The Aharonov-Bohm effect and scattering theory. (English) Zbl 0554.47003

Ruijsenaars, S. N. M.

Ann. Phys. 146, 1-34 (1983).

The scattering of electrons off a cylinder inside which there is a current-carrying solenoid is considered. There are two different viewpoints on the description of the excluded magnetic fields and they lead to different hamiltonians. The corresponding S-matrices and scattering cross sections are found and it is shown that they differ considerably for \(R/\lambda \to 0\) \((R=cylinder\) radius, \(\lambda =de\) Broglie wave length of the electron), hence an experiment could serve to verify one description and falsify the other one. The idealized situation \(R=0\) is also studied. The S-matrices are explicitely obtained, it is shown that the results are invariant under a large class of gauge transformations and an anomalous long-range behaviour of the wave and scattering operator is pointed out.

Reviewer: V.Georgescu

MSC:

47A40	Scattering theory of linear operators
81U20	\(S\)-matrix theory, etc. in quantum theory

Keywords:

Aharonov-Bohm effect; scattering of electrons off a cylinder inside which there is a current-carrying solenoid; excluded magnetic fields; S- matrices; scattering cross sections

References:

[1]	Aharonov, Y.; Bohm, D., Phys. rev., 115, 485-491, (1959)
[2]	Ehrenberg, W.; Siday, R.E., (), 8-21
[3]	Boyer, T.H., Phys. rev. D, 8, 1679-1692, (1973)
[4]	Möllenstedt, G.; Bayh, W., Phys. bl., 18, 299-305, (1962)
[5]	Boersch, H.; Hamish, H.; Grohmann, K., Z. phys., 169, 263-272, (1962)
[6]	Peshkin, M.; Talmi, I.; Tassie, L.J., Ann. phys. (N.Y), 12, 426-435, (1961)
[7]	DeWitt, B.S., Phys. rev., 125, 2189-2191, (1962)
[8]	Belinfante, F.J., Phys. rev., 128, 2832-2837, (1962)
[9]	Strocchi, F.; Wightman, A.S., J. math. phys., 15, 2198-2224, (1974)
[10]	Casati, G.; Guarneri, I., Phys. rev. lett., 42, 1579-1581, (1979)
[11]	Goldin, G.A.; Menikoff, R.; Sharp, D.H., J. math. phys., 22, 1664-1668, (1981)
[12]	Bocchieri, P.; Loinger, A., Nuovo cimento A, 47, 475-482, (1978)
[13]	Bocchieri, P.; Loinger, A.; Siragusa, G., Nuovo cimento A, 51, 1-17, (1979)
[14]	Bocchieri, P.; Loinger, A., Lett. nuovo cimento, 25, 476-478, (1979)
[15]	Bocchieri, P.; Loinger, A.; Siragusa, G., Nuovo cimento A, 56, 55, (1980)
[16]	Bocchieri, P.; Loinger, A., Nuovo cimento A, 59, 121-133, (1980)
[17]	Bocchieri, P.; Loinger, A., Nuovo cimento A, 66, 164-172, (1981)
[18]	Roy, S.M., Phys. rev. lett., 44, 111-114, (1980)
[19]	Bohm, D.; Hiley, B.J., Nuovo cimento A, 52, 295-308, (1979)
[20]	Klein, U., Lett. nuovo cimento, 25, 33-37, (1979)
[21]	Mignaco, J.A.; Novaes, C.A., Lett. nuovo cimento, 26, 453-456, (1979)
[22]	Klein, U., Acta phys. austr., 52, 269-276, (1980)
[23]	Breitenecker, M.; Grümm, H.-R., Nuovo cimento A, 55, 453-455, (1980)
[24]	Rothe, H.J., Nuovo cimento A, 62, 54-66, (1981)
[25]	Peshkin, M., Phys. rev. A, 23, (1981)
[26]	Zeilinger, A., Lett. nuovo cimento, 25, 333-336, (1979)
[27]	Bawin, M.; Burnel, A., Lett. nuovo cimento, 27, 4-6, (1980)
[28]	Rowe, E.G.P., Nuovo cimento A, 56, 16-20, (1980)
[29]	Greenberger, D.M., Phys. rev. D, 23, 1460-1462, (1981)
[30]	Klein, U., Phys. rev. D, 23, 1463-1465, (1981)
[31]	Lipkin, H.J., Phys. rev. D, 23, 1466-1467, (1981)
[32]	Kobe, D.H., Ann. phys., 123, 381-410, (1979)
[33]	Feinberg, E.L., Soviet phys. usp., 5, 753-760, (1963)
[34]	Henneberger, W.C., J. math. phys., 22, 116-117, (1981)
[35]	Kretzschmar, M., Z. phys., 185, 73-83, (1965)
[36]	Martin, C., Lett. math. phys., 1, 155-163, (1976)
[37]	Ruijsenaars, S.N.M., J. funct. anal., 48, 135-171, (1982)
[38]	Reed, M.; Simon, B., ()
[39]	Schulman, L.S., J. math. phys., 12, 304-308, (1971)
[40]	Grümm, H.-R., Acta phys. austr., 53, 113-131, (1981)
[41]	Bernstein, H.J.; Phillips, A.V., Sci. amer., 245, 94-109, (1981)
[42]	Ruijsenaars, S.N.M., Ann. phys. (N.Y), 132, 328-382, (1981)
[43]	Henneberger, W.C., Phys. rev. A, 22, 1383-1388, (1980)
[44]	Corinaldesi, E.; Rafeli, F., Amer. J. phys., 46, 1185-1187, (1978)
[45]	Purcell, K.M.; Henneberger, W.C., Amer. J. phys., 46, 1255-1256, (1978)
[46]	Morse, P.M.; Feshbach, H., (), Part II
[47]	Crichton, J.H., Nuovo cimento, 45A, 256-258, (1966)
[48]	Berends, F.A.; Ruijsenaars, S.N.M., Nucl. phys. B, 56, 507-524, (1973)
[49]	Reed, M.; Simon, B., ()
[50]	Reed, M.; Simon, B., ()
[51]	Ruijsenaars, S.N.M.; Bongaarts, P.J.M., Ann. inst. H. Poincaré (Grenoble) A, 26, 1-17, (1977)
[52]	()
[53]	Bocchieri, P.; Loinger, A., Lett. nuovo cimento, 30, 449-450, (1981)
[54]	Nagel, B., ()
[55]	Boersch, H.; Grohmann, K.; Hamisch, H.; Lischke, B.; Wohlleben, D., Lett. nuovo cimento, 30, 257-258, (1981)
[56]	Peshkin, M., Phys. rep., 80, 375-386, (1981)
[57]	Möllenstedt, G.; Schmid, H.; Lichte, H., Measurement of the phase shift between electron waves due to a magnetic flux enclosed in a metallic cylinder, (), 433-434, Hamburg