  In this paper we study the $L^p$-$L^q$ boundedness of the Fourier multipliers
in the setting where the underlying Fourier analysis is introduced with respect
to the eigenfunctions of an anharmonic oscillator $A$. Using the notion of a
global symbol that arises from this analysis, we extend a version of the
Hausdorff-Young-Paley inequality that guarantees the $L^p$-$L^q$ boundedness of
these operators for the range $1<p \leq 2 \leq q <\infty$. The boundedness
results for spectral multipliers acquired, yield as particular cases Sobolev
embedding theorems and time asymptotics for the $L^p$-$L^q$ norms of the heat
kernel associated with the anharmonic oscillator. Additionally, we consider
functions $f(A)$ of the anharmonic oscillator on modulation spaces and prove
that Linsk\u ii's trace formula holds true even when $f(A)$ is simply a nuclear
operator.
