American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: http://www.sciepub.com/journal/ajams Editor-in-chief: Mohamed Seddeek
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American Journal of Applied Mathematics and Statistics. 2015, 3(3), 111-130
DOI: 10.12691/ajams-3-3-5
Open AccessReview Article

On Theoretical Methodology of Nuclear Reactions and Outcome of Certain Physical Phenomena from Them. I. What Follows from the Analytical Properties of the One-channel and the Multichannel S-matrix?

V.S. Olkhovsky1,

1Institute for Nuclear Research of NASU, prospekt Nauki, 47, Kiev-03028, Ukraine

Pub. Date: June 29, 2015

Cite this paper:
V.S. Olkhovsky. On Theoretical Methodology of Nuclear Reactions and Outcome of Certain Physical Phenomena from Them. I. What Follows from the Analytical Properties of the One-channel and the Multichannel S-matrix?. American Journal of Applied Mathematics and Statistics. 2015; 3(3):111-130. doi: 10.12691/ajams-3-3-5

Abstract

Analytical structure of the non-relativistic unitary and non-unitary S-matrix are investigated for he cases of any interactions with any motion equations inside a sphere of radius a, enclosed by the centrifugal and rapidly decreasing (exponentially or by the Yukawian law or by the more rapidly decreasing) potentials. Some kinds of the symmetry conditions are imposed. The Schroedinger equation for r > a for the particle motion and the condition of the completeness of the correspondent wave functions are assumed. The connection of the obtained results with the causality is examined. Partially some analytical properties for the multi-channel S-matrix are investigated and the sum rules for mean compound-nucleus time delays and the density of compound-nucleus levels. Sometimes (as physical revelations of profound general methodic and in very good consistent accordance with the experiment) observable physical effects, such as parity violation enhancement and time resonances or explosions, are manifested. Finally a scientific program of future search is presented as a clear continuation and extension of the obtained results.

Keywords:
S-matrix condition of completeness of external wave functions external centrifugal and rapidly decreasing potentials causality time resonances or explosions parity violation enhancement

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References:

[1]  Heisenberg,W., Die “beobachtbaren Groessen” in der Theorie der Elementarteilchen, Zs.f.Phys.A Hadrons and Nuclei, 120(1943) 513, 673-702.
 
[2]  Heisenberg,W., Der mathamatische Rahman der Quantentheorie der Wellnfelder, Zs.Naturforsch. Teil A, 1(1946)608.
 
[3]  Hu, Ning, Application of the Heisenberg’s S-matrix theory for the resonance scattering and nuclear reactions, Phys.Rev., 74(1948)131-140.
 
[4]  Van Kampen,N., The S-matrix and the causality condition. I. The electromagnetic field., Phys.Rev., 89(1953)1072-1079.
 
[5]  Van Kampen,N., The S-matrix and the causality condition. II. Non-relativistic particles., 91 (1953)1267-1276.
 
[6]  Olkhovsky, V.S. and Tsekhmistrenko,Yu.V., On the analytical structure of the S-function of elastic scattering in the case of infinite potentials, Ukrainian Physical Journal, 6(1961)149- 156 (in Ukrainian, abstract in English; partially in English); see also Theor.and Math.Physics, 147(2006)539-551, Appendix 1, pp.547-549.
 
[7]  Olkhovsky,V.S., To the study of properties of the unitary and non-unitary S-matrix on the base of caysality and the completeness conditions for wave functions, Teor.Matem.Fiz, 20(1974) 211-222.
 
[8]  Nikolayev,M.V., and Olkhovsky,V.S., On the analytical Structure of the S-matrix for some classes of potentials, Teor.Mat.Fiz., 31(1977)214.
 
[9]  Nikolayev,M.V., and Olkhovsky,V.S., On Completeness of Radial Equation Solutions and S-matrix Singularities in case of Complex Potentials, Lett.al Nuovo Cimento, 8(1973)703-707.
 
[10]  Olkhovsky,V.S., and Zaichenko,A.K., Analytical properties and resonance structure of the S- matrix in case of non-central and parity-violating interactions, Nuovo Cim., A63(1981)155-170.
 
[11]  Baldo,M., and Olkhovsky,V.S., Analytical properties of the S-matrix for interactions with Yukawa-potential tails, Theor. Mathem.Phys., 147(2006)539-551
 
[12]  Olkhovsky,V.S., On the analytic properties of the S-matrix for the unknown interactions surrounded by centrifugal and rapidly decreasing potentials, Central Eur. J. Phys.,2011,v.9, N1,13-44.
 
[13]  Saavedra,I., Dispersion relations, “causal” inequalities and the completeness of the set of eigen-states in non-relativistic potential scattering, Nucl.Phys., 29(1962)137-152; Completeness, unitarity, and dispersion relations in potential scattering, Nucl.Phys., 87(1966)267-272.
 
[14]  R.Newton, Scattering Theory of Waves and Particles, McGRAW-HILL BOOK COMPANY, New Yorl–San Francisco–St.Louis–Toronto–London–Sydney, 1969.
 
[15]  Gasymov,M.G., On the expansion over eigen functions og non-self-adjoint boundary-value problem for the differential equation with a singularity in zero, Doklady AN SSSR, 165(1965)261-264.
 
[16]  Bertero,M., and Dillon,G., An outline of scattering theory for absorptive potentials, Nuovo Cim., A2(1971)1024-1038.
 
[17]  V.De Alfaro, and T.Redge, Potential scattering, North-Holland Publish.Company, Amsterdam, 1965.
 
[18]  E.C.Titchmarsh, The Theory of Functions, Oxford University Press, 1939.
 
[19]  Jost, R., Uber die falschen Nullstellen der Eigenwerte der S-matrix, Helv.Phys.Acta, XX (1947)256-266.
 
[20]  Ma, S., Phys.Rev., On a General Condition of Heisenberg foe the S-matrix, 71(1947)195-200.
 
[21]  I.S.Gradstein and I.M.Ryzhik: Tables of Integrals, Sums, Series, and Products (Moscow, Fizmatgiz,1963) [in Russian]; English translat.: Tables of Integrals, Sums, Series, and Products, Acad.Press, New York(1965).
 
[22]  A.S.Davydov, Theory of atomic nucleus, Fizmatgiz, Moscow (in Russian), 1958.
 
[23]  Simonius,M., Overlapping resonances and unitarity of the scattering S-matrix, Nucl.Phys., A218(1974)53-60.
 
[24]  Olkhovsky,V.S., Nucl.Phys., A425 (1984)445-457 D’Arrigo,A., Giardina,G., Olkhovsky,V.S., et al, Internat. J. Mod. Phys., E2(1993)809-820.
 
[25]  Smith, F.T., Phys.Rev., 118 (1960) 349; Phys.Rev.,119 (1960) 2098.
 
[26]  Lyuboshitz, V.L., Phys.Lett.,B, 72N1(1977)41; Lyuboshitz, V.L., Sov.J.Nucl.Phys., 27(1978)502; Lyuboshitz, V.L., JETP Lett.,28:1(1978)32;
 
[27]  Baz’, A.I.,JETP Lett.,27:2(1978)132-135}.
 
[28]  V.S.Olkhovsky, Time as a quantum observable, canonically conjugate to energy. Time analysis of quantum processes of tunneling and collisions (nuclear reactions) [LAP LAMBERT Academic publishing, Germany,177 pages, 2012].
 
[29]  Gosset, J., Gutbrod, H., Meyer, W. et al, Phys.Rev.,C16(1977)629.
 
[30]  Westfall, G., Sexstro, R., Poskanzer, A. et al, Phys.Rev.,C17 (1978)1368.
 
[31]  Cumming, J., Haustein, P.,Ruth, T. et al, Phys.Rev., C17(1978)1632.
 
[32]  Kaufman, S., Steinberg, E., Phys.Rev.,C17(1980)167.
 
[33]  Fortney, D., Porile, N., Phys.Rev.,C21(1980)2511.
 
[34]  Sandoval, A.,Gutbrod, H., Meyer, W. et al, Phys.Rev., C21(1980)1321.
 
[35]  Danielewicz, P., Pan, O., Phys.Rev.,C46(1992)2002.
 
[36]  Alba, R., Coniglione, R., Del Zoppo, A. et al,Phys. Lett., B322(1994)38.
 
[37]  Chen, K., Fraenkel, Z., Friedlander, G., et al, Phys.Rev.,166(1968)949.
 
[38]  Amsden, Antony A., Goldhaber, Alfred S., Harlow, Francis H., and Rayford, Nix J., Phys.Rev.,C17 (1978)2080-2096.
 
[39]  Izumo, Ko, Araseki, H., Progr. Theor. Phys., 69(1983)158-170.
 
[40]  Goto, T., Yamaguchi, K. and Sudo, N., Progr. Theor. Phys., 64(1980)1-17.
 
[41]  Olkhovsky,V.S., Dolinska,M.D., Omelchenko,S.A., Central Eur. J. Phys., 4(2006)1-18.
 
[42]  Olkhovsky, V.S., Dolinska, M.D., Omelchenko, S.A., Central Europ.J.Phys.,9:N4(2011) 1131-1133.
 
[43]  Westfall, G.D., Gosset, J., Johansen, J. et al, Phys.Rev.Lett., 18: 37(1976)1202.
 
[44]  Gosset, J., Kapusta, J.I.,Westfall, G.D., Phys.Rev.,C18(1978)844.
 
[45]  Prilepsky, Yu., Shmonin, V.,Acta Phys.Polon.B 18(1987)729.
 
[46]  Olkhovsky, V.S., Sov. J. Part. Nucl.,151(1984)30-148.