Analytic solutions of \(n\)-th order differential equations at a singular point (Q2778477)

The author considers the linear nonhomogeneous ordinary differential equation with complex time NEWLINE\[NEWLINELy(z)=g(z),\quad Ly=z^n\frac{d^ny}{dz^n}+z^{n-1}a_1(z) \frac{d^{n-1}y}{dz^{n-1}}+\dots+a_n(z)y,\tag{1}NEWLINE\]NEWLINE where the functions \(a_i(z)\), \(g(z)\) belong to the space \(A_0\) of functions holomorphic on the unit disc and continuous in its closure. He studies the solvability of the equation in the class of functions \(y(z)\) holomorphic in a neigborhood of zero. The author presents a method that gives a necessary and sufficient condition for the solvability of (1) in the space \(A_n\) of functions \(y\) holomorphic on the unit disc and continuous in its closure with all the derivatives up to the order \(n\). It turns out that 1), for a generic operator \(L\), (1) has a solution in \(A_n\) for any \(g\in A_0\); 2) in the general case, for a given operator \(L\), the solvability of (1) is always determined by a finite number of the Taylor coefficients of \(g\). The main results are proved by introducing appropriate Banach norms in the spaces \(A_0\), \(A_n\) and applying the Fredholm alternative. This yields that (1) is solvable in \(A_n\), if and only if \(L^*g=0\) (thus, to write down the solvability condition explicitly, it suffices to describe the cokernel of \(L\)). It turns out that the last equation can be written explicitly as a system of equations of a finite number of coefficients in the function \(g\). At the end of the paper, the author considers Bessel equations and writes down explicitly the solvability criterium \(L^*g=0\) for them. For the first one of them,he reproves the solvability criterium obtained earlier by \textit{L. M. Hall} [SIAM J. Math. Anal. 8, 778-784 (1977; Zbl 0366.34053)]. The question to find a criterium of solvability for a more general linear equation in the space of holomorphic functions near a singular point was stated by \textit{H. L. Turrittin} [Sympos. ordinary diff. Equations, Minneapolis, Minnesota 1972, Lect. Notes Math. 312 (Berlin, Springer-Verlag), 1-22 (1973; Zbl 0272.34044)] and answered by \textit{L. J. Grimm} and \textit{L. M. Hall} [J. Differ. Equations 18, 411-422 (1975; Zbl 0312.34045)] applying a method similar to that from the paper under review, but the cokernel of \(L\) (which gives the solvability criterium \(L^*g=0\)) was not described explicitly.