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

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##### Citation

Becker, S., Mirahmadi, M., Schmidt, B., Schatz, K., & Friedrich, B. (2017). Conditional
quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart.*
The European Physical Journal D: Atomic, Molecular and Optical Physics,* *71*(6):
149. doi:10.1140/epjd/e2017-80134-6.

Cite as: http://hdl.handle.net/11858/00-001M-0000-002C-91FB-F

##### 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 [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 $\kappa$, as it determines the loci of the
intersections (genuine and avoided) of the eigenenergy surfaces spanned by the
dimensionless interaction parameters $\eta$ and $\zeta$. It also encapsulates
the conditions under which analytic solutions to the two eigenproblems obtain
and provides the number of analytic solutions. At a given $\kappa$, the
anti-isospectrality occurs for single states only (i.e., not for doublets),
like C-QES holds solely for integer values of $\kappa$, 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.