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

$L^p$-$L^q$ Boundedness of Spectral Multipliers of the Anharmonic Oscillator