Algebraic-differential transformations of linear differential operators of arbitrary order and their spectral properties applicable to the inverse problem. I. The case of finite $\mathfrak N$

Linear differential operators $R$ of order $n$ from $C^n[0,1]$ into $C[0,1]$, i.e. without boundary conditions, are discussed. With $\lambda$ complex, let $Z^R_\lambda$ denote the linear space of all solutions $z(x)\in C^n[0,1]$ of the homogeneous equation $Rz=\lambda z$. We use die operator $R$ and certain of its spectral properties to obtain an operator $L$ analogous to $R$. Our main result is to obtain expressions defining a linear mapping $T_\lambda\colon Z_\lambda^R\to Z_\lambda^L$ (Theorem 2.6). The linear mappings $T_\lambda$ are meromorphically dependent on $\lambda$.
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