# Quantum integrable systems, non-skew-symmetric r-matrices and algebraic Bethe ansatz

## Abstract

We prove the integrability of the general quantum Hamiltonian systems governed by an arbitrary non-skew-symmetric, so(3)-valued, nondynamical classical r-matrix with spectral parameters. We consider the most interesting example of these quantum integrable systems, namely, the so(3) 'generalized Gaudin systems' in detail. In the case of an arbitrary r-matrix which is 'diagonal' in the sl(2) basis we calculate the spectrum and the eigenvalues of the corresponding Hamiltonians using the algebraic Bethe ansatz technique.

- Authors:

- Bogoliubov Institute for Theoretical Physics, Institute of Mathematics of NASU, Metrologichna Strasse, 14-b, Kiev 03143 (Ukraine)

- Publication Date:

- OSTI Identifier:
- 20929643

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 2; Other Information: DOI: 10.1063/1.2435085; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; EIGENFUNCTIONS; EIGENVALUES; HAMILTONIANS; INTEGRAL CALCULUS; LATTICE FIELD THEORY; R MATRIX; SO-3 GROUPS

### Citation Formats

```
Skrypnyk, T.
```*Quantum integrable systems, non-skew-symmetric r-matrices and algebraic Bethe ansatz*. United States: N. p., 2007.
Web. doi:10.1063/1.2435085.

```
Skrypnyk, T.
```*Quantum integrable systems, non-skew-symmetric r-matrices and algebraic Bethe ansatz*. United States. doi:10.1063/1.2435085.

```
Skrypnyk, T. Thu .
"Quantum integrable systems, non-skew-symmetric r-matrices and algebraic Bethe ansatz". United States.
doi:10.1063/1.2435085.
```

```
@article{osti_20929643,
```

title = {Quantum integrable systems, non-skew-symmetric r-matrices and algebraic Bethe ansatz},

author = {Skrypnyk, T.},

abstractNote = {We prove the integrability of the general quantum Hamiltonian systems governed by an arbitrary non-skew-symmetric, so(3)-valued, nondynamical classical r-matrix with spectral parameters. We consider the most interesting example of these quantum integrable systems, namely, the so(3) 'generalized Gaudin systems' in detail. In the case of an arbitrary r-matrix which is 'diagonal' in the sl(2) basis we calculate the spectrum and the eigenvalues of the corresponding Hamiltonians using the algebraic Bethe ansatz technique.},

doi = {10.1063/1.2435085},

journal = {Journal of Mathematical Physics},

number = 2,

volume = 48,

place = {United States},

year = {Thu Feb 15 00:00:00 EST 2007},

month = {Thu Feb 15 00:00:00 EST 2007}

}

DOI: 10.1063/1.2435085

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