Eigenfunction expansions for the Schrödinger equation with inverse-square potential

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

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