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

TL;DR

This paper demonstrates that in quantum systems with multiple scattering channels, the finite-volume energy spectrum does not uniquely determine the S-matrix, especially when time-reversal symmetry is broken, highlighting fundamental limitations in spectral reconstruction.

Contribution

It proves that spectral data from large finite boxes cannot fully reconstruct the S-matrix in multi-channel quantum systems, revealing inherent ambiguities.

Findings

Spectral data cannot distinguish between different S-matrices in multi-channel systems.

Time-reversal symmetry affects the uniqueness of spectral reconstruction.

Illustrations provided for simple 1+1 dimensional quantum systems.

Abstract

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.