15,921 research outputs found
On the independent subsets of powers of paths and cycles
In the first part of this work we provide a formula for the number of edges
of the Hasse diagram of the independent subsets of the h-th power of a path
ordered by inclusion. For h=1 such a value is the number of edges of a
Fibonacci cube. We show that, in general, the number of edges of the diagram is
obtained by convolution of a Fibonacci-like sequence with itself.
In the second part we consider the case of cycles. We evaluate the number of
edges of the Hasse diagram of the independent subsets of the h-th power of a
cycle ordered by inclusion. For h=1, and n>1, such a value is the number of
edges of a Lucas cube.Comment: 9 pages, 4 figure
Efimov trimers under strong confinement
The dimensionality of a system can fundamentally impact the behaviour of
interacting quantum particles. Classic examples range from the fractional
quantum Hall effect to high temperature superconductivity. As a general rule,
one expects confinement to favour the binding of particles. However,
attractively interacting bosons apparently defy this expectation: while three
identical bosons in three dimensions can support an infinite tower of Efimov
trimers, only two universal trimers exist in the two dimensional case. We
reveal how these two limits are connected by investigating the problem of three
identical bosons confined by a harmonic potential along one direction. We show
that the confinement breaks the discrete Efimov scaling symmetry and destroys
the weakest bound trimers. However, the deepest bound Efimov trimer persists
under strong confinement and hybridizes with the quasi-two-dimensional trimers,
yielding a superposition of trimer configurations that effectively involves
tunnelling through a short-range repulsive barrier. Our results suggest a way
to use strong confinement to engineer more stable Efimov-like trimers, which
have so far proved elusive.Comment: 8 pages, 4 figures. Typos corrected, published versio
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