一类具有n阶转向点的大参数奇摄动方程的特征值问题

A Class of Singularly Perturbed Eigenvalue Problem for Large Parameter with Turning Point of n-th Order

讨论一类具有n阶转向点的大参数奇摄动方程的特征值问题。首先,利用一个特殊的变换将原具有n阶转向点的大参数问题转化为便于讨论的问题;其次,求得了用特殊函数表示的渐近解,并研究所得到解的性态,指出了该解为具有n阶转向点的大参数奇摄动问题渐近解的一致有效性;再次,利用所给的边界条件求出了原问题的特征函数和特征值;最后,用定理形式论述相应的大参数问题的结果,并指出当n=1时与前人讨论的结果一致。

A class of possessing n-th order turning point eigenvalue problem of singular perturbation equation for large parameter is discussed. Firstly, substituting the original possessing n-th order turning point problem for large parameter is turned into studied problem; Secondly, the asymptotic solution using the characteristic function representation is obtained; Then the behavior of the received solution is studied, and this asymptotic solution of possessing n-th order turning point eigenvalue problem for large parameter is uniformly valid; And then, the characteristic functions and characteristic values for the original problem are solved by using the boundary conditions;Finally, the conclusion of corresponding large parameter problem is discussed using the theorem, and it is pointed out that it is the same with that of previous study when n is equal to one.