Eigenfunction expansions for the Schr\"odinger equation with inverse-square potential

We consider the one-dimensional Schr"odinger equation $-f\mathrm{"}+q_{\kappa }f=Ef$ on the positive half-axis with the potential $q_{\kappa }(r)=({\kappa }^2-1\mathrm{/}4)r^{-2}$. For each complex number $\vartheta$, we construct a solution $u_{\vartheta }^{\kappa }(E)$ 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 $u_{\vartheta }^{\kappa }(E)$ determine a unitary eigenfunction expansion operator $U_{\kappa,\vartheta}\colon L_2(0,\infty)\to L_2(\mathbb R,\mathcal V_{\kappa,\vartheta})$,where$\mathcal V_{\kappa,\vartheta}$isapositivemeasureon$\mathbb R$. We show that every self-adjoint realization of the formal differential expression $-\partial^2_r + q_\kappa®$fortheHamiltonianisdiagonalizedbytheoperator$U_{\kappa,\vartheta}$forsome$\vartheta\in\mathbb R$. Using suitable singular Titchmarsh-Weyl $m$-functions, we explicitly find the measures ${\mathcal{V}}_{\kappa ,\vartheta }$ and prove their continuity in $\kappa$ and $\vartheta$.