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We lay down the preliminary work to apply the Functional Analytic Approach toquasi-periodic boundary value problems for the Helmholtz equation. Thisconsists in introducing a quasi-periodic fundamental solution and the relatedlayer potentials, showing how they are used to construct the solutions ofquasi-periodic boundary value problems, and how they behave when we perform asingular perturbation of the domain. To show an application, we study anonlinear quasi-periodic Robin problem in a domain with a set of holes thatshrink to points.

The Functional Analytic Approach for quasi-periodic boundary value problems for the Helmholtz equation