Coupling constant dependence for the Schr\"odinger equation with an inverse-square potential

We consider the one-dimensional Schr"odinger equation $-f^{\mathrm{\prime }\mathrm{\prime }}+q_{\alpha }f=Ef$ on the positive half-axis with the potential $q_{\alpha }(r)=(\alpha -1\mathrm{/}4)r^{-2}$. It is known that the value $\alpha =0$ plays a special role in this problem: all self-adjoint realizations of the formal differential expression $-\partial^2_r

• q_\alpha®$fortheHamiltonianhaveinfinitelymanyeigenvaluesfor$\alpha<0$andatmostoneeigenvaluefor$\alpha\geq 0$. We find a parametrization of self-adjoint boundary conditions and eigenfunction expansions that is analytic in $\alpha$ and, in particular, is not singular at $\alpha =0$. Employing suitable singular Titchmarsh–Weyl $m$-functions, we explicitly find the spectral measures for all self-adjoint Hamiltonians and prove their smooth dependence on $\alpha$ and the boundary condition. Using the formulas for the spectral measures, we analyse in detail how the “phase transition” through the point $\alpha =0$ occurs for both the eigenvalues and the continuous spectrum of the Hamiltonians.