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We present a comprehensive, analytical treatment of the finite Kitaev chainfor arbitrary chemical potential. We derive the momentum quantizationconditions and present exact analytical formulae for the resulting energyspectrum and eigenstate wave functions, encompassing boundary and bulk states.In accordance with an analysis based on the winding number topologicalinvariant, and as expected from the bulk-edge correspondence, the boundarystates are topological in nature. They can have zero, exponentially small oreven finite energy. A numerical analysis confirms their robustness againstdisorder.

Exact eigenvectors and eigenvalues of the finite Kitaev chain and its topological properties