Integration of some chains of nonlinear difference equations by the method of the inverse spectral problem (Q1081038)

Consider the difference equation \[ ({\mathcal L}u)_ n=a_{n- 1}u_{n-1}+b_ nu_ n+a_ nu_{n+1},\;n=0,1,\dots,\tag{1} \] where \(a_ n\), \(b_ n\) are bounded operators in a Hilbert space \(H\) and \(u=(u_ n)_ 0^{\infty}\) is a vector sequence in \(H\). The paper is devoted to the study of the non-abelian Toda chain and to the finite scalar Toda chain. A system \[ \dot c_ n=2(c_ nd_{n+1}-d_ nc_ n)+[d_ 0,c_ n],\dot d_ n=2(c_ n-c_{n-1})+[d_ 0,d_ n],\;n=0,1,\dots; \tag{2} \] \(c_{-1}=0\); \(\cdot =d/dt\), with respect to \(c_ n=c_ n(t)\), \(d_ n=d_ n(t)\), \(t\in (0,\infty)\), whose values are bounded operators in \(H\), is called a non-abelian Toda chain. The author proves the existence and the uniqueness of the solution of (2) under the form \(c_ n=a_ 0^{- 1}...a^{-1}_{n-1}a^ 2_ na_{n-1}...a_ 0\), \(d_ n=a_ 0^{- 1}...a^{-1}_{n-1}b_ na_{n-1}...a_ 0\), \(n=1,2,...\); \(c_ 0=a^ 2_ 0\), \(d_ 0=b_ 0\). A similar result for the so called scalar Toda chain is given.