Spectral analysis of two-particle Hamiltonians with short-range interactions
https://doi.org/10.17586/2220-8054-2025-16-5-577-585
Abstract
We analyze the spectral characteristics of lattice Schrodinger operators, denoted as Hγλµ(K), K ∈ (−π, π] 3 , which represent a system of two identical bosons existing on Z 3 lattice. The model includes onsite and nearest-neighbor interactions, parameterized by γ, λ, µ ∈ R. Our study of Hγλµ(0) reveals an invariant subspace on which its restricted form, Hea λµ(0), is solely dependent on λ and µ. To elucidate the mechanisms of eigenvalue birth and annihilation for Hea λµ(0), we define a critical operator. A detailed criterion is subsequently developed within the plane spanned by λ and µ. This involves: (i) the derivation of smooth critical curves that mark the onset of criticality for the operator, and (ii) the proof of exact conditions for the existence of precisely α eigenvalues below and β eigenvalues above the essential spectrum, where α, β ∈ {0, 1, 2} and α + β ≤ 2.
Keywords
Two-particle system,
lattice Schrodinger operator,
essential spectrum,
bound states,
Fredholm determinant
Supporting agencies: The authors are grateful to the referee for valuable insights. Support for this work was provided by the Fundamental Science Foundation of Uzbekistan, Grant No. FL-9524115052.
About the Authors
M. O. Akhmadova
Samarkand State University
Uzbekistan
Uzbekistan
Mukhayyo O. Akhmadova
140104, Samarkand
M. A. Azizova
Samarkand State University
Uzbekistan
Uzbekistan
Mukammal A. Azizova
140104, Samarkand
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For citations:
Akhmadova M.O., Azizova M.A. Spectral analysis of two-particle Hamiltonians with short-range interactions. Nanosystems: Physics, Chemistry, Mathematics. 2025;16(5):577-585. https://doi.org/10.17586/2220-8054-2025-16-5-577-585
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