Pseudo-Differential Calculus on Homogeneous Trees

To study concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct in this paper a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on L ² and give adjoint and product formulas. Finally, we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian.     

Generalized spherical harmonics and exterior differentiation in weighted sobolev spaces

We generalize spherical harmonics expansions of scalar functions to expansions of alternating differential forms (‘q-forms’). To this end we develop a calculus for the use of spherical co-ordinates for q-forms and determine the eigen-q-forms of the Beltrami-operator on S^N?1 which replace the classical spherical harmonics. We characterize and classify homogeneous q-forms u which satisfy Δu = 0 on ?^N??{0} and determine Fredholm properties, kernel and range of the exterior derivative d acting in weighted L^p-spaces of q-forms (generalizing results of McOwen for the scalar Laplacian). These techniques and results are necessary prerequisites for the discussion of the low-frequency behaviour in exterior boundary value problems for systems occurring in electromagnetism and isotropic elasticity.     

An inverse spectral problem for the Laplacian

We propose an algorithm for the recovery of a potential from the knowledge of the eigenvalues of the Laplacian operator and the traces of its eigenfunctions. This inverse spectral problem is solved by recasting the operator as an infinite matrix and using transition matrices together with spectral projections on the boundary.     

Reliable computation of eigenvalues of the magnetostatic integral operator

Some variational problems in magnetostatics can be reformulated as eigenvalue problems for vector surface integral operators in appropriate function spaces, e.g., the magnetostatic integral operator is of considerable interest in the theory of permanent magnetization of compact bodies. In the case that the underlying surface is either a sphere, a spheroid, or a triaxial ellipsoid, explicit expressions for eigenvalues and eigenfunctions are well known. For the ellipsoid, these quantities are given in terms of Lamé functions and surface ellipsoidal harmonics. Since there is an apparent lack in literature we provide an new effective scheme for the reliable computation of these functions and of the corresponding eigenvalues of the magnetostatic operator.     

Perturbing a rectangular membrane with a restorative force: effects on eigenbvalues and eigenfunctions.

Series expansions are obtained for a rich subset of eigenvalues and eigenfunctions of an operator that arises in the study of rectangular membranes: the operator is the 2-D Laplacian with restorative force term and Dirichlet boundary conditions. Expansions are extracted by considering the restorative force term as a linear perturbation of the Laplacian; errors of truncation for these expansions are estimated. Theriteria defining the subset of eigenvalues and eigenfunctions that can

be studied depends only on the size and linearity of the perturbation. The results are valid for almost all rectangular domains.     

Spectral properties of the massless relativistic quartic oscillator

An explicit solution of the spectral problem of the non-local Schrödinger operator obtained as the sum of the square root of the Laplacian and a quartic potential in one dimension is presented. The eigenvalues are obtained as zeroes of special functions related to the fourth order Airy function, and closed formulae for the Fourier transform of the eigenfunctions are derived. These representations allow to derive further spectral properties such as estimates of spectral gaps, heat trace and the asymptotic distribution of eigenvalues, as well as a detailed analysis of the eigenfunctions. A subtle spectral effect is observed which manifests in an exponentially tight approximation of the spectrum by the zeroes of the dominating term in the Fourier representation of the eigenfunctions and its derivative.     

Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian, as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.     

Constantin and Iyer’s Representation Formula for the Navier–Stokes Equations on Manifolds

The purpose of this paper is to establish a probabilistic representation formula for the Navier–Stokes equations on compact Riemannian manifolds. Such a formula has been provided by Constantin and Iyer in the flat case of ? ⁿ or of T ⁿ . On a Riemannian manifold, however, there are several different choices of Laplacian operators acting on vector fields. In this paper, we shall use the de Rham–Hodge Laplacian operator which seems more relevant to the probabilistic setting, and adopt Elworthy–Le Jan–Li’s idea to decompose it as a sum of the square of Lie derivatives.     

Asymptotic property and convergence estimation for the eigenelements of the Laplace operator

In this article, we provide a rigorous derivation of asymptotic expansions for eigenfunctions and we establish convergence estimation for both eigenvalues and eigenfunctions of the Laplacian. We address the integral equation method to investigate the interplay between the geometry, boundary conditions and spectral properties of the eigenelements of the Laplace operator under deformation of the domain. The asymptotic formula and convergence estimation are tested by numerical examples.     

Gaussian kernel operators on white noise functional spaces

The Gaussian kernel operators on white noise functional spaces, including second quantization, Fourier-Mehler transform, scaling, renormalization, etc. are studied by means of symbol calculus, and characterized by the intertwining relations with annihilation and creation operators. The infinitesimal generators of the Gaussian kernel operators are second order white noise operators of which the number operator and the Gross Laplacian are particular examples.     

Asymptotic and Spectral Analysis of the Spatially Nonhomogeneous Timoshenko Beam Model

We develop spectral and asymptotic analysis for a class of nonselfadjoint operators which are the dynamics generators for the systems governed by the equations of the spatially nonhomogeneous Timoshenko beam model with a 2–parameter family of dissipative boundary conditions. Our results split into two groups. We prove asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues and each branch has only two points of accumulation: +∞ and —∞), and for their generalized eigenvectors. Our second main result is the fact that these operators are Riesz spectral. To obtain this result, we prove that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces. We also obtain the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators. The pencil asymptotics are essential for the proofs of the spectral results for the aforementioned dynamics generators.     

Heat kernel asymptotics on manifolds with conic singularities

The Laplacian acting onk-forms on a manifold with isolated conic singularities is not in general an essentially self-adjoint operator. The heat kernels for self-adjoint extensions of the Laplacian on these metric spaces are described as functions conormal to a manifold with corners. The heat kernel for a given self-adjoint extension is constructed from the Friedrichs heat kernel. The terms in the difference of the heat trace expansions are shown to supply information parametrizing the extension.     

A spectral-theoretic approach for the solution of Maxwell's equations in stratified media

In the present work, the problem of electromagnetic wave propagation in three-dimensional stratified media is studied. The method of decoupling the electric and magnetic fields is implemented, and the spectral approach is adopted, componentwise, to the vector equation involving the electric field. Operational calculus of self-adjoint, positive operators in suitable Hilbert spaces is used to solve the corresponding initial value problems. The spectral families of these operators for the cases of the whole space and of a finite layer are constructed. A discussion on the applicability of the obtained results to physical problems is also included. © 1998 B.G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Spectral approximation by L 2 discretization of the Laplace operator on manifolds of bounded geometry

The author considers a discretization of the p-form Laplacian on open complete Riemannian manifolds of bounded geometry. Following Dodziuk and Patodi [8], the eigenvalues below the essential spectrum together with their eigenforms are approximated by eigenvalues and eigencochains of a semicombinatorical Laplacian acting on L ₂-cochains. We obtain a similar result for a ray which is contained in the essential spectrum. An example of a manifold of bounded geometry which admits eigenvalues below the essential spectrum is constructed.     

A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics

A manifold with fibered cusp metrics X can be considered as a geometrical generalization of locally symmetric spaces of \mathbbQ{\mathbb{Q}}-rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find harmonic representatives of the de Rham cohomology H ^p (X). Similar to the situation of locally symmetric spaces, these representatives are computed by special values or residues of generalized eigenforms of the Hodge–Laplace operator on Ω ^p (X).     

Asymptotics of the spectrum and eigenfunctions of the magnetic induction operator on a compact two-dimensional surface of revolution

Magnetic fields in conducting liquids (in particular, magnetic fields of galaxies, stars, and planets) are described by the magnetic induction operator. In this paper, we study the spectrum and eigenfunctions of this operator on a compact two-dimensional surface of revolution. For large magnetic Reynolds numbers, the asymptotics of the spectrum is studied; equations defining the eigenvalues (quantization conditions) are obtained; and examples of spectral graphs near which these points are located are given. The spatial structure of the eigenfunctions is studied.     

Lévy Laplacians in Hida Calculus and Malliavin Calculus

Some connections between different definitions of Lévy Laplacians in the stochastic analysis are considered. Two approaches are used to define these operators. The standard one is based on the application of the theory of Sobolev–Schwartz distributions over the Wiener measure (Hida calculus). One can consider the chain of Lévy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Lévy Laplacian. Another approach to defining the Lévy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (Malliavin calculus). It is proved that the Lévy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Lévy Laplacians, but not with the classical Lévy Laplacian, under the embedding of the Sobolev space over the Wiener measure in the space of generalized functionals over this measure. It is shown which Lévy Laplacian in the stochastic analysis is connected with the gauge fields.     

β−type fractional Sturm‐Liouville Coulomb operator and applied results

In this article, β‐type fractional Sturm‐Liouville Coulomb operator is considered by Hilfer fractional derivative. Fundamental spectral theory is investigated for the aforementioned problem. In this context, it is shown that the operator is self‐adjoint, eigenfunctions correspond to the distinct eigenfunctions are orthogonal, and eigenvalues are real. Furthermore, applications of this problem are given by the Adomian decomposition method and the results are shown with visual graphs.     

Spectral Convergence of Quasi-One-Dimensional Spaces

We consider a family of non-compact manifolds X_ε (“graph-like manifolds”) approaching a metric graph X₀ and establish convergence results of the related natural operators, namely the (Neumann) Laplacian and the generalized Neumann (Kirchhoff) Laplacian on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions. As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also in other geometric situations. Communicated by Claude Alain Pillet Submitted: December 21, 2005 Accepted: January 30, 2006