Monotonicity of the eigenvalues of the two-particle Schrödinger operatoron a lattice
J. I. AbdullaevInstitute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Mirzo Ulugbek 81, Tashkent 100170, UzbekistanA. M. KhalkhuzhaevInstitute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Mirzo Ulugbek 81, Tashkent 100170, UzbekistanL.S. UsmonovSamarkand State University, University Boulevard 15, Samarkand 140104, Uzbekistan
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We consider the two-particle Schrdinger operator H(k), (k T 3 (-, ] 3 is the total quasimomentum of a system of two particles) corresponding to the Hamiltonian of the two-particle system on the three-dimensional lattice Z 3 . It is proved that the number N (k) N (k (1) , k (2) , k (3) ) of eigenvalues below the essential spectrum of the operator H(k) is nondecreasing function in each k (i) [0, ], i = 1, 2, 3. Under some additional conditions potential v, the monotonicity of each eigenvalue zn(k) zn(k (1) , k (2) , k (3) ) of the operator H(k) in k (i) [0, ] with other coordinates k being fixed is proved.
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