Abstract
We discuss the existence and completeness of scattering for one-dimensional systems with different spatial asymptotics at ±∞, for example −d 2/dx 2+V(x) whereV(x)=0 (resp. sinx) ifx<0 (resp.x>0). We then extend our results to higher dimensional systems periodic, except for a localised impurity, in all but one space dimension. A new method, “the twisting trick”, is presented for proving the absence of singular continuous spectrum, and some independent applications of this trick are given in an appendix.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Agmon, S.: Lectures on elliptic boundary value problems. Princeton, London, Toronto: Van Nostrand 1965
Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Sci. II2, 151–218 (1975)
Alsholm, P. K., Kato, T.: Scattering with long range potentials. Proc. Symp. Pure Math.23, 393–399 (1973)
Amrein, W., Georgescu, V.: On the characterization of bound states and scattering states in quantum mechanics. Helv. Phys. Acta46, 635–658 (1973)
Amrein, W., Martin, P., Misra, B.: On the asymptotic condition in scattering theory. Helv. Phys. Acta43, 313–344 (1970)
Avron, J., Herbst, I.: Spectral and scattering theory of Schrödinger operators related to the Stark effect. Commun. math. Phys.52, 239–254 (1977)
Birman, M. S., Solomjak, M. Z.: On estimates of singular numbers of integral operators III. Vestn. Leningr. Univ.24, 35–48 (1969)
Coddington, E., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill 1955
Combes, J. M., : An algebraic approach to scattering theory. (unpublished) (1970)
Combescure, M., Ginibre, J.: Scattering and local absorption for the Schrödinger operator. J. Funct. Anal. (to appear)
Davies, E. B.: Scattering from infinite sheets. Math. Proc. Camb. Phil. Soc.82, 327–334 (1977)
Deift, P.: Classical scattering theory with a trace condition. Princeton: Princeton University Press 1978
Deift, P.: Application of a commutator formula. Duke Math. J. (to appear)
Deift, P., Simon, B.: On the decoupling of finite singularities from the question of asymptotic completeness. J. Funct. Anal.23, 218–238 (1976)
Deift, P., Simon, B.: A time dependent approach to the completeness of multiparticle quantum systems. Comm. Pure Appl. Math.30, 573–583 (1977)
Deift, P., Trubowitz, E.: Inverse scattering on the line. Comm. Pure Appl. Math. (to appear)
Dinaburg, E., Sinai, Ya.: The one dimensional Schrödinger equation with a quasiperiodic potential. Funct. Anal. Appl.9, 8–21 (1975)
Dollard, J.: On the definition of scattering subspaces in non-relativistic quantum mechanics. J. Math. Phys.18, 229–232 (1977)
Eastham, M. S. P.: The spectral theory of periodic differential equations. Scottish Academic Press 1973
Enss V.: A note on Hunziker's theorem. Commun. math. Phys.52, 233–238 (1977)
Enss, V.: Asymptotic completeness for quantum mechanical potential scattering. Commun. math. Phys.61, 258–291 (1978)
Gross, H., Grümm, H. R., Narnhofer, H., Thirring, W.: Algebraic theory of Coulomb scattering. Acta Phys. Austr.40, 97–103 (1974)
Hepp, K.: Scattering theory in the Heisenberg ferromagnet. Phys. Rev.135, 95–97 (1972)
Hörmander, L.: The existence of wave operators in scattering theory. Math. Z.146, 69–91 (1976)
Jost, R.: The general theory of quantized fields. Providence, RI: American Mathematical Society 1965
Klein, O.: Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac. Z. Physik53, 157–165 (1929)
Kuroda, S.: Scattering theory for differential operators I. J. Math. Soc. Japan25, 75–104 (1973)
Lavine, R. B.: Scattering theory for long range potentials. J. Funct. Anal.5, 368–382 (1970)
Lax, P., Phillips, R. S.: Scattering theory. New York, London: Academic Press 1967
Levitan, B., Sargstan, I.: Introduction to spectral theory. Am. Math. Soc. Monograph. Transl.39, 1975
Magnus, W., Winkler, S.: Hill's equation. New York: Wiley 1966
Pearson, D.: General theory of potential scattering with absorption at local singularities. Helv. Phys. Acta47, 249–264 (1974)
Pearson, D.: A generalization of the Birman trace theorem. J. Funct. Anal.28, 182–186 (1978)
Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. New York, London: Academic Press 1972
Reed M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis. New York, London: Academic Press 1975
Reed, M., Simon, B.: Methods of modern mathematical physics. III. Scattering theory. New York, London: Academic Press 1979
Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York, London: Academic Press 1978
Ruelle, D.: A remark on bound states in potential scattering theory. Nuovo Cimento61A, 655–662 (1969)
Ruelle, D.: On the asymptotic condition in quantum field theory. Helv. Phys. Acta35, 147–163 (1962)
Ruijenaars, S., Bongaarts, P.: Scattering theory for one-dimensional step potentials. Ann. Inst. Henri Poincaré26A, 1–17 (1977)
Semenov, Yu.: Wave operators for the Schrödinger equation with strongly singular short range potentials. Lett. Math. Phys.1, 457–462 (1977)
Simon, B.: Lectures on trace ideal methods. In: London Mathematical Society Lecture Notes. Cambridge: Cambridge University Press 1979
Simon, B.: Geometric methods in multiparticle quantum systems. Commun. math. Phys.55, 259–274 (1977)
Simon, B.: N-body scattering in the two cluster region. Commun. math. Phys.58, 205–210 (1978)
Sinha, K.: On the absolutely and singularly continuous subspace in scattering theory. Ann. Inst. Henri Poincaré26A, 263–277 (1977)
Streater, R. F.: Spin wave scattering. In: Scattering theory in mathematical physics, 273–298. Dordrecht, Stuttgart: Reidel 1974
Wilcox, C.: Scattering states and wave operators in the abstract theory of scattering. J. Funct. Anal.12, 257–274 (1973)
Author information
Authors and Affiliations
Additional information
Communicated by J. Ginibre
Research supported by NSF grant MPS-75-11864
On leave from Mathematical Institute, Oxford OX1 3JP, England
Rights and permissions
About this article
Cite this article
Davies, E.B., Simon, B. Scattering theory for systems with different spatial asymptotics on the left and right. Commun.Math. Phys. 63, 277–301 (1978). https://doi.org/10.1007/BF01196937
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01196937