aBukhara State University bСамаркандский Профессиональный Колледж Железнодорожного Транспорта, ул. Ибн-Холдун 79, 140102, Самарканд, Узбекистан
Abstract:
In the paper the block operator matrix $H$ associated with the system of at most three quantum particles on a $\mathrm{d}$-dimensional lattice is considered. Spectrum of this operator is studied in detail. In particular, it is shown that the operator $H$ has at most four simple eigenvalues lying outside of the essential spectrum. Moreover, the resolvent of $H$ is founded.
Keywords:Block operator matrix, Fock space, annihilation and creation operators, generalized Friedrichs model, Fredholm's determinant, essential and discrete spectrum, resolvent.