2012/10/31 by Dmitry K. Gridnev
Mathematics · Physics and Astronomy · #Boson #Bound state #Conjecture #Discrete spectrum #Eigenvalues and eigenvectors #Hamiltonian (control theory) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #Spectrum (functional analysis) #Zero (linguistics) #math-ph #math.MP
paper · pdf · doi:10.1063/1.4800764
published as J. Math. Phys. 54, 042105 (2013)
arxiv created 2013/02/06 · openalex publication_date 2013/04/01 · arxiv updated 2013/04/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider a system of N pairwise interacting particles described by the Hamiltonian H, where σess(H) = [0, ∞) and none of the particle pairs has a zero energy resonance. The pair potentials are allowed to take both signs and obey certain restrictions regarding the fall off. It is proved that if N ⩾ 4 and none of the Hamiltonians corresponding to the subsystems containing N − 2 or less particles has an eigenvalue equal to zero then H has a finite number of negative energy bound states. This result provides a positive proof to a long-standing conjecture of Amado and Greenwood stating that four bosons with an empty negative continuous spectrum have at most a finite number of negative energy bound states. Additionally, we give a short proof to the theorem of Vugal'ter and Zhislin on the finiteness of the discrete spectrum and pose a conjecture regarding the existence of the “true” four-body Efimov effect.