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Spherical-separability of non-Hermitian Hamiltonians and pseudo-\(\mathcal{PT}\)-symmetry. (English) Zbl 1161.81364

Summary: Non-Hermitian but \(\mathcal{P}_{\varphi }\mathcal{T}_{\varphi }\)-symmetrized spherically-separable Dirac and Schrödinger Hamiltonians are considered. It is observed that the descendant Hamiltonians \(H_{r}, H_{\theta}\), and \(H_{\varphi}\) play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a \(\mathcal{P}_{\varphi}\mathcal{T}_{\varphi}\)-symmetrized \(H _{\varphi }\), we have shown that the conventional Dirac (relativistic) and Schrödinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction \(V(\theta )\neq 0\) in the descendant Hamiltonian \(H _{\theta }\) would manifest a change in the angular \(\theta \)-dependent part of the general solution too. Whilst some \(\mathcal{P}_{\varphi}\mathcal{T}_{\varphi}\)-symmetrized \(H_{\varphi}\) Hamiltonians are considered, a recipe to keep the regular magnetic quantum number \(m\), as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the \(\mathcal{PT}\)-symmetric ones (here the non-Hermitian \(\mathcal{P}_{\varphi}\mathcal{T}_{\varphi}\)-symmetric Hamiltonians) are nicknamed as pseudo-\(\mathcal{PT}\)-symmetric.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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