  In this note we study the $L^p-L^q$ boundedness of Fourier multipliers of
anharmonic oscillators, and as a consequence also of spectral multipliers, for
the range $1<p \leq 2 \leq q <\infty$. The underlying Fourier analysis is
associated with the eigenfunctions of an anharmonic oscillator in some family
of differential operators having derivatives of any order. Our analysis relies
on a version of the classical Paley-type inequality, introduced by H\"ormander,
that we extend in our nonharmonic setting.
