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On Essential and Discrete Spectrum of The The Sixth International Conference "Modern Problems of Nuclear Physics", September 19-22,2006 MPNP'1006 THEOREM 1. Let PeQl and A = ]0;TC[ (Ae]%;2n[): a) If C* 2s(2s-l)B, then the operator H2 has two BSs q>\ and (pi with corresponding energy levels zx < mA and z2 > MK; b) If C=2s(2s-1)2? then the operator H2 has only one BS cp with the energy level z <mk. THEOREM 2. Let Peft and A = ]0;TI[ (Ae]w;2w[): a) If C>-2s(2s-l)£ then the operator H2 has a two BSs q>\ and <pi with corresponding energy values zi <mAand z2 > MA; b) If C<-2s(2s-l)Z? then the operator H2 has only one BS ^ with energy value z<mK. For arbitrary values of the full quasi momentum A of the system, the operator H2 has no more than 2v+l BSs (taking the energy degeneration order into account) with the energy values lying outside the continuous spectrum. We proved the spectrum and BSs of system for s=l/2 and J>1/2 differently. Solwing the problem of differency the spectrum and BS,s of the system for the whole and the naif an whole values of s and for odd values and even values of s. References: 1. S.M.Tashpulatov. Theor.Math.Phys. 107. No. 1.544 (1996). 2. S.M.Tashpulatov. Theor.Math.Phys. V.107. tfo 2. P. 620-628. (1996). 3. S.M.Tashpulatov. Theor.Math.Phys. V.107. N° 2. P.629-634 (1996). 4. S.M.Tashpulatov. Theor.Math.Phys. V.125. M> 2. P.1539-1551 (2000). UZ0603135 ON ESSENTIAL AND DISCRETE SPECTRUM OF THE ENERGY OPERATOR OF TWO-MAGNON SYSTEMS IN THE THREE- DIMENSIONAL ISOTROPIC FERROMAGNETIC IMPURITY NON- HEISENBERG MODEL WITH NEAREST-NEIGHBOR INTERACTIONS Tashpulatov S.M. Institute of Nuclear Physics, Tashkent, Uzbekistan The use of films in various areas of physics and technology is of great interest in studying a localized impurity state (LIS) of a magnet. The LISs in a Heisenberg ferromagnet with ferromagnetic and antiferromagnetic impurities were investigated in many papers, where the situations with linear and cubic lattices were considered in details. It was shown that there are two LIS types in the linear case and three types in the cubic case. In [I], the case of a v -dimensional lattice was considered, and it was proved that there are at most three types of LISs (not counting the degeneracy multiplicities of their energy levels) in the v -dimensional case. It was shown that the number of types LISs in the system changes with varying parameters of the Hamiltonian, and the LIS domains were found. In this case, it turns out that the three types LISs in the system are respectively nondegenerate, v-fold degenerate, and (v-l)- fold degenerate. In [2], the LISs in a non-Heisenberg ferromagnet with ferromagnetic 221 Section II. Radiation Physics of Condensed Matter <{§•£ The Sixth International Conference "Modern Problems of Nuclear Physics", September 19-22, 2006 MPNF'2006 1NP-5Q and antiferromagnetic impurities with arbitrary values spin s are investigated in a v -dimensional lattice Z". In this work, we consider the energy operator of two-magnon systems in a three- dimensional isotropic ferromagnetic impurity non-Heisenberg model with coupling with between nearest neighbors with an arbitrary spin value s. We investigated essential and discrete spectrum of energy operator of two-magnon system in the non-Heisenberg isotropic ferromagnetic impurity model with arbitrary spin values s in the three-dimensional lattice Z3. The considering systems is consists of three body systems; two-magnons and one impurity spins. The Hamiltonian of the system in question has the form n=\ r n=l Here Jn>0 are the parameters of the multipole exchange interaction between the nearest- neighbor atoms in the lattice, J° are the atom-impurity multipole exchange interaction parameters, t = ±eJ}j = 1,2,..., y, where ej are unit coordinate vectors, i.e., the summation over x y z r ranges the nearest neighbors, and Sm = {S m\S m;S m) is the operator of the atomic spin of x y 1 magnitude s at a lattice point m. We set S^=S m±S m, where <S~ and S * are the respectively magnon creation and annihilation operators at the site m. Hamiltonian (1) acts in the symmetric Fock space F. We let t//0 denote the so-called z vacuum vector uniquely determined by the conditions S*y/0 -0 and S my/0 =si//0,\\y/0 ||=1. The closure of the space spanned by vectors {S~iy0} and {S~S~ysQ} is denoted the corresponding by Fi and F2. Finitness or infinitness of discrete spectrum of the system depends of number of two-body subsystems and has a virtual state at the edge of essential spectrum of three-particle system. If two or three two-body subsystems have virtual levels, then the discrete spectrum of three-body system is infinite, if only one of two-body subsystems is having a virtual level or is not a virtual levels, then the discrete spectrum of the system is finite. Therefore, we investigated one-magnon system with impurity spins, i.e. two-body subsystems consisting of one-magnon and impurity spins. There are two such subsystems. For investigated essential spectrum of the system we use of the results of work's [3,4] by T. Ichinose. We prove the essential spectrum of the system consists no more than four segments. References: 1. S.M. Tashpulatov, "One-magnon systems in an isotropic Heisenberg impurity model", Theoretical and Mathematical Physics, 2001,vol. 126, No 3, pp. 482-488. 2. S.M. Tashpulatov, "One-magnon systems in an non_Heisenberg ferromagnetic impurity model, Theoretical and Mathematical Physics, 2005,vol. 142, No 1, pp.71-78. 3. T. Ichinose, "Spectral properties of Tensor Products of Linear Operators. 1", Trans, of the American Math. Soc.,vol.235, pp.75-112.,1978. 4. T. Ichinose, "Spectral Properties of Tensor Products of Linear Operators, 11: The Approximate Point Spectrum and Kato Essential Spectrum, Trans. Of the American Math. Soc, vol. 237, pp.233-254., 1978. 222 Section II. Radiation Physics of Condensed Matter.
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