A Note on the Completeness of Generalized Eigenfunctions of the Hamiltonian of Reggeon Field Theory in Bargmann Space

Abstract

The aim of the present paper is to review some old results of Intissar (Commun Math Phys 113:263–297, 1987) and to study some new spectral properties of the operator $H_{\mu , \lambda } = \mu A^{*}A + i\lambda A^{*}(A + A^{*})A$ which characterizes the Reggeon field theory, where $\mu$, $\lambda$ are real parameters and $i^{2} = -1$. $\displaystyle {A\varphi := \frac{d\varphi }{dz}}$ and $\displaystyle {A^{*}\varphi := z\varphi ; z = x + i y; (x, y) \in \mathbb {R}^{2}}$ are the annihilation and the creation operators, satisfying the commutation relation $[A, A^{*}] = I$ in Bargmann space $\displaystyle { \mathcal {B} = \{\varphi : \mathbb {C} \longrightarrow \mathbb {C} \, \text {entire function} \,; \int _{\mathbb {C} }\mid \varphi (z)\mid ^{2}e^{-\mid z \mid ^{2}}dx dy < \infty \}}$. The Hamiltonian $H_{\mu ,\lambda }$ is non-Hermitian with respect to the above standard scalar product. The domain of the adjoint and anti-adjoint parts are not included in each other, nor is the domain of their commutator. Hence the question arises, whether the eigenfunctions of $H_{\mu , \lambda }$ form a complete basis? The main new results of this Note are the determination of the boundary conditions for the eigenvalue problem associated to $H_{\mu ,\lambda }$ and the proof of the completeness of the basis mentioned in the question above by transforming $H_{\mu ,\lambda }$ in the Hermitian form. In particular, $H_{\mu ,\lambda }$ belongs to the class of pseudo-Hermitian Hamiltonians in the Mostafazadeh’s (J Math Phys 43(1):205–214, 2002 or Int J Geom Methods Mod Phys 7:1191–1306, 2010) sense.