Eigenvalue Problem of Hamiltonian of Single Cooper Pair Box

Questions
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1

I am facing problem to determine the eigenvalue (energy) from the Hamiltonian of Cooper pair box,
H ?=4Ec (-i?/??-n)^2 ?(?)-Ej cos(?) ?(?)=Ek ?(?).

I have used ‘DSolve’ in Mathematica to find eigenfunction as follow,

DSolve[4Ec(-?‘’[?]+2?n?'[?]+n^2?[?])-EjCos[?]?[?]==Ek?[?],?[?],?]

{{?[?]??^?n? C[1]MathieuC[Ek/Ec,-Ej/2Ec,?/2]+?^?n? C[2]MathieuS[Ek/Ec,-Ej/2Ec,?/2]}}

But I can’t find any particular procedure to get eigenvalue Ek in terms of n. In some paper it is given as

Ek(n)=Ec MathieuCharacteristicA [1+k-(1+k) mod 2 + 2 n (-1)^k,-2Ej/Ec]

I am a newly Mathematica user so if anyone has the procedure to determine eigenvalue of Hamiltonian, please let me know.

2

You need to find what values of the energy give wavefunctions that satisfy the boundary conditions.