  In three dimensions, exotic new state of matter of self-bound ultradilute
quantum droplets can be realized in free space, when the mean-field attraction
(i.e., with mean-field energy $E_{\textrm{MF}}\propto-n^{2}$ at the density
$n$) is balanced by the repulsive beyond-mean-field quantum fluctuations (i.e.,
$E_{\textrm{BMF}}\propto n^{2+\gamma}$). The parameter $\gamma>0$ typically
takes the value $1/2$ if we consider the Lee-Huang-Yang (LHY) energy
functional, but it can vary when the beyond-LHY-effect becomes important or the
three-body interaction becomes dominant. Here, we theoretically investigate how
collective excitations of a three-dimensional quantum droplet are affected by
the parameter $\gamma$ and a weak harmonic trapping potential, both of which
could be tuned in experiments. We use both the approximate approach based on a
Gaussian variational ansatz and the exact numerical solution of the Bogoliubov
equations resulting from the linearized time-dependent extended
Gross-Pitaevskii equation. We show that one of the key features of quantum
droplets, i.e., the existence of the surface modes with dispersion relation
$\omega_{s}\propto k^{3/2}$ is very robust with respect to the changes either
in the parameter $\gamma$ or in the harmonic trapping potential. We predict the
excitation spectrum of the droplet realized by binary $^{39}$K mixtures under
the typical experimental conditions, which might be readily measured in current
cold-atom laboratories.
