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Free quantum motion on a branching graph. (English) Zbl 0749.47038

Summary: We consider the free motion of a quantum-particle on the graph consisting of three half-lines whose ends are connected. It is shown that the time evolution can be described by a Hamiltonian and the class of all admissible Hamiltonians is constructed using the theory of selfadjoint extensions. Three subclasses are discussed in detail:
(a) the junction,
(b) the wider four-parameter family with the wavefunctions continuous between two branches of the graph only,
(c) the Hamiltonian invariant under permutations of the branches.
For the class (c), generalization to the graphs consisting of \(n\) half- lines is given. The scattering problem of such a branching graph is also discussed.

MSC:

47N50 Applications of operator theory in the physical sciences
47A40 Scattering theory of linear operators
46N50 Applications of functional analysis in quantum physics
81T25 Quantum field theory on lattices
81U99 Quantum scattering theory
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[1] Bishop, J. D.; Licini, J. C.; Dolan, G. J., Appl. Phys. Lett., 46, 1000 (1985)
[2] Chandrasekhar, V.; Rooks, J. M.; Wind, S.; Prober, D. E., Phys. Rev. Lett., 55, 1610 (1985)
[3] Pannetier, B.; Chaussy, J.; Rammal, R., J. Physique Lett., 44, L 853 (1983)
[4] Umbach, C. P., Phys. Rev. Lett., 45, 386 (1986)
[5] Umbach, C. P.; Washburn, S.; Laibowitz, R. B.; Webb, R. A., Phys. Rev., B30, 4048 (1984)
[6] Webb, R. A., Physica, 140A, 175 (1986)
[7] Datta, S.; Bandyopadhyay, S., Phys. Rev. Lett., 58, 717 (1987)
[8] Shapiro, B., Phys. Rev. Lett., 50, 747 (1983)
[9] Buettiker, M.; Imry, Y.; Azbel, M. Ya., Phys. Rev., A30, 1982 (1984)
[10] Gefen, Y.; Imry, Y.; Azbel, M. Ya., Phys. Rev. Lett., 52, 129 (1984)
[11] Gefen, Y.; Imry, Y.; Azbel, M. Ya., Surface Science, 142, 203 (1984)
[12] Jauch, J. M., Foundation of Quantum Mechanics (1968), Addison-Wesley: Addison-Wesley Reading, Mass, Chap. 11 · Zbl 0166.23301
[13] Exner, P.; Šeba, P., JINR E2-87-18 (1987), Dubna
[14] Reed, M.; Simon, B., Methods of Modern Mathematical Physics. II Fourier Analysis. Self-Adjointness (1975), Academic Press: Academic Press New York · Zbl 0308.47002
[15] Albeverio, S.; Hoegh-Krohn, R., J. Oper. Theory, 6, 313 (1981)
[16] Albeverio, S.; Gesztesy, F.; Hoegh-Krohn, R.; Kirsch, W., J. Oper. Theory, 12, 101 (1984)
[17] Albeverio, S.; Hoegh-Krohn, R., Physica, 124A, 11 (1984)
[18] Bulla, W.; Gesztesy, F., J. Math. Phys., 26, 2520 (1985)
[19] Šeba, P., Contact Interaction in Quantum Mechanics, (PhD Thesis (1986), Charles University: Charles University Prague) · Zbl 0648.35026
[20] Kuperin, Yu. A.; Makarov, K. A.; Pavlov, B. S., Teor. Mat. Fiz., 63, 78 (1985)
[21] Dittrich, J.; Exner, P., J. Math. Phys., 26, 2000 (1985)
[22] Šeba, P., Czech. J. Phys., 36B, 455 (1986)
[23] Exner, P.; Šeba, P., Lett. Math. Phys., 12 (1986)
[24] Exner, P.; Šeba, P., JINR E2-86-693 (1986), Dubna
[25] Exner, P.; Šeba, P., Czech. J. Phys., B38, 1095 (1989)
[26] Dunford, N.; Schwartz, J. T., Linear Operators, Vol. II (1964), Interscience Publ: Interscience Publ New York
[27] Weidmann, J., Linear Operators in Hilbert Space (1980), Springer-Verlag: Springer-Verlag New York
[28] Exner, P.; Šeba, P., JINR E2-87-000 (1987), Dubna
[29] Temkin, H., Appl. Phys. Lett., 50, 413 (1987)
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