О свойствах определителя Фредгольма, ассоциированный с обобщенной модели Фридрихса на нецелочисленном решетке

In this paper we consider the generalized Friedrichs model A h (k), h > 0, k ∈ (−π/h;π/h] 3 ,
corresponding to the Hamiltonian of a system of non-conserved and at most two particles on the
non-integer lattice. The necessary and sufficient conditions for either z = 0 to be an eigenvalue of
A h (0) or operator A h (0) to have zero energy resonance are given, where 0 := (0,0,0). If the operator
A h (0) have a zero energy resonance or zero eigenvalue, then for any k ∈ T 3
h
the positivity of the
operator A h (−k)+l 1 ε h (k)I is established and the two-sided estimates for the Fredholm determinant
is given, where ε h (·) is a dispersion function of special form and l 1 > 0. In the case when the operator
A h (0) have a zero energy resonance the expansion for the Fredholm determinant is obtained.