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We prove uniqueness problems for meromorphic inner functions on the upperhalf-plane. In these problems we consider spectral data depending partially orfully on the spectrum, derivative values at the spectrum, Clark measure or thespectrum of the negative of a meromorphic inner function. Moreover we considerapplications of these uniqueness results to inverse spectral theory ofcanonical Hamiltonian systems and obtain generalizations of Borg-Levinsontwo-spectra theorem for canonical Hamiltonian systems and unique determinationof a Hamiltonian from its spectral measure under some conditions.

Uniqueness theorems for meromorphic inner functions and canonical systems