论文标题
二阶线性微分方程具有仅具有真实零的解决方案的基础
Second order linear differential equations with a basis of solutions having only real zeros
论文作者
论文摘要
令$ a $是有限订单的先验全部功能。我们表明，如果微分方程$ w''+aw = 0 $具有两个线性独立的解决方案，仅具有真实的零，则$ a $的顺序必须是奇数整数或一个奇数整数的一半。此外，从莱文和pfluger的意义上讲，$ a $完全定期增长。这些结果来自更通用的几何定理，该定理将所有零和极点位于真实轴上的对称局部同构归类为对称的局部同态，并且仅在有限的非零值上具有有限的许多奇异性。
Let $A$ be a transcendental entire function of finite order. We show that if the differential equation $w''+Aw=0$ has two linearly independent solutions with only real zeros, then the order of $A$ must be an odd integer or one half of an odd integer. Moreover, $A$ has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.
