Quasi-exact solvability beyond the sl(2) algebraization

Gómez‐Ullate, David; Kamran, Niky

doi:10.1134/s1063778807030118

Cited by 13 publications

(25 citation statements)

References 19 publications

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“…This is in agreement with the properties of the energy levels of the spin system formulations of both the planar pendulum and the Razavy Hamiltonians [12,17,18,21,38]. In either case, we obtain the conditions for quasi-analytic solvability (QES) as a trivial consequence of our approach, independent of previous algebraic work, see e.g., references [17,18,[39][40][41][42].…”

Section: Problems and Applications Reference Problems And Applicationsupporting

confidence: 72%

Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart

et al. 2017

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Abstract.We have subjected the planar pendulum eigenproblem to a symmetry analysis with the goal of explaining the relationship between its conditional quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces, established in our earlier work [Front. Phys. Chem. Chem. Phys. 2, 1 (2014)]. The present analysis revealed that this relationship can be traced to the structure of the tridiagonal matrices representing the symmetry-adapted pendular Hamiltonian, as well as enabled us to identify many more -40 in total to be exact -analytic solutions. Furthermore, an analogous analysis of the hyperbolic counterpart of the planar pendulum, the Razavy problem, which was shown to be also C-QES [Am. J. Phys. 48, 285 (1980)], confirmed that it is anti-isospectral with the pendular eigenproblem. Of key importance for both eigenproblems proved to be the topological index κ, as it determines the loci of the intersections (genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless interaction parameters η and ζ. It also encapsulates the conditions under which analytic solutions to the two eigenproblems obtain and provides the number of analytic solutions. At a given κ, the anti-isospectrality occurs for single states only (i.e., not for doublets), like C-QES holds solely for integer values of κ, and only occurs for the lowest eigenvalues of the pendular and Razavy Hamiltonians, with the order of the eigenvalues reversed for the latter. For all other states, the pendular and Razavy spectra become in fact qualitatively different, as higher pendular states appear as doublets whereas all higher Razavy states are singlets.

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Section: Problems and Applications Reference Problems And Applicationsupporting

confidence: 72%

Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart

et al. 2017

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“…In fact there are more general (than the Liealgebraically based) differential equations which do not possess a underlying Lie algebraic structure but are nevertheless quasi-exactly solvable (i.e., have exact polynomial solutions). [10][11][12] …”

Section: Hidden Lie Algebraic Structurementioning

confidence: 99%

On the solvability of the generalized hyperbolic double-well models

Agboola

2014

Journal of Mathematical Physics

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Articles you may be interested inWe present exact solutions for the Schrödinger equation with the hyperbolic doublewell potential V p q (x) = −V 0 sinh p (αx)/cosh q (αx). We show that the model preserves a finite dimensional polynomial space for some p and q. Thus using the Bethe ansatz method, we obtain closed form expressions for the spectrum and wavefunction, as well as the allowed parameter for the class V p 6 (x), which is contrary to a report in a recent article [C. A. Downing, J. Math. Phys. 54, 072101 (2013)]. We also discuss the hidden sl 2 algebraic structure of the class. C 2014 AIP Publishing LLC.

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“…Quasi-periodic behaviour However, for the same values of these parameters, (60), initial data can be chosen within the algebraic submanifold defined by (10). One such assignment is …”

Section: Generic (Chaotic) Behaviourmentioning

confidence: 99%

Two novel classes of solvable many-body problems of goldfish type with constraints

Calogero¹,

Gómez‐Ullate²

2007

J. Phys. A: Math. Theor.

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Two novel classes of many-body models with nonlinear interactions "of goldfish type" are introduced. They are solvable provided the initial data satisfy a single constraint (in one case; in the other, two constraints): i. e., for such initial data the solution of their initialvalue problem can be achieved via algebraic operations, such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials. Entirely isochronous versions of some of these models are also exhibited: i.e., versions of these models whose nonsingular solutions are all completely periodic with the same period.

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Quasi-exact solvability beyond the sl(2) algebraization

Cited by 13 publications

References 19 publications

Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart

Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart

On the solvability of the generalized hyperbolic double-well models

Two novel classes of solvable many-body problems of goldfish type with constraints

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