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Universality of the discrete spectrum asymptotics of the three-particle Schrodinger operator on a lattice

Аннотация:

In the present paper, we consider the Hamiltonian H(K), K ∈ T 3 := (−π; π] 3 of a system of three arbitrary quantum mechanical particles moving on the three-dimensional lattice and interacting via zero range potentials. We find a finite set Λ ⊂ T 3 such that for all values of the total quasi-momentum K ∈ Λ the operator H(K) has infinitely many negative eigenvalues accumulating at zero. It is found that for every K ∈ Λ, the number N(K; z) of eigenvalues of H(K) lying on the left of z, z < 0, satisfies the asymptotic relation lim z→−0 N(K; z)| log |z||−1 = U0 with 0 < U0 < ∞, independently on the cardinality of Λ. 

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