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On the Limiting absorption principle for a new class of Schrödinger Hamiltonians
Alexandre Martin1
1 Département de Mathématiques, Université de Cergy-Pontoise, 95000 Cergy-Pontoise, France
Confluentes Mathematici, Tome 10 (2018) no. 1, pp. 63-94.

We prove the limiting absorption principle and discuss the continuity properties of the boundary values of the resolvent for a class of form bounded perturbations of the Euclidean Laplacian Δ that covers both short and long range potentials with an essentially optimal behaviour at infinity. For this, we give an extension of Nakamura’s results (see [16]).

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DOI : 10.5802/cml.46
Classification : 35J10, 35P25, 35Q40, 35S05, 47B15, 47B25, 47F05
Mots clés : Schrödinger operators, Mourre theory, Limiting Absorption Principle
Alexandre Martin 1

1 Département de Mathématiques, Université de Cergy-Pontoise, 95000 Cergy-Pontoise, France
Licence : CC-BY-NC-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Alexandre Martin. On the Limiting absorption principle for a new class of Schrödinger Hamiltonians. Confluentes Mathematici, Tome 10 (2018) no. 1, pp. 63-94. doi : 10.5802/cml.46. https://cml.centre-mersenne.org/articles/10.5802/cml.46/

[1] W. Amrein; A. Boutet de Monvel; V. Georgescu C 0 -groups, commutator methods, and spectral theory of N-body Hamiltonians, Birkhäuser Verlag, 1996 | Zbl

[2] M. Ben-Artzi; A. Devinatz Spectral and scattering theory for the adiabatic oscillator and related potentials, Journal of Mathematical Physics, Volume 20 (1979) no. 4, pp. 594-607 | DOI | MR | Zbl

[3] A. Boutet de Monvel; V. Georgescu Boundary Values of the Resolvent of a Self-Adjoint Operator: Higher Order Estimates, Algebraic and Geometric Methods in Mathematical Physics, 1993 | Zbl

[4] A. Boutet de Monvel; V. Georgescu; J. Sahbani Higher Order Estimates in the Conjugate Operator Theory (1997) (see https://www.ma.utexas.edu/mp_arc/index-97.html)

[5] M. Combescure Spectral and scattering theory for a class of strongly oscillating potentials, Communications in Mathematical Physics, Volume 73 (1980) no. 1, pp. 43-62 | DOI | MR | Zbl

[6] M. Combescure; J Ginibre Spectral and scattering theory for the Schrödinger operator with strongly oscillating potentials, Annales de l’IHP Physique théorique, Volume 24 (1976) no. 1, pp. 17-30 | Numdam | Zbl

[7] H.L. Cycon; R.G. Froese; W. Kirsch; B. Simon Schrödinger operators, with applications to quantum mechanics and global geometry, Springer, 2008 (2nd corrected printing)

[8] A. Devinatz; R. Moeckel; P. Rejto A limiting absorption principle for Schrödinger operators with Von Neumann-Wigner type potentials, Integral Equations and Operator Theory, Volume 14 (1991) no. 1, pp. 13-68 | DOI | Zbl

[9] A. Devinatz; P. Rejto A limiting absorption principle for Schrödinger operators with oscillating potentials. Part I, Journal of Differential Equations, Volume 49 (1983) no. 1, pp. 29-84 | DOI | Zbl

[10] A. Devinatz; P. Rejto A limiting absorption principle for Schrödinger operators with oscillating potentials. Part II, Journal of Differential Equations, Volume 49 (1983) no. 1, pp. 85-104 | DOI | Zbl

[11] V. Georgescu; C. Gérard On the Virial Theorem in Quantum Mechanics, Comm. Math. Phys. (1999) | DOI | MR | Zbl

[12] V. Georgescu; M. Măntoiu On the spectral theory of Dirac type Hamiltonians, J. of Operator Theory, Volume 46 (2001), pp. 289-321 | MR | Zbl

[13] T. Jecko; A. Mbarek Limiting Absorption Principle for Schrödinger Operators with Oscillating Potentials, Documenta Mathematica, Volume 22 (2017), pp. 727-776 | Zbl

[14] E. Mourre Absence of Singular Continuous Spectrum for Certain Self-Adjoint Operators, Comm. Math. Phys., Volume 78 (1981), pp. 391-408 | DOI | MR | Zbl

[15] E. Mourre Opérateurs conjugués et propriétés de propagation, Comm. Math. Phys., Volume 91 (1983), pp. 279-300 | DOI | MR | Zbl

[16] S. Nakamura A remark on the Mourre theory for two body Schrödinger operators, J. Spectral Theory, Volume 4 (2015) no. 3, pp. 613-619 | DOI

[17] C.R. Putnam On commutators and Jacobi matrices, Proc. Amer. Math. Soc., Volume 7 (1956), pp. 1026-1030 | DOI | MR

[18] C.R. Putnam Commutation properties of Hilbert space operators and related topics, Springer, 1967

[19] M. Reed; B. Simon Methods of modern mathematical physics: Vol. 1, Functional Analysis, Academic Press, 1970

[20] M. Reed; B. Simon Methods of modern mathematical physics: Vol. 3, Scattering theory, Academic Press, 1970 | Zbl

[21] P. Rejto; M. Taboada A Limiting Absorption Principle for Schrödinger Operators with Generalized Von Neumann–Wigner Potentials I. Construction of Approximate Phase, Journal of Mathematical Analysis and Applications, Volume 208 (1997) no. 1, pp. 85-108 | DOI | Zbl

[22] P. Rejto; M. Taboada A Limiting Absorption Principle for Schrödinger Operators with Generalized Von Neumann–Wigner Potentials II. The Proof, Journal of Mathematical Analysis and Applications, Volume 208 (1997) no. 2, pp. 311-336 | DOI | Zbl

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