Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2004-05-12
Phys.Lett.A328:102-109,2004
Physics
High Energy Physics
High Energy Physics - Theory
11 pages, 1 figure
Scientific paper
10.1016/j.physleta.2004.05.063
To determine the Hilbert space and inner product for a quantum theory defined by a non-Hermitian $\mathcal{PT}$-symmetric Hamiltonian $H$, it is necessary to construct a new time-independent observable operator called $C$. It has recently been shown that for the {\it cubic} $\mathcal{PT}$-symmetric Hamiltonian $H=p^2+ x^2+i\epsilon x^3$ one can obtain $\mathcal{C}$ as a perturbation expansion in powers of $\epsilon$. This paper considers the more difficult case of noncubic Hamiltonians of the form $H=p^2+x^2(ix)^\delta$ ($\delta\geq0$). For these Hamiltonians it is shown how to calculate $\mathcal{C}$ by using nonperturbative semiclassical methods.
Bender Carl M.
Jones Hugh F.
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