arXiv:math/0604282v1 [math.SP] 12 Apr 2006 LOW ENERGY EFFECTS FOR A FAMILY OF FRIEDRICHS MODELS UNDER RANK ONE PERTURBATIONS SERGIO ALBEVERIO 1,2,3 , SAIDAKHMAT N. LAKAEV 4,5 , RAMIZA KH. DJUMANOVA 5 ABSTRACT. A family of Friedrichs models with rank one perturbations hµ(p),p ∈ (-π, π] 3 ,µ > 0 associated to a system of two particles on the lattice Z 3 is consid- ered. The existence of a unique strictly positive eigenvalue below the bottom of the es- sential spectrum of hµ(p) for all nontrivial values p ∈ (-π, π] 3 under the assumption that hµ(0) has either a zero energy resonance (virtual level) or a threshold eigenvalue is proved. Low energy asymptotic expansion for the Fredholm determinant associated to family of Friedrichs models is obtained. Subject Classification: Primary: 81Q10, Secondary: 35P20, 47N50 Key words and phrases: A family of Friedrichs models, free Hamiltonian, eigenvalue, zero energy resonance, Hilbert-Schmidt operator. 1. I NTRODUCTION In the present paper we consider a family of Friedrichs models under rank one pertur- bations associated to a system of two particles on the lattice Z 3 . The main goal of the paper is to give a thorough mathematical treatment of the spectral properties of a family of Friedrichs models with emphasis on low energy expansions for the Fredholm determinants associated to the family (see, e.g.[1, 4, 5, 9, 16, 26, 27, 30] for relevant discussions and [13, 21], [32] for the general study of the low-lying excitation spectrum for quantum systems on lattices). These kind of models have been discussed in quantum mechanics [8, 10], solid physics [24, 20, 22, 11] and in lattice field theory [19, 17, 18]. Threshold energy resonances (virtual levels) for the two-particle Schr¨ odinger operators have been studied in [1, 5, 4, 12, 16, 30]. The threshold expansions for the resolvent of two-particle Schr¨ odinger operators have been studied in [5, 6, 12, 15, 16, 26, 27, 30] and have been applied to the proof of the existence of Efimov’s effect in [5, 16, 26, 27, 29]. Similarly to the lattice Schr¨ odinger operators and in contrast to the continuous Schr¨ odinger operators the family of Friedrichs models h µ (p),p ∈ (−π,π] 3 ,µ> 0 depends paramet- rically on the internal binding p, the quasi-momentum, which ranges over a cell of the dual lattice and hence it has spectral properties analogous to those of lattice Schr¨ odinger operators. Let us recall that the spectrum and resonances of the original Friedrichs model and its generalizations have been studied and the finiteness of the eigenvalues lying below the bottom of the essential spectrum has been proven in [10, 8, 14, 31]. In [17, 18] a peculiar family of Friedrichs models was considered and the appearance of eigenvalues for values of the total quasi-momentum p ∈ (−π,π] d ,d =1, 2 of the system lying in a neighbourhood of some particular values of the parameter p has been proven. Date: December 8, 2017. 1