Second order linear differential equations with a basis of solutions having only real zeros

Abstract

Let $A$ be a transcendental entire function of finite order. We show that if the differential equation $w^{\prime \prime }+Aw=0$ 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.

46 pages, 5 figures. V2: some overall revision of v1

References in corpus (3)

• Dynamics of meromorphic functions with direct or logarithmic singularities
• Meromorphic functions with linearly distributed values and Julia sets of rational functions
• The Eremenko-Lyubich constant