KdV polynomials and \(\Lambda\)-operator
Let \(u(x)\) be a meromorphic function defined in a region of the complex plane \(C\). The author studies algebraic properties of the spectrum of the differential operator \(H(u) = - \partial^2 + u(x)\) with the help of the associated recursion operator \(\Lambda (u) = \partial^{-1} ({1 \over 2} u'(x) + u(x) \partial - {1 \over 4} \partial^3)\). This operator generates the infinite sequence of differential polynomials (KdV polynomials) by the recurrence formula \(Z_n (u) = \Lambda (u)Z_{n - 1} (u)\), \(n \in \mathbb{N}\), \(Z_0 (u) = 1\). The case when the \(C-\) vector space \(V(u) = \cup_{m \in \mathbb{Z}^+} CZ_m (u)\) has finite dimension is considered in detail. The author shows that in this case the essential part of the problem can be reduced to the study of some algebraic properties of \(\Lambda (u) \in \text{End} (V(u))\).
scientific article; zbMATH DE number 880179