I am considering the following problem: suppose we have that a smooth function that has the following property $p(x) \to 1$ as $|x| \to \pm \infty$ and that $f(x,k)$ is the solution of $(p(x)f')' + k^2f = 0$ (derivative wrt $x$). Is it true that $f$ belongs to the Hardy space $H^2_+$ which is defined to be the space of functions that are analytic wrt the variable $k$ in the UHP$:=\{z\in \mathbb{C}| Im(z)>0\}$, whose square integrals along all lines $Im(k) = constant$ are bounded by some constant.
We may assume that our solution $f(x,k)$ is analytic in $k$. How would one go about solving such a problem! If possible could i get some help with this problem or at least a pointer into which text books I should look into!
Many Thanks.