2018
Том 70
№ 11

All Issues

Dymarskii Ya. M.

Articles: 5
Article (Russian)

Submanifolds of compact operators with fixed multiplicities of eigenvalues

Bondar A. A., Dymarskii Ya. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1179-1189

The manifold of symmetric real matrices with fixed multiplicities of eigenvalues was considered for the first time by V. Arnold. In the case of compact real self-adjoint operators, analogous results were obtained by Japanese mathematicians D. Fujiwara, M. Tanikawa, and S. Yukita. They introduced a special local diffeomorphism that maps Arnold's submanifold to a flat subspace. The properties of the indicated diffeomorphism were further studied by Ya. Dymarskii. In the present paper, we describe the smooth structure of submanifolds of finite-dimensional and compact operators of the general form in which a selected eigenvalue is associated with a single Jordan block.

Article (Russian)

Manifolds of Eigenfunctions and Potentials of a Family of Periodic Sturm–Liouville Problems

Dymarskii Ya. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1042-1052

We consider a family of boundary-value problems in which the role of a parameter is played by a potential. We investigate the smooth structure and homotopic properties of the manifolds of eigenfunctions and degenerate potentials corresponding to double eigenvalues.

Article (Russian)

On the Manifolds of Eigenvectors of Linear and Quasilinear Finite-Dimensional Self-Adjoint Operators. II

Dymarskii Ya. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 296-301

We investigate the manifold of normalized eigenvectors of self-adjoint operators. We present the homotopic classification of typical quasilinear eigenvector problems based on the properties of this manifold.

Article (Ukrainian)

On the Manifolds of Eigenvectors of Linear and Quasilinear Finite-Dimensional Self-Adjoint Operators. I

Dymarskii Ya. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 156-167

We investigate the vector bundle of the manifold of normalized eigenvectors of self-adjoint operators and its stratification with respect to the numbers and multiplicities of eigenvalues.

Article (Russian)

On manifolds of eigenfunctions and potentials generated by a family of periodic boundary-value problems

Dymarskii Ya. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 771-781

We consider a family of boundary-value problems with some potential as a parameter. We study the manifold of normalized eigenfunctions with even number of zeros in a period, and the manifold of potentials associated with double eigenvalues. In particular, we prove that the manifold of normalized eigenfunctions is a trivial fiber space over a unit circle and that the manifold of potentials with double eigenvalues is a homotopically trivial manifold trivially imbedded into the space of potentials.