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Inverse scattering on the line with incomplete scattering data. (English) Zbl 1072.34010

Conca, Carlos (ed.) et al., Partial differential equations and inverse problems. Proceedings of the Pan-American Advanced Studies Institute on partial differential equations, nonlinear analysis and inverse problems, Santiago, Chile, January 6–18, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3448-7/pbk). Contemporary Mathematics 362, 1-11 (2004).
Consider the equation \[ -y''+V(x)y=k^2y,\; -\infty<x<\infty, \tag{1} \] where \(V(x)\) is real and \((1+| x| )| V(x)| \in L(-\infty,\infty)\). Let \(e(x,k)\) and \(g(x,k)\) are the Jost solutions for (1) such that \[ e(x,k)=\exp(ikx)(1+o(1))\text{ as }x\to +\infty,\quad g(x,k)=\exp(-ikx)(1+o(1))\text{ as } x\to -\infty. \] Then \(e(x,k)=a(k)g(x,-k)+b(k)g(x,k)\). The inverse problem of recovering the potential \(V(x)\) from the given \(b(k)\) is studied, and the nonuniqueness for this incomplete inverse problem is analysed.
For the entire collection see [Zbl 1052.35004].

MSC:

34A55 Inverse problems involving ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L25 Scattering theory, inverse scattering involving ordinary differential operators
47A40 Scattering theory of linear operators
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