Abstract
Let \(H_{P,\sigma }\) be the single-electron fiber Hamiltonians of the massless Nelson model at total momentum P and infrared cut-off \(\sigma >0\). We establish detailed regularity properties of the corresponding n-particle ground state wave functions \(f^n_{P,\sigma }\) as functions of P and \(\sigma \). In particular, we show that
where c is a numerical constant, \(\lambda _0\mapsto \delta _{\lambda _0}\) is a positive function of the maximal admissible coupling constant which satisfies \(\lim _{\lambda _0\rightarrow 0}\delta _{\lambda _0}=0\) and \(\chi _{[\sigma ,\kappa )}\) is the (approximate) characteristic function of the energy region between the infrared cut-off \(\sigma \) and the ultraviolet cut-off \(\kappa \). While the analysis of the first derivative is relatively straightforward, the second derivative requires a new strategy. By solving a non-commutative recurrence relation, we derive a novel formula for \(f^n_{P,\sigma }\) with improved infrared properties. In this representation \(\partial _{P^{j'}}\partial _{P^{j}}f^n_{P,\sigma }\) is amenable to sharp estimates obtained by iterative analytic perturbation theory in part II of this series of papers. The bounds stated above are instrumental for scattering theory of two electrons in the Nelson model, as explained in part I of this series.
Similar content being viewed by others
References
Abdesselam, A., Hasler, D.: Analyticity of the ground state energy for massless Nelson models. Commun. Math. Phys. 310, 511–536 (2012)
Dybalski, W., Pizzo, A.: Coulomb scattering in the massless Nelson model I. Foundations of two-electron scattering. J. Stat. Phys. 154, 543–587 (2014)
Dybalski, W., Pizzo, A.: Coulomb scattering in the massless Nelson model II. Regularity of ground states. Preprint arXiv:1302.5012v2
Faupin, J., Fröhlich, J., Schubnel, B.: Analyticity of the self-energy in total momentum of an atom coupled to the quantized radiation field. J. Funct. Anal. 267, 4139–4196 (2014)
Fröhlich, J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. H. Poincaré Sect. A (N.S.) 19, 1–103 (1973)
Fröhlich, J.: Unpublished notes
Fröhlich, J.: Existence of dressed one electron states in a class of persistent models. Fortschr. Phys. 22, 158–198 (1974)
Fröhlich, J., Pizzo, A.: Renormalized electron mass in non-relativistic QED. Commun. Math. Phys. 294, 439–470 (2010)
Jivulescu, M.A., Messina, A., Napoli, A., Petruccione, F.: Exact treatment of linear difference equations with noncommutative coefficients. Math. Methods Appl. Sci. 30, 2147–2153 (2007)
Jivulescu, M.A., Messina, A., Napoli, A.: General solution of a second order non-homogeneous linear difference equation with non-commutative coefficients. Appl. Math. Inf. Sci. 4, 1–14 (2010)
Könenberg, M., Matte, O.: The mass-shell in the semi-relativistic Pauli–Fierz model. Ann. Henri Poincaré 15, 863–915 (2014)
Pizzo, A.: One-particle (improper) states in Nelson’s massless model. Ann. Henri Poincaré 4, 439–486 (2003)
Puhlfürst, G.: Non-commutative recurrence relations for scattering amplitudes. Ph.D. Thesis, Ludwig-Maximilians-Universität München (2015)
Strominger, A.: Lectures on the infrared structure of gravity and gauge theory. Preprint arXiv:1703.05448
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Claude Alain Pillet.
Rights and permissions
About this article
Cite this article
Dybalski, W., Pizzo, A. Coulomb Scattering in the Massless Nelson Model III: Ground State Wave Functions and Non-commutative Recurrence Relations. Ann. Henri Poincaré 19, 463–514 (2018). https://doi.org/10.1007/s00023-017-0642-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-017-0642-6