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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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