# Note on Morita Inequality for Planar Noncommutative Inverted Oscillator

## TL;DR

This paper examines Morita's conjecture relating temperature bounds to chaos in a noncommutative two-dimensional oscillator, finding that noncommutativity prevents the classical Lyapunov-based chaos but still allows quantum temperature effects.

## Contribution

It extends Morita's conjecture analysis to a noncommutative 2D oscillator, showing classical chaos suppression but quantum effects induce an effective temperature.

## Key findings

- Noncommutativity suppresses classical chaos in the oscillator.
- Morita's temperature bound does not hold in the noncommutative classical case.
- Quantum effects generate an effective temperature despite classical chaos suppression.

## Abstract

A recent conjecture of Morita predicts a lower bound in temperature $T$ of a chaotic system, $T\geq (\hbar/2\pi)\Lambda$, $\Lambda$ being the Lyapunov exponent, which was demonstrated for a one dimensional inverse harmonic oscillator. In the present work we discuss the robustness of this demonstration in an extended version of the above model, where the inverse harmonic oscillator lives a in two dimensional noncommutative space. We show that, without noncommutativity, Morita's conjecture survives in an essentially unchanged way in two dimensions. However, if noncommutativity is switched on, the noncommutativity induced correction terms conspire to produce, in classical framework, a purely oscillating non-chaotic system without any exponential growth so that Lyapunov exponent is not defined. On the other hand, following Morita's analysis, we show that quantum mechanically an effective temperature with noncommutative corrections is generated. Thus Morita's conjecture is not applicable in the noncommutative plane. A dimensionless parameter $\sigma =m\alpha\theta^2$, (where $m, \alpha, \theta$ are the particle mass, coupling strength with inverse oscillator and the noncommutative parameter respectively) plays a crucial role in our analysis.