On an eigenvalue of an operator

The studied eigenvalue is the smallest one of the operator: Hsub(μ)=μAsup(*)A+ilambdaAsup(*)(A+Asup(*))A, in the orthogonal complement of the vacuum where Asup(*) and A are the creation and annihilation operators. Call this eigenvalue E(μ) as attention is focused on the dependence on μ. This eigenvalue exists and is a positive number for positive values of μ. Using the fact that the inverse of Hsub(μ) is a positive operator, it is proved that E extends to a positive, increasing, analytic function on the whole real line. In particular, E(0)not=0, contrary to what might have been expected from the fact that Asup(*)(A+Asup(*))A is formally self-adjoint. (orig.)