  Inspired by special and general relativistic systems that can have
Hamiltonians involving square roots, or more general fractional powers, in this
article we address the question how a suitable set of coherent states for such
systems can be obtained. This becomes a relevant topic if the semiclassical
sector of a given quantum theory wants to be analysed. As a simple setup we
consider the toy model of a deparametrised system with one constraint that
involves a fractional power of the harmonic oscillator Hamiltonian operator and
we discuss two approaches for finding suitable coherent states for this system.
In the first approach we consider Dirac quantisation and group averaging that
have been used by Ashtekar et. al. but only for integer powers of operators.
Our generalisation to fractional powers yields in the case of the toy model a
suitable set of coherent states. The second approach is inspired by coherent
states based on a fractional Poisson distribution introduced by Laskin, which
however turn out not to satisfy all properties to yield good semiclassical
results for the operators considered here and in particular do not satisfy a
resolution of identity as claimed. Therefore, we present a generalisation of
the standard harmonic oscillator coherent states to states involving fractional
labels, which approximate the fractional operators in our toy model
semiclassically more accurately and satisfy a resolution of identity. In
addition, motivated by the way the proof of the resolution of identity is
performed, we consider these kind of coherent states also for the polymerised
harmonic oscillator and discuss their semiclassical properties
