Analysis of the spectrum of a 2x2 operator matrix. Discrete spectrum asymptotics
Tulkin Husenovich Rasulov, Elyor B. Dilmurodov · Nanosystems Physics Chemistry Mathematics · 2020
We consider a 2 × 2 operator matrix Aµ, µ > 0 related with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles.We obtain an analog of the Faddeev equation and its symmetric version for the eigenfunctions of Aµ.We describe the new branches of the essential spectrum of Aµ via the spectrum of a family of generalized Friedrichs models.It is established that the essential spectrum of Aµ consists the union of at most three bounded closed intervals and their location is studied.For the critical value µ 0 of the coupling constant µ we establish the existence of infinitely many eigenvalues, which are located in the both sides of the essential spectrum of Aµ.In this case, an asymptotic formula for the discrete spectrum of Aµ is found.