On the Borel summability of WKB solutions of certain Schrodinger-type differential equations

JOURNAL OF APPROXIMATION THEORY

卷:265

Article

Nemes, Gergo¹ 

[1] Alfred Renyi Inst Math, Realtanoda Utca 13-15, H-1053 Budapest, Hungary

关键词: Asymptotic expansions;    Borel summability;    Error bounds;    Factorial series;    WKB analysis;   

DOI  :  10.1016/j.jat.2021.105562

来源: Elsevier

【 摘 要 】

A class of Schrodinger-type second-order linear differential equations with a large parameter u is considered. Analytic solutions of this type of equations can be described via (divergent) formal series in descending powers of u. These formal series solutions are called the WKB solutions. We show that under mild conditions on the potential function of the equation, the WKB solutions are Borel summable with respect to the parameter u in large, unbounded domains of the independent variable. It is established that the formal series expansions are the asymptotic expansions, uniform with respect to the independent variable, of the Borel re-summed solutions and we supply computable bounds on their error terms. In addition, it is proved that the WKB solutions can be expressed using factorial series in the parameter, and that these expansions converge in half-planes, uniformly with respect to the independent variable. We illustrate our theory by application to a radial Schrodinger equation associated with the problem of a rotating harmonic oscillator and to the Bessel equation. (C) 2021 The Author(s). Published by Elsevier Inc.Y

【 授权许可】

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