Sufficient conditions for convergence of antinormally ordered expansions of boson number operator functions f(a+a)

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Abstract
From previous work on ordering expansions of the boson number operator exp(- mu a+a) formulae are derived for normally and antinormally ordered forms of arbitrary boson number operator functions f(a+a) in terms of the Fourier transform g( lambda ) of f(x). When the antinormally ordered form f(a)(a+a) is applied to states belonging to the domain of definition of f(a+a), the result differs in general from the action of f(a+a) on these states or is not even defined. For the number operator eigenstates mod m) two criteria are derived which give conditions on the Fourier transform g( lambda ) ensuring validity of f(a)(a+a) mod m)=f(m) mod m). The general considerations are demonstrated by two examples for functions f.