MathDB

Generalized Gaussian decay conjecture for the harmonic oscillator

Let a,b>0a,b>0 with ab<1ab<1. Let \a0Ωa,b\a0\Omega_{a,b} be the function defined in equation

, and let $u(x,t)$ solve the Cauchy problem

with initial data u0u_0. Write

s=−tan⁡(4πt)4π.s=-\frac{\tan(4\pi t)}{4\pi}.

If u0∈E1(a,b)u_0\in E^1(a,b), then the generalized Gaussian decay conjecture.

u(x,t)∈E1(Ωa,b(s),Ωa,b(s)).u(x,t)\in E^1\bigl(\Omega_{a,b}(s),\Omega_{a,b}(s)\bigr).

This conjecture extends the corresponding conjecture of Kulikov et al. for the harmonic oscillator and is suggested by the preceding Gaussian-decay analysis and the proof of the evolution theorem. Its status is not resolved in the supplied source context.

Sources & referencesView supporting material

Primary source

Manish Chaurasia, “Gaussian decay for the Harmonic oscillator”, arXiv:2606.04635 (2026).

fractional-fourier-transformgaussian-decayharmonic-oscillator

Progress summary

Refreshed
Partially solved

A 2026 paper proves the conjecture only for a restricted class of initial data, so the general question remains open.

The conjecture asks whether harmonic-oscillator evolution preserves the stated Gaussian-decay bound with the time-dependent parameter s=−tan⁡(4πt)/(4π)s=-\tan(4\pi t)/(4\pi). A 2026 paper labels this statement Conjecture 2.4 and presents its analysis as evidence for validity, not as a complete proof.

2026 partial resolution

The paper proves the desired membership conclusion for the restricted class of functions covered by Theorem 1.7, and states that the bound is sharp there. It does not prove or disprove the full assertion for every u0∈E1(a,b)u_0\in E^1(a,b) with a,b>0a,b>0 and ab<1ab<1.

Current status (as of August 2026): The conjecture has a sharp partial result for the class in Theorem 1.7, but its full statement remains open.

Sources

Solutions 0

No solutions have been posted yet.