论文标题
关于某些Schrödinger-Type微分方程的WKB解决方案的Borel总结性
On the Borel summability of WKB solutions of certain Schrödinger-type differential equations
论文作者
论文摘要
考虑了具有较大参数$ u $的Schrödinger型二阶线性微分方程。可以通过（发散）正式系列以$ u $ $ u $的（发散）形式系列来描述这种类型的方程式的分析解决方案。这些正式的系列解决方案称为WKB解决方案。我们表明，在方程式的潜在功能的轻度条件下，WKB解决方案在自变量的大型无界域中相对于参数$ u $可总结。可以确定的是，正式的系列扩展是Borel重新点燃解决方案的独立变量均匀的渐近扩展，我们就其误差项提供了可计算的界限。另外，证明WKB解决方案可以使用参数中的阶段序列表示，并且这些扩展相对于自变量均匀地收敛于半平台。我们通过应用于与旋转谐波振荡器和贝塞尔方程相关的径向schrödinger方程来说明我们的理论。
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.
