Table 1 Boundary condition ( 3.50 ) with $\mathit{g}\mathbf{=}\mathbf{0}$

(3.20) and (3.50) with g = 0

(3.21) and (3.51)

z does not obey (3.20) or (3.50)

s + 1 + p + 1 + m − 1 eigenvalues

s + p + m eigenvalues

i.e. one extra eigenvalue 0

(3.20) and (3.50) with g = 0

(3.22) and (3.51)

z obeys (3.20) but not (3.50)

s + p + 1 + m − 1 eigenvalues

s + p + m eigenvalues

i.e. same eigenvalues

(3.20) and (3.50) with g = 0

(3.21) and (3.53)

z obeys (3.50) but not (3.20)

s + 1 + p + m − 1 eigenvalues

s + p + m eigenvalues

i.e. same eigenvalues

(3.20) and (3.50) with g = 0

(3.22) and (3.53)

z obeys both (3.20) and (3.50)

s + p + m − 1 eigenvalues

s + p + m eigenvalues

i.e. one less eigenvalue 0

(3.28) and (3.50) with g = 0

(3.29) and (3.51)

z does not obey (3.28) or (3.50), $a=\frac{-c(m)}{c(m-1)}$

s + p + 1 + m + 1 eigenvalues

s + p + m + 1 eigenvalues

i.e. one extra eigenvalue 0

(3.28) and (3.50) with g = 0

(3.31) and (3.51)

z obeys (3.28) but not (3.50), $a=\frac{-c(m)}{c(m-1)}$

s − 1 + p + 1 + m + 1 eigenvalues

s + p + m + 1 eigenvalues

i.e. same eigenvalues

(3.28) and (3.50) with g = 0

(3.29) and (3.53)

z obeys (3.50) but not (3.28), $a=\frac{-c(m)}{c(m-1)}$

s + p + m + 1 eigenvalues

s + p + m + 1 eigenvalues

i.e. same eigenvalues

(3.28) and (3.50) with g = 0

(3.31) and (3.53)

z obeys both (3.28) and (3.50), $a=\frac{-c(m)}{c(m-1)}$

s − 1 + p + m + 1 eigenvalues

s + p + m + 1 eigenvalues

i.e. one less eigenvalue 0

(3.28) and (3.50) with g = 0

(3.30) and (3.51)

z does not obey (3.28) or (3.50), $a\ne \frac{-c(m)}{c(m-1)}$

s + 1 + p + 1 + m eigenvalues

s + p + m + 1 eigenvalues

i.e. one extra eigenvalue 0

(3.28) and (3.50) with g = 0

(3.32) and (3.51)

z obeys (3.28) but not (3.50), $a\ne \frac{-c(m)}{c(m-1)}$

s + p + 1 + m eigenvalues

s + p + m + 1 eigenvalues

i.e. same eigenvalues

(3.28) and (3.50) with g = 0

(3.30) and (3.53)

z obeys (3.50) but not (3.28), $a\ne \frac{-c(m)}{c(m-1)}$

s + 1 + p + m eigenvalues

s + p + m + 1 eigenvalues

i.e. same eigenvalues

(3.28) and (3.50) with g = 0

(3.32) and (3.53)

z obeys both (3.28) and (3.50), $a\ne \frac{-c(m)}{c(m-1)}$

s + p + m eigenvalues

s + p + m + 1 eigenvalues

i.e. one less eigenvalue 0

(3.37) and (3.50) with g = 0

(3.38) and (3.51)

z does not obey (3.37) or (3.50)

s + p + 1 + m − 1 eigenvalues

s + p + m − 1 eigenvalues

i.e. one extra eigenvalue 0

(3.37) and (3.50) with g = 0

(3.39) and (3.51)

z obeys (3.37) but not (3.50)

s − 1 + p + 1 + m − 1 eigenvalues

s + p + m − 1 eigenvalues

i.e. same eigenvalues

(3.37) and (3.50) with g = 0

(3.38) and (3.53)

z obeys (3.50) but not (3.37)

s + p + m − 1 eigenvalues

s + p + m − 1 eigenvalues

i.e. same eigenvalues

(3.37) and (3.50) with g = 0

(3.39) and (3.53)

z obeys both (3.37) and (3.50)

s − 1 + p + m − 1 eigenvalues

s + p + m − 1 eigenvalues

i.e. one less eigenvalue 0