# T. H. Rasulov

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### On the finiteness of the discrete spectrum of a 3x3 operator matrix

Methods Funct. Anal. Topology **22** (2016), no. 1, 48-61

An operator matrix $H$ associated with a lattice system describing three particles in interactions, without conservation of the number of particles, is considered. The structure of the essential spectrum of $H$ is described by the spectra of two families of the generalized Friedrichs models. A symmetric version of the Weinberg equation for eigenvectors of $H$ is obtained. The conditions which guarantee the finiteness of the number of discrete eigenvalues located below the bottom of the three-particle branch of the essential spectrum of $H$ is found.

### The Faddeev equation and essential spectrum of a Hamiltonian in Fock space

Mukhiddin I. Muminov, Tulkin H. Rasulov

Methods Funct. Anal. Topology **17** (2011), no. 1, 47-57

A Hamiltonian (model operator) $H$ associated to a quantum system describing three particles in interaction, without conservation of the number of particles, is considered. The Faddeev type system of equations for eigenvectors of $H$ is constructed. The essential spectrum of $H$ is described by the spectrum of the channel operator.

### On the spectrum of a model operator in Fock space

Tulkin H. Rasulov, Mukhiddin I. Muminov, Mahir Hasanov

Methods Funct. Anal. Topology **15** (2009), no. 4, 369-383

A model operator $H$ associated to a system describing four particles in interaction, without conservation of the number of particles, is considered. We describe the essential spectrum of $H$ by the spectrum of the channel operators and prove the Hunziker-van Winter-Zhislin (HWZ) theorem for the operator $H.$ We also give some variational principles for boundaries of the essential spectrum and interior eigenvalues.

### The Efimov effect for a model operator associated with the Hamiltonian of a non conserved number of particles

Sergio Albeverio, Saidakhmat N. Lakaev, Tulkin H. Rasulov

Methods Funct. Anal. Topology **13** (2007), no. 1, 1-16

A model operator associated with the energy operator of a system of three non conserved number of particles is considered. The essential spectrum of the operator is described by the spectrum of a family of the generalized Friedrichs model. It is shown that there are infinitely many eigenvalues lying below the bottom of the essential spectrum, if a generalized Friedrichs model has a zero energy resonance.