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Closing the “green gap”: energy savings from the math of the landscape function

I recently returned from the 2025 Annual Meeting of the “Localization of Waves” collaboration (supported by the Simons Foundation, with additional related support from the NSF), where I…

What I also did not realize before this meeting is the role that recent advances in pure mathematics – and specifically, the development of the “landscape function” that was a primary focus of this collaboration – played in accelerating this transition. (where we have now suppressed all physical constants for simplicity), it turns out that the behavior of the eigenfunctions at various energy levels is controlled to a remarkable extent by the landscape function, defined to be the solution to the equation However, the landscape function makes a prediction for this density of states that is significantly more accurate in practice in these regimes, with a mathematical justification (up to multiplicative constants) of this accuracy obtained in this paper of David, Filoche, and Mayboroda.

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