Doi: http://dx.doi.org/10.15446/recolma.v49n1.54167

¹Universidad Sergio Arboleda, Bogotá, Colombia. Email: leonardo.cano@usa.edu.co

We show the existence and orthogonality of wave operators naturally associated to a compatible Laplacian on a complete manifold with a corner of codimension 2. In fact, we prove asymptotic completeness i.e. that the image of these wave operators is equal to the space of absolutely continuous states of the compatible Laplacian. We achieve this last result using time dependent methods coming from many-body Schrödinger equations.

Key words: Quantum scattering theory, Manifolds with corners, Wave operators, Many-body Schrödinger equations.

Demostramos la existencia y ortogonalidad de operadores de onda naturalmente asociados a un Laplaciano compatible sobre una variedad completa con una esquina de codimensión 2. De hecho, probamos su completitud asintótica, es decir que la imagen de esos operadores de onda es igual al espacio de estados absolutamente contínuos del Laplaciano compatible. Logramos esto último usando métodos dependientes del tiempo que provienen del estudio de operadores de Schrödinger de varios cuerpos.

Palabras clave: Teoría de dispersión cuántica, variedades con esquinas, operadores de onda, ecuaciones de Schrödinger de varios cuerpos.

Texto completo disponible en PDF

References

[1] M. F. Atiyah, V. K. Patodi, and I. M. Singer, `Spectral Asymmetry and Riemannian Geometry. I´, Math. Proc. Cambridge Philos. Soc. 77, (1975), 43-69.         [ Links ]

[2] U. Bunke and M. Olbrich, Scattering Theory for Geometrically Finite Groups, `Geometry, analysis and topology of discrete groups´, 2008, Vol. 6 of Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, p. 40-136.         [ Links ]

[3] L. Cano, Analytic Dilation on Complete Manifolds With Corners of Codimension 2, PhD thesis, Bonn University, 2011. pp. 1-117         [ Links ]

[4] L. Cano, `Mourre Estimates for Compatible Laplacians on Complete Manifolds with Corners of Codimension 2´, Ann. Glob. Anal. Geom. 43, (2013), 75-97.         [ Links ]

[5] G. Carron, `Déterminant relatif et la fonction Xi´, Amer. J. Math. 124, 2 (2002), 307-352.         [ Links ]

[6] C. Gordon, P. Perry, and D. Schueth, Isospectral and Isoscattering Manifolds: A Survey of Techniques and Examples, `Geometry, spectral theory, groups, and dynamics´, 2005, Vol. 387 of Contemp. Math., Amer. Math. Soc., Providence, USA, p. 157-179.         [ Links ]

[7] G. M. Graf, `Asymptotic Completeness for N-Body Short-Range Quantum Systems: A New Proof´, Comm. Math. Phys. 132, 1 (1990), 73-101.         [ Links ]

[8] C. R. Graham and M. Zworski, `Scattering Matrix in Conformal Geometry´, Invent. Math. 152, 1 (2003), 89-118.         [ Links ]

[9] L. Guillopé, `Théorie spectrale de quelques variétés à bouts´, Ann. Sci. École Norm. Sup. (4) 22, 1 (1989), 137-160.         [ Links ]

[10] A. Hassell, R. Mazzeo, and R. B. Melrose, `A Signature Formula for Manifolds With Corners of Codimension Two´, Topology 36, 5 (1997), 1055-1075.         [ Links ]

[11] W. Hunziker and I. M. Sigal, `Time-Dependent Scattering Theory of N-Body Quantum Systems´, Rev. Math. Phys. 12, 8 (2000a), 1033-1084.         [ Links ]

[12] W. Hunziker and I. M. Sigal, `The Quantum N-Body Problem´, J. Math. Phys. 41, 6 (2000b), 3448-3510.         [ Links ]

[13] R. Husseini, `Zur Spektraltheorie Verallgemeinerter Laplace-Operatoren Auf Mannigfaltigkeiten Mit Zylindrischen Enden´, Diplomarbeit, Rheinischen Friedrich-Wilhelms-Universität Bonn, (2005).         [ Links ]

[14] R. Mazzeo and A. Vasy, `Scattering Theory on SL(3)/SO(3): Connections with Quantum 3-Body Scattering´, Proc. Lond. Math. Soc. (3) 94, 3 (2007), 545-593.         [ Links ]

[15] R. B. Melrose, Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidian Spaces, `Spectral and scattering theory (Sanda, 1992)´, 0000, Vol. 161 of Lecture Notes in Pure and Appl. Math., Dekker, New York, USA, p. 85-130.         [ Links ]

[16] W. Müller, `On the L²-Index of Dirac Operators on Manifolds with Corners of Codimension Two I´, J. Differential Geom. 44, 1 (1996), 97-177.         [ Links ]

[17] W. Müller and G. Salomonsen, `Scattering Theory for the Laplacian on Manifolds with Bounded Curvature´, J. Funct. Anal. 253, 1 (2007), 158-206.         [ Links ]

[18] W. Müller and A. Strohmaier, `Scattering at Low Energies on Manifolds with Cylindrical Ends and Stable Systoles´, Geom. Funct. Anal. 20, 3 (2010), 741-778.         [ Links ]

[19] M. Reed and B. Simon, Methods of Modern Mathematical Physics. III, Academic Press, New York, USA, 1979. Harcourt Brace Jovanovich Publishers         [ Links ]

[20] S. Richard and R. Tiedra de Aldecoa, `Spectral Analysis and Time-Dependent Scattering Theory on Manifolds with Asymptotically Cylindrical Ends´, Rev. Math. Phys. 25, 2 (2013), 1350003, 40.         [ Links ]

[21] M. A. Shubin, `Spectral Theory of Elliptic Operators on Non-Compact Manifolds´, Paper on lectures Summer School on Semiclassical Methods, Nantes, (1991).         [ Links ]

[22] I. M. Sigal and A. Soffer, `The N-Particle Scattering Problem: Asymptotic Completeness for Short-Range Systems´, Ann. of Math. (2) 126, 1 (1987), 35-108.         [ Links ]

[23] I. M. Sigal and A. Soffer, `Long-Range Many-Body Scattering. Asymptotic Clustering for Coulomb-Type Potentials´, Invent. Math. 99, 1 (1990), 115-143.         [ Links ]

[24] M. E. Taylor, Partial Differential Equations I. Basic Theory, Vol. 115 of Applied Mathematical Sciences, Second edn, Springer, New York, USA, 2011.         [ Links ]

[25] A. Vasy, Geometry and Analysis in Many-Body Scattering, `Inside out: inverse problems and applications´, 2003, Vol. 47 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, UK, p. 333-379.         [ Links ]

[26] D. R. Yafaev, Mathematical Scattering Theory, Vol. 105 of Translations of Mathematical Monographs, American Mathematical Society, Providence, USA, 1992. General theory, Translated from the Russian by J. R. Schulenberger         [ Links ]

[27] D. Yafaev, `Radiation Conditions and Scattering Theory for N-Particle Hamiltonians´, Comm. Math. Phys. 154, 3 (1993), 523-554.         [ Links ]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

Doi: http://dx.doi.org/ @ARTICLE{RCMv49n1a06,
    AUTHOR  = {Cano G., Leonardo A.},
    TITLE   = {{Time Dependent Quantum Scattering Theory on Complete Manifolds with a Corner of Codimension 2}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2015},
    volume  = {49},
    number  = {1},
    pages   = {105--138}
}