Abstract
Let H(0) be a dilation-analytic three-particle Schrödinger operator with analytic continuation H(ϕ) (ϕ>0). Let a be zero or the energy of a two-particle bound state. Let-Δ (a) be the Laplace operator representing the kinetic energy of the relative motion of fragments scattered in channel a. By recent results, wave operators W (±, a, ϕ) with conjugates W† (±, a, ϕ) exist such that W (±, a, ϕ) W† (±, a, ϕ) is a projection P (a, ϕ) commuting with H (ϕ) while [H (ϕ)-a]W (±, a, ϕ) equals-W(±, a, ϕ) Δ (a) e2iϕ. This paper shows that the wave operators transform dilation-analytic functions of particle coordinates into dilation-analytic functions. Specifically, if the left shoulder of the spectrum of P (a,ϕ) H (ϕ) does not sweep across eigenvalues of H(ϕ) when α≤ϕ≤β, then W(-, a, ϕ) and W† (+, a, ϕ) are dilation analytic in [α, β]. If the right shoulder does not sweep across eigenvalues, W(+, a, ϕ) and W†(-, a, ϕ) are dilation analytic in [α,β]. A semisimple eigenvalue of H (ψ) embedded in the spectrum of P (a, ψ) H (ψ) does not prevent the wave operators from being dilation analytic in an interval [α, β] with ψ as an interior point.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys.22 (1971), 269–279.
E. Balslev and J. M. Combes, Spectral properties of many-body Schrödinger operators with dilation-analytic interactions, Comm. Math. Phys.22 (1971), 280–294.
J. M. Combes and L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys.34 (1973), 251–270.
R. Froese and I. Herbst, Exponential bounds and absence of positive eigenvalues for N-body Schrödinger operators, Comm. Math. Phys.87 (1982), 429–447.
T. Kato, “Perturbation Theory for Linear Operators”, Springer-Verlag, Berlin/Heidelberg/New York, 1966.
T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann.162 (1966), 258–279.
M. Reed and B. Simon, “Methods of Modern Mathematical Physics IV: Analysis of Operators”, Academic Press, New York, 1978.
F. Smithies, “Integral Equations”, Cambridge Univ. Press, Cambridge, 1958.
E. C. Titchmarsh, “The Theory of Functions”, 2nd ed., Oxford Univ. Press, Oxford, 1939.
C. van Winter, Fredholm equations on a Hilbert space of analytic functions, Trans. Amer. Math. Soc.162 (1971), 103–139.
C. van Winter, Complex dynamical variables for multiparticle systems with analytic interactions, I, J. Math. Anal. Appl.47 (1974), 633–670.
C. van Winter, Complex dynamical variables for multiparticle systems with analytic interactions, II, J. Math. Anal. Appl.48 (1974), 368–399.
C. van Winter, Invariant subspaces of analytic multiparticle Hamiltonians, J. Math. Anal. Appl.49 (1975), 88–123.
C. van Winter, Semigroups associated with analytic Schrödinger operators, J. Math. Anal. Appl.94 (1983), 377–405.
C. van Winter, Groups generated by analytic Schrödinger operators, J. Math. Anal. Appl.94 (1983), 406–434.
C. van Winter, The resolvent of a dilation-analytic three-particle system, J. Math. Anal. Appl.101 (1984), 195–267.
C. van Winter, Wave operators and similarity of certain non-normal one-parameter groups, Preprint Univ. of Kentucky, 1984.
Author information
Authors and Affiliations
Additional information
This work was supported in part by the National Science Foundation under grant DMS-8301096.
Rights and permissions
About this article
Cite this article
van Winter, C. Dilation-analytic wave operators for three particles. Integr equ oper theory 9, 387–451 (1986). https://doi.org/10.1007/BF01199352
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01199352