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Physics Letters A

On harmonic oscillators described by non-Hermitian operators and generalized Heisenberg relations

N. Debergh, J. Beckers, Franciszek Hugon Szafraniec

Published Mar 1, 2000• 6 citations
Publisher
DOI
10.1016/s0375-9601(00)00096-7

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After a few generalities we compare fundamental quantum mechanics applied to the harmonic oscillator with unusual oscillatorlike developments dealing with non-hermitian operators, this latter aspect exploiting in particular the property of subnormality. In this last context we can also restore the hermiticity of the Hamiltonian operator and discover a nice property of the new scalar product. General constructions of oscillatorlike Hamiltonians are also considered and new ideas connected with Heisenberg relations are presented.
In the present paper we study the deformed harmonic oscillator for the non-Hermitian operator [Formula: see text] where λ,θ are real positive parameters, since the parameters α,β,m are for the general case complex. For the case α=1,β=1 and mass m real, we find the eigenfunctions and eigenvalues of energy, the coherent states, the time evolution of the operators [Formula: see text] in the Heisenberg picture and the uncertainty relations. In this case the operator ℋ is Hermitian and PT-symmetric. Also for the case m complex α=1,β=1, the operator ℋ is non-Hermitian and no more PT symmetric, but CPT symmetric with real discrete positive spectrum and the CPT symmetry is preserved. In the general case α,β,m complex, for the non-Hermitian operator ℋ, we obtain complex spectrum and for the special values of the complex parameters α,β the spectrum is real discrete and positive and the CPT symmetry is preserved. The general problem of deformed oscillator for non hermitian operators can be applied to the Solid State Physics.