Vol 50, No 1 (2016)

The set of all linear transformations with a fixed Jordan structure J is a symplectic manifold isomorphic to the coadjoint orbit O(J) of the general linear group GL(N, C). Any linear transformation can be projected along its eigenspace onto a coordinate subspace of complementary dimension. The Jordan structure \(\tilde J\) of the image under the projection is determined by the Jordan structure J of the preimage; consequently, the projection \(O\left( J \right) \to O\left( {\tilde J} \right)\) is a mapping of symplectic manifolds.

It is proved that the fiber ℰ of the projection is a linear symplectic space and the map \(O\left( J \right)\tilde \to E \times O\left( {\tilde J} \right)\) is a birational symplectomorphism. Successively projecting the resulting transformations along eigensubspaces yields an isomorphism between O(J) and the linear symplectic space being the direct product of all fibers of the projections. The Darboux coordinates on O(J) are pullbacks of the canonical coordinates on this linear symplectic space.

Canonical coordinates on orbits corresponding to various Jordan structures are constructed as examples.

In the paper, we consider the question as to whether a unital full amalgamated free product of quasidiagonal C*-algebras is itself quasidiagonal. We give a sufficient condition for a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite-dimensional C*-algebra to be quasidiagonal. By applying this result, we conclude that the unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there exists a faithful tracial state on each of the two AF algebras such that the restrictions of these states to the common subalgebra coincide.

Self-adjoint commuting differential operators with polynomial coefficients are considered. These operators form a commutative subalgebra of the first Weyl algebra. New examples of commuting differential operators of rank 2 are found.

We prove the strong Suslin reciprocity law conjectured by A. B. Goncharov and describe its corollaries for the theory of scissor congruence of polyhedra in hyperbolic space. The proof is based on the study of Goncharov’s conjectural description of certain rational motivic cohomology groups of a field. Our main result is a homotopy invariance theorem for these groups.

In the paper, we consider the one-dimensional nonstationary Schrödinger equation with a potential slowly depending on time. It is assumed that the corresponding stationary operator depending on time as a parameter has a finite number of negative eigenvalues and absolutely continuous spectrum filling the positive semiaxis. A solution close at some moment to an eigenfunction of the stationary operator is studied. We describe its asymptotic behavior in the case where the eigenvalues of the stationary operator move to the edge of the continuous spectrum and, having reached it, disappear one after another.