Recently, I am working on PT symmetric non-hermitian lattices. A PT symmetric non-hermitian Hamiltonian gives real eigenvalue spectrum in the unbroken symmetry region whereas it gives imaginary eigenvalue spectrum in the broken symmetry region. Can anyone explain why non-hermitian term appears in a system? What is the meaning of 'loss and gain' in a system? What is the significance of imaginary part of energy eigenvalue?
Well the usual criteria of having real eigenvalues and orthogonal eigenvectors was that the hamiltonian was hermitian, ie. that it is equal to its transpose conjugate.
But if you care only about real eigenvalues, it was possible
to have non hermitian hamiltonians, called pseudo hermitian, and it is possible o orthogonalize with respect to some weight function or matrix sandwitched between the vectors.
It is then further notice that hamiltonians with symmetry under parity and time reversal symmetry also have real eigenvalues.
so the game seems to be how far you can fool the usual rules of QM using math, out of the traditional hermitian operator with orthonormal eignestates.
So far it is a mathematical physics game with not direct relavance to physics proper that I am convinced of, but its popular.
Loss or gain may mean in energy loss or gain? If so it would not be T invariant. This can also be lost with a magnetic fields.
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