Skip to main content



I. Linear Overdetermined Systems of Partial Differential Equations. Initial and Initial-Boundary Value Problems

Consider a linear partial differential operator A that maps a vector-valued function y = (y 1 , ...,y m ) into a vector-valued function f = (f 1 ,...,f l ) We assume at first that all the functions, as well as the coefficients of the differential operator, are defined in an open domain Ω in the n-dimensional Euclidean space ℝn, and that they are smooth (infinitely differentiable). A is called an overdetermined operator if there is a non-zero differential operator A′ such that the composition AA is the zero operator (and underdetermined if there is a non-zero operator A″ such that AA″ = 0). If A is overdetermined, then A′ f = 0 is a necessary condition for the solvability of the system A y = f with an unknown vector-valued function y.
P. I. Dudnikov, S. N. Samborski

II. Spectral Analysis of a Dissipative Singular Schrödinger Operator in Terms of a Functional Model

Historically, the first general method in the spectral analysis of non-selfadjoint differential operators was the Riesz integral, complemented by the refined technique of estimating the resolvent on the contours that divide the spectrum. Using this method, Lidskij (1962) proved the summation over groups (“with brackets”) of the spectral resolution of a general regular second order differential operator. Since then, the so-called “bases with brackets” have been studied extensively by his successors (see the references in Sadovnichij (1973)). Unfortunately, the arrangement of the “brackets”, that is, the combination into one group of the sets of eigenvectors and root vectors corresponding to some neighbouring points of the spectrum, is defined non-uniquely and, to a large extent, non-constructively. Hence, as a rule, the assertions concerning bases with brackets have the character of existence theorems.
B. S. Pavlov

III. Index Theorems

The theory of the index of elliptic operators has for a long time been developed in parallel within the framework of two branches of mathematics that, traditionally, are regarded as quite far apart. One of them is the the theory of elliptic equations and boundary value problems—in particular, the theory of singular integral equations. The other is topology and algebraic geometry, where very specific elliptic operators have been considered. A significant role in bringing these two domains together was played by Gel’fand (1960), who posed the problem of topological classification of elliptic operators, in particular, the computation of the index in topological terms. The latter was fully solved by Atiyah and Singer in 1963. The Atiyah-Singer theorem has generated a tremendous amount of interest, which has continued to this day and has exercised an immense influence on the subsequent development and convergence of the theory of differential equations and topology. Thus, for example, the necessity to extend the class of deformations of elliptic operators has led to new algebras of pseudodifferential operators (PDOs). In topology, the Atiyah-Singer theorem has stimulated the further development of K-theory.
B. V. Fedosov


Weitere Informationen