  The mobility edge, as a central concept in disordered models for
localization-delocalization transitions, has rarely been discussed in the
context of random matrix theory (RMT). Here we report a new class of random
matrix model by direct coupling between two random matrices, showing that their
overlapped spectra and un-overlapped spectra exhibit totally different scaling
behaviors, which can be used to construct tunable mobility edges. This model is
a direct generalization of the Rosenzweig-Porter model, which hosts ergodic,
localized, and non-ergodic extended (NEE) phases. A generic theory for these
phase transitions is presented, which applies equally well to dense, sparse,
and even corrected random matrices in different ensembles. We show that the
phase diagram is fully characterized by two scaling exponents, and they are
mapped out in various conditions. Our model provides a general framework to
realize the mobility edges and non-ergodic phases in a controllable way in RMT,
which pave avenue for many intriguing applications both from the pure
mathematics of RMT and the possible implementations of ME in many-body models,
chiral symmetry breaking in QCD and the stability of the large ecosystems.
