Multi-channel S-matrices from energy levels in finite boxes

We show that for a generic quantum mechanical system with more than one open scattering channel, it is not possible to fully reconstruct the theory's S-matrix from spectral information obtained in large finite volumes with periodic boundary conditions. Physically distinct S-matrices can have identical finite-volume spectra for large finite boxes of arbitrary sizes. If the theory is not time-reversal symmetric, there exists an uncountably infinite set of distinct S-matrices with the same spectra. If the theory respects time-reversal symmetry there exists a discrete set of S-matrices with identical energy levels for finite boxes. We illustrate the issue for simple quantum mechanical systems in 1+1 dimensions.