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

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.