Finiteness of Eigenvalues for Operator Matrices Arising in Quantum Mechanics
In this paper, an operator matrix of order three corresponding to a lattice system
with a non-conserved number of particles not exceeding three, arising in quantum mechanics is
considered as a linear, bounded, and self-adjoint operator acting in a Hilbert space. The essential
spectrum of the considered third-order operator matrix is investigated. The two-particle and three-
particle branches of the essential spectrum are identified. It is proved that, for an arbitrary value of
the spectral parameter, the discrete spectrum of the operator matrix is finite.
1. Tretter, C. Spectral theory of block operator matrices and applications. Imperial College Press, 2008.
2. Mogilner, A. I. Hamiltonians in solid state physics as multiparticle discrete Schrödinger operators: problems
and results. Advances in Sov. Math., 5 (1991), 139–194.
3. Friedrichs, K.O. Perturbation of spectra in Hilbert space. Amer. Math. Soc. Providence, Rhole Island, 1965.
4. Malishev, V. A.; Minlos, R. A. Linear infnite-particle operators. Translations of Mathematical Monographs.
143, AMS, Providence, RI, 1995.
5. Zhukov, Yu. V.; Minlos, R. A. Spectrum and scattering in the spin–boson model with no more than three
photons. Theoretical and Mathematical Physics, 103:1 (1995), 63–81.
6. Minlos R.A., Spohn H. The three-body problem in radioactive decay: the case of one atom and at most
two photons Topics in Statistical and Theoretical Physics, AMS Transl.-Series 2, 177 1996, P. 159–193.
7. Muminov, M.; Neidhardt, H.; Rasulov, T.; On the spectrum of the lattice spin-boson Hamiltonian for any
coupling: 1D case. J. Math. Phys., 56 (2015), 053507.
8. K. O. Friedrichs, Perturbation of Spectra in Hilbert Space, American Mathematical Society, Providence,
Rhode Island, 1965.
9. Reed, M.; Simon, B. Methods of modern mathematical physics. IV: Analysis of Operators. Academic Press,
New York, 1979.
10. Glazman, I. M. Direct methods of the qualitative spectral analysis of singular differential operators. J.: IPS
Trans., 1965.
11. Albeverio S.; Lakaev, S.N.; Rasulov T.H. On the spectrum of an Hamiltonian in Fock space. Discrete
spectrum asymptotics. J. Stat. Phys., 127:2 (2007), 191220.
12. Albeverio, S.; Lakaev, S.N.; Muminov, Z.I. Schrödinger Operators on Lattices. The Efimov Effect and
Discrete Spectrum Asymptotics. Ann. Henri Poincare, 5 (2004), 743–772.
13. Muminov, M. E.; Aliev, N. M. On the spectrum of a three-particle Schrödinger operator on a one-
dimensional lattice. Theoretical and Mathematical Physics, 171:3 (2012), 754–768.
Copyright (c) 2026 «ACTA NUUz»
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
.jpg)
1.png)
