In this work, we study Schrödinger type equations with domain in a complex space and their solutions in the classical as well as in the sense of the space of currents. For such an equation with a (possibly) complex parameter and a (possibly) nonzero potential, conditions are given under which an analogue of the Weyl's lemma holds, meaning that each solution within the space of currents is distributionally realizable (in a suitable sense) by a semismooth, or by a semi-real-analytic function. Motivated by the Schwartz–Gunning theorem on the holomorphy of a locally integrable function, conditions guaranteeing a regular current on a general domain to be realized by a function which is (locally) a solution of the Schrödinger equation are obtained. Also, generalized Helmholtz formulas are derived and utilized in studying some special classes of zero-degree currents in regard to the property of μ-harmonicity or weak-harmonicity.

在这项工作中，我们研究在复杂空间中具有域的Schrödinger型方程及其在经典空间和电流空间意义上的解。对于具有（可能）复杂参数和（可能）非零电势的等式，给出了满足Weyl引理的类似条件的条件，这意味着电流空间内的每个解都可以在分布上实现（在适当的意义上）通过半平滑或通过半实数分析函数。根据关于局部可积函数全纯的Schwartz-Gunning定理的动机，得到了一个条件，该条件保证了一般域上的规则电流可以通过函数（局部地是Schrödinger方程的解）来实现。也，μ谐波或弱谐波。