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

TL;DR

This paper constructs eigenfunction expansions for the one-dimensional Schr"odinger equation with inverse-square potential, analyzing solutions' analyticity, and explicitly determining the spectral measures for all self-adjoint realizations.

Contribution

It introduces a family of solutions analytic in the potential parameter and explicitly characterizes the spectral measures for all self-adjoint realizations of the operator.

Findings

Solutions are analytic in the potential parameter near the singular point

Eigenfunction expansion operators are constructed for the specified potential

Spectral measures are explicitly computed and shown to vary continuously with parameters

Abstract

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 $\vartheta$.