  This paper studies the reconstruction of Stekloff eigenvalues and the index
of refraction of an inhomogeneous medium from Cauchy data. The inverse spectrum
problem to reconstruct Stekloff eigenvalues is investigated using a new
integral equation for the reciprocity gap method. Given reconstructed
eigenvalues, a Bayesian approach is proposed to estimate the index of
refraction. Moreover, since it is impossible to know the multiplicities of the
reconstructed eigenvalues and since the eigenvalues can be complex, we employ
the recently developed spectral indicator method to compute Stekloff
eigenvalues. Numerical experiments validate the effectiveness of the proposed
methods.
