Recently I found one dimentional hermitian operator has equivalent complex operator under iso-spectral behaviour. In fact I am serching for a hermitian operator which does not have equivalent complex operator.
Try the unit operator (that is, in some sense, the only one there is).
@Olaussen
Please understand my question and give your detail views . Your views will help me completing the paper.Presently I am working in analytical and numerical results. Could you suggest a real operator which does not have complex counter part ?(1D ).
Prof B.Rath
Consider finite-dimensional operators (i.e., matrices). If a NxN matrix M have real eigenvalues, any similar transform of it, S^(-1) M S, where S can be any NxN complex matrix with non-zero determinant, will have the same eigenvalues. Only when a hermitian matrix M is proportional to the unit matrix does it remain hermitian under all possible similarity transforms.
You may say that the hermitian property is still there, only in some cases hidden because a weird basis is being used. Which is true. Any finite matrix which can be diagonalised with all eigenvalues real is similarity equivalent to hermitian matrices -- the diagonalised one D, and all unitary transforms of it, U* D U.
On the other hand, there are matrices M with all eigenvalues real, which are not hermitian. Because they cannot be diagonalised, but only transformed to a Jordan normal form. A simple example is the matrix [[0,0], [1, 0]].
Infinite-dimensional operators do add some extra concerns, which may be of relevance or not. One such concern is a correct definition of the property of being hermitian.
So, to answer you question, I cannot suggest any such operators (except those proportional to the unit operator) because there obviously are none. That is a triviality. Unless you have a deeper definition of the notion 'complex counter part'.
@Olaussen
Thank you for suggestion. Suppose one can not find similarity transformation , then suggest how to proceed with the problem. For example I have the complex hamiltonian and real hamiltonian . How to find similarity transformation under iso-spectra ?
Dr Kare Olaussen, I've a question for you: who are you and why your identity doesn't appear in the official web site of the Norwegian University of Science and Technology, nor Amna Noreen, PhD..?
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