发表机构
University of Science and Technology Beijing（北京科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Tumura--Clunie方法求解Hill方程的非振荡解，建立其与Liouvillian解的对应关系，并针对特定方程确定具有零点性质的非振荡解。
AI 中文摘要
我们考虑Hill方程 $f''-(\sum_{i=-\mathbf{l}}^{\mathbf{k}}b_{i}e^{iz})f=0$ ($†$)，其中 $\mathbf{k}\geq 1$ 和 $\mathbf{l}\geq 0$ 是整数，$b_{-\mathbf{l}}$，$\cdots$，$b_{\mathbf{k}}$ 是常数且 $b_{\mathbf{k}}\not=0$。我们指出，方程($†$)的满足 $\lambda(f)<\infty$ 的非振荡解类与方程 $x^2u''-(\sum_{i=-\mathbf{l}}^{\mathbf{k}}b_{i}x^{i})u=0$ ($‡$) 的Liouvillian解类之间存在完全对应关系。本文有双重目的。第一，与Kovacic算法寻找方程($‡$)的Liouvillian解相平行，我们发展Tumura--Clunie方法来寻找Hill方程高阶版本的非振荡解。第二，对于特定的Hill方程 $f''-(b_{\mathbf{k}}e^{\mathbf{k}z}+b_{\mathbf{s}}e^{\mathbf{s}z}+b_0)f=0$，其中 $\mathbf{k}>\mathbf{s}\geq 1$ 是整数且 $b_{\mathbf{k}}b_{\mathbf{s}}\not=0$，我们使用Tumura--Clunie方法确定具有额外零点性质的非振荡解 $f$。
英文摘要
Let $\varrho\neq -1$ be a complex number with modulus $|\varrho|=1$ and $K(x,y)=P_1(x)+P_2(y)$ for two polynomials $P_1$ and $P_2$. We consider the non-oscillatory solutions such that $λ(f)<\infty$ of the generalized Hill equation $f''-K(e^z,e^{-\varrho z})f=0$ ($\sharp$). When $\varrho=1$, the Hill equation $f''-K(e^z,e^{-z})f=0$ ($†$) is also written as $x^2u''-[K(x,x^{-1})-1/4]u=0$ ($‡$). We point out that there is a full correspondence between the non-oscillatory solutions of equation ($†$) and the Liouvillian solutions of equation ($‡$). Then this paper has two purposes. First, we show that if equation ($\sharp$) has a nonzero non-oscillatory solution, then $\varrho=1$. To this end, we solve entire solutions of a general Tumura--Clunie type differential equation. Second, for the particular Hill equation $f''-(e^{\mathbf{k}z}+b_{\mathbf{s}}e^{\mathbf{s}z}+b_0)f=0$, where $\mathbf{k}>\mathbf{s}\geq 1$ are integers and $b_{\mathbf{s}}\not=0$, we use Kovacic's algorithms to determine the non-oscillatory solutions with relatively few zeros.
Comments24 pages; a very simple proof for the main theorem in previous version
