Statistical Mechanics of a Quantum Harmonic Oscillator with Folded Gaussian Frequency
arXiv:2608.16617
2026
Memory
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper provides a concrete spectral mechanism: folding a Gaussian frequency through \(\omega=|\xi|\) creates a nonzero density at zero frequency, producing soft modes without a Lifshitz-tail mechanism. The transferable asset is that a continuum of stable modes with rates distributed near zero generates algebraic memory even though every individual mode decays exponentially. This can be implemented as an initialization and spectral regularization scheme for diagonal state-space or recurrent layers, giving controllable long-horizon memory while retaining pointwise stability. The decisive test is whether the measured impulse-response envelope follows the predicted \(t^{-1}\) law and whether its prefactor agrees with the folded-Gaussian density at zero.
Ideas from this paper
Unverified
2026
Initialize a stable diagonal state-space layer with decay rates \(\omega_i=|\xi_i|\), where \(\xi_i\sim\mathcal N(\mu,\sigma^2)\), instead of using a narrowly clustered rate distribution. The nonzero density of rates near zero creates a population of slow modes whose aggregate impulse response has an algebraic tail, enabling long-horizon memory while every finite-dimensional mode remains exponentially stable.
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