Schr\"odinger operators with complex singular potentials

Abstract

We study one-dimensional Schr\"{o}dinger operators $\mathrm{S}(q)$ on the space $L^{2}(\mathbb{R})$ with potentials $q$ being complex-valued generalized functions from the negative space $H_{unif}^{-1}(\mathbb{R})$. Particularly the class $H_{unif}^{-1}(\mathbb{R})$ contains periodic and almost periodic $H_{loc}^{-1}(\mathbb{R})$-functions. We establish an equivalence of the various definitions of the operators $\mathrm{S}(q)$, investigate their approximation by operators with smooth potentials from the space $L_{unif}^{1}(\mathbb{R})$ and prove that the spectrum of each operator $\mathrm{S}(q)$ lies within a certain parabola.