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

Abstract

We consider the one-dimensional Schrödinger equation $-f^{\prime \prime }+q_{\alpha }f=Ef$ 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 $-{\partial }_r^2+q_{\alpha }(r)$ for the Hamiltonian have infinitely many eigenvalues for $\alpha <0$ and at most one eigenvalue for $\alpha \ge 0$. 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.

48 pages, 6 figures, final version

References in corpus (2)

• On Schroedinger operators with inverse square potentials on the half-line
• A Gronwall-type Trigonometric Inequality