## 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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