Energy Levels of Classical Interacting Fields in a Finite Domain in 1+1 Dimension

TL;DR

This paper investigates how bound energy levels of two classical interacting fields in a finite 1+1 dimensional domain depend on system size and parameters, revealing critical sizes where instabilities occur.

Contribution

It provides a detailed analysis of energy eigenvalues and eigenfunctions for interacting fields in a finite domain, highlighting parameter-dependent stability thresholds.

Findings

Existence of critical box sizes for field instability

Dependence of energy levels on system parameters

Identification of stability and instability regimes

Abstract

We study the behavior of bound energy levels for the case of two classical interacting fields $\phi$ and $\chi$ in a finite domain (box) in (1 + 1) dimension on which we impose Dirichlet boundary conditions (DBC). The total Lagrangian contain a $\frac{\lambda}{4}\phi^4$ self-interaction and an interaction term given by $g \phi^2 \chi^2$. We calculate the energy eigenfunctions and its correspondent eigenvalues and study their dependence on the size of the box (L) as well on the free parameters of the Lagrangian: mass ratio $\beta = \frac{M^{2}_{\chi}}{M^{2}_{\phi}}$, and interaction coupling constants $\lambda$ and $g$. We show that for some configurations of the above parameters, there exists critical sizes of the box for which instability points of the field $\chi$ appear.