On the hydrogen symmetry

We construct O(4)-invariant hydrogen wave function in coordinate representation

The Kazhdan-Lusztig conjecture for finite W-algebras

We study the representation theory of finite W-algebras. After introducing parabolic subalgebras to describe the structure of W-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finite W-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors, and find that for W-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a `dual' analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infinite W-algebras.Comment: 11 page

Energy and orientation of Bloch type domain walls in magnatics with mixed anisotropy

Theoretical energy calculation has been carried out for domain walls (DW) of several types with different gyration angles of the magnetization vector in ferrite-garnet crystals with mixed anisotropy (presence of the uniaxial EK and cubic anisotropy EK). The DW orientations satisfying to the energy minimum have been determined. It is shown that at parameter v = KU/K₁, the 1-st type DW (the magnetization vector in the neighboring domains have the same positive projection onto the axis [111], the normal to the film surface) are transformed into the 2-nd type DW (the projections of opposite signs, the angle between magnetization vectors being distinct from 180°)