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

Abstract

We study the behavior of bound energy levels for the case of two classical interacting fields $\varphi$ and $\chi$ in a finite domain (box) in (1 + 1) dimension on which we impose Dirichlet boundary conditions (DBC). The total Lagrangian contain a $\frac{\lambda }{4}{\varphi }^4$ self-interaction and an interaction term given by $g{\varphi }^2{\chi }^2$. 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 $\beta =\frac{M_{\chi }^2}{M_{\varphi }^2}$, 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.

arXiv comment20 pages, LaTex, 4 ps-figs; revised version; few small parts changed

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