Abstract
Let \(H:\text {dom}(H)\subseteq \mathfrak {F}\to \mathfrak {F}\) be self-adjoint and let \(A:\text {dom}(H)\to \mathfrak {F}\) (playing the role of the annihilation operator) be H-bounded. Assuming some additional hypotheses on A (so that the creation operator A∗ is a singular perturbation of H), by a twofold application of a resolvent Kreı̆n-type formula, we build self-adjoint realizations \(\widehat H\) of the formal Hamiltonian H + A∗ + A with \(\text {dom}(H)\cap \text {dom}(\widehat H)=\{0\}\). We give an explicit characterization of \(\text {dom}(\widehat H)\) and provide a formula for the resolvent difference \((-\widehat H+z)^{-1}-(-H+z)^{-1}\). Moreover, we consider the problem of the description of \(\widehat H\) as a (norm resolvent) limit of sequences of the kind \(H+A^{*}_{n}+A_{n}+E_{n}\), where the An’s are regularized operators approximating A and the En’s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Kreı̆n’s resolvent formula and nonperturbative theory of renormalizable models in Quantum Field Theory; in particular, as an explicit example, we consider the Nelson model.
Article PDF
Similar content being viewed by others
References
Arai, A.: Analysis on Fock Spaces and Mathematical Theory of Quantum Fields. An Introduction to Mathematical Analysis of Quantum Fields. World Scientific, Singapore (2018)
Behrndt, J., Hassi, S., De Snoo, H.: Boundary Value Problems, Weyl Functions, and Differential Operators. Basel, Birkhäuser (2020)
Behrndt, J., Langer, M.: Dirichlet-to-Neumann Maps and Quasi Boundary Triples. In: Operator Methods for Boundary Value Problems. Cambridge Univ. Press, Cambridge, pp 121–160 (2012)
Cacciapuoti, C., Fermi, D., Posilicano, A.: On inverses of Kreı̆n’s \(\mathscr Q\)-functions. Rend. Mat. Appl. 39, 229–240 (2018)
Derkach, V., Hassi, S., Malamud, M., De Snoo, H.: Boundary Triplets and Weyl Functions. Recent Developments. In: Operator Methods for Boundary Value Problems. Cambridge Univ. Press, Cambridge, pp 161–220 (2012)
Griesemer, M., Linden, U.: Spectral theory of the Fermi polaron. Ann. Henri Poincaré 20, 1931–1967 (2019)
Griesemer, M., Wünsch, A.: Self-adjointness and domain of the fröhlich Hamiltonian. J. Math. Phys. 57(021902), 15 (2016)
Griesemer, M., Wünsch, A.: On the domain of the Nelson Hamiltonian. J. Math. Phys. 59(042111), 21 (2018)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976)
Kreı̆n, S.G., Petunin, Yu.I.: Scales of Banach spaces. Russ. Math. Surv. 21, 85–159 (1966)
Lampart, J.: A nonrelativistic quantum field theory with point interactions in three dimensions. Ann. Henri Poincaré 20, 3509–3541 (2019)
Lampart, J.: The Renormalised Bogoliubov-Fröhlich Hamiltonian. arXiv:1909.02430 (2019)
Lampart, J., Schmidt, J.: On Nelson-type Hamiltonians and abstract boundary conditions. Comm. Math. Phys. 367, 629–663 (2019)
Lampart, J., Schmidt, J., Teufel, S., Tumulka, R.: Particle creation at a point source by means of interior-boundary conditions. Math. Phys. Anal. Geom. 21(12), 37 (2018)
Mantile, A., Posilicano, A., Sini, M.: Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces. J. Differential Equations 261, 1–55 (2016)
Mantile, A., Posilicano, A.: Asymptotic completeness and S-Matrix for singular perturbations. J. Math. Pures Appl. 130, 36–67 (2019)
Moshinsky, M.: Boundary conditions for the description of nuclear reactions. Phys. Rev. 81, 347–352 (1951)
Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964)
Posilicano, A.: A Kreı̆n-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal. 183, 109–147 (2001)
Posilicano, A.: Self-adjoint extensions by additive perturbations. Ann. Sc. Norm. Super. Pisa Cl. Sci.(V) 2, 1–20 (2003)
Posilicano, A.: Boundary triples and Weyl functions for singular perturbations of self-adjoint operators. Methods Funct. Anal. Topology 10, 57–63 (2004)
Posilicano, A: Self-adjoint extensions of restrictions. Oper. Matrices 2, 483–506 (2008)
Schmidt, J.: On a direct description of pseudorelativistic Nelson Hamiltonians. J. Math. Phys. 60(102303), 21 (2019)
Schmidt, J.: The Massless Nelson Hamiltonian and its Domain. arXiv:1901.05751 (2019)
Spohn, H: Dynamics of Charged Particles and Their Radiation Field. Cambridge University Press, Cambridge (2004)
Stone, M.H.: Linear transformations in Hilbert space. American Mathematical Society New York (1932)
Thomas, L.E.: Multiparticle Schrödinger Hamiltonians with point interactions. Phys. Rev. D 30, 1233–1237 (1984)
Tretter, C.: Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London (2008)
Yafaev, D.R.: On a zero-range interaction of a quantum particle with the vacuum. J. Phys. A: Math. Gen. 25, 963–978 (1992)
Acknowledgments
The author thanks Jonas Lampart for some useful explanations, stimulating comments and bibliographic remarks.
Funding
Open access funding provided by Università degli Studi dell’Insubria within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Posilicano, A. On the Self-Adjointness of H+A∗+A. Math Phys Anal Geom 23, 37 (2020). https://doi.org/10.1007/s11040-020-09359-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-020-09359-x