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