On the Darboux transformation of \(N\)th order (Q1589054)

Let \(h_0\Psi(x)=E\Psi\), \(h_0\equiv -D^2+V(x)\), \(D\equiv {d\over dx}\), with \(x\in(a,b)\). \(V(x)\) is a sufficiently smooth real-valued function in \((a,b)\). A linear differential \(N\)th-order operator \(\widehat L^{(N)}\), with coefficient of \(D^N=1\), acting from \(T_0\equiv \{\Psi:(h_0-E) \Psi(x)=0\}\) to \(T_{N1}= \{\varphi:\varphi= \widehat L\Psi,\;\Psi\in T_0\}\) is a Darboux transformation of order \(N\), if \(\widehat L^{(N)}h_0= (h_0+A_N(x)\widehat L^{(N)})\), where \(A_N(x)\) is a sufficiently smooth function. It is established, that ``The action of any nontrivial operator \(\widehat L^{(N)}\) is equivalent to the resulting action of a certain chain of \(k(\leq N)\) Darboux transformations of first order''.