The Compactness of Right Inverse of Imaginary Part of Reggeon Field Theory Hamiltonian on Bargmann Space

The Hamiltonian of Reggeon field theory is defined by Hμ,λ=μA*A + iλA*(A+A*)A, where A and A* are the annihilation and creation operators satisfying [A,A*]=I and μ, λ are real parameters, and i2=−1. This operator acts on Bargmann space B where B is a Hilbert space of holomorphic square integrable functions with respect to the Gaussian-weighted Lebesgue measure. In this work, we consider the operator Hλ=iλA*(A+A*)A with maximum domain D(Hλ)={φ∈B;Hλφ∈B}. If we limit the domain to polynomials and take the closure of the obtained operator, we denote it by Hλmin, of which Hλ is obviously an extension. Contrary to what happens for μ≠0, it is well known that these two operators are different. The main purpose of the present work is to show that Hλ admits a right-inverse Kλ, i.e., HλKλ=I on negative imaginary axis and that Kλ is compact.