2020/04/19 by Lakaev, Saidakhmat N., Bach, Volker, Pedra, W. de Siqueira
#2010 Mathematics Subject Classification. Primary: 81Q10 #47B39 #FOS: Mathematics #Secondary: 47A10 #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2004.08813
We consider a wide class of the two-particle Schrödinger operators Hμ(k)=H0(k)+μV, μ>0, with a fixed two-particle quasi-momentum k in the d -dimensional torus \mathbbTd, associated to the Bose-Hubbard hamiltonian Hμ of a system of two identical quantum-mechanical particles (bosons) on the d- dimensional hypercubic lattice ℤ% d interacting via short-range pair potentials. We study the existence of eigenvalues of Hμ(k) below the threshold of the essential spectrum depending on the interaction energy μ>0 and the quasi-momentum k∈ \mathbbTd of particles. We prove that the threshold (bottom of the essential spectrum), as a singular point (a threshold resonance or a threshold eigenvalue), creates eigenvalues below the essential spectrum under perturbations of both the coupling constant μ>0 and the quasi-momentum k of the particles. Moreover, we show that if the threshold is a regular point, then it does not create any eigenvalues under small perturbations of the coupling constant μ>0 and the quasi-momentum k.