We consider expansions of eigenvalues and eigenvectors of models of quantum field theory. For a class of models known as generalized spin–boson model, we prove the existence of asymptotic expansions of the ground state and the ground state energy to arbitrary order. We need a mild but very natural infrared assumption, which is weaker than the assumption usually needed for other methods such as operator theoretic renormalization to be applicable. The result complements previously shown analyticity properties.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Abdesselam, A.: The ground state energy of the massless spin-boson model. Ann. Henri Poincaré 12(7), 1321–1347 (2011)
Abdesselam, A., Hasler, D.: Analyticity of the ground state energy for massless Nelson models. Commun. Math. Phys. 310(2), 511–536 (2012)
Arai, A.: A new asymptotic perturbation theory with applications to models of massless quantum fields. Ann. Henri Poincaré 15(6), 1145–1170 (2014)
Bach, V., Chen, T., Fröhlich, J., Sigal, I.M.: Smooth Feshbach map and operator-theoretic renormalization group methods. J. Funct. Anal. 203(1), 44–92 (2003)
Bach, V., Fröhlich, J., Pizzo, A.: Infrared-finite algorithms in QED: the groundstate of an atom interacting with the quantized radiation field. Commun. Math. Phys. 264(1), 145–165 (2006)
Bach, V., Fröhlich, J., Pizzo, A.: Infrared-finite algorithms in QED. II. The expansion of the groundstate of an atom interacting with the quantized radiation field. Adv. Math. 220(4), 1023–1074 (2009)
Bach, V., Fröhlich, J., Sigal, I.M.: Quantum electrodynamics of confined nonrelativistic particles. Adv. Math. 137(2), 299–395 (1998)
Bach, V., Fröhlich, J., Sigal, I.M.: Renormalization group analysis of spectral problems in quantum field theory. Adv. Math. 137(2), 205–298 (1998)
Bach, V., Fröhlich, J., Sigal, I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207(2), 249–290 (1999)
Barbaroux, J.-M., Chen, T., Vougalter, V., Vugalter, S.: On the ground state energy of the translation invariant Pauli–Fierz model. Proc. Am. Math. Soc. 136(3), 1057–1064 (2008)
Barbaroux, J.-M., Chen, T., Vougalter, V., Vugalter, S.: Quantitative estimates on the binding energy for hydrogen in non-relativistic QED. Ann. Henri Poincaré 11(8), 1487–1544 (2010)
Barbaroux, J.-M., Chen, T., Vugalter, S.: Binding conditions for atomic \(N\)-electron systems in non-relativistic QED. Ann. Henri Poincaré 4(6), 1101–1136 (2003)
Catto, I., Hainzl, C.: Self-energy of one electron in non-relativistic QED. J. Funct. Anal. 207(1), 68–110 (2004)
Fröhlich, J.: Existence of dressed one electron states in a class of persistent models. Fortschritte der Physik 22(3), 159–198 (1974)
Gérard, C.: On the existence of ground states for massless Pauli–Fierz Hamiltonians. Ann. Henri Poincaré 1(3), 443–459 (2000)
Griesemer, M., Hasler, D.G.: Analytic perturbation theory and renormalization analysis of matter coupled to quantized radiation. Ann. Henri Poincaré 10(3), 577–621 (2009)
Griesemer, M., Lieb, E.H., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145(3), 557–595 (2001)
Hainzl, C., Seiringer, R.: Mass renormalization and energy level shift in non-relativistic QED. Adv. Theor. Math. Phys. 6(5), 847–871 (2002)
Hasler, D., Herbst, I.: Convergent expansions in non-relativistic QED: analyticity of the ground state. J. Funct. Anal. 261(11), 3119–3154 (2011)
Hasler, D., Herbst, I.: Smoothness and analyticity of perturbation expansions in QED. Adv. Math. 228(6), 3249–3299 (2011)
Hasler, D., Herbst, I.: Ground states in the spin boson model. Ann. Henri Poincaré 12(4), 621–677 (2011)
Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 edition
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1978)
Spohn, H.: Ground state of a quantum particle coupled to a scalar Bose field. Lett. Math. Phys. 44(1), 9–16 (1998)
Communicated by Abdelmalek Abdesselam.
About this article
Cite this article
Bräunlich, G., Hasler, D. & Lange, M. On Asymptotic Expansions in Spin–Boson Models. Ann. Henri Poincaré 19, 515–564 (2018). https://doi.org/10.1007/s00023-017-0625-7