  We generate data-driven reduced order models (ROMs) for inversion of the one
and two dimensional Schr\"odinger equation in the spectral domain given
boundary data at a few frequencies. The ROM is the Galerkin projection of the
Schr\"odinger operator onto the space spanned by solutions at these sample
frequencies. The ROM matrix is in general full, and not good for extracting the
potential. However, using an orthogonal change of basis via Lanczos iteration,
we can transform the ROM to a block triadiagonal form from which it is easier
to extract $q$. In one dimension, the tridiagonal matrix corresponds to a
three-point staggered finite-difference system for the Schr\"odinger operator
discretized on a so-called spectrally matched grid which is almost independent
of the medium. In higher dimensions, the orthogonalized basis functions play
the role of the grid steps. The orthogonalized basis functions are localized
and also depend only very weakly on the medium, and thus by embedding into the
continuous problem, the reduced order model yields highly accurate internal
solutions. That is to say, we can obtain, just from boundary data, very good
approximations of the solution of the Schr\"odinger equation in the whole
domain for a spectral interval that includes the sample frequencies. We present
inversion experiments based on the internal solutions in one and two
dimensions.
