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arXiv:2605.28875

Quantum and Thermal Properties of the Klein-Gordon Inverted Harmonic Oscillator with Physical Applications

Kevin Hernández

quant-phSubmitted May 26, 2026• Announced May 27, 2026
v1 · May 26, 2026v2 · May 29, 2026
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Abstract

We develop a systematic framework for the quantum and thermal properties of a Klein-Gordon scalar field subject to an inverted harmonic potential $-{1\over2} m^2ω^2 x^2$. Starting from a non-Hermitian momentum substitution $P \to P - mωx$, we employ a symplectic phase-space rotation $V = \exp\!\left[-\tfracπ{8}(xp+px)\right]$ to map the system onto an analytically tractable effective harmonic oscillator evaluated at $xe^{iπ/4}$. This allows us to define a well-regulated partition function $Z(β,ω,m)$ and derive closed-form expressions for the free energy, entropy, and thermal correlation functions. We then apply this framework to three physical settings: (i) scalar field fluctuations during cosmological inflation, (ii) quantum fields near black-hole horizons, and (iii) order-parameter dynamics near second-order phase transitions in condensed matter. Our results unify previously scattered results in the literature and provide new predictions for the finite-temperature spectral density and entanglement entropy of unstable quantum systems.

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We develop a systematic framework for the quantum and thermal properties of a Klein-Gordon scalar field subject to an inverted harmonic potential −12m2ω2x2-{1\over2} m^2ω^2 x^2−21​m2ω2x2. Starting from a non-Hermitian momentum substitution P→P−mωxP \to P - mωxP→P−mωx, we employ a symplectic phase-space rotation V=exp⁡ ⁣[−π8(xp+px)]V = \exp\!\left[-\tfracπ{8}(xp+px)\right]V=exp[−8π​(xp+px)] to map the system onto an analytically tractable effective harmonic oscillator evaluated at xeiπ/4xe^{iπ/4}xeiπ/4. This allows us to define a well-regulated partition function Z(β,ω,m)Z(β,ω,m)Z(β,ω,m) and derive closed-form expressions for the free energy, entropy, and thermal correlation functions. We then apply this framework to three physical settings: (i) scalar field fluctuations during cosmological inflation, (ii) quantum fields near black-hole horizons, and (iii) order-parameter dynamics near second-order phase transitions in condensed matter. Our results unify previously scattered results in the literature and provide new predictions for the finite-temperature spectral density and entanglement entropy of unstable quantum systems.
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We develop an exact analytical framework for studying the nonequilibrium dynamics of interacting quantum harmonic oscillators coupled to a modelled thermal reservoir and driven by external classical fields. The total Hamiltonian is diagonalized via successive Bogoliubov transformations, enabling a nonperturbative treatment of dissipation and driving. For two coupled oscillators with an external source applied to the first, we derive explicit energy expressions and analyze their dependence on coupling and bath parameters. The phase-space structure is characterized through Husimi QQQ-functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to nnn interacting oscillators, with analytical energy expressions provided and the case n=3n=3n=3 examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.
An analytical study of the radial Schrodinger wave equation with magnetic field, nonlinear and inverted nonlinear isotropic harmonic oscillator potentials was carried out. A modified substitution technique was used to determine the energy eigenvalues of the nonlinear and inverted nonlinear oscillators while the Laplace and computational methods were used to determine the radial wave functions. The determination of the energies’ eigenvalues and the radial wave functions enabled the study to generate tables, plot graphs, and discuss the effect of the magnetic field, angular frequency, and nonlinear term on the thermodynamic functions and systems. Results of the study revealed that the energies’ eigenvalues of the inverted nonlinear oscillator fluctuates increasingly and decreasingly, while that of the nonlinear isotropic oscillator fluctuates decreasingly and degeneracies, completely removed. As the angular frequency is improved, expansion or increase is observed in the nonlinear and the inverse nonlinear cases in the radial wave functions. In the case of increase in the nonlinear term, saturation in the radial wave functions are observed in the inverted and non-inverted cases. For the effect of the increase in the magnetic field potential, initial decrease with saturation was observed in the both cases. The effect of the magnetic field and nonlinear term in both the nonlinear and inverted oscillators on the thermodynamic functions, attained saturation on the polyatomic molecules, while the effects of angular frequency is observed to enhance the thermodynamic functions. Generally, the contraction or saturation experienced with the increase of the magnetic field term is as a result of the small value of the electron charge, and the carbon (iv) oxide molecule is at higher potential than that of the water molecule.
It has been known for decades that black holes behave as thermodynamic objects and, as such, have an entropy, which was acquired in exact form as the Bekenstein-Hawking entropy. However, the origin of this entropy, and in particular its proportionality to the area of the horizon, remained unexplained. Motivated by this mystery, several authors examined the possibility of entanglement entropy as a source of black hole entropy, by considering a scalar ?eld in the ground state and\ntracing out the degrees of freedom inside an imaginary sphere, analogous to the interior of the black hole. The resulting entanglement entropy between the two regions turned out to scale with the area of their mutual boundary. In the simplest model the scalar ?eld is represented by a system of coupled harmonic oscillators. After an introduction to the phenomenon of quantum entanglement and the associated entanglement entropy, we will review the problem of two coupled oscillators in the ground state and derive the entanglement entropy using an approach in Fock space, as opposed to the standard method involving integrals in positon space. This will also\nallow us to look at some excited states of the system. Subsequently we will try to extend this approach to a system of N coupled harmonic oscillators and derive the entanglement entropy between an inner and outer region of oscillators.
In the framework of the Lindblad theory for open quantum systems, we derive closed analytical expressions of the Heisenberg and Schr\"odinger generalized uncertainty functions for a particle moving in a harmonic-oscillator potential. The particle is initially in an arbitrary correlated coherent state, and interacts with an environment at finite temperature. We describe how the quantum and thermal fluctuations contribute to the uncertainties in the canonical variables of the system and analyze the relative importance of these fluctuations in the evolution of the system. We show that upon contact with the bath, the system evolves from a quantum-dominated to a thermal-dominated state in a time that is of the same order as the decoherence time calculated in other models in the context of transitions from quantum to classical physics.