  In this article we improve some of the inverse spectral results proved by
Guillemin and Uribe in \cite{GU}. They proved that under some symmetry
assumptions on the potential $V(x)$, the Taylor expansion of $V(x)$ near a
non-degenerate global minimum can be recovered from the knowledge of the
low-lying eigenvalues of the associated Schr\"odinger operator in $\mathbb
R^n$. We prove some similar inverse spectral results using fewer symmetry
assumptions. We also show that in dimension 1, no symmetry assumption is needed
to recover the Taylor coefficients of $V(x)$. We establish our results by
finding some explicit formulas for wave invariants at the bottom of the well.
