Quasi-Hermitian quantum mechanics and a new class of user-friendly matrix Hamiltonians

Abstract

In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state $\psi$ is controlled, in Hilbert space $\mathcal{L}$, by a Hamiltonian $\mathfrak{h}$ which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces $\mathfrak{h}$ by its isospectral but non-Hermitian avatar $H={\Omega }^{-1}\mathfrak{h}\Omega$ with ${\Omega }^{†}\Omega =\Theta \ne I$. Although acting in another, manifestly unphysical Hilbert space $\mathcal{H}$, the amended Hamiltonian $H\ne H^{†}$ can be perceived as self-adjoint with respect to the amended inner-product metric $\Theta$. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of $H$ we introduce and describe a specific new family of Hamiltonians $H$ for which the metrics $\Theta$ become available in closed form.

12 pp

References in corpus (4)

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• A return to observability near exceptional points in a schematic PT-symmetric model
• Non-stationary non-Hermitian "wrong-sign'' quantum oscillators and their meaningful physical interpretation
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