Completeness Relation in Renormalized Quantum Systems

TL;DR

This paper proves that the fundamental completeness relation for eigenvectors in quantum mechanics remains valid even after renormalization of Hamiltonians with delta potentials in various geometries, with applications to sudden perturbations.

Contribution

It demonstrates the preservation of the completeness relation under renormalization of Hamiltonians with delta potentials in multiple spatial configurations.

Findings

Completeness relation holds for renormalized Hamiltonians with delta potentials.

Extension to multiple centers and delta interactions on curves is possible.

Application to sudden perturbations of delta potential support is provided.

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.