A Two-Particle Quantum System with Zero-Range Interaction
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 18, 17 p.

We study a two-particle quantum system given by a test particle interacting in three dimensions with a harmonic oscillator through a zero-range potential. We give a rigorous meaning to the Schrödinger operator associated with the system by applying the theory of quadratic forms and defining suitable families of self-adjoint operators. Finally we fully characterize the spectral properties of such operators.

Correggi, Michele 1

1 CIRM, Fondazione Bruno Kessler Via Sommarive 14 38100 Trento Italy
@article{SEDP_2008-2009____A18_0,
     author = {Correggi, Michele},
     title = {A {Two-Particle} {Quantum} {System} with {Zero-Range} {Interaction}},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:18},
     pages = {1--17},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2008-2009},
     language = {en},
     url = {http://archive.numdam.org/item/SEDP_2008-2009____A18_0/}
}
TY  - JOUR
AU  - Correggi, Michele
TI  - A Two-Particle Quantum System with Zero-Range Interaction
JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:18
PY  - 2008-2009
SP  - 1
EP  - 17
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://archive.numdam.org/item/SEDP_2008-2009____A18_0/
LA  - en
ID  - SEDP_2008-2009____A18_0
ER  - 
%0 Journal Article
%A Correggi, Michele
%T A Two-Particle Quantum System with Zero-Range Interaction
%J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
%Z talk:18
%D 2008-2009
%P 1-17
%I Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://archive.numdam.org/item/SEDP_2008-2009____A18_0/
%G en
%F SEDP_2008-2009____A18_0
Correggi, Michele. A Two-Particle Quantum System with Zero-Range Interaction. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 18, 17 p. http://archive.numdam.org/item/SEDP_2008-2009____A18_0/

[AGH-KH] S. Albeverio, F. Gesztesy, R. Hogh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, New-York, 1988. | MR | Zbl

[BC] J. Bellandi, E.S. Caetano Neto, The Mehler Formula and the Green Function of Multidimensional Isotropic Harmonic Oscillator, J. Phys. A: Math Gen. 9 (1976), 683-685. | MR | Zbl

[BF] F.A. Berezin, L.D. Faddeev, A Remark on Schrödinger Equation with a Singular Potential, Sov. Math. Dokl. 2 (1961), 372-375. | Zbl

[CDF] M. Correggi, G. Dell’Antonio, D. Finco, Spectral Analysis of a Two Body Problem with Zero-range Perturbation, J. Funct. Anal. 255 (2008), 502–531. | MR | Zbl

[DFigT] G. Dell’Antonio, R. Figari, A. Teta, Hamiltonians for Systems of N Particles Interacting through Point Interactions, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), 253–290. | Numdam | MR | Zbl

[DFinT] G. Dell’Antonio, D. Finco, A. Teta, Singularly Perturbed Hamiltonians of a Quantum Reyleigh Gas Defined as Quadratic Forms, Pot. Anal. 22 (2005), 229–261. | MR | Zbl

[F] E. Fermi, Sul Moto dei Neutroni nelle Sostanze Idrogenate, (in italian) Ricerca Scientifica 7 (1936), 13-52. | Zbl

[HS] P.R. Halmos, V.S. Sunder, Bounded Integral Operators on L 2 Spaces, Springer Verlag, New York, 1978. | MR | Zbl

[KP] R. Kronig, W.G. Penney, Quantum Mechanics of Electrons in Crystal Lattices, Proc. R. Soc. A 130 (1931), 499-513. | Zbl

[P] A. Posilicano, Self-adjoint Extensions of Restrictions, Oper. Matrices 2 (2008), 483–506. | MR

[S] B. Simon, Trace Ideals and Their Applications, Mathematical Surveys and Monographs 120, American Mathematical Society, Providence, RI, 2005. | MR | Zbl

[RS1] M. Reed, B. Simon, Methods of Modern Mathematical Physics. Vol I: Functional Analysis, Academic Press, San Diego, 1972. | Zbl

[RS3] M. Reed, B. Simon, Methods of Modern Mathematical Physics. Vol III: Scattering Theory, Academic Press, San Diego, 1975.

[RS4] M. Reed, B. Simon, Methods of Modern Mathematical Physics. Vol IV: Analysis of Operators, Academic Press, San Diego, 1978. | MR | Zbl