Related papers: On the spectrum of second order differential opera…
For a scalar elliptic self-adjoint operator on a compact manifold without boundary we have two-term asymptotics for the number of eigenvalues between zero and lambda when lambda tends to infinity, under an additional dynamical condition.…
We prove the validity of regularizing properties of a double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in Schauder spaces by…
An introductory review of algebraic classification of the Weyl tensor and algebraically special solutions in higher dimensions.
Using the method of similar operators we study an even order differential operator with periodic, semiperiodic, and Dirichlet boundary conditions. We obtain asymptotic formulas for eigenvalues of this operator and estimates for its spectral…
An algorithm is described giving effective determination of the second differential in the Adams spectral sequence. The algorithm is based on the notion of secondary derived functor, and on the explicit algebraic model of the groupoid…
We consider self-adjoint extensions of differential operators of the type $ (-\frac{d^2}{dr^2} + \frac{l(l+1)}{r^2})^3 $ on the real semi-axis for l=1,2 with two kinds of boundary conditions: first that nullify the value of a function and…
The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering theory, the spectral properties of this…
A new approach to normal operators in real Hilbert spaces is discussed, and a spectral representation is obtained, derived directly from the complex case. The results are then applied to quaternionic normal operators, regarded as a special…
We consider a large class of self-adjoint elliptic problem associated with the second derivative acting on a space of vector-valued functions. We present two different approaches to the study of the associated eigenvalues problems. The…
A class of the Newell-Whitehead-Segel equations (also known as generalized Fisher equations and Newell-Whitehead equations) is studied with Lie and "nonclassical" symmetry points of view. The classifications of Lie reduction operators and…
Based on the Liouville-Weyl definition of the fractional derivative, a new direct fractional generalization of higher order derivatives is presented. It is shown, that the Riesz and Feller derivatives are special cases of this approach.
The nature of so-called differential-algebraic operators and their approximations is constitutive for the direct treatment of higher-index differential-algebraic equations. We treat first-order differential-algebraic operators in detail and…
This paper is devoted to the study of the second-order variational analysis of spectral functions. It is well-known that spectral functions can be expressed as a composite function of symmetric functions and eigenvalue functions. We…
A computer-algebra aided method is carried out, for determining geometric objects associated to differential operators that satisfy the elliptic ansatz. This results in examples of Lame curves with double reduction and in the explicit…
In this paper the Weyl tensor is used to define operators that act on the space of forms. These operators are shown to have interesting properties and are used to classify the Weyl tensor, the well known Petrov classification emerging as a…
Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite…
We study refinements between spectral resolutions in an arbitrary II$_1$ factor $\M$ and obtain diffuse (maximal) refinements of spectral resolutions. We construct models of operators with respect to diffuse spectral resolutions. As an…
A method of generating differential operators is used to solve the spectral problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we find a linear differential operator with polynomial coefficients of the first order…
Lecture notes for one of the courses at the OPSFA Summerschool 6, July 11-15, 2016. All the results in these notes have appeared in the literature. Many special functions are eigenfunctions to explicit operators, such as difference and…
In this paper, we introduce some analytical techniques to solve some classes of second order differential equations. Such classes of differential equations arise in describing some mathematical problems in Physics and Engineering.