We study an inverse scattering problem for the Helmholtz equation on whole line. The goal of this paper is to obtain a Gelfand–Levitan–Marchenko (GLM)-type Jost solution that corresponds 1D differential operator. assume simplicity refraction index compact support. Using asymptotic behavior solutions with respect wave-number, we derive generalized Povzner–Levitan representation in space tempered...

Proofs that endomorphism (convolution) algebras are commutative most often depend upon identifying an anti-involution to interchange the order of factors, but which nevertheless acts as the identity on the algebra or suitable subalgebras. Silberger gave such an argument for the spherical Hecke algebras of p-adic reductive groups, for example. Apparently the first occurrence of the Gelfand-Kazhd...

In 2010, V. Futorny and S. Ovsienko gave a realization of $U(\mathfrak{gl}_n)$ as subalgebra the ring invariants certain noncommutative with respect to action $S_1\times S_2\times\cdots\times S_n$, where $S_j$ is symmetric group on $j$ variables. An interesting question what similar algebra would be in invariant product alternating groups. this paper we define such an algebra, denoted $\mathscr...

Let S be a connected and simply connected unimodular solvable Lie group, K a connected compact group acting on S as automorphisms. We call the pair (K;S) a Gelfand pair if the Banach -algebra L 1 (KnKnS=K) of allK-biinvariant integrable functions on K n S is commutative. The assumption that (K;S) is a Gelfand pair prescribes the structure of S. For example, if (K;S) is a Gelfand pair, then S is...

The quantum group version of the Bernstein-Gelfand-Gelfand resolution is used to construct a double complex of Uq(g)-modules with exact rows and columns. The locally finite dual of its total complex is identified with the de Rham complex for quantized irreducible flag manifolds. MSC: 17B37, 58B32