-symmetric quantum toboggans

Abstract

Among all the PT −symmetric potentials defined on complex coordinate contours C(s), the name "quantum toboggan" is reserved for those whose C(s) winds around a singularity and lives on at least two different Riemann sheets. An enriched menu of prospective phenomenological models is then obtainable via the mere changes of variables. We pay thorough attention to the harmonic oscillator example with a fractional screening and emphasize the role of the existence and invariance of its quasi-exact states for different tobogganic C(s).

Quantum toboggans with two branch points

“…Still, in our present extension of the results of papers [8,9] we demonstrated that this innovation may still lead to certain very nontrivial consequences, especially in the context of building phenomenological models in Quantum Mechanics or even beyond its scope like, say, in classical magnetohydrodynamics [15] etc.…”

“…By the way, it is also amusing to notice that in the limit of the large x and/or z, our final formula (15) degenerates to its single-spire predecessor of refs. [8,9] exemplified also here, in paragraph 3.2, by its sextic-oscillator illustration (10).…”

“…3 QT models with the single branch point 3.1 A tobogganic alternative to contour (6) The introduction of the concept of quantum toboggans (QT, [8]) can be perceived …”

“…Fortunately, one can again try to proceed in analogy with the above-outlined rectification recipe applied to the models with the single branch point x (BP ) = 0 [8,9].…”

Behavior of eigenvalues in a region of broken PT symmetry

“…For example, if ε = −3 (this is the complex PTsymmetric version of the Coulomb potential for which H = p 2 + i/x [30]), the contour loops around the origin exactly twice; it goes from an angle −2π to the angle 2π. Looping contours for other complex eigenvalue problems have been studied in the past and have been called "toboggan contours" [31]. In the PT -symmetric Coulomb case the contour is shown in Fig.…”