论文标题
精确的解决方案的精确解决方案和精确的WKB分析
Exact Solutions for the Singularly Perturbed Riccati Equation and Exact WKB Analysis
论文作者
论文摘要
在复杂域中，奇异的riccati方程是一阶非线性ode $ \ hbar \ hbar \ partial_x f = af^2 + bf + c $，其中$ \ hbar $是一个小的复杂参数。我们证明了在半Plane中，具有规定的渐近解决方案为$ \ hbar \至0 $的精确解决方案的存在和唯一定理。这些精确的解决方案是使用Borel-Laplace方法构建的。即，它们是正式发散$ \ hbar $ - 功率系列解决方案的Borel总结。作为一种应用，我们证明了具有合理潜力的复杂的一维schrödinger方程的精确WKB解决方案的存在和唯一性。
The singularly perturbed Riccati equation is the first-order nonlinear ODE $\hbar \partial_x f = af^2 + bf + c$ in the complex domain where $\hbar$ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as $\hbar \to 0$ in a halfplane. These exact solutions are constructed using the Borel-Laplace method; i.e., they are Borel summations of the formal divergent $\hbar$-power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.
