Geodesic
On the decay rate of solutions to the stationary Schr\"odinger equation with a potential depending on one variable
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Physics, Tome 225 (2023), pp. 115-122

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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.
Keywords: stationary Schrödinger equation, Sturm–Liouville operator, transformation operator, Landis problem, a priori estimate, Neumann series, Bessel function. 
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     author = {S. M. Sitnik},
     title = {On the decay rate of solutions to the stationary {Schr\"odinger} equation with a potential depending on one variable},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {115--122},
     publisher = {mathdoc},
     volume = {225},
     year = {2023},
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     url = {http://geodesic.mathdoc.fr/item/INTO_2023_225_a9/}
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S. M. Sitnik. On the decay rate of solutions to the stationary Schr\"odinger equation with a potential depending on one variable. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Physics, Tome 225 (2023), pp. 115-122. http://geodesic.mathdoc.fr/item/INTO_2023_225_a9/