×

Continuity of some operators arising in the theory of superoscillations. (English) Zbl 1402.81167

Summary: The study of superoscillations naturally leads to the analysis of a large class of convolution operators acting on spaces of entire functions. In particular, the key point is often the proof of the continuity of these operators on appropriate spaces. Most papers in the current literature utilize abstract methods from functional analysis to establish such continuity. In this paper, on the other hand, we rely on some recent advances in the study of entire functions, to offer explicit proofs of the continuity of such operators. To demonstrate the applicability and the flexibility of these explicit methods, we will use them to study the important case of superoscillations associated with quadratic Hamiltonians. The paper also contains a list of interesting open problems, and we have collected as well, for the convenience of the reader, some well-known results, and their proofs, on Gamma and Mittag-Leffler functions that are often used in our computations.

MSC:

81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aharonov, Y; Albert, D; Vaidman, L, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett., 60, 1351-1354, (1988) · doi:10.1103/PhysRevLett.60.1351
[2] Aharonov, Y; Colombo, F; Nussinov, S; Sabadini, I; Struppa, DC; Tollaksen, J, Superoscillation phenomena in \(SO(3)\), Proc. R. Soc. A, 468, 3587-3600, (2012) · Zbl 1371.81196 · doi:10.1098/rspa.2012.0131
[3] Aharonov, Y; Colombo, F; Sabadini, I; Struppa, DC; Tollaksen, J, Some mathematical properties of superoscillations, J. Phys. A, 44, 365304, (2011) · Zbl 1230.42004 · doi:10.1088/1751-8113/44/36/365304
[4] Aharonov, Y; Colombo, F; Sabadini, I; Struppa, DC; Tollaksen, J, On some operators associated to superoscillations, Complex Anal. Oper. Theory, 7, 1299-1310, (2013) · Zbl 1297.47095 · doi:10.1007/s11785-012-0227-9
[5] Aharonov, Y; Colombo, F; Sabadini, I; Struppa, DC; Tollaksen, J, On the Cauchy problem for the Schrödinger equation with superoscillatory initial data, J. Math. Pures Appl., 99, 165-173, (2013) · Zbl 1258.35172 · doi:10.1016/j.matpur.2012.06.008
[6] Aharonov, Y; Colombo, F; Sabadini, I; Struppa, DC; Tollaksen, J, Superoscillating sequences as solutions of generalized Schrödinger equations, J. Math. Pures Appl., 103, 522-534, (2015) · Zbl 1304.30036 · doi:10.1016/j.matpur.2014.07.001
[7] Aharonov, Y., Colombo, F., Sabadini, I., Struppa, D.C., Tollaksen, J.: On superoscillations longevity: a windowed Fourier transform approach. In: Struppa, D.C., Tollaksen, J. (eds.) Quantum Theory: A Two-Time Success Story, pp. 313-325. Springer, Berlin (2013) · Zbl 1282.81066
[8] Aharonov, Y; Colombo, F; Sabadini, I; Struppa, DC; Tollaksen, J, Evolution of superoscillatory data, J. Phys. A, 47, 205301, (2014) · Zbl 1290.32005 · doi:10.1088/1751-8113/47/20/205301
[9] Aharonov, Y; Colombo, F; Sabadini, I; Struppa, DC; Tollaksen, J, The mathematics of superoscillations, Mem. Am. Math. Soc., 247, v+107, (2017) · Zbl 1383.42002
[10] Aharonov, Y; Colombo, F; Sabadini, I; Struppa, DC; Tollaksen, J, Evolution of superoscillatory initial data in several variables in uniform electric field, J. Phys. A, 50, 185201, (2017) · Zbl 1372.78004 · doi:10.1088/1751-8121/aa66d9
[11] Aharonov, Y; Colombo, F; Sabadini, I; Struppa, DC; Tollaksen, J, Superoscillating sequences in several variables, J. Fourier Anal. Appl., 22, 751-767, (2016) · Zbl 1345.32001 · doi:10.1007/s00041-015-9436-8
[12] Aharonov, Y., Rohrlich, D.: Quantum Paradoxes: Quantum Theory for the Perplexed. Wiley-VCH Verlag, Weinheim (2005) · Zbl 1081.81001 · doi:10.1002/9783527619115
[13] Aharonov, Y; Vaidman, L, Properties of a quantum system during the time interval between two measurements, Phys. Rev. A, 41, 11-20, (1990) · doi:10.1103/PhysRevA.41.11
[14] Aharonov, Y., Sabadini, I., Tollaksen, J., Yger, A.: Classes of superoscillating functions. In: Quantum Studies: Mathematics and Foundations. https://doi.org/10.1007/s40509-018-0156-z · Zbl 1402.81122
[15] Aharonov, Y., Colombo, F., Struppa, D.C., Tollaksen, J.: Schrödinger evolution of superoscillations under different potentials. In Quantum Studies: Mathematics and Foundations. https://doi.org/10.1007/s40509-018-0161-2 · Zbl 1402.81121
[16] Aoki, T., Colombo, F., Sabadini, I., Struppa, D.C.: Continuity theorems for a class of convolution operators and applications to superoscillations (2016) (preprint) · Zbl 1411.35091
[17] Berry, MV; Morley-Short, S, Representing fractals by superoscillations, J. Phys. A Math. Theor., 50, 22lt01, (2017) · Zbl 1370.28003 · doi:10.1088/1751-8121/aa6fba
[18] Berry, MV; Anandan, JS (ed.); Safko, JL (ed.), Faster than Fourier, 55-65, (1994), Singapore
[19] Berry, M; Dennis, MR, Natural superoscillations in monochromatic waves in D dimension, J. Phys. A, 42, 022003, (2009) · Zbl 1161.35472 · doi:10.1088/1751-8113/42/2/022003
[20] Berry, MV; Popescu, S, Evolution of quantum superoscillations, and optical superresolution without evanescent waves, J. Phys. A, 39, 6965-6977, (2006) · Zbl 1122.81029 · doi:10.1088/0305-4470/39/22/011
[21] Buniy, R; Colombo, F; Sabadini, I; Struppa, DC, Quantum harmonic oscillator with superoscillating initial datum, J. Math. Phys., 55, 113511, (2014) · Zbl 1301.81054 · doi:10.1063/1.4901753
[22] Colombo, F; Gantner, J; Struppa, DC, Evolution of superoscillations for Schrödinger equation in uniform magnetic field, J. Math. Phys., 58, 092103, (2017) · Zbl 1372.81040 · doi:10.1063/1.4991489
[23] Colombo, F., Gantner, J., Struppa, D.C.: Evolution by Schrödinger equation of Ahronov-Berry superoscillations in centrifugal potential (2016) (submitted, preprint)
[24] Colombo, F., Sabadini, I., Struppa, D.C., Yger, A.: Superoscillating sequences and hyperfunctions (2017) (preprint)
[25] Colombo, F., Struppa, D.C., Yger, A.: Superoscillating sequences towards approximation in \(\cal{S}\) or \(\cal{S^{′ }}\)-type spaces and extrapolation. J. Fourier Anal. Appl. (2018). https://doi.org/10.1007/s00041-018-9592-8 · Zbl 1421.42003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.