diff --git a/Chapter-4/Chapter-4.tex b/Chapter-4/Chapter-4.tex
index 1882999..0cf5bee 100644
--- a/Chapter-4/Chapter-4.tex
+++ b/Chapter-4/Chapter-4.tex
@@ -1,4 +1,117 @@
 \chapter{Resonances}
 \label{chap:Resonances}
 
-[\ldots]
+\section{Zeldovich regularization}
+\label{sec:zeldovich}
+
+As an alternative to complex scaling, here we discuss Zeldovich regularization,
+where the known asymptotic forms of wave functions in configuration
+space are utilized to define a regularized inner-product.
+Using this, we can implement CA-EC in configuration space, without the complications
+of complex-scaled contours. As $\Ip(p)<0$ for resonance states,
+the wave function is ever increasing and not square integrable.
+They are said to belong in a ``Rigged Hilbert Space''. Inner products and
+normalization factors of Gamow states can be calculated by inserting a regularization
+factor inside integrals which are otherwise divergent \cite{delaMadrid:2008sr},
+allowing us to work directly in the RHS.
+
+Let,
+%
+\begin{equation}
+    \label{eq:zeldovich}
+    \braket{\phi_1}{\phi_2}
+    =\lim_{\mu\to 0}\int_{0}^{\infty}e^{-\mu r^2}\phi_1(r)\phi_2(r)\,dr
+\end{equation}
+%
+Now assume that after some $r>R$, the wave functions becomes approximately equal to
+their asymptotic forms, $N_1 e^{ip_1r}$ and $N_2 e^{ip_2r}$ respectively. Then the
+integral can be broken into two.
+%
+\begin{spliteq}
+    \braket{\phi_1}{\phi_2} 
+    & \approx \lim_{\mu\to 0}\int_{0}^{R}e^{-\mu r^2}\phi_1(r)\phi_2(r)\,dr \\
+    & \quad +N_1N_2\lim_{\mu\to 0}
+    \int_{R}^{\infty}e^{-\mu r^2}e^{ip_1r}e^{ip_2r}\,dr \\
+\end{spliteq}
+%
+where $N_{1,2}$ can be determined from $N=\phi(r)/e^{ipr}$ for any $r>R$. Using
+results from \cite{kukulin1989theory} for the second term, and taking the limit in
+the first term, this simplifies as follows.
+%
+\begin{equation}
+    \label{eq:easy}
+    \braket{\phi_1}{\phi_2} \approx \int_{0}^{R}\phi_1(r)\phi_2(r)\,dr
+    +\frac{\ii N_1 N_2 e^{\ii(p_1+p_2)R}}{p_1+p_2}
+\end{equation}
+%
+If $\phi_1(r)$ and $\phi_2(r)$ are solutions to finite-ranged potentials with ranges
+$a_1$ and $a_2$, this result can be made exact by taking $R=\max\{a_1,a_2\}$.
+For other potentials, a sufficiently large $R$ has to be considered such that
+$V(r)\approx 0$ for all $r>R$.
+
+Now, we have all the necessary tools for performing Galerkin projection on to the
+reduced basis, to arrive at a smaller generalized eigenvalue problem. Rest of
+the procedure would continue similarly.
+
+\subsection{RHA-EC for Zeldovich method}
+
+RHA-EC is straightforward in this configuration space and we can directly construct
+free wave functions of the form $e^{\ii p r}$ and include them in the basis.
+
+\subsection{CA-EC for Zeldovich method}
+
+\ny{I need to check this math again. For now, please ignore this section.}
+
+There will be no saving of additional memory usage by implementing CA-EC in
+un-complex-scaled bases. Since RHA-EC already works better than CA-EC and is much
+easier to implement, there is no reason to go with CA-EC,
+other than for sanity checks, for which this section is dedicated.
+
+\begin{lemma}
+    For bound states, complex conjugation is equivalent to analytically rotating
+    the function by an angle $2\phi$.
+\end{lemma}
+
+Consider a bound state training point $\ket{\psi}$.
+%
+\begin{equation}
+    \ket{\psi}=\int_{0}^{\infty} dp\,p^2\, u(p)\ket{p}
+\end{equation}
+%
+Now complex-scale the integration contour by an angle $-\phi$ (i.e., $\phi$ in the
+clockwise direction).
+%
+\begin{equation}
+    \ket{\psi}=\int_{-\phi} dp\,p^2\, u(p)\ket{p}
+\end{equation}
+%
+This $u(p)$ is the reduced radial wave function that we obtain when we solve for
+eigenstates with complex scaling. Now, consider the complex-conjugated wave function
+on this contour.
+%
+\begin{equation}
+    \ket{\Tilde{\psi}}=\int_{-\phi} dp\,p^2\, u^*(p)\ket{p}
+\end{equation}
+%
+Since $\ket{\psi}$ is a bound state, $u(p)$ is real on the positive real axis.
+Therefore, Schwarz reflection principle can be invoked to say $u^*(p)=u(p^*)$.
+%
+\begin{spliteq}
+    \ket{\Tilde{\psi}}&=\int_{-\phi} dp\,p^2\, u(p^*)\ket{p}\\
+    &=\int_{-\phi} dp\,p^2\, u(pe^{2i\phi})\ket{p}
+\end{spliteq}
+%
+Since $u(pe^{2\ii\phi})$ is an analytic function, we can complex-scale the
+contour back to the positive real axis.
+%
+\begin{equation}
+    \ket{\Tilde{\psi}}=\int_{0}^{\infty} dp\,p^2\, u(pe^{2\ii\phi})\ket{p}
+\end{equation}
+%
+The end result $\ket{\Tilde{\psi}}$ is simply the original state $\ket{\psi}$
+rotated by an angle $2\phi$ on the complex $p$-plane.
+
+Therefore, the equivalent of complex-conjugated vectors for the Zeldovich method
+can be obtained by solving the Schr\"odinger's equation along a complex-scaled contour
+($re^{-2\ii\phi}$) for some arbitrary angle $\phi$. As mentioned above, this method
+is very inefficient in the Zeldovich implementation compared to RHA-EC.