Abstract
Using the formalism of the effective Hamiltonian, we consider bound states in a continuum (BIC). They are nonhermitian effective Hamiltonian eigenstates that have real eigenvalues. It is shown that BICs are orthogonal to open channels of the leads, i.e., disconnected from the continuum. As a result, BICs can be superposed to a transport solution with an arbitrary coefficient and exist in a propagation band. The one-dimensional Aharonov-Bohm rings that are opened by attaching single-channel leads to them allow exact consideration of BICs. BICs occur at discrete values of the energy and magnetic flux; however, it’s realization strongly depends on the way to the BIC point.
Similar content being viewed by others
References
J. von Neumann and E. Wigner, Phys. Z. 30, 465 (1929).
F. H. Stillinger and D. R. Herrick, Phys. Rev. A 11, 446 (1975).
R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer, Berlin, 1982; Mir, Moscow, 1969).
H. Friedrich and D. Wintgen, Phys. Rev. A 32, 3231 (1985); Phys. Rev. A 32, 3964 (1985).
F. Capasso, C. Sirtori, J. Faist, et al., Nature 358, 565 (1992).
M. Robnik, J. Phys. A: Math. Gen. 19, 3845 (1986).
L. S. Cederbaum, R. S. Friedman, V. M. Ryaboy, and N. Moiseyev, Phys. Rev. Lett. 90, 13001 (2003).
T. V. Shahbazyan and M. E. Raikh, Phys. Rev. B 49, 17 (1994); Phys. Rev. B 49, 123 (1994).
O. Olendski and L. Mikhailovska, Phys. Rev. B 66, 35331 (2002).
M. L. Ladron de Guevara, F. Claro, and P. A. Orellana, Phys. Rev. B 67, 195335 (2003).
I. Rotter and A. F. Sadreev, Phys. Rev. E 69, 66201 (2004); Phys. Rev. E 71, 046204 (2005).
A. F. Sadreev, E. N. Bulgakov, and I. Rotter, JETP Lett. 82, 498 (2005).
A. F. Sadreev, E. N. Bulgakov, and I. Rotter, J. Phys. A 38, 10 647 (2005).
K. D. Rowe and P. J. Siemens, J. Phys. A: Math. Gen. 38, 9821 (2005).
G. Ordonez, K. Na, and S. Kim, Phys. Rev. A 73, 022113 (2006).
A. F. Sadreev, E. N. Bulgakov, and I. Rotter, Phys. Rev. B 73, 235342 (2006).
B. Wunsch and A. Chudnovskiy, Phys. Rev. B 68, 245317 (2003).
P. A. Orellana, M. L. Ladron de Guevara, and F. Claro, Phys. Rev. B 70, 233315 (2004).
J.-B. Xia, Phys. Rev. B 45, 3593 (1992).
D. L. Pursey and T. A. Weber, Phys. Rev. A 52, 3932 (1995).
V.I. Smirnov, A Course of Higher Mathematics, 10th ed. (Nauka, Moscow, 1974; Pergamon, Oxford, 1964), Vol. 3, Part 1.
C. Texier, J. Phys. A: Math. Gen. 35, 3389 (2002).
C. Texier and M. Büttiker, Phys. Rev. B 67, 245410 (2003).
S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge Univ. Press, Cambridge, 1995).
A. F. Sadreev and I. Rotter, J. Phys. A 36, 11 413 (2003).
F. M. Dittes, Phys. Rep. 339, 215 (2000).
J. Okołowicz, M. Płoszajczak, and I. Rotter, Phys. Rep. 374, 271 (2003).
P. Exner, P. Seba, A. F. Sadreev, et al., Phys. Rev. Lett. 80, 1710 (1998).
T. Chakraborty and P. Pietiläinen, Phys. Rev. B 50, 8460 (1994); Phys. Rev. B 52, 1932 (1995).
R. Berkovits, F. von Oppen, and J. W. Kantelhardt, Europhys. Lett. 68, 699 (2004).
Author information
Authors and Affiliations
Additional information
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Bulgakov, E.N., Pichugin, K.N., Sadreev, A.F. et al. Bound states in the continuum in open Aharonov-Bohm rings. Jetp Lett. 84, 430–435 (2006). https://doi.org/10.1134/S0021364006200057
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0021364006200057