Landau-Lifshitz's conjecture about the motion of a quantum mechanical particle under the inverse square potential

Abstract

Landau and Lifshitz [4, Section 35] conjectured that for an arbitrary $k\in \mathbb{R}$, there exists the motion of a quantum mechanical particle under the inverse square potential $k|x|^{-2}$, $x \in \mathbb{R}^3$. When $k$ is negative and $| k |$ is very large, the inverse square potential becomes very deep and generates the very strong attractive force, and hence a quantum mechanical particle is likely to fall down to the origin (the center of the inverse square potential). Therefore this conjecture (Landau-Lifshitz's conjecture) seems to be wrong at first sight. We however prove Landau-Lifshitz's conjecture by showing that there exists a selfadjoint extension for the Schr\"odinger operator with the inverse square potential $-\Delta+k|x|^{-2}$ in $\mathbb{R}^N\ (N\geq 2)$ and that the spectrum of the selfadjoint extension is bounded below for an arbitrary $k\in \mathbb{R}$. We thus give the affirmative and complete answer to Landau-Lifshitz's conjecture in $\mathbb{R}^N\ (N\geq 2)$.