  In this thesis, we study a quantization condition in relation to the
solvability of Schr\"{o}dinger equations. This quantization condition is called
the SWKB (supersymmetric Wentzel-Kramers-Brillouin) quantization condition and
has been known in the context of supersymmetric quantum mechanics for decades.
The main contents of this thesis are recapitulated as follows: the foundation
and the application of the SWKB quantization condition. The first half of this
thesis aims to understand the fundamental implications of this condition based
on extensive case studies. It turns out that the exactness of the SWKB
quantization condition indicates the exact solvability of a system via the
classical orthogonal polynomials. The SWKB quantization condition provides
quantizations of energy, which we call the direct problem of the SWKB. We
formulate the inverse problem of the SWKB: the problem of determining the
superpotential from a given energy spectrum. The formulation successfully
reconstructs all conventional shape-invariant potentials from the given energy
spectra. We further construct novel solvable potentials, which are
classical-orthogonal-polynomially quasi-exactly solvable, by this formulation.
We further demonstrate several explicit solutions of the Schr\"{o}dinger
equations with the classical-orthogonal-polynomially quasi-exactly solvable
potentials, whose family is referred to as a harmonic oscillator with
singularity functions in this thesis. In one case, the energy spectra become
isospectral, with several additional eigenstates, to the ordinary harmonic
oscillator for special choices of a parameter. By virtue of this, we formulate
a systematic way of constructing infinitely many potentials that are strictly
isospectral to the ordinary harmonic oscillator.
