  We study a particular class of families of multi-dimensional lattice
Schr\"o\-dinger operators with deterministic (including quasi-periodic)
potentials generated by the "hull" given by an orthogonal series over the Haar
wavelet basis on the torus, of arbitrary dimension, with expansion coefficients
considered as independent parameters. In the strong disorder regime, we prove
Anderson localization for generic operator families, using a variant of the
Multi-Scale Analysis, and show that all localized eigenfunctions are unimodal
and feature uniform exponential decay away from their respective localization
centers. Using the Klein--Molchanov argument and a variant of the Minami
estimate for deterministic potentials, we prove the simplicity of the spectrum
in our model.
  NOTE: This text completes our earlier manuscript (math-ph/0907.1494),
originally uploaded in 2009 and revised in 2011, which is kept in
\textbf{arXiv} in a reduced form, merely to avoid broken references in earlier
works. Compared to [\texttt{math-ph/0907.1494}], we add the results on
unimodality of the eigenstates, uniform dynamical localization, and simplicity
of p.p. spectra.
  Compared to earlier versions of this preprint, the presentation has been
adapted to the future extension of the main results (uniform localization,
unimodality of the eigenfunctions) to the multi-particle Anderson Hamiltonians
with a nontrivial interaction between the particles, which we plan to publish
in a forthcoming paper.
