Abstract
We solve explicitly and without approximation the problem of a quantum-mechanical particle inR 3 subjected to point interactions that are periodic inR 3 with periodicity of the typeZ, Z 2, andZ 3. In the first case we get a model of an infinite straight polymer, in the second case we get a model of a monomolecular layer and in the third case we get a model of a crystal. In all three cases the unit cell of the Bravais lattice is allowed to contain any finite number of interaction sites (atomes), placed arbitrarily and with arbitrary interaction strength. In the case: one interaction site per unit cell we find explicit formulas for the resonance bands and energy bands and their corresponding wavefunctions.
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References
Wannier, G.H.: Elements of solid state theory. Cambridge: Cambridge University Press 1959
Sutherland, B.: Exact solutions of some non separable band problems. Phys. Rev. Letters42, 915 (1979)
Breit, G.: Phys. Rev.71, 215–231 (1947)
Blatt, J.M., Weisskopf, V.F.: Theoretical nuclear physics, p. 74. New York: Wiley and Sons 1952
Thomas, L.H.: Phys. Rev.47, 903–909 (1935)
Huang, K., Yang, C.N.: Phys. Rev.105, 767–775 (1957)
Huang, K., Yang, C.N., Luttinger, J.M.: Phys. Rev.105, 776–784 (1957)
Lee, T.D., Huang, K., Yang, C.N.: Phys. Rev.106, 1134–1145 (1957)
Wu, T.T.: Phys. Rev.115, 1390–1404 (1959)
Huang, K.: Statistical mechanics. New York: Wiley and Sons 1963
Danilov, G.S.: J.E.T.P.13, 349–355 (1961)
Faddeev, L.D., Minlos, R.A.: J.E.P.P.14, 1315–1316 (1962)
Flamand, G.: Cargese lectures in theoretical physics, pp. 247–287. New York: Gordon and Breach 1965
Albeverio, S., Fenstad, J.E., Høegh-Krohn, R.: Singular perturbation and non-standard analysis. To appear in Trans. Am. Math. Soc.
Grossmann, A., Høegh-Krohn, R., Mebkhout, M.: A class of explicitly soluble, many-center Hamiltonians for one-particle quantum mechanics in two and three dimensions. CNRS, CPT, Marseille, preprint to appear in J. Math. Phys.
Thomas, L.E.: Scattering from point interactions. To appear in J. Math. Phys.
Avron, J.E., Grossmann, A., Rodriguez, R.: Ann. Phys.103, 47 (1977)
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Communicated by J. Ginibre
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Grossmann, A., Høegh-Krohn, R. & Mebkhout, M. The one particle theory of periodic point interactions. Commun.Math. Phys. 77, 87–110 (1980). https://doi.org/10.1007/BF01205040
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DOI: https://doi.org/10.1007/BF01205040