7 problems

GAL simultaneous-relation conjecture. Regardless of whether $a+b+f+g$ is odd or even, relations $(2.17)$ and $(2.18)$ are both valid.

Half-integer GAL partner-relation conjecture. Relations $(2.17)$ and $(2.18)$ should hold simultaneously whenever at least one of the four parameters $a,b,f,g$ is a half-integer.

GAL integer-to-half-integer correspondence conjecture. Relations $(2.17)$ and $(2.18)$ should hold simultaneously when $a,b,f,g$ are integers, and the band-edge eigenvalues of the…

GAL partner-potential conjecture. The following relationships hold:

Finite-band-gap conjecture. Every GAL potential whose four parameters $a,b,f,g$ are integers has a finite number of band gaps.

Spectral-polynomial identity. The two polynomials are equal:

Potential-count invariance conjecture.