$$L^p$$ - $$L^q$$ Boundedness of Fourier Multipliers Associated with the Anharmonic Oscillator

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 \(1<p \le 2 \le q <\infty \). 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 Linskĭi’s trace formula holds true even when f(A) is simply a nuclear operator.