Geodesic
Inverse source problem for the 1-D Schr\"odinger equation
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 41, Tome 393 (2011), pp. 5-11

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We consider the inverse problem of determining a source in the dynamical Schrödinger equation $iu_t-u_{xx}+q(x)u=w(t)a(x)$, $0$, with Dirichlet boundary conditions and zero initial condition. From the measurement $u_x(0,t)$, $0$, we recover unknown $a(x)$ provided $q(x)$ and $w(t)$ are given. We describe also how to recover $a(x)$ and $q(x)$ from the measurements at the both boundary points. 
@article{ZNSL_2011_393_a0,
     author = {S. A. Avdonin and V. S. Mikhaylov},
     title = {Inverse source problem for the {1-D} {Schr\"odinger} equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--11},
     publisher = {mathdoc},
     volume = {393},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_393_a0/}
}
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S. A. Avdonin; V. S. Mikhaylov. Inverse source problem for the 1-D Schr\"odinger equation. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 41, Tome 393 (2011), pp. 5-11. http://geodesic.mathdoc.fr/item/ZNSL_2011_393_a0/