A class of exactly solvable models for the Schrödinger equation

Downing, C. A.

doi:10.2478/s11534-013-0301-6

Cited by 4 publications

(2 citation statements)

References 30 publications

(42 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also do not discuss scattering by multiple electrostatic barriers [66,67] or by magnetic barriers [68][69][70], but our method can be generalized to account for the presence of vector potentials.…”

Section: Introductionmentioning

confidence: 99%

Optimal traps in graphene

et al. 2015

View full text Add to dashboard Cite

We transform the two-dimensional Dirac-Weyl equation, which governs the charge carriers in graphene, into a nonlinear first-order differential equation for scattering phase shift, using the so-called variable-phase method. This allows us to utilize the Levinson theorem, relating scattering phase shifts of a slow particle to its bound states, to find zero-energy bound states created electrostatically in realistic structures. These confined states are formed at critical potential strengths, which leads us to posit the use of "optimal traps" to combat the chiral tunneling found in graphene: this could be explored experimentally with an artificial network of point charges held above the graphene layer. We also discuss scattering on these states and find that the s states create a dominant peak in the scattering cross section as the energy tends towards the Dirac point energy, suggesting a dominant contribution to the resistivity.

show abstract

Section: Introductionmentioning

confidence: 99%

Optimal traps in graphene

et al. 2015

View full text Add to dashboard Cite

show abstract

“…which is also important, for example, in systems with potentials decaying like the inverse square of distance [63] such as a repulsive antidot confinement potential in a 2D electron gas [64] and quantum rings [65]. One only needs to make modifications in some places to the angular momentum quantum number in the solutions presented here.…”

Section: Coulomb Problem With a Regularized Magnetic Fieldmentioning

confidence: 99%

Magnetic quantum dots and rings in two dimensions

Downing

Portnoi

2016

Phys. Rev. B

View full text Add to dashboard Cite

We consider the motion of electrons confined to a two-dimensional plane with an externally applied perpendicular inhomogeneous magnetic field, both with and without a Coulomb potential. We find that as long as the magnetic field is slowly decaying, bound states in magnetic quantum dots are indeed possible. Several example cases of such magnetic quantum dots are considered in which one can find the eigenvalues and eigenfunctions in closed form, including two hitherto unknown quasi-exactly-solvable models treated with confluent and biconfluent Heun polynomials. It is shown how a modulation of the strength of the magnetic field can exclude magnetic vortexlike states, rotating with a certain angular momenta and possessing a definite spin orientation, from forming. This indicates one may induce localization-delocalization transitions and suggests a mechanism for spin separation.

show abstract