  We consider the one-dimensional Schr\"odinger equation $-f"+q_\kappa f = Ef$
on the positive half-axis with the potential
$q_\kappa(r)=(\kappa^2-1/4)r^{-2}$. For each complex number $\vartheta$, we
construct a solution $u^\kappa_\vartheta(E)$ of this equation that is analytic
in $\kappa$ in a complex neighborhood of the interval $(-1,1)$ and, in
particular, at the "singular" point $\kappa = 0$. For $-1<\kappa<1$ and real
$\vartheta$, the solutions $u^\kappa_\vartheta(E)$ determine a unitary
eigenfunction expansion operator $U_{\kappa,\vartheta}\colon L_2(0,\infty)\to
L_2(\mathbb R,\mathcal V_{\kappa,\vartheta})$, where $\mathcal
V_{\kappa,\vartheta}$ is a positive measure on $\mathbb R$. We show that every
self-adjoint realization of the formal differential expression $-\partial^2_r +
q_\kappa(r)$ for the Hamiltonian is diagonalized by the operator
$U_{\kappa,\vartheta}$ for some $\vartheta\in\mathbb R$. Using suitable
singular Titchmarsh-Weyl $m$-functions, we explicitly find the measures
$\mathcal V_{\kappa,\vartheta}$ and prove their continuity in $\kappa$ and
$\vartheta$.
