01

NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2021, 12 (6), P. 657–663

Monotonicity of the eigenvalues of the two-particle Schrödinger operatoron a lattice

J. I. Abdullaev – Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Mirzo Ulugbek 81, Tashkent 100170; Samarkand State University, University Boulevard 15, Samarkand 140104, Uzbekistan; jabdullaev@mail.ru
A. M. Khalkhuzhaev – Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Mirzo Ulugbek 81, Tashkent 100170; Samarkand State University, University Boulevard 15, Samarkand 140104, Uzbekistan; ahmad_x@mail.ru
L. S. Usmonov – Samarkand State University, University Boulevard 15, Samarkand 140104, Uzbekistan; u.lochinbek@bk.ru

We consider the two-particle Schrödinger operator H(k), (k∈T3≡(-π, π]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 Z3. It is proved that the numberN(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, π], i = 1, 2, 3. Under some additional conditions potential vˆ, the monotonicity of each eigenvalue zn(k)≡zn(k(1), k(2), k(3)) of the operator H(k) in k(i)≡[0, π] with other coordinates k being fixed is proved.

Keywords: two-particle Schrödinger operator, Birman-Schwinger principle, total quasimomentum, monotonicity of the eigenvalues.

PACS 02.30.Tb

DOI 10.17586/2220-8054-2021-12-6-657-663

Download

Comments are closed.