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

## Abstract

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 constant, the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model H for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of H below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.

- Authors:

- Faculty of Sciences, Universiti Teknologi Malaysia, 81310 Skudai (Malaysia)
- Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin (Germany)
- Faculty of Physics and Mathematics, Bukhara State University, M. Ikbol str. 11, 200100 Bukhara (Uzbekistan)

- Publication Date:

- OSTI Identifier:
- 22403151

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COUPLING CONSTANTS; EIGENVALUES; HAMILTONIANS; PHOTONS; RADIATIVE DECAY; SPECTRA

### Citation Formats

```
Muminov, M., E-mail: mmuminov@mail.ru, Neidhardt, H., E-mail: neidhard@wias-berlin.de, and Rasulov, T., E-mail: rth@mail.ru.
```*On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case*. United States: N. p., 2015.
Web. doi:10.1063/1.4921169.

```
Muminov, M., E-mail: mmuminov@mail.ru, Neidhardt, H., E-mail: neidhard@wias-berlin.de, & Rasulov, T., E-mail: rth@mail.ru.
```*On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case*. United States. doi:10.1063/1.4921169.

```
Muminov, M., E-mail: mmuminov@mail.ru, Neidhardt, H., E-mail: neidhard@wias-berlin.de, and Rasulov, T., E-mail: rth@mail.ru. Fri .
"On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case". United States. doi:10.1063/1.4921169.
```

```
@article{osti_22403151,
```

title = {On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case},

author = {Muminov, M., E-mail: mmuminov@mail.ru and Neidhardt, H., E-mail: neidhard@wias-berlin.de and Rasulov, T., E-mail: rth@mail.ru},

abstractNote = {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 constant, the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model H for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of H below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.},

doi = {10.1063/1.4921169},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 5,

volume = 56,

place = {United States},

year = {2015},

month = {5}

}

DOI: 10.1063/1.4921169

Other availability

Save to My Library

You must Sign In or Create an Account in order to save documents to your library.