  We show a transition from a bound state to a continuum resonance in a shallow
quantum well (QW) by electrostatic gating to bend the conduction band edge.
This bound state-continuum resonance (BSCR) transition is particularly relevant
in topological quantum computing platforms where shallow InAs QWs are used. We
predict the observed capacitance jump and the parallel metal-insulator
transition accompanied the BSCR transition. An experimental puzzle in shallow
InAs QWs is the mobility drop at an electron density smaller than expected for
the bound-state second subband occupation. We explain this puzzle as a result
of intersubband scattering involving a level-broadened continuum resonance,
mediated by screened Coulomb impurities.
