Sur une valeur propre d'un opérateur (Q790403)

Let A and \(A^*\) be the annihilation and creation operators on the Bargmann space. The present paper studies analytic dependence of a special eigenvalue of the operator \(H_{\mu}=\mu A^*A+iA^*(A+A^*)A,\) considered on the orthocomplement of the vacuum state. A key point is that for \(\mu>0\) the inverse \(K_{\mu}\) of \(H_{\mu}\) can be considered as an integral operator of Hilbert Schmidt type with strictly positive kernel \(N_{\mu}\) on a suitable weighted \(L^ 2\)-space. \(N_{\mu}\) decreases as \(\mu\) increases. Moreover one can extend analytically the map \(\mu \to N_{\mu}\) over the whole complex plane in such a way that \(N_{\mu}\) is positive on the whole real line. According to the Jentzsch theorem, a continuous version of the Perron- Frobenius theorem, for real \(\mu\), the spectral radius \(\sigma\) (\(\mu)\) of \(N_{\mu}\) is a simple positive eigenvalue. The function \(\sigma\) (\(\mu)\) is decreasing and becomes analytic. This proves that for \(\mu>0\), \(E(\mu):=1/\sigma(\mu)\) is a smallest eigenvalue of \(H_{\mu}\), and that the function \(E(\mu)\) admits a positive, increasing analytic extension of the whole real line. In particular, \(E(0)\neq 0,\) contrary to what might have been expected from the fact that \(A^*(A+A^*)A\) is formally self- adjoint.