…
Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems,…
Let $G$ be a compact Lie group and $P_{e,a}(G)=C([0,1]\to G~|~\gamma(0)=e, \gamma(1)=a)$ be the pinned path space with a pinned Brownian motion measure $\nu_{\lambda,a}$ defined by the heat kernel $p(\lambda^{-1}t,x,y)$, where $\lambda$ is…
This paper constructs translation invariant operators on L2(R^d), which are Lipschitz continuous to the action of diffeomorphisms. A scattering propagator is a path ordered product of non-linear and non-commuting operators, each of which…
We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is…
On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We…
We solve the following problem: to describe in geometric terms all differential operators of the second order with a given principal symbol. Initially the operators act on scalar functions. Operator pencils acting on densities of arbitrary…
The aim of this work is to provide a geometric characterization of the positive Radon measures $\mu$ with compact support on the plane such that the associated Cauchy transform defines a compact operator from $L^2(\mu)$ to $L^2(\mu).$ It…