  A one-dimensional quantum harmonic oscillator perturbed by a smooth compactly
supported potential is considered. For the corresponding eigenvalues
$\lambda_n$, a complete asymptotic expansion for large $n$ is obtained, and the
coefficients of this expansion are expressed in terms of the heat invariants. A
sequence of trace formulas is obtained, expressing regularised sums of integer
powers of eigenvalues $\lambda_n$ in terms of the heat invariants.
