zbMATH — the first resource for mathematics

Asymptotic properties of generalized eigenfunctions for three body Schrödinger operators. (English) Zbl 0774.35064
Continuing his previous work [Commun. Math. Phys. 146, No. 2, 241-258 (1992; Zbl 0748.35026)], the author considers the generalized eigenfunctions of the Schrödinger operator for three particles with binary interactions. He proves that the large-distance asymptotic behavior of those generalized eigenfunctions that describe an initial state of two clusters agrees with that derived by less rigorous arguments by R. G. Newton [Ann. Phys. 74, 324-351 (1972)]. He thereby rigorously derives expressions for elements of the \(S\) matrix that correspond to the collision of a particle with a bound state of two others.

35Q40 PDEs in connection with quantum mechanics
35P25 Scattering theory for PDEs
47A40 Scattering theory of linear operators
81U10 \(n\)-body potential quantum scattering theory
Full Text: DOI
[1] Gérard, C.: Sharp propagation estimates for N-particle systems. Duke Mach. J.67, 483–515 (1992) · Zbl 0760.35049
[2] Isozaki, H.: Structures of S-matrices for three body Schrödinger operators. Commun. Math. Phys.146, 241–258 (1992) · Zbl 0748.35026
[3] Isozaki, H.: Differentiability of generalized Fourier transforms associated with Schrödinger operators. J. Math. Kyoto Univ.25, 789–806 (1985) · Zbl 0612.35004
[4] Jensen, A. and Kato, T.: Spectral properties of Schrödinger operators and time decay of the wave functions. Duke Math. J.46, 583–611 (1979) · Zbl 0448.35080
[5] Mercuriev, S.P.: On the three-body Coulomb scattering problem. Ann. Phys.130, 395–426 (1980) · Zbl 0483.35066
[6] Newton, R.G.: The asymptotic form of the three-particle wave functions and the cross sections. Ann. Phys.74, 324–351 (1972)
[7] Newton, R.G.: The three particle S-matrix. J. Math. Phys.15, 338–343 (1974)
[8] Nuttal, J.: Asymptotic form of the three-particle scattering wave functions for free incident particles. J. Math. Phys.12, 1896–1899 (1971)
[9] Robert, D.: Autour de l’approximation semiclassique. Basel-Boston: Birkhäuser 1983
[10] Saito, Y.: Spectral representation for Schrödinger operators with long-range potentials. Lect. Notes in Math.727, Berlin, Heidelberg, New York: Springer 1979 · Zbl 0414.47012
[11] Sigal, I.M. and Soffer, A.: Local decay and propagation estimates for time-dependent and time-independent Hamiltonians. Preprint 1988
[12] Skibsted, E.: Propagation estimates for N-body Schrödinger operators. Commun. Math. Phys.142, 67–98 (1991) · Zbl 0760.35035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.