论文标题
确切可解决的非线性特征值问题
Exactly solvable nonlinear eigenvalue problems
论文作者
论文摘要
研究了一类二阶半透明微分方程的非线性特征值问题。非线性特征值定义为产生分离溶液的初始条件。半透明方程可以将一次集成到一阶非线性方程中，例如ricatti方程。结果表明，这些半跨性方程的非线性特征值问题等同于线性特征值问题。他们共享完全相同的特征值。这两个问题中的征量密切相关。准确地解决了量子力学中（一半）谐波振荡器等效的非线性特征值问题。这是第一个可解决的非线性特征值问题。还研究了一些扩展方程的非线性特征值问题。
The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied. A nonlinear eigenvalue is defined as the initial condition which gives rise a separatrix solution. A semi-transcendental equation can be integrated once to a first order nonlinear equation, e.g., the Ricatti equation. It is shown that the nonlinear eigenvalue problems of these semi-transcendental equations are equivalent to linear eigenvalue problems. They share the exactly same eigenvalues. The eigensolutions in the two problems are closely related. The nonlinear eigenvalue problem equivalent to the (half) harmonic oscillator in quantum mechanics is solved exactly. This is the first solvable nonlinear eigenvalue problem. The nonlinear eigenvalue problems of some extended equations are also studied.
