Quantum knots

Abstract

It is known that besides the usual unitary mappings $\Omega =1/{\Omega }^{†}$ between the equivalent representations of the physical Hilbert space of Quantum Mechanics (often, Fourier transformations), the generalized non-unitary maps $\Omega \ne 1/{\Omega }^{†}$ 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 $\mathcal{C}$ 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 $\mathcal{C}$. Our construction evokes several new open questions in physics ($\mathcal{P}\mathcal{T}-$symmetric wave packets at a single energy?) as well as in mathematics (a three-Hilbert-space generalized formulation of Quantum Mechanics?).

18 pp, 3 figs

References in corpus (20)

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• Spiked potentials and quantum toboggans
• Quantum toboggans
• Quantum toboggans with two branch points

Cited by in corpus (5)

• Discrete PT-symmetric models of scattering
• Identification of observables in quantum toboggans
• Topology-controlled spectra of imaginary cubic oscillators in the large-L approach
• Supersymmetric quantum mechanics living on topologically nontrivial Riemann surfaces
• Planarizable Supersymmetric Quantum Toboggans