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

Reduced order models for spectral domain inversion: Embedding into the continuous problem and generation of internal data