Monotonicity of the eigenvalues of the two-particle Schrӧdinger operator on a lattice

Аннотация

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