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Construction of the half-line potential from the Jost function. (English) Zbl 1071.34006

Inverse Probl. 20, No. 3, 859-876 (2004); corrigendum 20, No. 4, 1355 (2004).
Let \(f_l(k,x)\) and \(f_r(k,x)\) be the left and the right Jost solutions, respectively, to the Schrödinger equation \[ \Psi^{\prime\prime}(k,x)+k^2\Psi(k,x)=V(x)\Psi(k,x), \;\;x\in \mathbb{R}, \eqno{(1)} \] with \[ V(x)=c\delta(x)+U(x), \] where \(c\) is a real constant, \(\delta(x)\) is the Dirac delta-distribution and \(U(x)\) is a real-valued potential vanishing for \(x<0\) and such that \(\int_0^{\infty}| U(x)| (1+x)dx<\infty\). The left and the right Jost solutions of (1) are specified by \[ f_l(k,x)=e^{ikx}[1+o(1)], \;\;x\to +\infty, \]
\[ f_r(k,x)=e^{-ikx}[1+o(1)], \;\;x\to -\infty. \] The following data sets are introduced: \[ {\mathcal D}:=\{f_l(k,0): \;k\in \mathbb{R}\}, \;\;{\mathcal D}^{\prime}:= \{f^{\prime}_l(k,0^-): k\in \mathbb{R}\}, \]
\[ {\mathcal E}:=\{| f_l(k,0)| : \;k\in \mathbb{R}\}, \;\;{\mathcal E}^{\prime}:= \{| f^{\prime}_l(k,0^-)| : k\in \mathbb{R}\}. \] The author investigates what data uniquely determine the potential \(V(x)\). For example, he shows that the data set \({\mathcal E}\cup c\) uniquely determines \(V\).

MSC:

34A55 Inverse problems involving ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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