Repository: Freie Universität Berlin, Math Department

Conditional Quasi-Exact Solvability of the Quantum Planar Pendulum and its Anti-Isospectral Hyperbolic Counterpart

Becker, S. and Mirahmadi, M. and Schmidt, B. and Schatz, K. and Friedrich, B. (2017) Conditional Quasi-Exact Solvability of the Quantum Planar Pendulum and its Anti-Isospectral Hyperbolic Counterpart. Eur. J. Phys. D, 71 (6). p. 149.

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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 [Frontiers in Physical Chemistry and Chemical Physics 2, 1-16, (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 - forty 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 [American Journal of Physics 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.

Item Type:Article
Subjects:Physical Sciences > Chemistry > Physical Chemistry
Physical Sciences > Physics > Mathematical & Theoretical Physics > Quantum Mechanics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics > BioComputing Group
ID Code:2037
Deposited By: BioComp Admin
Deposited On:28 Feb 2017 11:09
Last Modified:12 Jun 2017 08:05

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