arXiv subjects
Publications and source records attributed to V. M. Buchstaber.
In the present paper we give the general solution of the functional equation $$ (f(x)+g(y)+h(z))^2 = F(x)+G(y)+H(z), x+y+z=0 $$ which is related to the exact factorized ground-state wave function for the quantum one-dimensional problem of three different particles with pairwise interaction.
If A is an algebra of functions on X, there are many cases when X can be regarded as included in Hom(A,C) as the set of ring homomorphisms. In this paper the corresponding results for the symmetric products of X are introduced. It is shown that the symmetric product Sym^n(X) is included in Hom(A,C) as the set of those functions that satisfy equations generalising f(xy)=f(x)f(y). These equations are related to formulae introduced by Frobenius and, for the relevant A, they characterise linear maps on A that are the sum of ring homomorphisms. The main theorem is proved using an identity satisfied by partitions of finite sets.
The theory of indices of Morse--Bott vector fields on a manifold is constructed and the famous localization problem for the transfer map is solved on its base in the present paper. As a consequence, we obtained addition theorems for the universal Pontrjagin characteristic classes in cobordisms. These results gave us a possibility to complete the construction, which was begun more than twenty years ago, of the universal characteristic classes' theory.
A new ansatz is presented for a Lax pair describing systems of particles on the line interacting via (possibly nonsymmetric) pairwise forces. Particular cases of this yield the known Lax pairs for the Calogero-Moser and Toda systems, as well as their relativistic generalisations. The ansatz leads to a system of functional equations. Several new functional equations are described and the general analytic solution to some of these is given. New integrable systems are described.
The general analytic solution to the functional equation $$ ϕ_1(x+y)= { { \biggl|\matrix{ϕ_2(x)&ϕ_2(y)\crϕ_3(x)&ϕ_3(y)\cr}\biggr|} \over { \biggl|\matrix{ϕ_4(x)&ϕ_4(y)\crϕ_5(x)&ϕ_5(y)\cr}\biggr|} } $$ is characterised. Up to the action of the symmetry group, this is described in terms of Weierstrass elliptic functions. We illustrate our theory by applying it to the classical addition theorems of the Jacobi elliptic functions and the functional equations $$ ϕ_1(x+y)=ϕ_4(x)ϕ_5(y)+ϕ_4(y)ϕ_5(x) $$ and \[ Ψ_1(x+y)=Ψ_2(x+y) ϕ_2(x)ϕ_3(y) +Ψ_3(x+y) ϕ_4(x)ϕ_5(y). \]
We consider generalizations of Dunkl's differential-difference operators associated with groups generated by reflections. The commutativity condition is equivalent to certain functional equations. These equations are solved in many cases. In particular, solutions associated with elliptic curves are constructed. In the $A_{n-1}$ case, we discuss the relation with elliptic Calogero-Moser integrable $n$-body problems, and discuss the quantization ($q$-analogue) of our construction.