Eigenvalues of Positive Definite Integral Operators on Unbounded Intervals

Let k(x, y) be the positive definite kernel of an integral operator on an unbounded interval of ℝ. If k belongs to class defined below, the corresponding operator is compact and trace class. We establish two results relating smoothness of k and its decay rate at infinity along the diagonal with the decay rate of the eigenvalues. The first result deals with the Lipschitz case; the second deals with the uniformly C¹ case. The optimal results known for compact intervals are recovered as special cases, and the relevance of these results for Fourier transforms is pointed out.     

On the Spectrum of the Reflection Operator on Conical Surfaces

We study the mapping properties of the reflection operator on a conical surface. This allows us to derive regularity results for the solution of the radiosity equation on conical surfaces in a scale of weighted Sobolev spaces. To motivate the calculations we first study the operator on a cylinder. Here we estimate the asymptotic behavior of the spectrum of the reflection operator by partial integration. This method works also for the conical case, but first we have to find a simple representation for some hypergeometric functions.     

On the discrete version of Picone's identity

For a real number p with 1<p we consider the first eigenvalues of the p-Laplacian on graphs, and estimates for the solutions of p-Laplace equations on graphs. We provide a discrete version of Picone's identity and its application. More precisely, we prove a Barta-type inequality for graphs with boundary. Finally, we provide a discrete version of the anti-maximum principle.     

Spectrum of the Kerzman–Stein Operator for Model Domains

For a domain , the Kerzman-Stein operator is the skew-hermitian part of the Cauchy operator acting on L²(b), which is defined with respect to Euclidean measure. In this paper we compute the spectrum of the Kerzman-Stein operator for three domains whose boundaries consist of two circular arcs: a strip, a wedge, and an annulus. We also treat the case of a domain bounded by two logarithmic spirals.     

On the existence of nonnegative solutions to an integral equation with applications to boundary value problems

By the use of a Guo-Krasnoselskii theorem in cones, existence of positive eigenvalues yielding nonnegative or positive solutions to an integral equation is studied. The results are applied to a variety of boundary value problems concerning ordinary differential equations.     

Shimura invariant differential operators and their eigenvalues

Let be the covariant Cauchy-Riemann operator and the covariant holomorphic differential operator on a line bundle over a Hermitian symmetric space . We study the Shimura invariant differential operators defined via and . We find the eigenvalues of a family of the Shimura operators and of the generators. Received October 2, 1999 / Published online October 30, 2000     

Approximation of the eigenvalues of a fourth order differential equation with non-smooth coefficients

The eigenvalues of a fourth order, generalized eigenvalue problem in one dimension, with non-smooth coefficients are approximated by a finite element method, introduced in an earlier work by the author and A. Lutoborski, in the context of a similar source problem with non-smooth coefficients. Error estimates for the approximate eigenvalues and eigenvectors are obtained, showing a better performance of this method, when applied to eigenvalue approximation, compared to a standard finite element method with arbitrary mesh.     

THE EIGENVALUE DISTRIBUTION OF A HUBERT-SCHMIDT-TYPE OPERATOR

Abstract

We prove the following theorem in answer to a question raised by P Nowosad and R Tovar in [3]. If K is a kernel operator on L²(x,u) with kernel K(x, y) if P(x): = U_X |K(x, y)|² d μ(y))½ and Q(x): = (U_X |K (y, x)|² d μ(y))½ and if _x PQdμ < ∞, then σ|λ_i|² < ∫_X PQd μ where (λ_i) is the se = zuence of eigenvalues of K.     

Birkhoff's theorem and multidimensional numerical range

We show that, under certain conditions, Birkhoff's theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections. Using this result, we study the relation between the spectrum of a self-adjoint operator A and its multidimensional numerical range. It turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator A. Every collection of eigenvalues which can be obtained by the Rayleigh-Ritz formula generates an extreme point of the multidimensional numerical range. However, it may also have other extreme points.     

Splitting of resonant and scattering frequencies under shape deformation

It is well known that the main difficulty in solving eigenvalue problems under shape deformation relates to the continuation of multiple eigenvalues of the unperturbed configuration. These eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues, and the splitting may only become apparent at high orders in their Taylor expansion. In this paper, we address the splitting problem in the evaluation of resonant and scattering frequencies of the two-dimensional Laplacian operator under boundary variations of the domain. By using surface potentials we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions that are of Fredholm type with index 0. We then proceed from the generalized Rouché's theorem to investigate the splitting problem.     

Spectrum of the 1‐Laplacian and Cheeger's Constant on Graphs

We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1 ? Laplacian Δ₁. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the structure of the solutions, the minimax characterization of eigenvalues, the multiplicity theorem, etc. The eigenvalues as well as the eigenvectors are computed for several elementary graphs. The graphic feature of eigenvalues are also studied. In particular, Cheeger's constant, which has only some upper and lower bounds in linear spectral theory, equals to the first nonzero Δ₁ eigenvalue for connected graphs.     

Properties of an integral operator preserved under pointwise multiplication of its kernel by a function

Let K be a kernel that determines an integral operator on some space of functions, and let H be a function. This paper investigates conditions under which certain properties of the integral operator determined by K (especially compactness properties) also hold for the integral operator determined by HK.     

On products and line graphs of signed graphs, their eigenvalues and energy

In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended p-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs. For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. We give exact results for those signed planar, cylindrical and toroidal grids which are Cartesian products of signed paths and cycles.We also treat the eigenvalues and energy of the line graphs of signed graphs, and the Laplacian eigenvalues and Laplacian energy in the regular case, with application to the line graphs of signed grids that are Cartesian products and to the line graphs of all-positive and all-negative complete graphs.     

The energy change of weighted graphs

The energy of an (edge)-weighted graph is the sum of the absolute values of the eigenvalues of its (weighted) adjacency matrix. We study how the energy of a weighted graph changes when the weights change. We give some sufficient conditions so that the energy of a weighted graph increases when the positive weight increases. We also characterize some classes of weighted graphs satisfying these sufficient conditions.     

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.     

A census of boundary cubic rooted planar maps

In this paper, boundary cubic rooted planar maps are investigated and exact enumerative formulae are given. First, an enumerative formula for boundary cubic inner-forest maps with the size (number of edges) as a parameter is derived. For the special case of boundary cubic inner-tree maps, a simple formula with two parameters is presented. Further, according to the duality, a corresponding result for outer-planar maps is obtained. Finally, some results for boundary cubic planar maps and general planar maps are obtained. Furthermore, two known Tutte's formulae are easily deduced in the paper.     

The integral graphs with index 3 and exactly two main eigenvalues

A graph is called integral if the spectrum of its adjacency matrix has only integral eigenvalues. An eigenvalue of a graph is called main eigenvalue if it has an eigenvector such that the sum of whose entries is not equal to zero. In this paper, we show that there are exactly 25 connected integral graphs with exactly two main eigenvalues and index 3.     

On the regularity of solutions to integral equations with nonsmooth kernels on a union of open intervals

The behaviour of a solution to a Fredholm integral equation of the second kind on a union of open intervals is examined. The kernel of the corresponding integral operator may have diagonal singularities, information about them is given through certain estimates. The weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described.     

Eigenvalue Distribution of Positive Definite Kernels on Unbounded Domains

We study eigenvalues of positive definite kernels of L² integral operators on unbounded real intervals. Under the assumptions of integrability and uniform continuity of the kernel on the diagonal the operator is compact and trace class. We establish sharp results which determine the eigenvalue distribution as a function of the smoothness of the kernel and its decay rate at infinity along the diagonal. The main result deals at once with all possible orders of differentiability and all possible rates of decay of the kernel. The known optimal results for eigenvalue distribution of positive definite kernels in compact intervals are particular cases. These results depend critically on a 2-parameter differential family of inequalities for the kernel which is a consequence of positivity and is a differential generalization of diagonal dominance.     

On Some Integral Equations in Hilbert Space with an Application to the Theory of Elasticity

A Volterra type integral equation in a Hilbert space with an additional linear operator L and a spectral parameter depending on time is considered. If the parameter does not belong to the spectrum of L unconditional solvability of the considered problem is proved. In the case where the initial value of the parameter coincides with some isolated point of the spectrum of the operator L sufficient conditions for solvability are established. The obtained results are applied to the partial integral equations associated with a contact problem of the theory of elasticity.