Investigation of the Number of Negative Eigenvalues of a Third-Order Operator Matrix on a Noninteger Lattice
Annotatsiya
In this paper, a family of block operator matrices $${{\mathcal{A}}_{{\text{h}}}}(K),$$ $$K \in {{( - \pi {\text{/h}};\pi {\text{/h}}]}^{3}}$$ , associated with the Hamiltonian of a system with a non-conserved number of particles not exceeding three on a non-integer lattice $${{({\text{h}}\mathbb{Z})}^{3}}$$ with step $${\text{h}} > 0$$ , is considered. It is established that the operator $${{\mathcal{A}}_{{\text{h}}}}({\mathbf{0}}),$$ $${\mathbf{0}}: = (0,0,0),$$ has a finite number of negative eigenvalues if the corresponding generalized Friedrichs model has a zero eigenvalue. It is shown that the operator $${{\mathcal{A}}_{{\text{h}}}}({\mathbf{0}})$$ possesses an infinite number of negative eigenvalues accumulating at zero (the Efimov effect) if the generalized Friedrichs model has a zero-energy resonance. An asymptotic formula is obtained for the number $${{N}_{{\text{h}}}}(z)$$ of eigenvalues of the operator $${{\mathcal{A}}_{{\text{h}}}}({\mathbf{0}})$$ lying below $$z,$$ $$z \leqslant 0$$ as the spectral parameter $$z \to - 0.$$