Asymptotics of the eigenvalues of a discrete Schrödinger operator with zero-range potential

We consider a?family of discrete Schr?dinger operators , . These operators are associated with the Hamiltonian of a?system of two identical quantum particles (bosons) moving on the -dimensional lattice , , and interacting by means of a?pairwise zero-range (contact) attractive potential 0$ SRC=http://ej.iop.org/images/1064-5632/76/5/A05/tex_im_2611_img7.gif/>. It is proved that for any there is a?number 0$ SRC=http://ej.iop.org/images/1064-5632/76/5/A05/tex_im_2611_img9.gif/> which is a?threshold value of the coupling constant; for \mu(k)$ SRC=http://ej.iop.org/images/1064-5632/76/5/A05/tex_im_2611_img10.gif/> the operator , , has a?unique eigenvalue placed to the left of the essential spectrum. The asymptotic behaviour of is found as and as and also as for every value of the quasi-momentum belonging to the manifold , where .

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