  Observationally, the X-ray spectrum ($0.5-10$ keV) of low-level accreting
neutron stars (NSs) can generally be well fitted by the model with two
components, i.e, a thermal soft X-ray component plus a power-law component.
Meanwhile, the fractional contribution of the power-law luminosity $\eta$
($\eta\equiv L^{\rm power\ law}_{\rm 0.5-10\rm keV}/L_{\rm 0.5-10\rm keV}$)
varies with the X-ray luminosity $L_{\rm 0.5-10\rm keV}$. In this paper, we
systematically investigate the origin of such X-ray emission within the
framework of the advection-dominated accretion flow (ADAF) around a weakly
magnetized NS, in which the thermal soft X-ray component arises from the
surface of the NS and the power-law component arises from the ADAF itself. We
test the effects of the viscosity parameter $\alpha$ in the ADAF and
thermalized parameter $f_{\rm th}$ (describing the fraction of the ADAF energy
released at the surface of the NS as thermal emission) on the relation of
$\eta$ versus $L_{\rm 0.5-10\rm keV}$. It is found that $\eta$ is nearly a
constant ($\sim$ zero) with $L_{\rm 0.5-10\rm keV}$ for different $\alpha$ with
$f_{\rm th}=1$, which is inconsistent with observations. Meanwhile, it is found
that a change of $f_{\rm th}$ can significantly change the relation of $\eta$
versus $L_{\rm 0.5-10\rm keV}$. By comparing with a sample of non-pulsating
NS-LMXBs probably dominated by low-level accretion onto NSs, it is found that a
small value of $f_{\rm th} \lesssim 0.1$ is needed to match the observed range
of $\eta \gtrsim 10\%$ in the diagram of $\eta$ versus $L_{\rm 0.5-10\rm keV}$.
Finally, we argue that the small value of $f_{\rm th} \lesssim 0.1$ implies
that the radiative efficiency of NSs with an ADAF accretion may not be as high
as the predicted result previously of $\epsilon \sim {\dot M GM\over
R_{*}}/{\dot M c^2}\sim 0.2$ despite the existence of the hard surface.
