Spectral analysis of two-particle Hamiltonians with short-range interactions
Аннотация
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 H γλµ (0) reveals an invariant subspace on which its restricted form, H ea λµ (0) , is solely dependent on λ and µ . To elucidate the mechanisms of eigenvalue birth and annihilation for H ea 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 .