L. Amour, B. Grébert, and J. Guillot, A mathematical model for the Fermi weak interactions, pp.37-57, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00115559

W. Amrein, A. Boutet-de-monvel, and V. Georgescu, C 0 -groups, commutator methods and spectral theory of N -body Hamiltonians, 1996.

A. Arai, Essential Spectrum of a Self-Adjoint Operator on an Abstract Hilbert Space of Fock Type and Applications to Quantum Field Hamiltonians, Journal of Mathematical Analysis and Applications, vol.246, issue.1, pp.189-216, 2000.
DOI : 10.1006/jmaa.2000.6782

A. Arai and M. Hirokawa, On the Existence and Uniqueness of Ground States of a Generalized Spin-Boson Model, Journal of Functional Analysis, vol.151, issue.2, pp.455-503, 1997.
DOI : 10.1006/jfan.1997.3140

W. Aschbacher, J. Barbaroux, J. Faupin, and J. Guillot, Spectral Theory for a Mathematical Model of the Weak Interaction: The Decay of the Intermediate Vector Bosons W ??, II, Annales Henri Poincar??, vol.38, issue.2
DOI : 10.1007/s002220100159
URL : https://hal.archives-ouvertes.fr/hal-01291782

V. Bach, J. Fröhlich, and A. Pizzo, Infrared-Finite Algorithms in QED: The Groundstate of an Atom Interacting with the Quantized Radiation Field, Communications in Mathematical Physics, vol.264, issue.1, pp.145-165, 2006.
DOI : 10.1007/s00220-005-1478-3

V. Bach, J. Fröhlich, and I. M. Sigal, Renormalization Group Analysis of Spectral Problems in Quantum Field Theory, Advances in Mathematics, vol.137, issue.2, pp.205-298, 1998.
DOI : 10.1006/aima.1998.1733

J. Barbaroux, J. Faupin, and J. , Guillot Spectral properties for Hamiltonians of weak interactions , Operator Theory: Advances and Applications, pp.11-36, 2016.

J. Barbaroux, J. Faupin, and J. , Spectral theory near thresholds for weak interactions with massive particles, Journal of Spectral Theory, vol.6, issue.3, pp.505-555, 2016.
DOI : 10.4171/JST/131
URL : https://hal.archives-ouvertes.fr/hal-00957524

J. Barbaroux and J. Guillot, Limiting absorption principle at low energies for a mathematical model of weak interaction: The decay of a boson, Comptes Rendus Mathematique, vol.347, issue.17-18, pp.17-18, 2009.
DOI : 10.1016/j.crma.2009.07.014

J. Barbaroux, M. Dimassi, and J. Guillot, Quantum electrodynamics of relativistic bound states with cutoffs. II, Mathematical Results in Quantum Mechanics, pp.9-14, 2002.
DOI : 10.1090/conm/307/05262
URL : https://hal.archives-ouvertes.fr/hal-00008352

J. Barbaroux, M. Dimassi, and J. Guillot, QUANTUM ELECTRODYNAMICS OF RELATIVISTIC BOUND STATES WITH CUTOFFS, Journal of Hyperbolic Differential Equations, vol.9, issue.02, pp.271-314, 2004.
DOI : 10.1017/CBO9781139644167
URL : https://hal.archives-ouvertes.fr/hal-00008352

J. Bony and J. Faupin, Resolvent smoothness and local decay at low energies for the standard model of non-relativistic QED, Journal of Functional Analysis, vol.262, issue.3, pp.850-888, 2012.
DOI : 10.1016/j.jfa.2011.10.006
URL : https://hal.archives-ouvertes.fr/hal-00994365

J. Bony and D. Häfner, Low frequency resolvent estimates for long range perturbations of the euclidean Laplacian, Mathematical Research Letters, vol.17, issue.2, pp.303-308, 2010.
DOI : 10.4310/MRL.2010.v17.n2.a9
URL : https://hal.archives-ouvertes.fr/hal-00384726

J. Bony and D. Häfner, Local energy decay for several evolution equations on asymptotically euclidean manifolds, Annales scientifiques de l'??cole normale sup??rieure, vol.45, issue.2, pp.311-335, 2012.
DOI : 10.24033/asens.2166
URL : https://hal.archives-ouvertes.fr/hal-00994364

T. Chen, J. Faupin, J. Fröhlich, and I. M. Sigal, Local Decay in Non-Relativistic QED, Communications in Mathematical Physics, vol.4, issue.3, pp.543-583, 2012.
DOI : 10.1017/CBO9780511535178
URL : https://hal.archives-ouvertes.fr/hal-01279968

J. Derezi´nskiderezi´nski and C. Gérard, ASYMPTOTIC COMPLETENESS IN QUANTUM IN FIELD THEORY: MASSIVE PAULI???FIERZ HAMILTONIANS, Reviews in Mathematical Physics, vol.42, issue.04, pp.383-450, 1999.
DOI : 10.1007/BF02102107

J. Derezi´nskiderezi´nski and C. Gérard, Spectral and Scattering Theory of Spatially Cut-Off P (?)

J. Faupin, J. S. Møller, and E. Skibsted, Second Order Perturbation Theory for Embedded Eigenvalues, Communications in Mathematical Physics, vol.10, issue.1, pp.193-228, 2011.
DOI : 10.1142/S0129055X9800032X
URL : https://hal.archives-ouvertes.fr/hal-01279964

J. Fröhlich, M. Griesemer, and I. M. Sigal, Spectral Theory for the Standard Model of Non-Relativistic QED, Communications in Mathematical Physics, vol.38, issue.5, pp.613-646, 2008.
DOI : 10.1017/CBO9780511535178

V. Georgescu and C. Gérard, On the Virial Theorem in Quantum Mechanics, Communications in Mathematical Physics, vol.208, issue.2, pp.275-281, 1999.
DOI : 10.1007/s002200050758

V. Georgescu, C. Gérard, and J. S. Møller, Commutators, C0-semigroups and resolvent estimates, Journal of Functional Analysis, vol.216, issue.2, pp.303-361, 2004.
DOI : 10.1016/j.jfa.2004.03.004
URL : https://doi.org/10.1016/j.jfa.2004.03.004

V. Georgescu, C. Gérard, and J. S. Møller, Spectral Theory of Massless Pauli-Fierz Models, Communications in Mathematical Physics, vol.38, issue.1, pp.29-78, 2004.
DOI : 10.1063/1.531974
URL : https://hal.archives-ouvertes.fr/hal-00096143

J. Glimm and A. Jaffe, Quantum field theory and statistical mechanics, Birkhuser Boston Inc. Expositions, 1977.

W. Greiner, Relativistic quantum mechanics. Wave equations, 2000.
DOI : 10.1007/978-3-662-04275-5

W. Greiner and B. Müller, Gauge Theory of Weak Interactions, 2000.

J. Guillot, Spectral theory of a mathematical model in quantum field theory for any spin, Contemp. Math, vol.640, pp.13-37, 2015.
DOI : 10.1090/conm/640/12842

W. Hunziker, I. M. Sigal, and A. Soffer, Minimal escape velocities, Communications in Partial Differential Equations, vol.61, issue.11-12, pp.11-12, 1999.
DOI : 10.1007/BF02099172
URL : http://arxiv.org/pdf/math-ph/0002013

D. Jackson, Fourier Series and Orthogonal Polynomials, Carus Monograph Series, 1971.
DOI : 10.5948/upo9781614440062

D. Kastler, IntroductionàIntroductionà l'Electrodynamique Quantique, 1960.

T. Kato, Perturbation Theory for Linear Operators [35] ´ E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, New York Inc. Comm. Math. Phys, vol.78, pp.81-391, 1966.
DOI : 10.1007/978-3-662-12678-3

A. Pizzo, One-particle (improper) States in Nelson???s Massless Model, Annales Henri Poincar??, vol.4, issue.3, pp.439-486, 2003.
DOI : 10.1007/s00023-003-0136-6
URL : https://link.springer.com/content/pdf/10.1007%2Fs00023-003-0136-6.pdf

M. Reed and B. Simon, Methods of modern mathematical physics, I and II, 1972.

S. Schweber, An Introduction to Relativistic Quantum Field Theory, 1961.
DOI : 10.1119/1.1942132

T. Takaesu, Essential spectrum of a fermionic quantum field model, Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol.19, issue.04, p.1450024, 2014.
DOI : 10.1007/978-3-662-02753-0

B. Thaller, The Dirac Equation. Texts and Monographs in Physics, 1992.

S. Weinberg, The Quantum Theory of Fields Vol. I, 2005.

S. Weinberg, The Quantum theory of fields, II, 2005.

J. Barbaroux, C. Cpt, and F. Marseille, E-mail address: barbarou@univ-tln.fr Jérémy Faupin, Institut Elie Cartan de Lorraine, France. E-mail address: jeremy.faupin@univ-lorraine