MR Yorimasa OSHIME Yorimasa OSHIME Funkcialaj Ekvacioj. Serio Internacia 59 2016 289--301 http://fe.math.kobe-u.ac.jp/FE/FullPapers/59-2/59_289.pdf http://www.ams.org/mathscinet-getitem?mr=MR Consider any essentially self-adjoint Schrödinger operator $S_0=-\triangle+q_1(x)+q_2(x)$, $\mathrm{Dom}(S_0)=C^\infty_0(\mathbf{R}^N)$ where $q_1(x)\in L^2_{loc}$ satisfies $q_1(x)\geq 0$ and $q_2(x)\in L^2_{loc}$ is a $(-\triangle)-$bounded real-valued multiplication operator with bound less than 1. Let now $q_3(x)$ be a rapidly oscillating potential, e.g., $q_3(x)=|x|^3\sin|x|^5$ or $(1+|x|^2)^{-1}e^{|x|}\cos(e^{|x|})$ with a singularity near $|x|=\infty$. In this paper, it is guaranteed that the perturbation $T_0=S_0+q_3(x)$ is also essentially self-adjoint. Moreover, their Friedrichs extensions $T$ and $S$ have the same essential spectrum, i.e., $\sigma_{ess}(T)=\sigma_{ess}(S)$. In fact, we study these problems more generally, i.e., for complex-valued potentials. Perturbation, Oscillating potentials, Essential self-adjointness, Essential spectrum. Primary 35J10; secondary 35P15. 59-289 2016 Yorimasa OSHIME Yorimasa OSHIME

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