Theoretical and Mathematical Physics, 135(3): 849–871 (2003) ESSENTIAL AND DISCRETE SPECTRA OF THE THREE-PARTICLE SCHR ¨ ODINGER OPERATOR ON A LATTICE S. N. Lakaev ∗ and M. ´ E. Muminov ∗ We consider the system of three quantum particles (two are bosons and the third is arbitrary ) interacting by attractive pair contact potentials on a three-dimensional lattice. The essential spectrum is described. The existence of the Efimov effect is proved in the case where either two or three two-particle subsystems of the three-particle system have virtual levels at the left edge of the three-particle essential spectrum for zero total quasimomentum (K = 0). We also show that for small values of the total quasimomentum (K = 0), the number of bound states is finite. Keywords: essential spectrum, virtual level, channel operator, discrete spectrum, Weyl inequality, Hilbert– Schmidt operator 1. Introduction A remarkable result in the spectral analysis of the three-particle continuous Schr¨ odinger operator is the so-called Efimov effect: in a system of three particles interacting via short-range pair potentials, if none of the three two-particle subsystems has bound states with negative energy but at least two of them have a resonance with zero energy, then this three-particle system has infinitely many three-particle bound states with negative energy that accumulate as the energy tends to zero. This effect was first discovered by Efimov [1]. Since then, this problem has been studied in a number of works [2], [3]. A rigorous mathematical proof of the existence of the Efimov effect was first given in [4] and then in [5]–[7]. Yafaev [4] proved the existence of the Efimov effect using Faddeev’s method of integral equations. On the other hand, Ovchinnikov and Sigal [5] established the Efimov effect by an interesting variational method for the three-particle system consisting of two heavy (2H) and one light (1L) particle, assuming that only the H–L subsystems interacting via spherically symmetric pair potentials have resonances with zero energy. Further, using the variational method in [5], Tamura [6] proved the existence of the Efimov effect without restrictions on the masses of particles interacting via pair potentials (not necessarily spherically symmetric) in the case where all two-particle subsystems have resonances with zero energy. In the models of solid state physics [7], [8] as well as in lattice field theory [9], the so-called discrete Schr¨ odinger operators arise; they are the lattice analogues of the ordinary three-particle Schr¨ odinger oper- ator in the continuous space. It is natural to expect that the Efimov effect also arises for such operators because, as shown by thorough analysis [4], the “long-wavepart” of the spectrum of the Schr¨odinder oper- ator is responsible for this effect and this part is the same in both the lattice and the continuous cases. At the physical level, this was shown in [7], [8]. In [10], a result was formulated concerning the Efimov effect in the case of three arbitrary quantum particles interacting via pair contact attractive potentials for a fixed value of the total system quasimo- mentum, but the proof was only sketched. In [11], a system of three identical quantum particles (bosons) * Samarkand State University, Samarkand, Uzbekistan, e-mail: samgu@sam.silk.glas.apc.org. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 135, No. 3, pp. 478–503, June, 2003. Original article submitted July 23, 2002. 0040-5779/03/1353-0849$25.00 c  2003 Plenum Publishing Corporation 849