Commuting functions of the position and momentum observables on locally compact Abelian groups (Q1119885)

Let G be an arbitrary locally compact abelian group, \(\Gamma\) its dual group and \({\mathcal F}: L^ 2(G)\to L^ 2(G)\) denotes the Fourier transformation. For \(g\in G\) a function f: \(G\to {\mathbb{C}}\) is called essentially g-invariant, if \(f(x+g)=f(x)\) for almost every \(x\in G\). For a subset \(K\subset G\), f is said to be essentially K-invariant, if it is essentially y-invariant for every \(y\in K\) (a similar definition is given for subsets of the group \(\Gamma)\). The main results are the following: Theorem. Suppose \(h\in L^{\infty}(G)\) and \(g\in L^{\infty}(\Gamma)\). The following conditions are equivalent: 1) f\({\mathcal F}^{-1}(g{\mathcal F}(\phi))={\mathcal F}^{-1}[g{\mathcal F}(f\phi)]\) for all \(\phi \in L^{\infty}(G)\), i.e. the linear operators \(\Phi (f)\phi =f\phi\), \(\psi (g)\phi ={\mathcal F}^{-1}(g{\mathcal F}(\phi))\), \(\Phi\),\(\psi\) : \(L^{\infty}(G)\to L^{\infty}(G)\) commute; 2) \(K_ g\supset K_ f^{\perp}\), where \(K_ g=\{y\in \Gamma:\) g is essentially y-invariant\(\}\), \(K_ f=\{x\in G:\) f is essentially x- invariant\(\}\) and \(K^{\perp}\) denotes the annihilator of a subset \(K\subset G\) in \(\Gamma\) ; 3) there is a closed subgroup \(K\subset G\) such that f is essentially K- invariant and g is essentially \(K^{\perp}\)-invariant. The case \(G={\mathbb{R}}\) is considered in an article of \textit{P. Busch}, \textit{T. P. Schonbek} and \textit{F. E. Schroeck} [J. Math. Phys. 28, 2866-2872 (1987)].