Abstract
The scattering of plane waves and of point source pulses by irregularities in a discrete lattice model of the Schrödinger equation is considered. Closed form expressions are derived for the scattered wave function in terms of lattice Green's functions in the case that a finite number of lattice points or “bonds” are defective. The scattered wave function appears in the form of the ratio of two determinants. While in continuum scattering theory the scatterer must have some symmetry, perhaps spherical, cylindrical or elliptical, in order to allow separation of variables in the basic scattering differential equation, such symmetries are not necessary for the construction of scattered wave functions on discrete lattices. When the number of irregularities becomes large, the determinants in the solution of the scattering problem become large.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. M. Nussenzveig,J. Math. Phys. 10:82, 125 (1969).
G. F. Koster and J. C. Slater,Phys. Rev. 95:1167 (1954);96:1208 (1954).
A. A. Maradudin, E. W. Montroll, G. H. Weiss, and I. P. Ipatova,Theory of Lattice Vibrations in the Harmonic Approximation, Academic Press, New York (1971).
I. M. Lifshitz,Nuovo Cimento Suppl. (10)3:716 (1956).
E. W. Montroll, inApplied Combinatorial Mathematics, E. F. Beckenbach, ed., Wiley, New York (1964), Chapter IV, p. 96.
E. W. Montroll and H. Scher,J. Stat. Phys. 9:101 (1973).
E. W. Montroll,Comm. Pure Appl. Math. 5:415 (1952).
E. W. Montroll, inEnergetics in Metallurgical Phenomenon, Gordon and Breach, New York (1967), Vol. 3, p. 123.
G. A. Baker, Jr.,Advances in Theoretical Physics, Academic Press (1965), Vol. I, p. 1.
A. A. Maradudin, E. W. Montroll, G. H. Weiss, R. Herman, and H. W. Miles,Acad. Roy. Belg. C. Sci. Mem. Coll. in 4°(2)14(7):1–176 (1960).
T. Wolfrom and J. Callaway,Phys. Rev. 130:2207 (1963).
T. Morita and T. Horiguchi,J. Phys. A 5:67 (1972).
T. Morita and T. Horiguchi,Tables of Lattice Green's Functions for Simple Cubic Lattices, Tohuku University (1971).
S. Inawashiro, S. Katsura, and Y. Abe,J. Math. Phys. 14:560 (1973).
G. S. Joyce, J. Phys. A5:L69 (1972);Phil. Proc. Soc. London 232:583 (1972).
E. W. Montroll, inProc. 3rd Berkeley Symposium on Math. Stat. and Prob. (1956), Vol. III, p. 299.
R. Jost and A. Pais,Phys. Rev. 82:840 (1951).
Author information
Authors and Affiliations
Additional information
This research was supported by DARPA and monitored by Air Force System Command, Rome Air Development Center, under Contract F30602-72-C-0494.
Rights and permissions
About this article
Cite this article
Montroll, E.W., West, B.J. Scattering of waves by irregularities in periodic discrete lattice spaces. I. Reduction of problem to quadratures on a discrete model of the Schrödinger equation. J Stat Phys 13, 17–42 (1975). https://doi.org/10.1007/BF01012597
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01012597