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Consider a partial differential equation that is of the following form: \begin{equation} (-\partial_x^2+g(x))f(x, t)=i\partial_tf(x, t) \end{equation} where $g(x)$ is a real function. Suppose that $f(x, t=0)=f_1(x)$ and $f(x, t=T)=f_2(x)$. Is it possible to reconstruct $g(x)$ from $f_1(x)$, $f_2(x)$ and $T$? If so, how to do it and is the solution unique?

This question can also be formulated in terms of the Green's function of this problem. Here the Green's function $G(x, x', t)$ is defined such that \begin{equation} (-\partial_x^2+g(x))G(x, x', t)=i\partial_tG(x, x', t) \end{equation} with the initial condition $G(x, x' ,t=0)=\delta(x-x')$. Then the above problem becomes: from $f_1(x)$, $f_2(x)$ and $T$, is it possible to reconstruct $G(x, x', t)$? If so, how to do it and is the solution unique?

Such an inverse problem is usually ill-posed, but now there are some constraints on $G(x, x', t)$ (i.e. $G(x, x', t)$ is the solution of the above initial-value problem), maybe it is possible to find such a $G(x, x', t)$ almost uniquely.

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  • $\begingroup$ certainly not; think of $f_1$ and $f_2$ as vectors that are linearly related by an unknown matrix $G$; two vectors can obviously not determine a full matrix; for example, if $f_1$ would be an eigenfunction of $G$ you would only know one single eigenvalue. $\endgroup$ Commented Jun 12, 2018 at 17:15
  • $\begingroup$ @CarloBeenakker Thanks for the comment. Usually this is not possible, but now that there is some constraint on $G$ (i.e. $G$ is the Green's function of that equation), maybe it becomes possible. One (probably inappropriate) analog is: given a single eigenfunction of the operator on the LHS of the above differential equation, say, $f(x)$, is it possible to reconstruct $g(x)$? Usually one would think it is not possible, but now this is possible: $g(x)=\lambda+\frac{\partial_x^2f(x)}{f(x)}$ with $\lambda$ the corresponding eigenvalue. In this way, $g(x)$ is uniquely determined up to a constant. $\endgroup$ Commented Jun 12, 2018 at 18:23

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One can regard your problem as probing a medium by a known source $f_1$ generated at time $t=0$, recording the medium response $f_2$ at time $t=T$ (called the data) and the goal is the recovery of the medium property $g(x)$.

If your data consists of $G(x,x',T)$ for all $x,x'$ then it is equivalent to having data generated by all possible sources $f_1$ and that inverse problem is easier than the problem where you have data from a single $f_1$. I assume you are interested in the single $f_1$ data problem.

Of course $f_1$ must be a non-zero function otherwise $f_2$ carries no information about the medium. I believe there are no results for a (single) general $f_1$ but there are results (at least for hyperbolic PDEs) when $f_1$ is an impulsive source such as $f_1(x)=\delta(x)$ or (for bounded domain problems) when $f_1(x)>0$ for all $x$ in the domain.

In inverse problems, typically one studies problems where data consists of measurements on the boundary since usually one is unable to "reach" the interior of the medium. Your problem uses interior measurements. I suggest two references as good starting points. The first one is for your PDE but for "boundary data" (measurement on the boundary) and the second one is with your data (that is measurement at $t=T$) but for the heat equation.

(i) Reconstructing the potential for the 1D Schr¨odinger equation from boundary measurements by Avdonin, Mikhaylov, Ramdani in IMA Journal of Mathematical Control and Information, Oxford University Press (OUP), 2014, 31 (1), pp.137-150. ff10.1093/imamci/dnt009ff.ffhal-00804268f .

(ii) Section 9.1 in Inverse Problems for Partial Differential Equations by Victor Isakov.

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