Solutions of \(f''+A (z) f=0\) in the unit disc having Blaschke sequences as the zeros
The zero sequences of the nontrivial solutions of \(f''+ A(z)f= 0\) are studied, where \(A(z)\) is analytic in \(D= \{z:|z|< 1\}\). The author shows that if \(A(z)\) is analytic in \(D\) satisfying at least one of the conditions \[ \iint_D|A(z)|^{1/2} d\sigma<\infty\quad\text{and}\quad \iint_D |A(z)|(1-|z|)\,d\sigma< \infty, \] then the equation is Blaschke-oscillatory. On the other hand, he obtains the following result. Let \(\{z_n\}\) be an infinite interpolating Blaschke sequence of nonzero points in \(D\). Then, the following assertions hold. (1) For each function \(F\), analytic in \(D\), there exists a function \(A_F(z)\), analytic in \(D\) and depending on \(F\), such that \(f''+ A_F(z)f= 0\) possesses a solution \(f_F\) having zeros precisely at the points \(z_n\). (2) There are uncountably many functions \(F\), analytic in \(D\), each resulting in a different function \(A_F(z)\) satisfying the properties stated in (1).
scientific article; zbMATH DE number 5011335
scientific article