Spectral theory of Volterra-composition operators

Given a Lebesgue measurable self-map of the interval [0, 1], the Volterra- composition operator is defined as We develop the spectral theory of these operators. In particular, for a class of natural symbols , finiteness of the spectrum is characterized and formulae for the trace and the convergence exponent of eigenvalues are provided. The positivity of the spectrum as well as the analyticity of the eigenfunctions are also treated. The theory of entire functions as well as solving some Cauchy Problems will play a fundamental role in this theory. We also supply some examples of symbols to which the theory can be applied and, in particular, eigenvalues and eigenfunctions are computed explicitly. Partially supported by Plan Nacional I+D+I grant no. MTM2006-09060, Junta de Andalucía FQM-260 and P06-FQM-02225.     

Spectral theory of left definite difference operators

This paper is concerned with the spectral theory for the second-order left definite difference boundary value problems. Existence of eigenvalues of boundary value problems is proved, numbers of their eigenvalues are calculated and fundamental spectral results are obtained.     

Spectral analysis of higher order differential operators with unbounded coefficients

Higher even order linear differential operators with unbounded coefficients are studied. For these operators the eigenvalues of the characteristic polynomials fall into distinct classes or clusters. Consequently the spectral properties, deficiency indices and spectra, of the underlying differential operators are superpositions of the contributions from the individual clusters. These results are based on a quantitative improvement of Levinson's Theorem. Our methods will also be applicable to other classes of linear differential operators.     

A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems

We consider a simple system modelling phase transition phenomena with long term interactions. It is shown that any solution converges with growing time to a single stationary state. To this end, a non-smooth version of the celebrated Simon-Lojasiewicz theorem is proved.     

Spectral properties of difference and differential operators in weighted spaces

We describe the spectrum of a difference operator (a weighted shift operator). We obtain applications to finding spectra of differential operators in weighted function spaces.     

Traveling waves for delayed non-local diffusion equations with crossing-monostability

This paper is concerned with the traveling waves for a class of delayed non-local diffusion equations with crossing-monostability. Based on constructing two associated auxiliary delayed non-local diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space using the traveling wave fronts of the auxiliary equations, the existence of traveling waves is proved by Schauder’s fixed point theorem. The result implies that the traveling waves of the delayed non-local diffusion equations with crossing-monostability are persistent for all values of the delay τ?0.     

Spectral and oscillatory properties of a linear pencil of fourth-order differential operators

The paper deals with the spectral and oscillatory properties of a linear operator pencilA ? λB, where the coefficient A corresponds to the differential expression (py″)″ and the coefficient B corresponds to the differential expression ?y″ + cry. In particular, it is shown that all negative eigenvalues of the pencil are simple and, under some additional conditions, the number of zeros of the corresponding eigenfunctions is related to the serial number of the corresponding eigenvalue.     

Weighted eigenvalue problems for Hessian equations

In this paper we consider weighted eigenvalue problems for fully nonlinear elliptic equations involving Hessian operators. In particular we consider a singular weight, which behaves like a Hardy potential and we prove the existence of weak eigenfunctions.     

Spectral properties of some regular boundary value problems for fourth order differential operators

In this paper we consider the problem $\begin{gathered} y^{iv} + p_2 (x)y'' + p_1 (x)y' + p_0 (x)y = \lambda y,0 < x < 1, \hfill \\ y^{(s)} (1) - ( - 1)^\sigma y^{(s)} (0) + \sum\limits_{l = 0}^{s - 1} {\alpha _{s,l} y^{(l)} (0) = 0,} s = 1,2,3, \hfill \\ y(1) - ( - 1)^\sigma y(0) = 0, \hfill \\ \end{gathered} $ where λ is a spectral parameter; p _j (x) ∈ L ₁(0, 1), j = 0, 1, 2, are complex-valued functions; α _s;l , s = 1, 2, 3, $l = \overline {0,s - 1} $ , are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α _3,2 + α _1,0 ≠ α _2,1. It is proved that the system of root functions of this spectral problem forms a basis in the space L _p (0, 1), 1 < p < ∞, when α _3,2+α _1,0 ≠ α _2,1, p _j (x) ∈ W ₁ ^j (0, 1), j = 1, 2, and p ₀(x) ∈ L ₁(0, 1); moreover, this basis is unconditional for p = 2.     

Support properties and Holmgren's uniqueness theorem for differential operators with hyperplane singularities

Let W be a finite Coxeter group acting linearly on Rn. In this article we study the support properties of a W-invariant partial differential operator D on Rn with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W. In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W. We show that conv(suppDf)=conv(suppf) holds for all compactly supported smooth functions f so that conv(suppf) is W-invariant. The main tools in the proof are Holmgren's uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented.     

On the solution of the non-local parabolic partial differential equations via radial basis functions

In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered. The approximate solution is found using the radial basis functions collocation method. There are some difficulties in computing the solution of the time dependent partial differential equations using radial basis functions. If time and space are discretized using radial basis functions, the resulted coefficient matrix will be very ill-conditioned and so the corresponding linear system cannot be solved easily. As an alternative method for solution, we can use finite-difference methods for discretization of time and radial basis functions for discretization of space. Although this method is easy to use but an accurate solution cannot be provided. In this work an efficient collocation method is proposed for solving non-local parabolic partial differential equations using radial basis functions. Numerical results are presented and are compared with some existing methods.     

Spectral analysis of fourth order differential operators III

This paper extends the results of the two previous papers in several directions. For one we allow slower decay of the coefficients, but higher order differentiability. For this an expansion for the diagonalizing transformations is derived. Secondly unbounded coefficients are permitted. This requires further transformations in order to achieve Levinson's form, but also a modification with the usual M‐matrix approach. While the standard results carry over to weakly singular coefficients, very singular coefficients will generally lead to discrete spectra (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)