ORIGINAL RESEARCH article
Front. Phys.
Sec. Interdisciplinary Physics
Volume 13 - 2025 | doi: 10.3389/fphy.2025.1695365
This article is part of the Research TopicRecent Mathematical and Theoretical Progress in Quantum MechanicsView all 8 articles
Completeness Relation in Renormalized Quantum Systems
Provisionally accepted
Fatih Erman1*
Osman Teoman Turgut2- 1Izmir Institute of Technology, Urla, Türkiye
- 2Bogazici Universitesi, Istanbul, Türkiye
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In this work, we show that the completeness relation for the eigenvectors, which is an essential assumption of quantum mechanics, remains true if the Hamiltonian, having a discrete spectrum, is modified by a delta potential (to be made precise by a renormalization scheme) supported at a point in two and three-dimensional compact manifolds or Euclidean spaces. The formulation can be easily extended to N center case, and the case where delta interaction is supported on curves in the plane or space. We finally give an interesting application for sudden perturbation of the support of the delta potential.
Keywords: Completeness relation, Dirac δ interactions, point interactions, Green's function, renormalization, Schr¨odinger operators, Resolvent, Compact manifolds
Received: 29 Aug 2025; Accepted: 02 Oct 2025.
Copyright: © 2025 Erman and Turgut. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
* Correspondence: Fatih Erman, fatih.erman@gmail.com
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