arXiv subjects
Publications and source records attributed to Iveta Semorádová.
Diverging eigenvalues in domain truncations of Schrödinger operators with complex potentials are analyzed and their asymptotic formulas are obtained. Our approach also yields asymptotic formulas for diverging eigenvalues in the strong coupling regime for the imaginary part of the potential.
The large-N expansion technique is tested via an anomalous, soft-core potential which admits the tunneling through its central barrier. The precision of the approximation is found sensitive to the asymptotic component of the interaction. Once chosen in the most common harmonic-oscillator form, and once complemented by the short range part represented by the general power-law anharmonicity $\sim |x|^α$, we found that the latter power-law spike may be well approximated by an elementary logarithmic function, in the limit of the smallest $α\to 0$ at least. In such a model, the large-N method is found applicable and offering still an efficient and user-friendly method of the solution of the Schrödinger equation.
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.
Linear square-well Schrödinger equation endowed with a singular logarithmic spike in the origin is studied. The study is methodical, motivated by the problem of non-gausson states $ψ_n(x)$, $n \neq 0$ generated by nonlinear Schrödinger equations. Once the state-dependent self-interaction term is chosen logarithmic, $\sim -g\,\ln[ψ^*_n(x)ψ_n(x)]$, the nonlinear model develops the puzzling logarithmic (i.e., weakly singular) repulsive barriers near the nodal zeros of $ψ_n(x)$ at $n \neq 0$. In our linearized approach the weak-coupling regime is shown reliably described by the routine Rayleigh-Schrödinger perturbation theory. It even provides the first-order picture of the spectrum in closed-form. Beyond the weak-coupling regime an amendment of the unperturbed Hamiltonian is recommended. Finally, an analytic insight into the nature of the singularity at $x=0$ is obtained, in a non-perturbative setting, after the change of variables $x=\exp y$.
During the recent developments of quantum theory it has been clarified that the observable quantities (like energy or position) may be represented by operators (with real spectra) which are manifestly non-Hermitian. The mathematical consistency of the resulting models of stable quantum systems requires a reconstruction of an alternative, amended, physical inner product of states. We point out the less known fact that for more than one observable the task is not always feasible. The difficulty is re-analyzed and its elementary linear-algebraic interpretation and treatment are outlined.
The overall principles of what is now widely known as PT-symmetric quantum mechanics are listed, explained and illustrated via a few examples. In particular, models based on an elementary local interaction V(x) are discussed as motivated by the naturally emergent possibility of an efficient regularization of an otherwise unacceptable presence of a strongly singular repulsive core in the origin. The emphasis is put on the constructive aspects of the models. Besides the overall outline of the formalism we show how the low-lying energies of bound states may be found in closed form in certain dynamical regimes. Finally, once these energies are found real we explain that in spite of a manifest non-Hermiticity of the Hamiltonian the time-evolution of the system becomes unitary in a properly amended physical Hilbert space.