The Casimir Energy in a Separable Potential

TL;DR

This paper investigates the Casimir energy in a simplified model using a separable potential, demonstrating the equivalence of two calculation methods and analyzing the convergence of approximations.

Contribution

It introduces a toy model with a separable potential to examine Casimir energy calculations and compares two traditional methods, highlighting their correctness and convergence properties.

Findings

Both Greens function and phase shift methods are correct and equivalent in the model.

The convergence of the Born approximation is studied and related to computational challenges.

The model provides insights applicable to more complex systems.

Abstract

The Casimir energy is the first-order-in-\hbar correction to the energy of a time-independent field configuration in a quantum field theory. We study the Casimir energy in a toy model, where the classical field is replaced by a separable potential. In this model the exact answer is trivial to compute, making it a good place to examine subtleties of the problem. We construct two traditional representations of the Casimir energy, one from the Greens function, the other from the phase shifts, and apply them to this case. We show that the two representations are correct and equivalent in this model. We study the convergence of the Born approximation to the Casimir energy and relate our findings to computational issues that arise in more realistic models.