We prove that the bound state energies of the two-dimensional massive Dirac operator with dipole type potentials accumulate with exponential rate at the band edge. In fact we prove a corresponding formula of De Martino et al. [Phys. Rev. Lett. 112(18), 186603 (2014)].
REFERENCES
1.
D. I.
Abramov
and I. V.
Komarov
, “Weakly bound states of a charged particle in a finite-dipole field
,” Theor. Math. Phys.
13
(2
), 1090
–1098
(1972
).2.
J.-C.
Cuenin
and H.
Siedentop
, “Dipoles in graphene have infinitely many bound states
,” J. Math. Phys.
55
(12
), 122304
(2014
).3.
A.
De Martino
, D.
Klöpfer
, D.
Matrasulov
, and R.
Egger
, “Electric-dipole-induced universality for Dirac fermions in graphene
,” Phys. Rev. Lett.
112
(18
), 186603
(2014
).4.
T. M.
Dunster
, “Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter
,” SIAM J. Math. Anal.
21
(4
), 995
–1018
(1990
).5.
J. D.
Jackson
, Classical Electrodynamics
, 1st ed. (John Wiley & Sons, Inc.
, New York-London-Sydney
, 1962
).6.
N. N.
Khuri
, A.
Martin
, and T.-T.
Wu
, “Bound states in n dimensions (especially n = 1 and n = 2)
,” Few-Body Syst.
31
, 83
–89
(2002
).7.
W.
Kirsch
and B.
Simon
, “Corrections to the classical behavior of the number of bound states of Schrödinger operators
,” Ann. Phys.
183
(1
), 122
–130
(1988
).8.
N. W.
McLachlan
, Theory and Application of Mathieu Functions
(Clarenden Press
, Oxford
, 1947
).9.
S.
Rademacher
, “Energieniveaus von Dipolen in Graphen
,” Master’s thesis, Mathematisches Institut der Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333 München, May 2015.10.
M.
Reed
and B.
Simon
, Analysis of Operators
, 1st ed. Methods of Modern Mathematical Physics
Vol. 4
(Academic Press
, New York
, 1978
).11.
E.
Shargorodsky
, “On negative eigenvalues of two-dimensional Schrödinger operators
,” Proc. London Math. Soc.
108
(3
), 441
–483
(2013
).12.
G. N.
Watson
, A Treatise on the Theory of Bessel Functions
, 1st ed. (Cambridge University Press
, Cambridge
, 1922
).© 2016 Author(s).
2016
Author(s)
You do not currently have access to this content.
