On the decay rate of solutions to the stationary Schrödinger equation with a potential depending on one variable

In 1982, E. M. Landis posed the problem of exact estimates for the exponential decay rate of solutions to the stationary Schrödinger equation. A few years later, this problem in its original formulation was solved by the Voronezh mathematician V. Z. Meshkov. He constructed an example of a solution that decreases superlinearly at infinity, which gives a negative answer to the original question in Landis' problem. In this paper, we prove that for some potentials of a special form, nevertheless, the answer to the question in Landis' problem may be positive. Some generalizations and modern results in this direction are also presented.