Periodic effective potentials for spin systems and new exact solutions of the one-dimensional Schr�dinger equation for the energy bands

Заславский, О. Б.; Ulyanov, Vladimir V.

doi:10.1007/bf01028652

Cited by 19 publications

(11 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In classical mechanics this is the Zhukovsky-Volterra gyrostat [1], [16]. Note moreover that a similar Hamiltonian was exploited to describe spin systems with anisotropy [24]. We thus see that the Heun operator pencil on the su(2) algebra is equivalent to the Hamiltonian of a generalized Euler top (or quantum Zhukovskii-Volterra gyrostat).…”

Section: Su(2)mentioning

confidence: 88%

Heun algebras of Lie type

Crampé

Vinet²,

Zhedanov

2019

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For su(2), this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtchouk operator is identified as the Hamiltonian of the quantum analogue of the Zhukovski-Voltera gyrostat. For su(1, 1), one obtains the Heun algebras attached to the Meixner, Meixner-Pollaczek and Laguerre polynomials. These Heun algebras are shown to be isomorphic the the Hahn algebra. Focusing on the harmonic oscillator algebra ho leads to the Heun-Charlier algebra. The connections to orthogonal polynomials are achieved through realizations of the underlying Lie algebras in terms of difference and differential operators. In the su(1, 1) cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn and Confluent Heun operators respectively.

show abstract

Section: Su(2)mentioning

confidence: 88%

Heun algebras of Lie type

Crampé

Vinet²,

Zhedanov

2019

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The relation between the Hamiltonian of a quantum Euler top in a magnetic field built from sl 2 (R) and the Heun operator has been recently studied in [Tu16] (see also [WZ95]). It is a quantum version of the Zhukovsky-Volterra gyrostat of classical mechanics [Ba09,LOZ06], and arises in spin systems with anisotropy [ZU87]. Considering the realization of the Askey-Wilson algebra given in Example 1, it is straightforward to derive a q-deformed analog of the Euler top [Tu16, eq.…”

Section: Applicationsmentioning

confidence: 99%

Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

Baseilhac

Pimenta

2019

Nuclear Physics B

View full text Add to dashboard Cite

An operator of Heun-Askey-Wilson type is diagonalized within the framework of the algebraic Bethe ansatz using the theory of Leonard pairs. For different specializations and the generic case, the corresponding eigenstates are constructed in the form of Bethe states, whose Bethe roots satisfy Bethe ansatz equations of homogeneous or inhomogenous type. For each set of Bethe equations, an alternative presentation is given in terms of 'symmetrized' Bethe roots. Also, two families of on-shell Bethe states are shown to generate two explicit bases on which a Leonard pair acts in a tridiagonal fashion. In a second part, the (in)homogeneous Baxter T-Q relations are derived. Certain realizations of the Heun-Askey-Wilson operator as second q-difference operators are introduced. Acting on the Q-polynomials, they produce the T-Q relations. For a special case, the Q-polynomial is identified with the Askey-Wilson polynomial, which allows one to obtain the solution of the associated Bethe ansatz equations. The analysis presented can be viewed as a toy model for studying integrable models generated from the Askey-Wilson algebra and its generalizations. As examples, the q-analog of the quantum Euler top and various types of three-sites Heisenberg spin chains in a magnetic field with inhomogeneous couplings, three-body and boundary interactions are solved. Numerical examples are given. The results also apply to the time-band limiting problem in signal processing.MSc: 81R50; 81R10; 81U15.

show abstract

“…H 2 is known to describe isotropic paramagnets in two dimensions and a similar Hamiltonian has been studied in ref. [26]. Indeed, by a S 2 rotation of angle π/2, one obtains the Zaslavskii-Ul'yanov Hamiltonian H 3 [26] in zero external magnetic field…”

Section: The Symmetries Of An Euler Topmentioning

confidence: 99%

“…[26]. Indeed, by a S 2 rotation of angle π/2, one obtains the Zaslavskii-Ul'yanov Hamiltonian H 3 [26] in zero external magnetic field…”

Section: The Symmetries Of An Euler Topmentioning

confidence: 99%

AnSU(2) analogue of the Azbel-Hofstadter Hamiltonian

Floratos

Nicolis

1998

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

Motivated by recent findings due to Wiegmann and Zabrodin, Faddeev and Kashaev concerning the appearence of the quantum U q (sl(2)) symmetry in the problem of a Bloch electron on a two-dimensional magnetic lattice, we introduce a modification of the tight binding Azbel-Hofstadter Hamiltonian that is a specific spin−S Euler top and can be considered as its "classical" analog. The eigenvalue problem for the proposed model, in the coherent state representation, is described by the S−gap Lamé equation and, thus, is completely solvable. We observe a striking similarity between the shapes of the spectra of the two models for various values of the spin S. †

show abstract

Periodic effective potentials for spin systems and new exact solutions of the one-dimensional Schr�dinger equation for the energy bands

Cited by 19 publications

References 12 publications

Heun algebras of Lie type

Heun algebras of Lie type

Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

AnSU(2) analogue of the Azbel-Hofstadter Hamiltonian

Contact Info

Product

Resources

About