Different faces of harmonic
                    oscillator

Turbiner, A. V.

doi:10.1090/crmp/025/39

Cited by 3 publications

(6 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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: 83%

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

et al. 2017

View full text Add to dashboard Cite

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.

show abstract

Section: Problems and Applications Reference Problems And Applicationsupporting

confidence: 83%

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

et al. 2017

View full text Add to dashboard Cite

show abstract

“…In this case the perturbative corrections ε n remain the same as in continuous case, the corrections ϕ n = φ n (b)|0 > remain polynomials in x with coefficients changed accordingly. The case of the perturbed harmonic oscillator (3.1) (continuous, on uniform lattice and on exponential lattice) was studied in some details in [83,84].…”

Section: Algebraic Perturbations Of Exactly-solvable Problemsmentioning

confidence: 99%

One-dimensional quasi-exactly solvable Schrödinger equations

2016

View full text Add to dashboard Cite

Quasi-Exactly Solvable Schrödinger Equations occupy an intermediate place between exactly-solvable (e.g. the harmonic oscillator and Coulomb problems etc) and non-solvable ones. Mainly, they were discovered in the 1980ies. Their major property is an explicit knowledge of several eigenstates while the remaining ones are unknown. Many of these problems are of the anharmonic oscillator type with a special type of anharmonicity. The Hamiltonians of quasi-exactly-solvable problems are characterized by the existence of a hidden algebraic structure but do not have any hidden symmetry properties. In particular, all known one-dimensional (quasi)-exactly-solvable problems possess a hidden sl(2, R)− Lie algebra. They are equivalent to the sl(2, R) Euler-Arnold quantum top in a constant magnetic field.Quasi-Exactly Solvable problems are highly non-trivial, they shed light on the delicate analytic properties of the Schrödinger Equations in coupling constant, they lead to a non-trivial class of potentials with the property of Energy-Reflection Symmetry. The Lie-algebraic formalism allows us to make a link between the Schrödinger Equations and finite-difference equations on uniform and/or exponential lattices, it implies that the spectra is preserved. This link takes the form of quantum canonical transformation. The corresponding isospectral spectral problems for finite-difference operators are described. The underlying Fock space formalism giving rise to this correspondence is uncovered. For a quite general class of perturbations of unperturbed problems with the hidden Lie algebra property we can construct an algebraic perturbation theory, where the wavefunction corrections are of polynomial nature, thus, can be found by algebraic means.In general, Quasi-Exact-Solvability points to the existence of a hidden algebra formalism which ranges from quantum mechanics to 2-dimensional conformal field theories.

show abstract

“…For similar results formulated in terms of operators acting in Fock spaces and the notion of isospectral discretization see Turbiner et al ([19]- [24]), and [17] for further discussions.…”

Section: Exact Solvability and Spectral Properties Of Discrete Superimentioning

confidence: 92%

“…The formulas (5.13) remain the same (with J i → J i ) and all commutation relations are preserved, as are polynomial solutions. For similar results formulated in terms of operators acting in Fock spaces and the notion of isospectral discretization see Turbiner et al ([19]- [24]), and [17] for further discussions.…”

Section: Discrete Generalized Coulomb Potentialsmentioning

confidence: 92%

“…Umbral calculus has recently been used explicitly [17], or implicitly [18]- [21], to provide discrete representations of canonical commutation relations, specially in the context of exactly solvable and quasi exactly solvable quantum systems [22]- [24]. Linear differential equations have been discretized in a symmetry preserving manner using commuting difference operators [25]- [27].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Umbral calculus, difference equations and the discrete Schrödinger equation

Levi

Tempesta

Winternitz

2004

Journal of Mathematical Physics

View full text Add to dashboard Cite

We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schrödinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space-time. In this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case. Moreover, systems that are integrable, superintegrable or exactly solvable preserve these properties in the discrete case.

show abstract

Different faces of harmonic oscillator

Abstract: Harmonic oscillator in Fock space is defined. Isospectral as well as polynomiality-of-eigenfunctions preserving, translation-invariant discretization of the harmonic oscillator is presented. Dilatationinvariant and polynomiality-of-eigenfunctions preserving discretization is also given.

Cited by 3 publications

References 3 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

One-dimensional quasi-exactly solvable Schrödinger equations

Umbral calculus, difference equations and the discrete Schrödinger equation

Contact Info

Product

Resources

About