Mathematical Notes, vol. 73, no. 4, 2003, pp. 521–528. Translated from Matematicheskie Zametki, vol. 73, no. 4, 2003, pp. 556–564. Original Russian Text Copyright c 2003 by S. N. Lakaev, T. Kh. Rasulov. A Model in the Theory of Perturbations of the Essential Spectrum of Multiparticle Operators S. N. Lakaev and T. Kh. Rasulov Received February 15, 2001; in final form, February 6, 2002 Abstract—We find the continuous spectrum of the model Schr¨ odinger operator acting in the direct sum of Hilbert spaces of n-particle states, n =0 , 1 , 2 , 3. Key words: multiparticle operator, Schr¨ odinger operator, n-particle state, essential spectrum, Fredholm determinant, perturbation theory, Hilbert space, Fock space. In his well-known paper, Friedrichs [1] treats the operator of multiplication by an independent variable as an unperturbed operator and chooses the integral operator for the perturbation. This model was subsequently called the model of the theory of perturbations of continuous spectra, since the continuous (essential) spectrum of a self-adjoint operator remains unchanged under a compact perturbation. The spin–boson Hamiltonian similar to the standard three-particle Hamiltonian was dealt with in [2], where an absolutely continuous spectrum and related states were studied. In the present paper, we consider a model operator in a subspace of Fock space which is a lattice analog of the spin–boson Hamiltonian [2] and of the three-particle Hamiltonian on a lattice [3, 4]. We describe the position and structure of the essential spectrum (i.e., the two-particle and three- particle essential spectra are singled out). We define the “regularized Fredholm determinant” whose zeros coincide with the eigenvalues of this operator lying outside the essential spectrum. Suppose that T ν is the ν -dimensional torus, (T ν ) 2 = T ν × T ν is the Cartesian product, L 2 (T ν ) is the Hilbert space of square-integrable functions defined on T ν , L s 2 ((T ν ) 2 ) is the Hilbert space of square-integrable symmetric functions, and C 1 is the one-dimensional complex space. Suppose that the operator A acts in the Hilbert space H = C 1 ⊕ L 2 (T ν ) ⊕ L s 2 ((T ν ) 2 ) by the formula A   f 0 f 1 (x) f 2 (x,y)   =   u 0 f 0 +  b(t)f 1 (t) dt b(x)f 0 + u(x)f 1 (x)+  b(t)f 2 (x,t) dt 1 2 (b(x)f 1 (y)+ b(y)f 1 (x))+ w(x,y)f 2 (x,y)   . (1) Here f =(f 0 ,f 1 ,f 2 ) ∈H , u 0 is a fixed real number, b(x), u(x) are real analytic functions on the torus T ν , and w(x,y) is a real analytic symmetric function on (T ν ) 2 . Here and elsewhere, an integral without limits denotes integration over the whole range of variables of integration. Lemma 1. The operator A acting in the Hilbert space H by formula (1) is bounded and self- adjoint. Proof. The proof of Lemma 1 is obvious.  For any x ∈ T ν , we define the following regular function in C \ [m w (x); M w (x)]: Δ 1 (x,z)= u(x) - z - 1 2  b 2 (t) w(x,t) - z dt, 0001-4346/2003/7334-0521$25.00 c 2003 Plenum Publishing Corporation 521