We have developed a formalism that includes both quasibound states with real
energies and quantum resonances within the same theoretical framework, and that
admits a clean and unambiguous distinction between these states and the states
of the embedding continuum. States described broadly as 'quasibound' are
defined as having a connectedness (in the mathematical sense) to true bound
states through the growth of some parameter. The approach taken here builds on
our earlier work by clarifying several crucial points and extending the
formalism to encompass a variety of continuous spectra, including those with
degenerate energy levels. The result is a comprehensive framework for the study
of quasibound states. The theory is illustrated by examining several cases
pertinent to applications widely discussed in the literature