7,761 research outputs found
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
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
Twisted TMDs in the small-angle limit: exponentially flat and trivial bands
Recent experiments discovered fractional Chern insulator states at zero
magnetic field in twisted bilayer MoTe [C23,Z23] and WSe [MD23]. In
this article, we study the MacDonald Hamiltonian for twisted transition metal
dichalcogenides (TMDs) and analyze the low-lying spectrum in TMDs in the limit
of small twisting angles. Unlike in twisted bilayer graphene Hamiltonians, we
show that TMDs do not exhibit flat bands. The flatness in TMDs for small
twisting angles is due to spatial confinement by a matrix-valued potential. We
show that by generalizing semiclassical techniques developed by Simon [Si83]
and Helffer-Sj\"ostrand [HS84] to matrix-valued potentials, there exists a wide
range of model parameters such that the low-lying bands are of exponentially
small width in the twisting angle, topologically trivial, and obey a harmonic
oscillator-type spacing with explicit parameters.Comment: 25 pages, comments are welcom
- …