Decomposição espectral de Hs (R n ) aplicada à equação de Schrödinger fracionária assintoticamente linear

Abstract

In the present work, we are interested in studying the fractional Schrödinger equation, given by (−∆)su + V (x)u = g(x, u), x ∈ R n , s ∈ (0, 1), through the use of an abstract linking theorem applied to the energy functional associated with such equation. We will adopt some hypotheses related to g, and primarily, to V , and, to obtain weak solutions to this problem, we will study the spectrum of Ss = (−∆)s + V and decompose the fractional Sobolev spaces Hs (R n ) using Spectral Theory. We will construct the basic tools of Spectral Theory (including the proof of the Spectral Theorem for unbounded self-adjoint operators), as well as provide results related to the spectrum of Ss that have not been found in the literature, and also a new decomposition of Hs (R n ) in its fractional version that has not been found in the literature.