Casimir effect with an unstable mode

TL;DR

This paper explores the Casimir effect in a (1+1)-dimensional model with a critical mode, analyzing how condensate and fluctuation forces contribute and compete, with a focus on elliptic function solutions and force characteristics.

Contribution

It introduces models with condensate solutions described by elliptic functions and investigates the interplay of condensate and fluctuation forces in the Casimir effect.

Findings

Condensate force is always repulsive.

Two sources of Casimir force are identified: from fluctuations and the condensate.

Method for approximate solutions is developed and its applicability is demonstrated.

Abstract

We consider the Casimir effect in a (1+1)-dimensional model with a critical mode. Such a mode gives rise to a condensate described by the nonlinear Gross-Pitaevskii equation. In the condensate, there are two sources of the Casimir force; one is the conventional one resulting from the fluctuations, the other follows from the condensate. We consider three simple models that allow for condensate solutions in terms of elliptic Jacobi functions. We also investigate a method for obtaining approximate solutions and show its range of applicability. In all three examples we compute the condensate energy. In one example with a finite interval with Robin boundary conditions on one side and Dirichlet conditions on the other side, we calculate the vacuum energy and the Casimir force. There is a competition between the forces from the condensate and the fluctuations. We mention that the force from the condensate is always repulsive.