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Completeness Relation in Renormalized Quantum Systems

Fatih Erman, O. Teoman Turgut

TL;DR

The paper proves that the completeness relation for a discrete-spectrum Schrödinger operator remains valid when a renormalized $\delta$-interaction is added at a point on 2D/3D manifolds or Euclidean space. It employs a Krein-type resolvent formalism, renormalization of the coupling, and pole-interlacing arguments to construct the full Green’s function $G(x,y|E)$ and determine new eigenvalues $E_k^*$ from $\Phi(E)=0$, with explicit expressions for the new eigenfunctions $\psi_k(x)$. Orthogonality and completeness are established via contour integration and residue calculations, yielding $\sum_{n=0}^{\infty} \overline{\psi_n(x)}\psi_n(y) = \delta(x-y)$ and an explicit integral-kernel form for the renormalized Hamiltonian. The work also provides a Sudden Approximation application for time-dependent relocation of the delta center, along with extensions to multi-center and curve-supported interactions, highlighting a practical framework for singular point interactions in quantum systems.

Abstract

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.

Completeness Relation in Renormalized Quantum Systems

TL;DR

The paper proves that the completeness relation for a discrete-spectrum Schrödinger operator remains valid when a renormalized -interaction is added at a point on 2D/3D manifolds or Euclidean space. It employs a Krein-type resolvent formalism, renormalization of the coupling, and pole-interlacing arguments to construct the full Green’s function and determine new eigenvalues from , with explicit expressions for the new eigenfunctions . Orthogonality and completeness are established via contour integration and residue calculations, yielding and an explicit integral-kernel form for the renormalized Hamiltonian. The work also provides a Sudden Approximation application for time-dependent relocation of the delta center, along with extensions to multi-center and curve-supported interactions, highlighting a practical framework for singular point interactions in quantum systems.

Abstract

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 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.
Paper Structure (6 sections, 4 theorems, 53 equations, 3 figures)

This paper contains 6 sections, 4 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Discrete Spectrum Modified by a $\delta$ Interactions
  3. Orthogonality Relation
  4. Completeness Relation
  5. Application: Sudden Approximation in the Case of Time-dependent Center
  6. Acknowledgements

Key Result

Proposition 2.3

Let $\phi_k(x)$ be the eigenfunction of $H_0$ associated with the eigenvalue $E_k$. Then, the (new) energy eigenvalue $E_{k}^{*}$ of $H$, is found from the unique solution of the equation which lies in between $E_{k-1}$ and $E_k$, if $\phi_k(a) \neq 0$ for this particular $k$. If for this particular choice of $k$, we have $\phi_k(a)=0$, the corresponding energy eigenvalue does not change, i.e., $

Figures (3)

  • Figure 1: The contours $\Gamma_n$ along each simple pole $E_{n}^{*}$ with counterclockwise orientation.
  • Figure 2: The contour $\Gamma_{snake}$
  • Figure 3: The contours $\Gamma_{dual}^{n}$ along each simple pole $E_{n}$ with clockwise orientation.

Theorems & Definitions (20)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 4.1
  • ...and 10 more