Abstract
We obtain asymptotic formulas for the solutions of the one-dimensional Schrödinger equation − y″ +q(x)y = 0 with oscillating potential q(x)=x β P(x 1+α)+cx −2 as x→ +∞. The real parameters α and β satisfy the inequalities β − α ≥ −1, 2α − β > 0 and c is an arbitrary real constant. The real function P(x) is either periodic with period T, or a trigonometric polynomial. To construct the asymptotics, we apply the ideas of the averaging method and use Levinson’s fundamental theorem.
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Translated from Matematicheskie Zametki, vol. 80, no. 2, 2006, pp. 240–250.
Original Russian Text Copyright © 2006 by P. N. Nesterov.
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Nesterov, P.N. Construction of the asymptotics of the solutions of the one-dimensional Schrödinger equation with rapidly oscillating potential. Math Notes 80, 233–243 (2006). https://doi.org/10.1007/s11006-006-0132-5
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DOI: https://doi.org/10.1007/s11006-006-0132-5