Characterizing Time-Frequency Localization for Electronic-Wavefunction Basis Sets

The Dirac basis for discretely sampled data is completely localized in
the temporal (spatial) domain; meanwhile, the Fourier basis is completely
localized in the frequency (wavenumber) domain and completely delocalized in
time (space). In the last century, Haar, Daubechies and others introduced
basis sets between these extremes which are partially localized in each
domain. Although these new bases offer extensive freedom,
optimizing the choice of wavelet for a particular application
remains an open question. If one wanted to quantify how localized a single
wavelet is in each of these domains, the obvious thing would be to measure the
variance in the appropriate basis functions; however, this fails to
characterize a wavelet basis set as a whole. We provide
a quantitative measure for choosing a basis set by describing how close
it is to the Dirac basis, the Fourier basis, or in fact any
basis set, bounded by natural localization extrema. We discuss
an application to electronic wavefunctions on the Fibonacci chain.