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Resonance theory in atom-surface scattering. (English) Zbl 0682.47003
Summary: We study the problem of analytic extension of the resolvent for Hamiltonians arising in scattering of atoms by a quantum surface. We prove that the resolvent extends holomorphically to some regions of the lower half plane with isolated singularities called Landau resonances which are branch points of the resolvent. We study also the effect of impurities on the singularities of the resolvent and show that the presence of impurities adds poles to the Landau resonances.

47A40 Scattering theory of linear operators
81U99 Quantum scattering theory
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[1] [A] Alber, H. D.: A quasiperiodic boundary value problem for the Laplacian and the continuation of its resolvent. Proc. Roy. Soc. Edinburgh,82A, 251–272 (1979) · Zbl 0402.35033
[2] [B.C] Balslev, E., Combes, J. M.: Spectral theory of many-body Schrödinger operators with dilation analytic interactions. Comun. Math. Phys.22, 280–294 (1971) · Zbl 0219.47005
[3] [B.P] Bros, J., Pesenti, D.: Fredholm resolvents of meromorphic kernels with complex parameters: A Landau singularity and the associated equations of typeU in a non-holomorphic case. J. Math. Pures Appl.62, 215–252 (1983) · Zbl 0531.35067
[4] [C.F.K.S.] Cycon, H. L., Froese, R. G., Kirsch, W., Simon, B.: Schrödinger operators. Texts and Monographs in Physics. Berlin, Heidelberg, New York: Springer 1987 · Zbl 0619.47005
[5] [F.K] Fukuda, T., Kobayashi, T.: A local isotopy lemma. Tokyo J. Math.5, 31–36 (1982) · Zbl 0514.58034
[6] [F.H] Froese, R. G., Hislop, P.: Spectral analysis of second order elliptic operators on non-compact manifolds. Duke Math. J.58, 103–128 (1989) · Zbl 0687.35060
[7] [Ge] Gerber, R. G.: Molecular scattering from surfaces: Theoretical methods and results. Chem. Review 29–79 (1987)
[8] [G] Guillopé, L.: Théorie spectrale de quelques variétés à bouts. Ann. E.N.S. t22, 137–160 (1989) · Zbl 0682.58049
[9] [F.F.L.P] Fotiadi, D., Froissart, M., Lascoux, J., Pham, F.: Applications of an isotopy theorem. Topology4, 159–191 (1965) · Zbl 0173.09301
[10] [Hil] Hironaka, H.: Stratification and flatness. Proceedings of the Nordic Summer School. Oslo (1976)
[11] [Hi2] Hironaka, H.: Bimeromorphic smoothing of complex analytic space. Acta Math. Vietnamica. Tome2 (1977) · Zbl 0407.32006
[12] [H] Hislop, P.: Spectral analysis of non-compact manifolds using commutator methods. Séminaire E.D.P. Ecole Polytechnique (1988) · Zbl 0657.58045
[13] [K] Kobayashi, T.: On the singularities of the solution to the Cauchy problem with singular data in the complex domain. Math. Ann.269, 217–234 (1984) · Zbl 0571.35013
[14] [L] Leray, J.: Le calcul différentiel et intégral sur une variété analytique complexe. Bull. S.M.F.87, 81–180 (1959) · Zbl 0199.41203
[15] [Me] Mercier, D. J.: Théorémes de régularité de type Nilsson. Thése de doctorat de l’Université de Nice (1984)
[16] [Mi] Milnor, J.: Singular points of complex hypersurfaces. Princeton, NJ: Princeton Univ. Press 1968 · Zbl 0184.48405
[17] [M] Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391–408 (1981) · Zbl 0489.47010
[18] [Mo] Moiseyev, N.: Complex scaling applied to trapping of atoms and molecules on solid surfaces. In: Lecture Notes in Physics, vol. 325. Berlin, Heidelberg, New York: Springer 1989
[19] [P] Pham, F.: Introduction à l’étude topologique des singularités de Landau. Mémorial des sciences mathématiqueso164. Paris: Gauther-Villars 1967
[20] [Re-Si] Reed, M., Simon, B., Methods of modern mathematical physics IV. Analysis of operators. London: Academic Press 1978 · Zbl 0401.47001
[21] [Si] Simon, B.: Trace ideals and their applications. London Math. Soc. Lect. Notes vol. 35, Cambridge: Cambridge Univ. Press (1979) · Zbl 0423.47001
[22] [Sk] Skriganov, M. M.: Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators. Proceedings of the Stekloc Institute of Mathematics no2 (1987) · Zbl 0615.47004
[23] [V] Vaillant, J.: Ramification d’intégrales holomorphes. J. Math. pures et appl.65, 343–402 (1986) · Zbl 0611.32011
[24] [Va] Vainberg, B. R.: On the analytic properties of the resolvent of a certain class of operator pencils. Math. Sb.77, 199 (1968)
[25] [W] Wilcox, C.: Scattering theory for diffraction gratings. Applied Mathematical Sciences no4.6 Berlin, Heidelberg, New York: Springer 1980 · Zbl 0541.76001
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