Coherent states for continuous spectrum operators with non-normalizable fiducial states

TL;DR

This paper develops a method to construct coherent states from non-normalizable fiducial states by regularizing divergences, and applies it to systems with continuous spectra like free particles and inverted oscillators.

Contribution

It introduces a regularization approach for building coherent states from non-normalizable states, extending the formalism to continuous spectrum systems.

Findings

Constructed coherent states for free particle and inverted oscillator

Demonstrated regularization technique for non-normalizable fiducial states

Applicable to other systems with non-normalizable states

Abstract

The problem of building coherent states from non-normalizable fiducial states is considered. We propose a way of constructing such coherent states by regularizing the divergence of the fiducial state norm. Then, we successfully apply the formalism to particular cases involving systems with a continuous spectrum: coherent states for the free particle and for the inverted oscillator $(p^2 - x^2)$ are explicitly provided. Similar ideas can be used for other systems having non-normalizable fiducial states.