Quantum Inverted Harmonic Oscillator

Updated 25 December 2025

Quantum inverted harmonic oscillators are exactly solvable models describing particles in a repulsive quadratic potential, featuring continuous spectrum and non-normalizable eigenstates.
The model elucidates quantum instability and chaos through exponential spreading of wave packets and the analysis of out-of-time-order correlators.
Its rich group-theoretical symmetry under su(1,1) enables exact computation of Lyapunov exponents, quantum anomalies, and applications in quantum optics and gravitational physics.

The quantum inverted harmonic oscillator (IHO) is a fundamental, exactly solvable model describing a particle in a quadratic, unbounded, repulsive potential. The system arises naturally in diverse fields, ranging from quantum chaos diagnostics and quantum optics to black-hole thermodynamics and mesoscopic physics. Its Hamiltonian is defined as

$H_{\mathrm{IHO}} = \frac{p^2}{2m} - \frac{1}{2}m\omega^2 x^2,$

where $x$ and $p$ denote canonical conjugate operators and $\omega > 0$ encodes the instability rate. The repulsive sign of the potential leads to striking features: continuous real spectrum, exponentially growing/decaying solutions, non-normalizable eigenstates, and a deep connection to classical and quantum instabilities. The model is analytically tractable, supports a rich group-theoretical structure, and underpins the analysis of quantum-to-classical correspondence, out-of-time-order correlators, tunneling, and quantum anomalies.

1. Hamiltonian Structure, Quantization, and Fundamental Properties

The IHO is constructed by analytic continuation of the harmonic oscillator: $\omega \mapsto i\Omega$ , with $\Omega>0$ real. Canonical quantization proceeds via $[x,p]=i\hbar$ , and quadratic algebraic methods allow definition of ladder operators and displacement operators, albeit with significant modifications due to the potential's sign (Bhattacharyya et al., 2020, Zerimeche et al., 2022).

The eigenvalue equation,

$\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} - \frac{1}{2}m\omega^2 x^2\right]\psi_E(x) = E\psi_E(x),$

admits two linearly independent solutions for every $E\in\mathbb{R}$ , which are parabolic cylinder functions. The spectrum is continuous, with generalized eigenfunctions delta-normalized or bi-orthonormal in a suitable metric or distributional scheme (Bagarello, 2022, Flores, 2016, Amaouche et al., 2022).

Unlike the regular oscillator, the IHO’s eigenstates are not square-integrable on $\mathbb{R}$ , reflecting the absence of a ground state and physical non-normalizability. Nonetheless, a full spectral resolution can be constructed via distributional or weighted Hilbert-space methods (Bagarello, 2022).

A group-theoretical underpinning emerges via the $\mathfrak{sl}(2,\mathbb{R}) \sim \mathfrak{so}(2,1)$ algebra generated by quadratic combinations of $x$ and $p$ (Subramanyan et al., 2020, Sundaram et al., 21 Feb 2024), with the IHO Hamiltonian, dilatation (Berry–Keating) operator, and squeeze operators as generators.

2. Wave Packet Evolution, Quantum Instability, and Ehrenfest Time

Under the IHO Hamiltonian, initially localized states such as Gaussian wave packets evolve with rapid and anisotropic spreading. The propagator,

$K(x, t \mid x_0, 0) = \sqrt{\frac{\Omega}{2\pi i\hbar\sinh(\Omega t)}} \exp\left[\frac{i}{\hbar}S[x, x_0; t]\right],$

where $S[x,x_0;t]$ is an explicit classical action, determines the full quantum evolution (Golovinski, 2019). For a Gaussian initial state, both the mean position $\langle x(t) \rangle$ and the quantum width $\Sigma(t)$ evolve analytically: \begin{align*} \langle x(t) \rangle &= x_0\cosh(\Omega t) + \frac{p_0}{\Omega}\sinh(\Omega t) + \frac{1}{\Omega} \int_0^t F(s)\sinh[\Omega(t-s)]ds, \ \Sigma^2(t) &= 2\sigma^{2\cosh^2(\Omega} t) + \frac{\hbar^{2}{2\Omega^{2\sigma^{2}\sinh^2(\Omega}}} t). \end{align*} This reflects exponential growth of both the centroid (classically) and the quantum uncertainty (wave-packet spreading), signifying quantum instability.

The Ehrenfest time, $T_E \sim (1/\lambda)\ln(S/\hbar)$ with $\lambda$ the classical Lyapunov exponent, sets the timescale on which quantum-classical correspondence persists before quantum effects dominate (Wang et al., 2022).

Mean photon number and phase-space representations (e.g., the Husimi $Q$ function) further illustrate this instability: the $Q$ -function stretches along the unstable manifold and contracts along the stable one until the "quantum boundary" is reached.

3. Quantum Chaos, OTOCs, and Scrambling

The IHO provides an exactly solvable laboratory for quantum instability and chaos diagnostics, especially through out-of-time-order correlators (OTOCs):

$C(t) = -\langle [\hat{x}(t), \hat{p}(0)]^2 \rangle.$

For an initial coherent state,

$C(t) \sim e^{2\omega t},$

demonstrating that the OTOC’s exponential growth rate is twice the classical Lyapunov exponent ( $\mathrm{EGR} = 2\lambda$ ) (Wang et al., 2022). This doubling universally marks the quantum signature of classical saddle-type instability.

In the pure quadratic IHO, the OTOC modulus may remain unity for certain operator choices (quasi-scrambling), but a cubic perturbation induces genuine information scrambling with stretched exponential decay (Bhattacharyya et al., 2020).

Circuit complexity for time-evolved operators grows exponentially and then linearly, mirroring the instability and saturation regimes familiar in many-body chaotic systems. The quantum Lyapunov spectrum is exactly computable, with eigenvalues $\lambda_{\pm} = \pm\Omega$ pairing as in classical phase space (Bhattacharyya et al., 2020, Choudhury et al., 2022).

The non-equilibrium regime, accessible via Schwinger–Keldysh techniques, enables analytic calculation of OTOCs and the extraction of a quantum Lyapunov exponent under drive or quench, separating three regimes: early oscillatory, intermediate exponential growth, and saturation (Choudhury et al., 2022).

4. Spectral Theory, Tunneling, and Quasinormal Modes

The IHO's continuous spectrum is dual to that of the super-critical inverse square potential, mediated by the Berry–Keating $(xp+px)/2$ Hamiltonian (Sundaram et al., 21 Feb 2024). Canonical and similarity transforms connect the spectral and scattering properties of these models.

The analytic $S$ -matrix,

$\mathcal{S}(E) = \frac{1}{\sqrt{2\pi}\Gamma(\tfrac{1}{2} - iE/\hbar\omega)} \begin{pmatrix} e^{-i\pi/4}e^{-\pi E/2\hbar\omega} & e^{i\pi/4}e^{\pi E/2\hbar\omega} \ e^{i\pi/4}e^{\pi E/2\hbar\omega} & e^{-i\pi/4}e^{\pi E/2\hbar\omega} \end{pmatrix}$

yields transmission and reflection amplitudes and exposes the presence of quasinormal modes (QNMs) with complex energies $E_n = -i\hbar\omega(n+\frac{1}{2})$ , dictating quantized time decay rates (Subramanyan et al., 2020). The RG flow of boundary conditions in the dual model exhibits limit-cycle behavior, reflecting a quantum anomaly in the underlying (hidden) scale invariance (Sundaram et al., 21 Feb 2024).

Tunneling across the inverted potential can be computed in the Wigner-function formalism; tunneling and reflection weights are fully accounted for by classical phase-space flows for quadratic Hamiltonians (Flores, 2016).

A periodically driven IHO demonstrates modulation of tunneling rates; external periodic drive alters the pre-exponential factor and encodes an ADK-type correction in the tunneling probability (Golovinski, 2019).

5. Coherent States, Pseudo-Hermiticity, and Biorthogonality

The question of appropriate eigenstates and coherent-state structures for the IHO is addressed via pseudo-Hermitian and distributional frameworks (Zerimeche et al., 2022, Bagarello, 2022, Amaouche et al., 2022). The IHO Hamiltonian is not PT-symmetric but can be mapped (via a Dyson scaling operator) to an anti–PT–symmetric harmonic oscillator, under which a positive-definite metric operator $\eta$ ensures time invariance of the norm.

Inverted coherent states, defined as eigenstates of generalized ladder operators, minimize the position–momentum uncertainty and propagate along classical hyperbolic trajectories with exponentially diverging uncertainty (Zerimeche et al., 2022, Amaouche et al., 2022). These states admit a displacement-operator construction and bi-resolution of the identity in extended Hilbert spaces.

A Swanson-type or pseudo-bosonic representation allows construction of biorthonormal systems and coherent states for the IHO, all within the standard inner-product structure extended to Schwartz distributions; no ad hoc metric is required (Bagarello, 2022).

6. Physical Realizations, Open Quantum Dynamics, and Applications

The IHO appears as an effective Hamiltonian in numerous contexts:

Quantum Hall systems: The saddle potential projected onto the lowest Landau level maps directly to the IHO; time-resolved scattering at point contacts can probe QNMs in transport (Subramanyan et al., 2020).
Black-hole and cosmological physics: The close analogy with the Rindler Hamiltonian near event horizons provides insight into Hawking–Unruh effect and quantum decay (Subramanyan et al., 2020).
Open quantum systems: Coupling to a harmonic oscillator bath yields a Heisenberg–Langevin equation featuring both deterministic exponential evolution and stochastic Brownian diffusion, integrating quantum noise and dissipative effects (Golovinski, 2019).
Quantum optics: The IHO structure underpins squeeze operators, gain–loss systems, and non-Hermitian symmetry realizations (Zerimeche et al., 2022).
Quantum anomalies: The connection to the inverse square potential and limit-cycle RG flow provides a model system for studying quantum anomalies and Efimov physics (Sundaram et al., 21 Feb 2024).

The model also serves as a quantum analog for classical exponential instability, phase-space shearing, and breakdown of semiclassical approximations—the latter controlled by logarithmically $\hbar$ -dependent Ehrenfest times (Wang et al., 2022).

7. Group Theoretical Symmetry and Hidden Scaling Structure

The IHO, the inverse square potential, and the Berry–Keating Hamiltonian are unified by their embedding in an $\mathfrak{su}(1,1)$ spectrum-generating algebra (Subramanyan et al., 2020, Sundaram et al., 21 Feb 2024). This algebraic perspective enables mapping among models, exposes latent (albeit anomalously broken) scale invariance in the IHO, and allows construction of coherent-state representations and analysis of quantum anomalies via the structure of the RG flows.

This symmetry underlies the universality of exponential growth rates, the limit-cycle structure in the RG flow, and the quadratic Hamiltonian’s reduction to classical Liouville or Koopman–von Neumann evolution in phase space (Flores, 2016, Sundaram et al., 21 Feb 2024).

The quantum inverted harmonic oscillator thus provides a paradigmatic setting for the exact analysis of quantum instability, open-system diffusion, anomalous symmetry breaking, and quantum-classical correspondence. It supports a universal theoretical toolkit relevant for quantum chaos, condensed matter, open-system dynamics, and quantum field theory (Golovinski, 2019, Bhattacharyya et al., 2020, Wang et al., 2022, Subramanyan et al., 2020, Sundaram et al., 21 Feb 2024).