Local research of manifolds generated by families of self-adjoint operators

We consider V.I. Arnold’s manifold of self-adjoint operators with fixed multiplicity of eigenvalues and K. Uhlenbeck’s manifold of eigenvectors. Our aim is to consider the local analysis and the connection between these manifolds. We present the topological description of the spectrum perturbation problem, specifically the finite-multiple eigenvalue splitting problem. For investigation of manifolds, we use the local diffeomorphism introduced by D. Fujiwara, M. Tanikawa, and Sh. Yukita.