# Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices

@article{Muminov2020AnalysisOT, title={Analysis of the discrete spectrum of the family of \$3 \times 3\$ operator matrices}, author={Mukhiddin I. Muminov and Tulkin Husenovich Rasulov and Nargiza A. Tosheva}, journal={arXiv: Mathematical Physics}, year={2020} }

We consider the family of $3 \times 3$ operator matrices ${\bf H}(K),$ $K \in {\Bbb T}^3:=(-\pi; \pi]^3$ associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set $\Lambda \subset {\Bbb T}^3$ to prove the existence of infinitely many eigenvalues of ${\bf H}(K)$ for all $K \in \Lambda$ when the associated Friedrichs model has a zero energy resonance. It is found that…

## One Citation

### N ov 2 02 0 DISCRETE EIGENVALUES OF A 2 × 2 OPERATOR MATRIX

- 2020

## References

SHOWING 1-10 OF 38 REFERENCES

### Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

- Mathematics, Physics
- 2003

Abstract.
The Hamiltonian of a system of three quantum mechanical particles moving
on the three-dimensional lattice
$$ \mathbb{Z}^3 $$
and interacting via zero-range attractive
potentials is…

### On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics

- Mathematics
- 2005

AbstractA model operator H associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The location of…

### The Efimov effect. Discrete spectrum asymptotics

- Mathematics
- 1993

AbstractWe study a three-particle Schrödinger operatorH for which none of the two-particle subsystems has negative bound states and at least two of them have zero energy resonances. We prove that…

### ON THE THEORY OF THE DISCRETE SPECTRUM OF THE THREE-PARTICLE SCHRÖDINGER OPERATOR

- Mathematics
- 1974

We investigate the discrete spectrum of the Schr?dinger operator H for a system of three particles. We assume that the operators h?, ? = 1,?2,?3, which describe the three subsystems of two particles…

### Threshold analysis for a family of 2×2 operator matrices

- Mathematics
- 2019

We consider a family of 2× 2 operator matricesAμ(k), k ∈ T3 := (−π, π]3, μ > 0, acting in the direct sum of zeroand one-particle subspaces of a Fock space. It is associated with the Hamiltonian of a…

### On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case

- Physics, Mathematics
- 2014

A lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling…

### Essential and Discrete Spectra of the Three-Particle Schrödinger Operator on a Lattice

- Physics, Mathematics
- 2003

We consider the system of three quantum particles (two are bosons and the third is arbitrary) interacting by attractive pair contact potentials on a three-dimensional lattice. The essential spectrum…

### On the number of eigenvalues of a matrix operator

- Mathematics
- 2011

We consider a matrix operator H in the Fock space. We prove the finiteness of the number of negative eigenvalues of H if the corresponding generalized Friedrichs model has the zero eigenvalue (0 =…

### Asymptotics of the Discrete Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice

- Mathematics, Physics
- 2003

We consider the Hamiltonian Hμ(K) of a system consisting of three bosons that interact through attractive pair contact potentials on a three-dimensional integer lattice. We obtain an asymptotic value…

### On the spectral properties of Hamiltonians without conservation of the particle number

- Mathematics
- 2002

We consider quantum systems with variable but finite number of particles. For such systems we develop geometric and commutator techniques. We use these techniques to find the location of the…