December 2010
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Day December 2, 2010

Metric operators for non-Hermitian quadratic su(2) Hamiltonians

Paulo E. G. Assis

A class of non-Hermitian quadratic su(2) Hamiltonians that fulfil an anti-linear symmetry is constructed. If unbroken this anti-linear symmetry leads to a purely real spectrum and the Hamiltonian can be mapped to a Hermitian counterpart by, amongst other possibilities, a similarity transformation. Here Lie-algebraic methods which were used to investigate the generalised Swanson Hamiltonian is used to construct a class of quadratic Hamiltonians that allow for such a simple mapping to the Hermitian counterpart. While for the linear su(2) Hamiltonian every Hamiltonian of this type can be mapped to a Hermitian counterpart by a transformation which is itself an exponential of a linear combination of su(2) generators, the situation is more complicated for quadratic Hamiltonians. The existence of finite dimensional representations for the su(2) Hamiltonian, as opposed to the su(1,1) studied before, allows for comparison with explicit diagonalisation results for finite matrices. The possibility of more elaborate similarity transformations, including quadratic exponents, is also discussed. Finally, the similarity transformations are compared with the analogue of Swanson’s method of diagonalising the problem.
Quantum Physics (quant-ph); Mathematical Physics (math-ph)