The Heat and Schr¨odinger Equation in Weighted Spaces
DOI:
https://doi.org/10.15381/pesquimat.v27.i2.27401Keywords:
Heat equation, Schr¨odinger operators, weighted spaces, locally uniform spaces, analytical semigroup, fractional power spacesAbstract
This article analyzes the solution of the heat equation and the Schrödinger equation in Sobolev space with weight in RN. With weights ρ in the class Rρ1,ρ2 it is proven that the heat equation hs a unique solution u(t):=S(t)u0, where {S(t) := eΔt}t≥0 is the analytical semigroup generated by the elliptic operator second-order linear −Δ realized in the Banach space Lqρ(RN). We also prove thath the Schrödinger operator −Δ − V (x)I, with potentials V in locally uniform sapces in RN generates an anlytical semigroup SV (t) := e(Δ+V (x)I)t that preserves order in Lqρ(RN) and has the same fractional power spaces of −Δ.
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Copyright (c) 2024 Nancy Moya Lázaro, Teodoro Sulca Paredes, Gladys Chancan Rojas
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