Coupling constant dependence for the Schrödinger equation with an inverse-square potential

A. G. Smirnov

Published 2020-01-17Version 1

We consider the one-dimensional Schr\"odinger equation $-f''+q_\alpha f = Ef$ on the positive half-axis with the potential $q_\alpha(r)=(\alpha-1/4)r^{-2}$. It is known that the value $\alpha=0$ plays a special role in this problem: all self-adjoint realizations of the formal differential expression $-\partial^2_r + q_\alpha(r)$ for the Hamiltonian have infinitely many eigenvalues for $\alpha<0$ and at most one eigenvalue for $\alpha\geq 0$. For each complex number $\vartheta$, we construct a solution $\mathcal U^\alpha_\vartheta(E)$ of this equation that is entire analytic in $\alpha$ and, in particular, is not singular at $\alpha = 0$. For $\alpha<1$ and real $\vartheta$, the solutions $\mathcal U^\alpha_\vartheta(E)$ determine a unitary eigenfunction expansion operator $U_{\alpha,\vartheta}\colon L_2(0,\infty)\to L_2(\mathbb R,\mathcal V_{\alpha,\vartheta})$, where $\mathcal V_{\alpha,\vartheta}$ is a positive measure on $\mathbb R$. We show that each operator $U_{\alpha,\vartheta}$ diagonalizes a certain self-adjoint realization $h_{\alpha,\vartheta}$ of the expression $-\partial^2_r + q_\alpha(r)$ and, moreover, that every such realization is equal to $h_{\alpha,\vartheta}$ for some $\vartheta\in\mathbb R$. Employing suitable singular Titchmarsh--Weyl $m$-functions, we explicitly find the spectral measures $\mathcal V_{\kappa,\vartheta}$ and prove their smooth dependence on $\alpha$ and $\vartheta$. Using the formulas for the spectral measures, we analyse in detail how the transition through the point $\alpha=0$ occurs for both the eigenvalues and the continuous spectrum of $h_{\alpha,\vartheta}$.