This paper studies second order elliptic equations in both divergence and non-divergence forms with measurable complex valued principle coefficients and measurable complex valued potentials. The PDE operators can be considered as generalized Schrödinger operators. Under some sufficient conditions, we prove existence, uniqueness, and regularity estimates in Sobolev spaces for solutions to the equations. We particularly show that the non-zero imaginary parts of the potentials are the main mechanisms that control the solutions. Our results can be considered as limiting absorption principle for Schrödinger operators with measurable coefficients and they could be useful in applications. The approach is based on the perturbation technique that freezes the potentials. The results of the paper not only generalize known results but also provide a key ingredient for the study of $ L^p $-diffusion phenomena for dissipative wave equations.

中文翻译：

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具有复值势的椭圆型方程的存在唯一性和正则性理论

本文研究了具有可测量的复值主系数和可测量的复值势的二阶椭圆方程，具有发散和非发散形式。PDE运算符可以视为广义Schrödinger运算符。在某些足够的条件下，我们证明Sobolev空间中存在性，唯一性和正则性估计，用于求解方程。我们特别表明，电势的非零虚部是控制解的主要机制。我们的结果可以被认为是具有可测量系数的Schrödinger算子的极限吸收原理，在应用中可能有用。该方法基于冻结电位的摄动技术。