On the Behaviour of Eigenvalues in Adiabatic Processes

The energy values of a system are, as is well known, the eigenvalues of a Hermitian matrix which for the sake of simplicity we assume to be finite- 

To show this, we count the free real parameters of an n-dimensional Hermitian matrix with and without double eigenvalues. The difference between these numbers will give us the number of the parameters ki, /C2,. .. by whose adjustment a degeneracy of the eigenvalues can be obtained. 

As is well known, one can bring each Hermitian matrix into diagonal form by a unitary matrix Up  

When we deal with a real Hermitian matrix, the only change from the previous case is that everywhere real orthogonal matrices must be put in the place of the unitary matrices. The number of free parameters of a real orthogonal matrix of n 

In the figure the eigenvalues are given as a function of k (the drawn curves). In this, the original value k is replaced by a new variable k + c, so that for /c = 0 the two curves run parallel. 

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