2013/12/10 by Motohiro Sobajima, Sobajima, Motohiro, Shuji Watanabe +1
Mathematics · Physics and Astronomy · #47B25 #81Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1312.2774
openalex publication_date 2013/12/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Landau and Lifshitz [4, Section 35] conjectured that for an arbitrary k∈ ℝ, there exists the motion of a quantum mechanical particle under the inverse square potential k|x|-2, x ∈ ℝ3. 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 Schrödinger operator with the inverse square potential -Δ+k|x|-2 in ℝN (N≥ 2) and that the spectrum of the selfadjoint extension is bounded below for an arbitrary k∈ ℝ. We thus give the affirmative and complete answer to Landau-Lifshitz's conjecture in ℝN (N≥ 2).