Asymptotics of the eigenvalues of a discrete Schr?dinger operator with zero-range potential

摘要

We consider a family of discrete Schr?dinger operators Hμ(k), k ∈ G ? T_ d. These operators are associated with the Hamiltonian Hμ of a system of two identical quantum particles (bosons) moving on the d-dimensional lattice Z_ d, d > 3, and interacting by means of a pairwise zero-range (contact) attractive potential μ > 0. It is proved that for any k 2 G there is a number μ(k) > 0 which is a threshold value of the coupling constant; for μ > μ(k) the operator Hμ(k), k 2 G ~ T_ d, has a unique eigenvalue z(μ, k) placed to the left of the essential spectrum. The asymptotic behaviour of z(μ, k) is found as μ → μ(k) and as μ → +∞ and also as k → k* for every value of the quasi-momentum k* = k* (μ) belonging to the manifold {k ~ G: μ(k) = μ