In this thesis, we study two extensions of the WKB approximation. (i) We show that the
WKB quantization rule is suitable for the Andreev bound states in nonuniform superconductors,
and (ii) we prove a general approximate quantization rule ∫

^{ʀE}_{ʟE} k

_{0} dx = (N + ½)π for the bound
states in the potential well of the nth-order Schrödinger equations e

^{-iπn/2}∇

^{n}_{X}ψ = [E - ∆(x)]ψ, where k

_{0} = (E - ∆)

^{1/n} with N ∈ ℕ

_{0}. n is an even natural number and L

_{E} and R

_{E} are the
boundary points between the classically forbidden regions and the allowed region.

Firstly, we demonstrate that the WKB quantization rule is suitable for the Andreev bound
states in nonuniform superconductors. We consider nonhomogeneous superconductivity gap
functions ∆(x) in superconductors with the Bogoliubov quasiparticle energy E, the...[

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In this thesis, we study two extensions of the WKB approximation. (i) We show that the
WKB quantization rule is suitable for the Andreev bound states in nonuniform superconductors,
and (ii) we prove a general approximate quantization rule ∫

^{ʀE}_{ʟE} k

_{0} dx = (N + ½)π for the bound
states in the potential well of the nth-order Schrödinger equations e

^{-iπn/2}∇

^{n}_{X}ψ = [E - ∆(x)]ψ, where k

_{0} = (E - ∆)

^{1/n} with N ∈ ℕ

_{0}. n is an even natural number and L

_{E} and R

_{E} are the
boundary points between the classically forbidden regions and the allowed region.

Firstly, we demonstrate that the WKB quantization rule is suitable for the Andreev bound
states in nonuniform superconductors. We consider nonhomogeneous superconductivity gap
functions ∆(x) in superconductors with the Bogoliubov quasiparticle energy E, the Fermi level E

_{F}, and the total momentum P at E

_{F}+E. The Andreev bound states in the well of slowly varying ❘∆(x)❘ profile are studied, and the well may also be induced by the phase variation of ∆(x) for
massless Dirac fermions. By applying the WKB method to the Bogoliubov-de Gennes equation,
we obtain two main results: (i) For E

_{F} ∼ 0, the bound states are determined by ∫

^{ʀE}_{ʟE} ❘pdx❘ = (½ + n')πħ , where n' ∈ ℕ

_{0}, and L

_{E} and R

_{E} are the boundary points between the classically
allowed region and forbidden regions, and (ii) for E

_{F} >> E and ❘∆(x)❘, the bound states are
given by ∫

^{ʀE}_{ʟE} ❘p ± p

_{F}❘ dx = (½ + n')πħ with small p

_{y(z)}. Empirical quantization conditions are
provided for broader parameter regions. In addition to applying the traditional WKB method,
we also develop a generalized WKB method to tackle semimetals with parabolic dispersion
relationships. The applications of our results are discussed, for example, Dirac π-junctions or
non-chiral Majorana wires, SNS junctions, the excitation threshold, and the tunneling rate in
NSN junctions. In the π-junctions, the Majorana zero modes correspond to the zero-point energy
in the WKB formalism. This observation may provide insights into the Majorana bound state in
a vortex and the Majorana fermions in high-energy physics.

Secondly, we prove a general approximate quantization rule ∫

^{ʀE}_{ʟE} k

_{0} dx = (N + ½)π for
the bound states in the potential well of the nth-order Schrödinger equations e

^{-iπn/2}∇

^{n}_{X}ψ = [E - ∆(x)]ψ, where k

_{0} = (E - ∆)

^{1/n} with N ∈ ℕ

_{0}. The only hypothesis is that all exponentially growing components are negligible, which is appropriate for not narrow wells. We
discuss applications including the Schrödinger equation and Bogoliubov-de Gennes equation.
For instance, the Bogoliubov-de Gennes equation of the semimetal with parabolic dispersion
relationships is transformed into the mentioned differential equation with n = 4. Therefore, our
results greatly generalize the WKB approximation.

Overall, this thesis provides a formula to describe the Andreev bound states in nonuniform
superconductors and expands the scope of the WKB approximation to a broader range of ordinary
differential equations.

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