Research PapersNo Access

SCATTERING THEORY FOR LATTICE OPERATORS IN DIMENSION d ≥ 3

JEAN BELLISSARD

Georgia Institute of Technology, School of Mathematics, Atlanta GA 30332-0160, USA

Search for more papers by this author

and

HERMANN SCHULZ-BALDES

Department Mathematik, Universität Erlangen-Nürnberg, D-91054 Erlangen, Germany

Search for more papers by this author

https://doi.org/10.1142/S0129055X12500201Cited by:25 (Source: Crossref)

Abstract

This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension d ≥ 3, the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula, the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in the presence of embedded eigenvalues and threshold singularities.

Keywords:

AMSC: 35P25, 47A40

References

W. O. Amrein , A. Boutet de Monvel and V. Georgescu , C0-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians ( Birkhäuser , Basel , 1996 ) . Crossref, Google Scholar
V. Bach, W. de Siqueira Pedra and S. Lakaev, Bounds on the pure point spectrum of lattice Schödinger operators, mp-arc 11-161 (October, 15, 2011) . Google Scholar
B. Blackadar , K-Theory for Operator Algebras , 2nd edn. ( Cambridge Univ. Press , Cambridge , 1998 ) . Google Scholar
D. Bollé, Ideas and Methods in Quantum and Statistical Physics, eds. S. Albeverioet al. (Cambridge Univ. Press, Cambridge, 1992) pp. 173–196. Google Scholar
O. Bratteli and D. Robinson , Operator Algebras and Quantum Statistical Mechanics 1 ( Springer , Berlin , 1979 ) . Crossref, Google Scholar
M. S. Birman and D. R. Yafaev, St. Petersburg Math. J. 6, 453 (1995). Google Scholar
D. Bolleet al., Phys. Rev. Lett. 56, 900 (1986). Crossref, Web of Science, ADS, Google Scholar
K. M. Case and M. Kac, J. Math. Phys. 14, 594 (1973). Crossref, Web of Science, ADS, Google Scholar
C. De Carvalho and H. Nussenzveig, Phys. Rep. 364, 83 (2002). Crossref, Web of Science, ADS, Google Scholar
A. Connes , Non-Commutative Geometry ( Academic Press , San Diego , 1994 ) . Google Scholar
E. Doron, U. Smilansky and A. Frenkel, Phys. Rev. Lett. 65, 3072 (1990). Crossref, Web of Science, ADS, Google Scholar
E. N. Economou , Green's Functions in Quantum Physics , 3rd edn. ( Springer , New York , 2005 ) . Google Scholar
N. E. Firsova, Theoret. Math. Phys. 62, 130 (1985). Crossref, Web of Science, Google Scholar
C. Gerard and F. Nier, J. Math. Kyoto Univ. 38, 595 (1998). Crossref, Google Scholar
D. Gieseker , H. Knörrer and E. Trubowitz , The Geometry of Algebraic Fermi Curves ( Academic Press , Boston , 1993 ) . Google Scholar
D. B. Hinton, M. Klaus and J. K. Shaw, Siam J. Math. Anal. 22, 754 (1991). Crossref, Web of Science, Google Scholar
P. Hislop and I. Sigal , Introduction to Spectral Theory: With Applications to Schrödinger Operators ( Springer , New York , 1996 ) . Crossref, Google Scholar
A. Jensen, Comm. Math. Phys. 82, 435 (1981). Crossref, Web of Science, ADS, Google Scholar
A. Jensen and T. Kato, Duke Math. J. 46, 583 (1979). Crossref, Web of Science, Google Scholar
J. Kellendonk and S. Richard, J. Phys. A 39, 14397 (2006). Crossref, ADS, Google Scholar
J. Kellendonk and S. Richard, J. Phys. A 41, 295207 (2008). Crossref, Google Scholar
J. Kellendonk and S. Richard, Math. Phys. Elect. J. 14, 22 (2008). Web of Science, Google Scholar
J. Kellendonk and S. Richard, On the wave operators and Levinson's theorem for potential scattering in ℝ3, preprint (2010) . Google Scholar
M. Kohmoto, T. Koma and S. Nakamura, The spectral shift function, Friedel sum rule, and Levinson theorem (November 25, 2011) , arXiv:1111.5939 . Google Scholar
P. Kuchment and B. Vainberg, Comm. Math. Phys. 268, 673 (2006). Crossref, Web of Science, ADS, Google Scholar
N. Levinson, On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase, Kgl. Danske Videnskabernes Selskab Mat.-fys. Medd. 25 (1949) 3–29; Accessible in Selected Papers of Norman Levinson, eds. John A. Nohel and David H. Sattinger (Birkhäuser, Boston, Basel, Berlin, 1998), pp. 164–192 . Google Scholar
N. I. Muskhelishvili , Singular Integral Equations ( Noordhoff , Groningen , 1953 ) . Google Scholar
R. G. Newton , Scattering Theory of Waves and Particles , 2nd edn. ( Springer , New York , 1982 ) . Crossref, Google Scholar
R. G. Newton, J. Math. Phys. 32, 551 (1991). Crossref, Web of Science, ADS, Google Scholar
L. I. Nicolaescu , An Invitation to Morse Theory ( Springer , New York , 2007 ) . Google Scholar
M. Reed and B. Simon , Methods of Modern Mathematical Physics III: Scattering Theory ( Academic Press , New York , 1979 ) . Google Scholar
T. Sakai , Riemannian Geometry ( AMS , Providence , 1996 ) . Crossref, Google Scholar
T. Y. Tsang and T. A. Osborn, Nucl. Phys. A 247, 43 (1975). Crossref, Web of Science, ADS, Google Scholar
L. Van Hove, Phys. Rev. 89, 1189 (1953). Crossref, Web of Science, ADS, Google Scholar
D. R. Yafaev , Mathematical Scattering Theory: General Theory ( AMS , Providence , 1992 ) . Crossref, Google Scholar