On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space

Details On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space

2009-08-10

arxiv.org/abs/0908.1321v3

J. Phys. A:Math. Theor. 43(2010)125203

Physics

Quantum Physics

29 pages, revtex, minor changes, version to appear in Journal of Physics A(v3)

Scientific paper

10.1088/1751-8113/43/12/125203

A class of pseudo-hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark(Zeeman) effect with complex electric(magnetic) field, non-hermitian general quadratic form of N boson(fermion) operators, symmetric and asymmetric XXZ spin-chain in complex magnetic field, non-hermitian Haldane-Shastry spin-chain and Lipkin-Meshkov-Glick model.

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