  We investigate the connections between Weyl-Titchmarsh-Kodaira theory for
one-dimensional Schr\"odinger operators and the theory of $n$-entire operators.
As our main result we find a necessary and sufficient condition for a
one-dimensional Schr\"odinger operator to be $n$-entire in terms of square
integrability of derivatives (w.r.t. the spectral parameter) of the Weyl
solution. We also show that this is equivalent to the Weyl function being in a
generalized Herglotz-Nevanlinna class. As an application we show that perturbed
Bessel operators are $n$-entire, improving the previously known conditions on
the perturbation.
