Two-fermion lattice Hamiltonian with first and second nearest-neighboring-site interactions
S. N. Lakaev, А. К. Мотовилов, Saidakbar Kh. Abdukhakimov
Abstract
Abstract We study the Schrödinger operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>K</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> the fixed quasimomentum of the particles pair, associated with a system of two identical fermions on the two-dimensional lattice <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> with first and second nearest-neighboring-site interactions of magnitudes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>μ</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> , respectively. We establish a partition of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> </mml:math> plane so that in each its connected component, the Schrödinger operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential spectrum and above its top. Moreover, we establish a sharp lower bound for the number of isolated eigenvalues of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> in each connected component.