In this paper, we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and that the analytic solution develops a singularity in finite time. We also show that for a large class of solutions of the Schrödinger equation, superoscillating behavior at any given time implies superoscillating behavior at any other time.
REFERENCES
1.
Y.
Aharonov
, D.
Albert
, and L.
Vaidman
, Phys. Rev. Lett.
60
, 1351
–1354
(1988
).2.
Y.
Aharonov
, F.
Colombo
, S.
Nussinov
, I.
Sabadini
, D. C.
Struppa
, and J.
Tollaksen
, Proc. R. Soc. A
468
, 3587
–3600
(2012
).3.
Y.
Aharonov
, F.
Colombo
, I.
Sabadini
, D. C.
Struppa
, and J.
Tollaksen
, J. Phys. A
44
, 365304
(2011
).4.
Y.
Aharonov
, F.
Colombo
, I.
Sabadini
, D. C.
Struppa
, and J.
Tollaksen
, Complex Anal. Operator Theory
7
, 1299
–1310
(2013
).5.
Y.
Aharonov
, F.
Colombo
, I.
Sabadini
, D. C.
Struppa
, and J.
Tollaksen
, J. Math. Pures Appl.
99
, 165
–173
(2013
).6.
Y.
Aharonov
, F.
Colombo
, I.
Sabadini
, D. C.
Struppa
, and J.
Tollaksen
, Quantum Theory: A Two-time Success Story
(Springer
, 2013
), pp. 313
–325
.7.
Y.
Aharonov
, F.
Colombo
, I.
Sabadini
, D. C.
Struppa
, and J.
Tollaksen
, “Superoscillating sequences as solutions of generalized Schrödinger equations
,” J. Math. Pures Appl.
(published online).8.
Y.
Aharonov
, F.
Colombo
, I.
Sabadini
, D. C.
Struppa
, and J.
Tollaksen
, J. Phys. A
47
, 205301
(2014
).9.
Y.
Aharonov
, F.
Colombo
, I.
Sabadini
, D. C.
Struppa
, and J.
Tollaksen
, “The mathematics of superoscillations
,” preprint (2014
).10.
Y.
Aharonov
and D.
Rohrlich
, Quantum Paradoxes: Quantum Theory for the Perplexed
(Wiley-VCH Verlag
, Weinheim
, 2005
).11.
Y.
Aharonov
and L.
Vaidman
, Phys. Rev. A
41
, 11
–20
(1990
).12.
M. V.
Berry
, in Quantum Coherence and Reality; in Celebration of the 60th Birthday of Yakir Aharonov
, edited J. S.
Anandan
and J. L.
Safko
(World Scientific
, Singapore
, 1994
), pp. 55
–65
.13.
M.
Berry
and M. R.
Dennis
, J. Phys. A
42
, 022003
(2009
).14.
M. V.
Berry
and S.
Popescu
, J. Phys. A
39
, 6965
–6977
(2006
).15.
C. A.
Berenstein
and D. C.
Struppa
, Publ. RIMS, Kyoto Univ.
24
, 783
–810
(1988
).16.
L.
Ehrenpreis
, Fourier Analysis in Several Complex Variables
(Wiley Interscience
, New York
, 1970
).17.
R. P.
Feynman
, Rev. Mod. Phys.
20
, 367
(1948
).18.
R. P.
Feynman
and A. R.
Hibbs
, Quantum Mechanics and Path Integrals
(McGraw-Hill
, New York
, 1965
).19.
I. M.
Gelfand
and A. M.
Yaglom
, J. Math. Phys.
1
, 48
(1960
).20.
E. W.
Montroll
, Commun. Pure Appl. Math.
5
, 415
(1952
).21.
22.
B. A.
Taylor
, Entire Functions and Related Parts of Analysis
, in Proceedings of Symposia in Pure Mathematics, La Jolla, CA, 1966
(American Mathematical Society
, Providence, RI
, 1968
), pp. 468
–474
.© 2014 AIP Publishing LLC.
2014
AIP Publishing LLC
You do not currently have access to this content.