Slepian eigenvalues as tunnelling rates

We calculate the eigenvalues of an integral operator associated with Prolate Spheroidal Wave Functions (or Slepian functions) by interpreting them as tunnelling probabilities in an analogous quantum problem. Doing so allows us to extend a well-known approximation due to Slepian so that it applies outside an important transition region where these eigenvalues pass from being near unity to being near zero. Study of the eigenvalues has traditionally been associated with problems arising in signal analysis and optics but have more recently found relevance in quantifying the channel strengths available to Multiple-In Multiple-Out (MIMO) radio communication. The approach presented promises easier generalisation to the broader range of geometries possible in the latter context. •Eigenvalues of integral equation are interpreted as tunnelling rates.•Resulting approximation uniformly extends a classical result due to Slepian.•Other classical approximations from Fuchs, Widom achieved as limits.•Approach used allows future extension to more general geometries.•Eigenvalue distribution gives degree-of-freedom estimation for holographic surfaces.