十阶势Schroedinger方程的精确解和近似解
Exact and approximate solutions to Schroedinger's equation with decatic potentials
摘要：
分析了一维薛定谔方程十次多项式势精确解的存在性。在势参数一定的条件下，证明了十次多项式势$V(x)=ax^{10}+bx^8+cx^6+dx^4+ex^2$，$a>0$是精确可解的.通过研究一类多项式系数线性微分方程的多项式解，给出了相应多项式解依赖于能量的充分必要条件.它还表明，这些多项式满足四项递推关系，其实根是确切的能量本征值。此外，它表明，这些多项式产生相应的薛定谔方程的本征函数的解决方案。本文还用渐近迭代法对势参数的任意值作了进一步的分析。
英文摘要：
The one-dimensional Schroedinger's equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential's parameters, we show that the decatic polynomial potential $V(x)=ax^{10}+bx^8+cx^6+dx^4+ex^2$, $a>0$ is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schroedinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.
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