Reference of this post (page no 6 from equation 39)
The time averaged total energy, $\bar E$, has the following $\varepsilon$ expansion in $D$ dimension: \begin{equation} \bar{E}=\varepsilon^{2-D}\frac{E_0}{2\lambda}+ \varepsilon^{4-D}E_1\,, \quad {\rm where}\quad E_0=\int d^{D}\zeta\,S^2\,, \tag{1} \end{equation} In $D=2$ $\bar E$ tends to a constant, and for $D<2$ it goes to zero as $\varepsilon\to 0$. This also implies that the core energy of a QB in dimensions $D>2$ should exhibit a minimum for some frequency $\omega_{\rm m}$. In fact, from Eq.1 one immediately finds \begin{equation} \omega_{\rm m}^2=1-\varepsilon_{\rm m}^2= > 1-\frac{1}{2\lambda}\frac{(D-2)E_0}{(4-D)E_1}\,. \end{equation} The above result can only be taken as an indication of the minimum even if $\varepsilon_{\rm m}\ll1$. The numerical values of $E_0$ and $E_1$ will be given for the fundamental solutions in $D=2$ and $D=3$ in case of spherical symmetry in Section iii.
The problem I got from the above expresson is, how $D<2$ as as $\varepsilon\to 0$? and how will I explain the limit of dimension of this equation? I mean after what valoue of D the above equation would notwork.
Please ask if any thing needed to explain. Thanks in advance.