اگر راهنمای بسته amsmath را ببینید چنین نوشته:
Note: Certain equation environments wrap their contents in an unbreakable box, with the consequence that neither \displaybreak nor \allowdisplaybreaks will have any effect on them. These include split, aligned, gathered, and alignedat.
بنابراین \allowdisplaybreaks روی split داخل equation تأثیری نخواهید داشت. اگر از محیط align به همراه \notag استفاده کنید مشکل برطرف خواهد شد.
کد کامل:
\documentclass[aip,jmp,amsmath,amssymb,preprint,reprint,author-year,author-numerical]{revtex4-1}
\usepackage{graphicx}
\usepackage{dcolumn}
\usepackage{ptext}
\usepackage{amsmath}
\usepackage{bm}
\usepackage{subfigure}
\allowdisplaybreaks[1]
\begin{document}
\preprint{AIP/123-QED}
\begin{align}
&E_z^{(4)}(r,\theta)=\sum_{n=1,3,5}^\infty{A_n^{(4)}}\sin(n\theta){K_n}(h_4{r}) \notag \\
&+\sum_{n=0,2,4}^\infty{B_n^{(4)}}\cos(n\theta){K_n}(h_4{r})+\sum_{n=1,3,5}^\infty{S_n^{(4)}}\sin(n\theta){I_n}(h_4{r}) \notag \\
&+\sum_{n=0,2,4}^\infty{T_n^{(4)}}\cos(n\theta){I_n}(h_4{r}) \notag \\
&B_z^{(4)}(r,\theta)=\sum_{n=2,4,6}^\infty{C_n^{(4)}}\sin(n\theta){K_n}(h_4{r}) \notag \\
&+\sum_{n=1,3,5}^\infty{D_n^{(4)}}\cos(n\theta){K_n}(h_4{r})+\sum_{n=2,4,6}^\infty{U_n^{(4)}}\sin(n\theta){I_n}(h_4{r}) \notag \\
&+\sum_{n=1,3,5}^\infty{V_n^{(4)}}\cos(n\theta){I_n}(h_4{r}) \notag \\
\end{align}
Taking into account the Eqs.(13-14) the continuity of $E_t$ for a certain point $(r_m,\theta_m)$ on the boundaries are written as
\begin{align}
&E_t^{(1)}(r_m,\theta_m)= \notag \\
&\frac{ik_z}{h_1^2}\bigg\lbrace\sum_{n=1,3,5}^\infty{A_n^{(1)}}[h_1f(\theta_m)\sin(n\theta_m)J'_n(h_1{r_m})+ \notag \\
&(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)J_n(h_1{r_m})] \notag \\
&+\sum_{n=0,2,4}^\infty{B_n^{(1)}}[h_1f(\theta_m)\cos(n\theta_m)J'_n(h_1{r_m}) \notag \\
&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)J_n(h_1{r_m})] \notag \\
&+\sum_{n=1,3,5}^\infty{S_n^{(1)}}[h_1f(\theta_m)\sin(n\theta_m)N'_n(h_1{r_m})+ \notag \\
&(\frac{n}{r_m})g(\theta_m)\cos(n\theta_m)N_n(h_1{r_m})] \notag \\
&+\sum_{n=0,2,4}^\infty{T_n^{(1)}}[h_1f(\theta_m)\cos(n\theta_m)N'_n(h_1{r_m}) \notag \\
&-(\frac{n}{r_m})g(\theta_m)\sin(n\theta_m)N_n(h_1{r_m})] \notag \\
&+\sum_{n=2,4,6}^\infty(\frac{\omega}{k_zc}){C_n^{(1)}}[(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)J_n(h_1{r_m}) \notag \\
&-h_1g(\theta_m)\sin(n\theta_m)J'_n(h_1{r_m})] \notag \\
&+\sum_{n=1,3,5}^\infty(\frac{\omega}{k_zc}){D_n^{(1)}}[(-\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)J_n(h_1{r_m}) \notag \\
&-h_1g(\theta_m)\cos(n\theta_m)J'_n(h_1{r_m}] \notag \\
&+\sum_{n=2,4,6}^\infty(\frac{\omega}{k_zc}){U_n^{(1)}}[(\frac{n}{r_m})f(\theta_m)\cos(n\theta_m)N_n(h_1{r_m}) \notag \\
&-h_1g(\theta_m)\sin(n\theta_m)N'_n(h_1{r_m})] \notag \\
&+\sum_{n=1,3,5}^\infty(\frac{\omega}{k_zc}){V_n^{(1)}}[(-\frac{n}{r_m})f(\theta_m)\sin(n\theta_m)N_n(h_1{r_m}) \notag \\
&-h_1g(\theta_m)\cos(n\theta_m)N'_n(h_1{r_m}]\bigg\rbrace \notag \\
\end{align}
%\begin{thebibliography}{}
%\end{thebibliography}
\end{document}
نتیجه:
نکته: به خاطر محدودیت در کاراکترهای جواب مجبور شدم قسمتی از کد را حذف کنم.
