2008/02/11 by Miloslav Znojil · 2 citations
Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Topological Materials and Phenomena #quant-ph
paper · pdf · doi:10.1016/j.physleta.2008.02.016
published as Phys. Lett. A 372 (2008) 3591 - 3596 · 18 pp, 3 figs
arxiv created 2008/02/11 · openalex publication_date 2008/02/15 · arxiv updated 2009/12/01 · openalex created_date 2020/07/02 · openalex updated_date 2026/07/28
It is known that besides the usual unitary mappings Ω= 1/Ω^† between the equivalent representations of the physical Hilbert space of Quantum Mechanics (often, Fourier transformations), the generalized non-unitary maps Ω≠ 1/Ω^† can also help to simplify the analysis. We adapt the standard Dirac's notation and recollect the Buslaev's and Grecchi's repulsive quartic oscillator Hamiltonian as an example. Then we propose the whole new class of the models of the similar type, characterized by a complexification of the path \cal C of the (obviously, not observable!) "coordinates". An exactly solvable potentialless Schrödinger equation is finally chosen for illustration. In it, the dynamical (i.e., in our example, confining) role of the traditional potentials V(x) is shown to be taken over by the mere topologically nontrivial shape of \cal C. Our construction evokes several new open questions in physics (\cal PT-symmetric wave packets at a single energy?) as well as in mathematics (a three-Hilbert-space generalized formulation of Quantum Mechanics?).