Stability Criterion for Superfluidity based on the Density Spectral Function

We study a stability criterion hypothesis for superfluids in terms of the the local density spectral function $I_n (r, \omega)$ applicable both to homogeneous and inhomogeneous systems. We evaluate the local density spectral function in the presence of a one-dimensional repulsive/attractive external potential within the Bogoliubov theory using solutions of the tunneling problem. We also evaluate the local density spectral function using an orthogonal basis, and calculate the autocorrelation function $C_n (r,t)$. When superfluids flow below a threshold, we find that in the $d$-dimensional system, $I_n (r, \omega) \propto \omega^{d}$ in the low-energy regime and $C_n (r, t) \propto 1/t^{d+1}$ in the long-time regime hold. When superfluids flow with the critical current, on the other hand, we find $I_n (r, \omega) \propto \omega^{\beta}$ in the low-energy regime and $C_n (r,t) \propto 1/t^{\beta+1}$ in the long-time regime with $\beta < d$. These results support the stability criterion hypothesis recently proposed.