This monograph considers the interplay between spectral and oscillatory properties of both finite and infinite systems of linear ordinary differential operators. These can be written as single differential equations with matrix-valued and (bounded) operator-valued coefficients, respectively. The authors have been substantial contributors to this field, and this book places their work in the context of related work of others, which is described in detailed bibliographical comments at the end of each chapter. These comments also include a discussion of the relationship of this work to work on other topics such as nonlinear differential equations and partial differential equations.
The classical Sturm oscillation theorem is generalized to differential equations of arbitrary even order with operator coefficients on finite or infinite intervals, giving a result which includes the Morse index theorem as a special case. The authors also give a generalization of classical comparison and alternation theorems and factorization theorems, thus generalizing results of Frobenius, M. G. Krein, Heinz and Rellich. They also establish an analogue of the oscillation theorem for discrete levels in gaps of the continuous spectrum for equations of arbitrary even or odd order. Some results are new even in the classical case of scalar-valued coefficients. Also proved are results on the dependence of eigenvalues and the greatest lower bound of the essential spectrum on the variable end-point of a finite or semi-infinite interval for semi-bounded differential operators of arbitrary even order with operator coefficients.
These results are used in the proof of the oscillation theorems. The emphasis is on the application of operator theory to oscillation theory, but a comparison is also included with a topological approach of V. I. Arnold [Funct. Anal. Appl. 19, 251–259 (1985; Zbl 0606.58017)]. The bibliographical comments on the last two chapters, “Self-adjoint extensions of systems of differential equations of arbitrary order on an infinite interval in the absolutely indefinite case”, and “Discrete levels in spectral gaps of perturbed Schrödinger and Hill operators”, total nearly 100 pages, and, in the case of the last chapter, include much material on applications, including applications to theoretical physics.
The book is well written, and a list of symbols and the index prove useful. A substantial number of open questions is also included. Although addressed primarily to the research community, the book could also be used as a graduate textbook. The English translation contains substantial additions to the Russian original, especially to the bibliographical comments and to the bibliography, which now contains 941 items, including 209 monographs, and which gives details of available translations of all items.