During the recent developments of quantum theory it has been clarified that the observable quantities (like energy or position) may be represented by operators Λ (with real spectra) which are manifestly non-Hermitian in a preselected "friendly" Hilbert space H (F ) . The consistency of these models is known to require an upgrade of the inner product, i.e., mathematically speaking, a transition H (F ) → H (S) to another, "standard" Hilbert space.We prove that whenever we are given more than one candidate for an observable (i.e., say, two operators Λ 0 and Λ 1 ) in advance, such an upgrade need not exist in general.
Anharmonic oscillator is considered using an unusual, logarithmic form of the anharmonicity. The model is shown connected with the more conventional power-law anharmonicity ∼ |x| α in the limit α → 0. An efficient and user-friendly method of the solution of the model is found in the large−N expansion technique. 1 znojil@ujf.cas.cz 2 semoradova@ujf.cas.cz
Singular repulsive barrier V (x) = −g ln(|x|) inside a square well is interpreted and studied as a linear analogue of the state-dependent interaction L ef f (x) = −g ln[ψ * (x)ψ(x)] in nonlinear Schrödinger equation. In the linearized case, Rayleigh-Schrödinger perturbation theory is shown to provide a closed-form spectrum at the sufficiently small g or after an amendment of the unperturbed Hamiltonian. At any spike-strength g, the model remains solvable numerically, by the matching of wave functions. Analytically, the singularity is shown regularized via the change of variables x = exp y which interchanges the roles of the asymptotic and central boundary conditions. 1 znojil@ujf.cas.cz 2 semoradova@ujf.cas.cz
Abstract. We explore the Klein-Gordon equation in the framework of crypto-Hermitian quantum mechanics. Solutions to common problems with probability interpretation and indefinite inner product of the Klein-Gordon equation are proposed.
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