Geodesic
The Riemann--Hilbert problem for a one-dimensional Schrodinger operator with a potential in the form of a sum of a parabola and a finite potential
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 53, Tome 521 (2023), pp. 240-258

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The paper is devoted to the study of the Riemann–Hilbert problem for the Schrodinger operator $L=-\frac{d^2}{dx^2}-\frac{x^2}{4}+q(x)$ with a potential as the sum of a parabola (with branches down) and a smooth finite potential $q(x)$. The constructed Riemann–Hilbert problem can be considered as a construction of a direct scattering problem for a given operator. 
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     title = {The {Riemann--Hilbert} problem for a one-dimensional {Schrodinger} operator with a potential in the form of a sum of a parabola and a finite potential},
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V. V. Sukhanov. The Riemann--Hilbert problem for a one-dimensional Schrodinger operator with a potential in the form of a sum of a parabola and a finite potential. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 53, Tome 521 (2023), pp. 240-258. http://geodesic.mathdoc.fr/item/ZNSL_2023_521_a12/