Khanmamedov A. Kh.
On the inverse scattering problem for the one-dimensional Schrödinger equation with growing potential
Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1390-1402
We consider a one-dimensional Schrödinger equation on the entire axis whose potential rapidly decreases at the left end and infinitely increases at the right end. By the method of transformation operators, we study the inverse scattering problem. We establish conditions for the scattering data under which the inverse problem is solvable. The basic Marchenko-type integral equations are investigated and their unique solvability is established.
On conditions for the discreteness of the spectrum of a semiinfinite Jacobi matrix with zero diagonal
Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 285–288
We establish sufficient conditions for the discreteness of the spectrum of a second-order self-adjoint difference operator generated by a semiinfinite Jacobi matrix with zero principal diagonal.
Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1144 – 1152
Using the inverse scattering transform, we investigate an initial boundary-value problem with zero boundary condition for the Toda lattice. We prove the existence and uniqueness of a rapidly decreasing solution and determine a class of initial data that guarantees the existence of a rapidly decreasing solution.