The inverse problem for ordinary differential operators on the line.

*(English)*Zbl 0579.34008In this interesting article the author investigates the linear differential operator (n\(\geq 2)\)
\[
(1)\quad P=D^ n+q_{n-2}D^{n- 2}+...+q_ 0,\quad D=(1/i)d/dx
\]
\(q_ j\in L^ 1(R)\) for \(j=0,1,...,n-2\), being generally not self-adjoint. The inverse problem for the operator (1) on the line is the problem of determining the coefficients \(q_ 0,q_ 1,...,q_{n-2}\) from the knowledge of the asymptotic behaviour in x of the eigenfunctions \(\psi_ j(x,z)\), \(j=1,2,...,n\), which are a solution of the equation \(P\psi =z^ n\psi\), \(z\in C\). This family is meromorphic with respect to z in \(C-C_ 1\) where \(C_ 1\) is a union of lines through the origin. The scattering data describes the singularities in \(C-C_ 1\) and the jumps across the rays of \(C_ 1\). Let N be a nonnegative integer. For a dense open set \(Q^ d_ N\) in the Banach space Q of coefficients satisfying \(\sum^{n-2}_{j=0}\int_{R}(1+| x|)^{2N+2n-2}| q_ j(x)| dx\) the singularities in \(C-C_ 1\) are a finite set of simple poles, and the data on \(C_ 1\) is an \(n\times n\) matrix-valued function \(\nu\) from the class \(C^ N\) on each ray. If two elements of \(Q^ d_ N\) have the same scattering data, then they are equal. The function \(\nu\) satisfies various algebraic constraints, and there are also certain winding-number constraints relating \(\nu\) and the discrete data.

It is also shown that there may be introduced a metric space SD of formal scattering data, whose elements satisfy the algebraic and winding-number constraints mentioned above. There is a dense open subset of SD for which every element is the scattering data for an operator (1).

It is also shown that there may be introduced a metric space SD of formal scattering data, whose elements satisfy the algebraic and winding-number constraints mentioned above. There is a dense open subset of SD for which every element is the scattering data for an operator (1).

Reviewer: S.Stanek