Dymarskii Ya. M.
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.
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.
On the Manifolds of Eigenvectors of Linear and Quasilinear Finite-Dimensional Self-Adjoint Operators. II
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.
On the Manifolds of Eigenvectors of Linear and Quasilinear Finite-Dimensional Self-Adjoint Operators. I
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.
Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1194–1199
For nonlinear equations of a special type, in the case of double degeneration of a linearized problem, we prove the existence of unbounded branches of solutions originating at a bifurcation point.
Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 901-909
In the case of double degeneration of a linearized problem, we study the points of bifurcation of the null solutions of nonlinear equations of special type.
On manifolds of eigenfunctions and potentials generated by a family of periodic boundary-value problems
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.