### ON THE ESSENTIAL SPECTRUM OF A QUADRATIC OPERATOR MATRIX OF ORDER 4

#### Abstract

In the present paper, we precisely describe the location of the essential spectrum of a quadratic operator matrix A4 of order 4 associated to a system describing four particles in interaction, without conservation of the number of particles, in the quasi-momentum representation. It is also established that the essential spectrum of A4 consists of no more than seven bounded closed intervals.

#### Full Text:

PDF#### References

C.Tretter, Spectral Theory of block Operator Matrices and Applications, Imperial College Press, 2008.

H. Spohn. Ground states of the spin – boson Hamiltonians. Comm. Math. Phys., 123 (1989), 277 – 304.

R.A. Minlos, H. Spohn. The three – body problem in radiactive decay: The case of one atom and at most two photons, Topics in statistical and theoretical physics, American Mathematical Society Translation Series 2, 177 (1996), 159 – 193.

M. Muminov, H. Neidhardt, T. Rasulov. On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case. J.Math.Phys., 56 (2015), 053507.

T.Kh. Rasulov. Branches of the essential spectrum of the lattice spin-boson model with at most two photons. Theoretical and Mathematical Physics, 186:2 (2016), 251 – 267.

T.H. Rasulov, M.I. Muminov, M. Hasanov. On the spectrum of a model operator in Fock space, Methods of Functional Analysis and Topology 15(4) (2009), 369 – 383.

T.H. Rasulov. Investigations of the essential spectrum of a Hamiltonian in Fock space, Applied Mathematics and Information Sciences 4(3) (2010), 395 – 412.

T.H. Rasulov. On the finiteness of the discrete spectrum of 3×3 operator matrix. Methods of Funct. Anal. Topology. 22:1(2016), 48 – 61.

M.I. Muminov, T.Kh. Rasulov. An eigenvalue multiplicity formula for the Schur complement of a 3×3 block operator matrix. Siberian Math. J., 56:4 (2015), 878 – 895.

T.Rasulov, C.Tretter. Spectral inclusion for unbounded diagonally dominant n×n operator matrices. Rocky Mountain Journal of mathematics, 48:1 (2018), 279 – 324.

### Refbacks

- There are currently no refbacks.

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

CC BY-SA