## Inversion of reflectivity data for nondecaying potentials.(English)Zbl 0956.34073

Given the 1D Schrödinger equation $\psi''(k,x)+k^2\psi(k,x)=V(x)\psi(k,x),\qquad x\in{\mathbb{R}},$ where the potential $$V$$ is real and satisfies, for a constant $$c\geq 0$$, $\int_{-\infty}^0(1+|x|)|V(x)|dx+\int_0^\infty(1+x)|V(x)-c^2|dx<\infty,$ let $$f_l(k,x)$$ and $$f_r(k,x)$$ be the Jost solutions defined by $e^{-i\gamma x}f_l(k,x)=1+o(1),\;x\to+\infty;\qquad e^{ikx}f_r(k,x)=1+o(1),\;x\to-\infty,$ and $$L(k)$$, $$T_l(k)$$, $$R(k)$$ and $$T_r(k)$$ the scattering coefficients defined by $e^{-ikx}T_l(k)f_l(k,x)=1+L(k)e^{-2ikx}+o(1), \quad x\to-\infty,$
$e^{i\gamma x}T_r(k)f_r(k,x)=1+R(k)e^{2i\gamma x}+o(1), \quad x\to+\infty,$ where $$\gamma=(k^2-c^2)^{1/2}$$ having nonnnegative imaginary part. Using subscripts $$1,2$$ for quantities pertaining to the potentials $$V_1(x)=V(x)H(-x)$$ and $$V_2(x)=V(x)H(x)$$, where $$H(x)$$ is the Heaviside unit step function, as well as circles in the complex plane containing the values of $$F(k)=1-R_1(k)L_2(k)$$ for $$k\in{\mathbb{R}}$$, various three measurement uniqueness results are derived. For instance, if $$V_1,V_2$$ do not have bound states and $$V_1\not\equiv 0$$, then $$V_2$$ is uniquely determined by $$c>0$$ and $$\{R_1(k),|L(k)|,|L_2(k)|\}$$ for $$k>c$$. A numerical method for computing $$V_2$$ is provided and implemented.

### MSC:

 34L25 Scattering theory, inverse scattering involving ordinary differential operators 81U40 Inverse scattering problems in quantum theory 34A55 Inverse problems involving ordinary differential equations
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