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exhibits the change not only in the azimuthal part but also in the angular θ -part. 

## 4 Preservation of the magnetic quantum num- ber m and isospectrality 

To keep the magnetic quantum number as is (i.e., m = 0 , ± 1 , ± 2 , · · · ), one may consider the azimuthal part of the general solution to be of the form 

where F ( ϕ ) satisfies the single-valuedness condition F ( ϕ ) = F ( ϕ + 2 π ). 

Under such setting, the corresponding eigenvalue equation in (21) (with primes denoting derivatives with respect to ϕ ) reads 

In this case F ( ϕ ) would serve as a generating function for the sought after azimuthal potential V$_{eff}$ ( ϕ ) and shapes the form of the azimuthal solution Φ$_{m}$ ( ϕ ). As an illustrative example, a generating function F ( ϕ ) = cos ϕ would imply 

which is indeed a non-Hermitian and P ϕ T $_{ϕ}$-symmetric, P ϕ T $_{ϕ}$V$_{eff}$ ( ϕ ) = V$_{eff}$ ( ϕ ). 

However, one may wish to follow the other way around and consider a P ϕ T $_{ϕ}$- symmetric V$_{eff}$ ( ϕ ) and solve (46) for F ( ϕ ). In this manner, V$_{eff}$ ( ϕ ) would now serve as a generating function for F ( ϕ ) and consequently a generating function for Φ$_{m}$ ( ϕ ). An immediate example is in order. Consider 

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