We obtain a method to transform an minimization problem of the quadratic form corresponding to the Schr\"odinger operator in the Euclidean space to a variational problem of a tuple of functions defined on several regions. The method is based on a characterization of elements of the orthogonal complement of $H^1_0(\Omega)$ in $H^1(\Omega)$ as weak solutions to an elliptic partial differential equation on a region $\Omega$ with a bounded boundary.
Computability and Beltrami fields in Euclidean space
An elliptic partial differential equation and its application
Analysis of the diffuse domain approach for a bulk-surface coupled PDE system
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