含时Schroedinger方程、Riccati方程和Airy函数
The time-dependent Schroedinger equation, Riccati equation and Airy functions
摘要：
我们用Airy函数构造了$iψ_{t}+{1/4}ψ_{xx}\pm tx^{2}ψ=0$形式的Schroedinger方程的Green函数（或Feynman传播子），并在坐标和动量表象下求解了Cauchy初值问题.首先详细研究了量子参量振荡器的一种特殊情况，并给出了相应的变系数非线性薛定谔方程的特解。格林函数是明确给出的艾里函数和相应的过渡振幅的超几何函数。量子参量振荡器的一般情况，然后考虑在类似的方式。建立了跃迁振幅的群论意义及其与巴格曼函数的关系。
英文摘要：
We construct the Green functions (or Feynman's propagators) for the Schroedinger equations of the form $iψ_{t}+{1/4}ψ_{xx}\pm tx^{2}ψ=0$ in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schroedinger equations with variable coefficients are also found. A special case of the quantum parametric oscillator is studied in detail first. The Green function is explicitly given in terms of Airy functions and the corresponding transition amplitudes are found in terms of a hypergeometric function. The general case of quantum parametric oscillator is considered then in a similar fashion. A group theoretical meaning of the transition amplitudes and their relation with Bargmann's functions is stablished.
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