A class of Schrödinger-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 Schrödinger equation associated with the problem of a rotating harmonic oscillator and to the Bessel equation.

一类大参数薛定ding型二阶线性微分方程 $ü$被认为。这类方程的解析解可以通过（发散）形式级数来描述$ü$。这些正式的系列解决方案称为WKB解决方案。我们证明，在温和条件下，方程的势函数，WKB解对于参数而言是Borel可加的$ü$在自变量的大型无界域中。可以确定，形式级数展开是Borel重新求和的解的渐近展开，相对于自变量而言是均匀的，并且我们提供了它们的误差项的可计算范围。另外，证明了可以使用参数中的阶乘级数来表达WKB解，并且这些展开关于自变量均匀地收敛在半平面中。我们通过将其应用于与旋转谐波振荡器问题相关的径向Schrödinger方程和Bessel方程来说明我们的理论。