c Allerton Press, Inc., 2010. ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2010, Vol. 54, No. 12, pp. 27–37. c M.E. Muminov and U.R. Shodiev, 2010, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2010, No. 12, pp. 32–43. Original Russian Text
Spectral Properties of a Hamiltonian of a Four-Particle System on a Lattice M. E. Muminov* and U. R. Shodiev** Samarkand State University, Universitetskii bul’v. 15, Samarkand 140101, Republic of Uzbekistan Received April 10, 2009
Abstract—We consider the Hamiltonian of a system of four arbitrary quantum particles with two-particle contact (noncompact) interaction potentials on a three-dimensional lattice perturbed by three-particle contact potentials. We describe the location of the essential spectrum of the ¨ Schrodinger operator corresponding to a four-particle system. DOI: 10.3103/S1066369X10120030 Key words and phrases: four-particle Hamiltonian, Schrodinger ¨ operator, essential spectrum, compact operator.
1. INTRODUCTION ¨ In the nonrelativistic quantum mechanics, it is necessary to study the n-particle Schrodinger operator on a lattice, where two-particle and three-particle potentials are taken as perturbations [1]. A number of papers on the spectral theory of linear operators [2–6] are devoted to the study of spectral ¨ properties of three-particle and multiparticle Schrodinger operators with two-particle potentials (in absence of three-particle potentials) in the Euclidean space and on a lattice, respectively [7–11]. ¨ The essential spectrum of the Schrodinger operator in the continuous space is, generally speaking, a ¨ half-line [2, 4], and the essential spectrum of the Schrodinger operator on a lattice is a union of segments (the spectra of sub-Hamiltonians) [8, 9, 11]. In [11], a description was given of the location of the ¨ essential spectrum of the lattice Schrodinger operator H(K) corresponding to the Hamiltonian of a four-particle system interacting with two-particle short-range potentials on a three-dimensional lattice (in absence of three-particle interactions). In this paper, we consider a four-particle system in the case when in the system there are two¨ particle and three-particle interactions. We find the location of the essential spectrum of the Schrodinger 3 operator H(K) depending on the values of the total quasi-momentum K ∈ [−π, π) corresponding to a four-particle system. After determination of possible sub-Hamiltonians of a four-particle system and decomposition of these sub-Hamiltonians into direct integral, we describe their spectra. Then, using methods similar to the methods from [11, 12], we prove that the essential spectrum of the operator H(K) is the union of the spectra of the sub-Hamiltonians of the four-particle system. It turns out that, under a perturbation by three-particle interaction potentials, to the spectrum of the operator H(K) (with contact potentials), some segments are added. The spectra of the sub-Hamiltonians of the corresponding subsystems, without three-particle systems contained in the essential spectrum, do not change. The paper consists of four sections. In Section 2, we describe the Hamiltonian of the system in question and formulate the main result. In Section 3, we describe the spectra of the sub-Hamiltonians of a four-particle system and prove that these spectra are subsets of the essential spectrum of the operator H(K). In Section 4, the inverse inclusion is proved, i.e., that the essential spectrum of the * **
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operator H(K) is contained in the union of the spectra of the channel operators. By σ(A) and σess (A), we denote, respectively, the spectrum and the essential spectrum of an operator A, and by · and (·, ·) the norm and the scalar product in the corresponding Hilbert space respectively. 2. COORDINATE AND MOMENTUM REPRESENTATIONS ¨ OF THE SCHRODINGER OPERATOR AND THE MAIN RESULTS Let Z 3 be a three-dimensional lattice, and let 2 ((Z 3 )4 ) be the Hilbert space of quadratically summable functions defined on Z 3 . 0 of a system of four arbitrary quantum particles on the lattice Z 3 Consider the free Hamiltonian H defined as the bounded self-adjoint operator on 2 ((Z 3 )4 ) acting as follows: 0 = H
1 1 1 1 Δx + Δx + Δx + Δx , 2m1 1 2m2 2 2m3 3 2m4 4
where mi is the mass of the ith particle, Δx1 = Δ ⊗ I ⊗ I ⊗ I, Δx2 = I ⊗ Δ ⊗ I ⊗ I, Δx3 = I ⊗ I ⊗ Δ ⊗ I, Δx4 = I ⊗ I ⊗ I ⊗ Δ, and Δ is the lattice Laplacian. The operator Δ is a difference operator which describes the transfer of a particle from a lattice node to an adjacent node, i.e., + s)], ψ(x) [ψ(x) − ψ(x ∈ 2 (Z 3 ). (Δψ)(x) = |s|=1
In terms of the coordinate representation, the Hamiltonian of a system of four arbitrary quantum particles with two-particle and three-particle contact interactions acts on 2 ((Z 3 )4 ) by the formula =H 0 − H
i
