On the Darboux transformation of the second order differential operator of Fuchsian type on the Riemann sphere
The main purpose of the present paper is to clarify some analytical properties of the Darboux transformation of the second order ordinary differential operator \(L(P)=D^ 2-P(x)\), \(D=d/dx\), of Fuchsian type on the Riemann sphere \({\mathbb{P}}_ 1\). For brevity it is assumed that P(x) is of the form \((*)\quad P(x)=\sum^{n}_{j=1}\alpha_ j(x-a_ j)^{- 2}.\) The contents of the paper are as follows. In section 1 preliminary considerations about the Darboux transformation are made. Thus, if \({\mathbb{F}}\) is the set of all L(P) such that P(x) is of the form (*) and the ratio \(y_ 2(x)/y_ 1(x)\) of the fundamental system of solutions of \(L(P)y=y''-P(x)y=0\) is a rational function, then it is shown that the Darboux transformation L(P;\(\zeta\),Y) of L(P) is of Fuchsian type on \({\mathbb{P}}_ 1\) for any \(\zeta \in {\mathbb{P}}_ 1\) if and only if L(P)\(\in {\mathbb{F}}\). In section 2, the Darboux transformation is investigated in connection with the isomonodromic deformation. In section 3, the relation between Darboux transformation and the Lenard relation is discussed. Section 4 is devoted to the construction of the space of rational function solutions of the higher order KdV equation \(\partial u(x,\xi)/\partial \xi =X_ n(u(x;\xi)),\) where \(\xi\) is a complex variable and \(X_ n\) is given by the Lenard relation.
second order ordinary differential operator
scientific article; zbMATH DE number 4169077