  We prove uniqueness problems for meromorphic inner functions on the upper
half-plane. In these problems we consider spectral data depending partially or
fully on the spectrum, derivative values at the spectrum, Clark measure or the
spectrum of the negative of a meromorphic inner function. Moreover we consider
applications of these uniqueness results to inverse spectral theory of
canonical Hamiltonian systems and obtain generalizations of Borg-Levinson
two-spectra theorem for canonical Hamiltonian systems and unique determination
of a Hamiltonian from its spectral measure under some conditions.
