Geodesic
To the question of Schröedinger operator kernel resolvent asymptotics construction in the three one-dimensional quantum particles scattering problem interacting by finite repulsive pair potentials
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 45, Tome 438 (2015), pp. 95-103 
A. M. Budylin; S. B. Levin. To the question of Schröedinger operator kernel resolvent asymptotics construction in the three one-dimensional quantum particles scattering problem interacting by finite repulsive pair potentials. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 45, Tome 438 (2015), pp. 95-103. http://geodesic.mathdoc.fr/item/ZNSL_2015_438_a6/
@article{ZNSL_2015_438_a6,
     author = {A. M. Budylin and S. B. Levin},
     title = {To the question of {Schr\"oedinger} operator kernel resolvent asymptotics construction in the three one-dimensional quantum particles scattering problem interacting by finite repulsive pair potentials},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {95--103},
     year = {2015},
     volume = {438},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_438_a6/}
}
TY  - JOUR
AU  - A. M. Budylin
AU  - S. B. Levin
TI  - To the question of Schröedinger operator kernel resolvent asymptotics construction in the three one-dimensional quantum particles scattering problem interacting by finite repulsive pair potentials
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2015
SP  - 95
EP  - 103
VL  - 438
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2015_438_a6/
LA  - ru
ID  - ZNSL_2015_438_a6
ER  - 
%0 Journal Article
%A A. M. Budylin
%A S. B. Levin
%T To the question of Schröedinger operator kernel resolvent asymptotics construction in the three one-dimensional quantum particles scattering problem interacting by finite repulsive pair potentials
%J Zapiski Nauchnykh Seminarov POMI
%D 2015
%P 95-103
%V 438
%U http://geodesic.mathdoc.fr/item/ZNSL_2015_438_a6/
%G ru
%F ZNSL_2015_438_a6

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The present work aims at announcing a new approach to a construction of the asymptotics (at infinity in configuration space) of the Schrödinger operator resolvent kernel asymptotics in the scattering problem of three one-dimensional quantum particles interacting by the finite pair repulsive potentials. Within the framework of this approach the asymptotics of Schrödinger operator absolutely continuum spectrum eigenfunctions can be constructed explicitly. We should emphasize that the restriction of the consideration for the case of finite pair potentials does not lead to a simplification of the problem in its essence as the potential of the interaction of all three particles remains non-decreasing at infinity but allows to put aside a certain number of technical details.

[1] A. M. Budylin, V. S. Buslaev, “Reflection operator and their applications to asymptotic investigations of semiclassical integral equations”, Advances in Soviet Math., 7, no. 6, AMS, Providence, RI, 1991, 79–103 | MR

[2] L. D. Faddeev, Mathematical aspects of the three-body problem of the quantum scattering theory, Daniel Davey and Co., Inc., 1965 | MR

[3] V. S. Buslaev, S. B. Levin, P. Neittaanmäki, T. Ojala, “New approach to numerical computatio of the eigenfunctions of the continuous spectrum of three-particle Schrödinger operator: I. One-dimensional particles, short-range pair potentials”, JPhysA, 2010 | MR

[4] V. S. Buslaev, S. B. Levin, “Asymptotic Behavior of the Eigenfunctions of the Many-particle Shrödinger Operator. I. One-dimentional Particles”, Spectral theory of differential operators, Amer. Math. Soc. Transl. Ser. 2, 225, 2008, 55–71 | MR | Zbl

[5] A. M. Budylin, V. S. Buslaev, “Kvaziklassicheskaya asimptotika rezolventy integralnogo operatora svërtki s sinus-yadrom na konechnom intervale”, Adv. Sov. Math., 7, Amer. Math. Soc., 1995, 107–157 | MR

[6] E. Mourre, “Absence of singular continuous spectrum for certain self-adjoint operators”, Commun. MathPhys, 78 (1981), 391–408 | DOI | MR | Zbl

[7] P. Perry, I. M. Sigal, B. Simon, “Spectral analysis of $N$-body Schrodinger operators”, Annals of Mathematics, 114 (1981), 519–567 | DOI | MR | Zbl

[8] D. R. Yafaev, Matematicheskaya teoriya tasseyaniya, SPbGU, 1994

[9] I. M. Gelfand, N. Ya. Vilenkin, Nekotorye primeneniya garmonicheskogo analiza. Osnaschënnye gilbertovy prostranstva, Obobschënnye funktsii, 4, FM, 1961

[10] K. Moren, Metody gilbertova prostranstva, Mir, 1965 | MR

[11] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, v. 3, Teoriya Rasseyaniya, Mir, 1982 | MR