On the monodromy of the ramified Cauchy problem
Let \(a(x,D)=\sum_{\mid\alpha\mid\leq m}a_\alpha(x)D^\alpha\) be a linear differential operator of order \(m\) with coefficients \(a_\alpha(x)\) which are holomorphic in a neighbourhood of the origin of \(\mathbb C^{n+1}\). For a \((n+1)\)-tuple \(x=(x_0,\ldots,x_n)\) the corresponding \(n\)-tuple \((x_1,\ldots,x_n)\) will be denoted by \(x'\). Suppose that \(a_{(m,0,\ldots,0)}=1\) and that \(a(x,D)\) is an operator with multiple characteristics of constant multiplicity, meaning that there exist holomorphic function germs \(\lambda_i(x;\xi')\) at \((0;1,0,\ldots,0)\) and integers \(m_i \geq 1\) such that we can write the principal symbol \(g(x,\xi)\) of \(a(x,D)\) as \(g(x,\xi)=\prod_{i=1}^d(\xi_0 - \lambda_i(x,\xi'))^{m_i}\) with \(\lambda_i(0;1,0,\ldots,0) \neq \lambda_j(0;1,0,\ldots,0)\) if \(i\neq j\). Then the following result is due to \textit{E. Leichtnam} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 23, 369--443 (1990; Zbl 0717.35018)]. Denote \(S=\{x_0=0\}\) and \(T=\{x_0=x_1=0\}\). Let \(\Omega\) be an open connected neigbourhood of the origin of \(\mathbb C^{n+1}\) such that \(\Omega \cap S\) is connected and let \(v\) and \(w_h\), \(0\leq h < m\), be holomorphic functions on the universal covering of \(\Omega \setminus K\) and \(\Omega \cap S \setminus T\), respectively. Then there exists an open connected neighbourhood \(\Omega' \subseteq \Omega\) of the origin of \(\mathbb C^{n+1}\) and a unique solution \(u\) of the ramified Cauchy problem \(a(x,D)u(x)=v(x)\), \(D_0^hu(x)=w_h(x)\) for \(x_0=0\), \(0\leq h < m\), that is holomorphic on the universal covering of \(\Omega' \setminus K\). Here \(K=\bigcup_{i=1}^dK_i\) with \(K_i\) the characteristic hypersurfaces issuing from \(T\). Leichtnam also proved that, if \(v\) and \(w_h\) are of finite determination, then also the solution \(u\) is of finite determination; and if the monodromy of \(v\) is solvable, then the monodromy of \(u\) is solvable. In this paper the author studies the eigenvalues of the monodromy of the solution in more detail and relates them to those of the data of the problem. The solvability of the monodromy of the solution is an easy consequence of his results. In order to prove these facts the author first specifies the spectrum of the monodromy in the setting of the theorem of Hamada-Takeuchi [\textit{Y. Hamada} and \textit{A. Takeuchi}, C. R. Acad. Sci., Paris, Sér. I 295, 377--380 (1982; Zbl 0505.35015) and Taniguchi Symp., Hyperbolic equations and related topics (Katata/Kyoto, 1984), 51--62 (1986; Zbl 0726.35002)].
scientific article; zbMATH DE number 1975946