"On the location of zeros of solutions to w"" + Aw = 0 for certain entire functions A"
"Consider the differential equation $(\*)$ $w\sp{\prime\prime}$ + $Aw$ = 0 where $A$ is of the form $A$($z$) = $\sum\sbsp{j=1}{m}Q\sb{j}(z)$exp$P\sb{j}(z)$ with $Q\sb{j}$ and $P\sb{j}$ polynomials in $z$. We study the location in the complex plane of the zeros of solutions to $(\*)$. Under mild hypotheses on the $P\sb{j}$'s and $Q\sb{j}$'s, we show that there exist both ""zero-scarce"" regions and ""zero-rich"" regions. We call an unbounded region in the complex plane ""zero-scarce"" if every nontrivial solution of $(\*)$ has only finitely many zeros in the region. In contrast, a region is called ""zero-rich"" if there exists a nontrivial solution of $(\*)$ with infinitely many zeros in the region."
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign