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Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart
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 , as it determines the loci of the intersections
(genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless
interaction parameters and . It also encapsulates the conditions
under which analytic solutions to the two eigenproblems obtain and provides the
number of analytic solutions. At a given , the anti-isospectrality
occurs for single states only (i.e., not for doublets), like C-QES holds solely
for integer values of , 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
Transcript of My Father’s Heroics
This story is an excerpt from a longer interview that was collected as part of the Launching through the Surf: The Dory Fleet of Pacific City project. In this story, Sid Fisher recounts how his father, Walt Fisher, saved him from rolling his dory
From the chiral model of TBG to the Bistritzer--MacDonald model
We analyse the splitting of exact flat bands in the chiral model of the
twisted bilayer graphene (TBG) when the coupling of the full
Bistritzer--MacDonald model is taken into account. The first-order perturbation
caused by the potential the same for both bands and satisfies
interesting symmetries, in particular it vanishes on the line defined by the
points. The splitting of the flat bands is governed by the quadratic term
which vanishes at the points
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