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The inverse problem for ordinary differential operators on the line. (English) Zbl 0579.34008
In 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).
Reviewer: S.Stanek

##### MSC:
 34A55 Inverse problems involving ordinary differential equations 34L99 Ordinary differential operators 34E10 Perturbations, asymptotics of solutions to ordinary differential equations
##### Keywords:
scattering data; winding-number constraints
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