2022/11/08 by Raúl E. Curto, Curto, Raúl E., Philipp J. di Dio +5 · 1 citation
Physics and Astronomy · Mathematics · #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics #Advanced Chemical Physics Studies
paper · pdf · doi:10.48550/arxiv.2211.04416
We study the time-dependent moments and associated polynomials arising from the partial differential equation ∂t f = νΔf + g⋅∇ f + h⋅ f, and consider in detail the dual equation. For the heat equation we find that several non-negative polynomials which are not sums of squares become sums of squares under the heat equation in finite time. We show that every non-negative polynomial in ℝ[x,y,z]≤ 4 becomes a sum of squares in finite time under the heat equation. We solve the problem of moving atoms under the equation ∂t f = g⋅∇ f + h⋅ f with f0 = μ0 being a finitely atomic measure. The time evolution μt = ∑i=1k ci(t)⋅ δxi(t) of the atom positions xi(t) are described by the transport term g⋅∇ and the time-dependent coefficients ci(t) have an explicit solution depending on xi(t), h, and div g.