Constructing Barut-Girardello coherent states for the isotonic oscillator in the DOOT approach

In this work, we study the quantum system of the isotonic oscillator from the perspective of the diagonal operator ordering technique (DOOT). Within this framework, we construct the associated Barut-Girardello and Gazeau-Klauder coherent states. We examine their mathematical properties using reproducing kernels and compute the expectation values of observables that characterize the system and its relevant physical features. Further, we perform the quantization of main classical variables in the complex plane. Then, by exploring the thermal behavior of the physical system in the constructed coherent states, we analyze the properties of mixed states described by a canonical density operator. We also obtain the corresponding Glauber-Sudarshan P-representation.

💡 Research Summary

This paper presents a comprehensive study of the isotonic oscillator quantum system through the lens of the Diagonal Operator Ordering Technique (DOOT). The isotonic oscillator, a deformation of the standard harmonic oscillator with a singular 1/x^2 potential, possesses an underlying su(1,1) Lie algebraic structure. The primary objective is to construct and analyze associated coherent states, specifically Barut-Girardello (BG) and Gazeau-Klauder (GK) types, within this advanced operator framework.

The work begins by establishing the quantum Hamiltonian and demonstrating its connection to the su(1,1) algebra generators (K+, K-, K0). Their action on the Fock space basis |n, γ⟩ is defined, where γ is the Bargmann index related to the potential parameter. The core methodological innovation lies in applying DOOT rules to this system. DOOT generalizes techniques like integration within an ordered product (IWOP) by allowing diagonal ordering of operators paired with special functions. The Hilbert space is decomposed into even and odd subspaces (He and Ho) spanned by states |2n, γ⟩ and |2n+1, γ⟩, respectively. For each subspace, the vacuum state projector |0,γ⟩⟨0,γ| is derived in a closed form involving the confluent hypergeometric function 1F1 under the DOOT ordering symbol ##.

Using these projectors, two distinct families of Barut-Girardello coherent states (BGCS) are explicitly constructed: even BGCS |z,γ⟩e and odd BGCS |z,γ⟩o. These states are defined as eigenstates of the lowering operator K- and are expressed in terms of the raising operator K+ and the 1F1 function. Their normalization factors and overlap functions between two different states are calculated, revealing their dependence on the same hypergeometric function. The continuity of these states with respect to the complex label z is proven.

A fundamental property of coherent states—the resolution of the identity—is rigorously established for both families. By employing the inverse Mellin transform, appropriate positive weight functions We(|z|²,γ) and Wo(|z|²,γ) are found, ensuring that integrals of the projectors |z,γ⟩⟨z,γ| over the complex plane yield the identity operators on their respective Hilbert subspaces. This proves the overcompleteness of the constructed BGCS sets.

The paper then proceeds to evaluate the expectation values of the su(1,1) generators within these coherent state bases, linking the algebraic properties to physical averages. The study further delves into statistical mechanics. The thermal behavior of the system is analyzed by formulating the canonical density operator for a mixed state. Using the BGCS representation, the authors compute the Husimi Q-function, a quasi-probability distribution. Finally, they achieve a significant result by deriving the Glauber-Sudarshan P-representation for the density operator. This P-representation provides a diagonal expansion in terms of the coherent states and is crucial for characterizing non-classical features of the quantum system.

In conclusion, the research successfully demonstrates the power and utility of the DOOT formalism. It provides a mathematically rigorous framework for constructing and analyzing coherent states for the non-trivial isotonic oscillator system. The work connects algebraic structure, state construction, functional analysis (reproducing kernels, completeness), and physical applications (quantization, thermal properties) into a cohesive whole, offering a blueprint for studying coherent states in other quantum systems with similar algebraic foundations.