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@ -7,8 +7,16 @@ A summary of all acronyms is documented in Table \ref{tab:acro}.
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\hline
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\hline
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Acronym & Abbreviation \\
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Acronym & Abbreviation \\
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\hline
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\hline
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AC1 & Acronym 1 \\
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DVR & Discrete variable representation \\
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AC2 & Acronym 2 \\
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EC & Eigenvector continuation \\
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FFT & Fast Fourier transform \\
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FVEC & Finite-volume eigenvector continuation \\
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GEVP & Generalized eigenvalue problem \\
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GPU & Graphics processing unit \\
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IRAM & Implicitly restarted Arnoldi method \\
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MOR & Model-order reduction \\
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PT & Perturbation theory \\
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RBM & Reduced-basis method \\
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\hline
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\hline
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\end{longtable}
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\end{longtable}
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@ -1,4 +1,4 @@
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\chapter{Mathematica identities}
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\chapter{Mathematical identities}
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A summary of mathematical identities used in derivations is given below.
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A summary of mathematical identities used in derivations is given below.
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@ -1,4 +1,113 @@
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\chapter{Formalism}
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\chapter{Background}
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\label{chap:Formalism}
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\label{chap:Background}
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[\ldots]
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\section{Partial-wave decomposition}
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\label{sec:partial_wave}
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For spherical symmetric two-body potentials $V(r)$, where r is the distance between the particles, the Hamiltonian has an $SO(3)$ symmetry. Consequently, the time-independent Schr\"odinger equation,
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\begin{equation}
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\left[{-}\frac{1}{2\mu}\nabla^2 + V(r)\right]\psi(\vec{x}) = E \, \psi(\vec{x}) \, ,
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\label{eq:SE}
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\end{equation}
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can be separated into radial and angular parts by writing out the Laplacian operator in spherical coordinates,
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\begin{equation}
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\nabla^2 \equiv \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right) + \frac{1}{r^2} \left[ \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \right] \, ,
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\end{equation}
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which gives the angular eigen-equation,
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\begin{equation}
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{-}\left[ \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \right] Y^m_l(\theta,\phi) = l(l+1) \, Y^m_l(\theta,\phi) \, ,
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\end{equation}
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having the well-known spherical harmonics $Y^m_l$ as eigenfunctions and $l(l+1)$ as corresponding eigenvalues. Substituting this back into Eq.~\eqref{eq:SE} gives the radial Schr\"odinger equation,
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\begin{equation}
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\left[ \frac{\dd ^2}{\dd r^2} - \frac{l(l+1)}{r^2} + 2 \mu V(r) - p^2 \right] u(r) = 0 \, ,
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\label{eq:SE_rad}
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\end{equation}
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where $p=\sqrt{2 \mu E}$ and $u(r)$ is the \emph{reduced} radial wave function implicitly defined via,
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\begin{equation}
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\psi(r,\theta,\phi) = R(r) Y^m_l(\theta,\phi) = \frac{u(r)}{r} Y^m_l(\theta,\phi) \, .
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\end{equation}
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\subsection{Free solutions}
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In a region of zero-interaction, that is $V(r)=0$, Eq.~\eqref{eq:SE_rad} has solutions that are given by some linear combination of Ricatti-Hankel functions,
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\begin{equation}
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u(r) \sim a \, \hat{h}^-_l(pr) + b \, \hat{h}^+_l(pr) \, ,
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\label{eq:free_wf}
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\end{equation}
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where $\hat{h}^+_l(pr)$ and $\hat{h}^-_l(pr)$ are the Ricatti-Hankel functions of the first and second kind respectively. These Ricatti-Hankel functions can be defined using the familiar spherical Bessel functions,
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\begin{subalign}
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\hat{h}^+_l(z) &= \ii z \, h^{(1)}_l(z) \\
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\hat{h}^-_l(z) &= {-}\ii z \, h^{(2)}_l(z)
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\end{subalign}
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When $V(r)=0$ everywhere, we can fix the coefficients $a$ and $b$ to satify the boundary condition $u(r) \to 0$ for $r \to 0$. This gives the ``free solutions'' of the the radial Schr\"odinger equation,
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\begin{equation}
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u(r) = \hat{j}_l(pr) = \frac{i}{2} \left[\hat{h}^-_l(pr) - \hat{h}^+_l(pr)\right] \, .
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\end{equation}
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$\hat{j}_l(pr)$ are called the Ricatti-Bessel functions and they form a complete basis for any given $l$.
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\subsection{The radial basis}
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Next, we define the partial-wave ``radius'' state $\ket{r,l,m}$ such that,
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\begin{equation}
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\braket{r,l,m | \psi_{l,m}} = u(r) \, ,
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\end{equation}
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where $\psi_{l,m}$ is a state with definite angular momentum quantum numbers $(l,m)$ and $u(r)$ is the corresponding reduced radial wave function.
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This leads to the relation,
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\begin{equation}
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\braket{\vec{x} | r,l,m} = \frac{\delta (x - r)}{r} Y_l^m(\hat{x}) \, .
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\end{equation}
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Now, consider a free solution $\ket{E,l,m}$ with energy $E=p^2/\mu$. Its overlap with the radius state $\ket{r,l',m'}$ is,
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\begin{spliteq}
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\braket{r,l',m' | E,l,m} &= \int d^3 x \, \braket{r,l',m' | \vec{x}} \braket{\vec{x} | E,l,m} \\
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&= \iiint x^2 \, dx \, d(\cos{\theta}) \, d\phi \, \frac{\delta (x - r)}{r} Y_{l'}^{m'}(\theta,\phi) \left[\ii^l \sqrt{\frac{2\mu}{\pi p}} \frac{1}{r} \hat{j}_l(pr) Y_l^m(\theta,\phi) \right] \\
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&= \ii^l \sqrt{\frac{2\mu}{\pi p}} \hat{j}_l(pr) \, \delta_{l,l'} \, \delta_{m,m'}
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\end{spliteq}
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where the relation
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\begin{equation}
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\braket{\vec{x} | E,l,m} = \ii^l \sqrt{\frac{2\mu}{\pi p}} \frac{1}{r} \hat{j}_l(pr) Y_l^m(\theta,\phi)
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\end{equation}
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is used~\citep[Eq. 11.8]{taylor}.
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\section{Hankel transform}
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We can transform between the ``partial-wave momentum space'' and the usual partial-wave configuration space using the Hankel transforms. We adopt the ``symmetric'' convention given by,
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\begin{subalign}
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u(k) &= \sqrt{\frac{2}{\pi}} \int_0^\infty dr \, \hat{j}_l(kr) u(r) \, ,\\
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u(r) &= \sqrt{\frac{2}{\pi}} \int_0^\infty dk \, \hat{j}_l(kr) u(k) \, . \label{eq:inv_hankel}
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\end{subalign}
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To verify the constant pre-factors, we show that the normalization is preserved under this transform. We know that in configuration space, the radial part of the wave function obeys the normalization condition,
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\begin{equation} \begin{split}
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1 &= \int_0^\infty \, dr \, r^2 \, R^*(r) \, R(r) \\
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&= \int_0^\infty \, dr \, u^*(r) \, u(r)
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\end{split} \end{equation}
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And in momentum space, it obeys,
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\begin{equation}
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1 = \int_0^\infty dk \, u^*(k) \, u(k)
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\end{equation}
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Now, we show that we end up with a properly normalized $u(r)$ in configuration space,
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given we start with a normalized $\Tilde{u}(k)$ in momentum space.
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\begin{equation} \begin{split}
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\int_0^\infty \, dr \, u^*(r) u(r)
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&= \frac{2}{\pi} \int_0^\infty \int_0^\infty \int_0^\infty
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dr \, dk \, dk' \,
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\hat{j}_l(kr) \, \hat{j}_l(k'r) \, u^*(k) \, u(k') \\
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&= \int_0^\infty \int_0^\infty
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dk \, dk' \,
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\delta(k-k') \, u^*(k) \, u(k') \\
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&= \int_0^\infty
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dk \, u^*(k) \, u(k) \\
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&= 1 \\
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\end{split} \end{equation}
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where we have used Eq.~\eqref{eq:inv_hankel} in the first step
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and evaluated the integral w.r.t. $r$ using \cite[p183]{taylor}
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$$\int_0^\infty dr \, \hat{j}_l(kr) \, \hat{j}_l(k'r) = \frac{\pi}{2} \delta(k-k')$$
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in the second step.
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@ -1,4 +1,136 @@
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\chapter{Finite volume}
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\chapter{Eigenvector continuation}
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\label{chap:Finite_volume}
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\label{chap:Eigenvector_continuation}
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\section{Introduction to EC}
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First introduced by \cite{Frame:2017fah}, eigenvector continuation (EC) is a powerful method to tackle computationally expensive quantum mechanical problems, despite being cheap and simple to implement. Similar in spirit to Rayleigh-Schr\"odinger perturbation theory (also known simply as perturbation theory), the method aims to approximate solutions to a Hamiltonian of interest, given exact solutions to a set of neighbouring Hamiltonians differing by small deviations, although EC has been shown to greatly outperform PT for terms of convergence~\citep{Frame:2017fah,Sarkar:2020mad}. Recent work~\citep{Bonilla:2022rph,Melendez:2022kid} has shown that EC as a particular reduced-basis method (RBM) falls within a larger class of model-order reduction (MOR) techniques.
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Specifically, for a Hamiltonian with parametric dependence $H(c)$, EC enables robust extrapolations to a given target point $c_*$ from ``training data'' away from that point by exploiting information contained in eigenvectors. It can be said that the essence of system is ``learned'' via the construction of a highly effective (nonorthogonal) basis, leading to a variational calculation of the states of interest, or equivalently, projection of the target Hamiltonian onto a small subspace for rapid diagonalization. The latter approach can be boiled down to constructing projected Hamiltonian and norm matrices (denoted as $H_\text{EC}$ and $N_\text{EC}$, respectively) and solving the generalized eigenvalue problem which has the form $H_\text{EC}\ket{\psi} = \lambda N_\text{EC}\ket{\psi}$.
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As said above, EC works by obtaining eigenstates of a Hamiltonian $H(c)$ with a parametric dependence on a parameter $c$ for several values of that parameter. \footnote{For simplicity, we assume here that there is only one scalar parameter and note that the extension to multiple parameters is straightforward~\citep{Konig:2019adq}.}
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The set of parameters $\{c_i\}$ used for this step is referred to as ``training points,'' and the corresponding set of ``training vectors'' $\{\ket{\psi(c_i)}\}$ are used to construct an effective basis within which the problem is subsequently solved for one or more target values of the parameter $c$.
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For typical applications of EC, this procedure reduces the dimension of the problem from a large Hilbert space to the small subspace spanned by the training vectors, thereby leading to a vast reduction of the computational cost for each target evaluation. The projection onto this small subspace involves constructing the following Hamiltonian and norm matrices:
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%
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\begin{align}
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\label{eq:EC-H}
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\big(H_{\text{EC}}\big)_{ij} &= \braket{\psi(c_i)|H(c_*)|\psi(c_j)} \,, \\
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\label{eq:EC-N}
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\big(N_{\text{EC}}\big)_{ij} &= \braket{\psi(c_i) | \psi(c_j)} \,.
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\end{align}
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%
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Next, EC involves solving the generalized eigenvalue problem,
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%
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\begin{equation}
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H_{\text{EC}} \ket{\psi(c_*)_{\text{EC}}}
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= E_{\text{EC}} \, N_{\text{EC}} \ket{\psi(c_*)_{\text{EC}}} \,,
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\label{eq:EC-GEVP}
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\end{equation}
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%
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where we denote the target point as $c_*$.
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It can be shown that Eq.~\eqref{eq:EC-GEVP} is equivalent to the variational principle, by phrasing the problem as an optimization problem where the Rayleigh quotient,
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\begin{equation}
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E_{\text{EC}} = \frac{\bra{\psi(c_*)_{\text{EC}}} H_{\text{EC}} \ket{\psi(c_*)_{\text{EC}}}}{\bra{\psi(c_*)_{\text{EC}}} N_{\text{EC}} \ket{\psi(c_*)_{\text{EC}}}} \, ,
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\end{equation}
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is minimized with respect to $\ket{\psi(c_*)_{\text{EC}}}$~\citep{ghojogh2023eigenvalue}.
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However, considering numerical stability, it is safer to avoid solving the GEVP by orthonormalizing the basis $\{\ket{\psi(c_i)}\}$ beforehand. This is due to the fact that for near-singular $N_{\text{EC}}$, Eq.~\eqref{eq:EC-GEVP} is an ill-posed problem, as it is the case when too many training points are sampled from too narrow a region, leading to a near-redundant basis. Orthonormalization can be cheaply carried out using an algorithm such as Gram–Schmidt while dropping any redundant basis vectors in the process. Now, with respect to this new basis, $N_{\text{EC}}=1$, thereby reducing the GEVP to the ordinary eigenvalue problem,
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\begin{equation}
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H_{\text{EC}} \ket{\psi(c_*)_{\text{EC}}}
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= E_{\text{EC}} \ket{\psi(c_*)_{\text{EC}}} \, .
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\end{equation}
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\ny{Mention extremal eigenvalues.}
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\section{A cheap error estimation for EC}
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\subsection{Introduction and proof}
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Here a cheap uncertainty estimation is derived as an alternative for bootstrapping as it involves re-sampling the training points and solving the generalized eigenvalue problem each time. The proposed formula is similar to the one shown in~\cite{sarkar2021selflearning}.
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Let $E^\text{EC}$ and $\ket{\psi^\text{EC}}$ be an EC extrapolated eigenvalue and an eigenvector respectively for the target value of the parameter $c=c_*$.
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Now consider the expansion of $\ket{\psi^\text{EC}}$ in $\{\ket{\psi_i}\}$, the exact energy eigenbasis of the target Hamiltonian $H(c_*)$, henceforth simply called $H_*$.
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\begin{align}
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\ket{\psi^\text{EC}} &= \sum_i {a_i\ket{\psi_i}} \\
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\left(H_*-E^\text{EC}\right)\ket{\psi^\text{EC}} &= \sum_i {a_i\left(E_i-E^\text{EC}\right)\ket{\psi_i}} \\
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\norm{\left(H_*-E^\text{EC}\right)\ket{\psi^\text{EC}}} &= \sqrt{\sum_i {\abssym{a_i}^2 \, \abssym{E_i-E^\text{EC}}^2}}
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\label{eq:sd}
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\end{align}
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If the coefficients $\abssym{a_i}$ had a Gaussian-like distribution against $E_i$,
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it would have a sharp peak for a good extrapolation and a spread-out distribution
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otherwise. Right hand side of eq. \ref{eq:sd} gives the standard deviation for that distribution.
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Therefore,
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\begin{equation}
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\sigma = \norm{(H_*-E^\text{EC})\ket{\psi^\text{EC}}}
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\label{eq:formula}
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\end{equation}
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Eq. \ref{eq:formula} is a cheap calculation as it only involves one matrix-vector multiplication,
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followed by a vector-vector inner product. Obviously, one would need access to the extrapolated eigenvector $\ket{\psi^\text{EC}}$ and the full Hamiltonian $H_*$. I
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claim that quantity $\sigma$ is a good uncertainty quantifier for EC (or any eigenvector approximation algorithm).
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\subsection{Example from FVEC paper}
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Note how in fig. \ref{fig:FVEC_error}, the exact spectrum is properly
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enclosed by the error bands, as opposed to the ``bootstrap'' method, which
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failed to do so. However, the large widths of the bands may not give FVEC
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method the credit is deserves.
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\begin{figure}
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\centering
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\includegraphics[width=\textwidth]{Chapter-3/FVEC_error.png}
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\caption{Extrapolation error for the two-body system in FVEC paper.}
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\label{fig:FVEC_error}
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\end{figure}
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\subsection{Example of a bad EC model}
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\label{sec:AC_error}
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In fig. \ref{fig:AC_error}, the topmost band veers off course at smaller
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volumes due to an avoided crossing as the model does not include
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enough training states.
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Unfortunately, the error bands fail to account for that error. However,
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these error bands describe the other energy level involved in the
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avoided crossing (not shown here).
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\begin{figure}
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\centering
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\includegraphics[width=\textwidth]{Chapter-3/AC_error.png}
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\caption{Extrapolation error for as two-body system with an avoided
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crossing corresponding to a narrow resonance.}
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\label{fig:AC_error}
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\end{figure}
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\subsection{Alternative interpretation: Upper bounds of error}
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Let,
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\begin{equation}
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\braket{\psi^\text{EC} | \psi^\text{EC}}=\sum_i {\abssym{a_i}^2}=1
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\end{equation} \\
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Define $E'$ to be the \emph{closest} exact eigenvalue to the extrapolated value $E^\text{EC}$. That is,
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\begin{equation}
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\abssym{E'-E^\text{EC}} \leq \abssym{E_i-E^\text{EC}}
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\end{equation}
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for all $i$.
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Typically this is the eigenvalue of interest, but could also be a
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rogue eigenvalue that comes close and misguide the extrapolation
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in bad EC models, such as the one shown in fig. \ref{fig:AC_error}.
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Now, continuing from eq. \ref{eq:sd},
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\begin{align}
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\norm{\left(H_*-E^\text{EC}\right)\ket{\psi^\text{EC}}} &= \sqrt{\sum_i {\abssym{a_i}^2 \, \abssym{E_i-E^\text{EC}}^2}} \\
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&\geq \sqrt{\sum_i {\abssym{a_i}^2 \, \abssym{E'-E^\text{EC}}^2}} \\
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&= \abssym{E'-E^\text{EC}} \sqrt{\sum_i {\abssym{a_i}^2}} \\
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&= \abssym{E'-E^\text{EC}}
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\end{align}
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||||||
|
That is, $\sigma$ is also an upper bound for the extrapolation error for a good EC model.
|
||||||
|
If not, we can at least guarantee that the error bars would enclose \emph{at least one}
|
||||||
|
exact eigenvalue. \\
|
||||||
|
|
||||||
|
|
||||||
[\ldots]
|
|
||||||
|
|
|
||||||
Binary file not shown.
|
After Width: | Height: | Size: 75 KiB |
|
|
@ -1,14 +1,209 @@
|
||||||
\chapter{Resonances}
|
\chapter{Resonances}
|
||||||
\label{chap:Resonances}
|
\label{chap:Resonances}
|
||||||
|
|
||||||
\section{Zeldovich regularization}
|
\section{Introduction}
|
||||||
\label{sec:zeldovich}
|
\label{sec:res_intro}
|
||||||
|
|
||||||
As an alternative to complex scaling, here we discuss Zeldovich regularization,
|
\section{Complex-scaling method}
|
||||||
where the known asymptotic forms of wave functions in configuration
|
\label{sec:CSM}
|
||||||
space are utilized to define a regularized inner-product.
|
|
||||||
|
For most physically relevant cases, we are work with finite-ranged potentials, which are rigorously constrained such that,
|
||||||
|
\begin{subalign}
|
||||||
|
\lim_{r \to 0} r^2 V(r) &= 0 \, , \\
|
||||||
|
\lim_{r \to \infty} r^3 V(r) &= 0 \, . \label{eq:finite_ranged}
|
||||||
|
\end{subalign}
|
||||||
|
It can be shown that the condition Eq.~\eqref{eq:finite_ranged} implies that the wave function asymptotically converges to Eq.~\eqref{eq:free_wf} as $r \to \infty$. Then we can write,
|
||||||
|
\begin{equation}
|
||||||
|
u(r) \xrightarrow[r \rightarrow \infty]{} \frac{\ii}{2} \left[ \hat{h}^-_l(pr) - s_l(p) \hat{h}^+_l(pr) \right] \, .
|
||||||
|
\label{eq:asymptotic_wf}
|
||||||
|
\end{equation}
|
||||||
|
Here, $s_l(p)$ can be interpreted as the partial-wave $S$ matrix, since it appears as the ratio between the outgoing and incoming waves. Equation~\eqref{eq:asymptotic_wf} implies that wherever $s_l(p)$ has a pole, we have
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:pobc}
|
||||||
|
u(r) \xrightarrow[r \rightarrow \infty]{} \ii^l \gamma \, \hat{h}^+_l(pr) \,,
|
||||||
|
\end{equation}
|
||||||
|
where $\gamma$ is the asymptotic normalization constant and the $\ii^l$ factor is conventional. An illustration of the analytic structure of the $S$ matrix and its poles is shown in Fig.~\ref{fig:s_matrix}.
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\begin{figure}
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=0.65\textwidth]{Chapter-4/s_matrix.pdf}
|
||||||
|
\caption{The analytic structure of the $S$ matrix indicating its poles in the $p$ plane corresponding to bound states, virtual states, resonances, and antiresonances (capturing resonances).
|
||||||
|
Note especially how bound states lie on the positive imaginary axis while resonances are located in the \nth{4} quadrant.}
|
||||||
|
\label{fig:s_matrix}
|
||||||
|
\end{figure}
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
|
||||||
|
As shown in in Fig.~\ref{fig:s_matrix}, bound states have associated imaginary momenta $p=\ii\kappa$ with real $\kappa>0$, whereas resonances are described by complex $p$ with $\Ip(p)<0$. This means that asymptotically, bound state wave functions decay exponentially with $r$ (hence the term ``bound'') whereas resonance wave functions grow exponentially and are therefore---like scattering states but in some sense even more so---not square-integrable; \ie, they do not correspond to normalizable states in the ordinary Hilbert space. While the rigged Hilbert space construction offers a rigorous mathematical formalism to deal with this difficulty (see, for example, \cite{delaMadrid:2012aa} for an introduction), for practical calculations there exists a much simpler alternative.
|
||||||
|
|
||||||
|
The so-called (uniform) complex-scaling method~\citep{Aguilar:1971ve,Balslev:1971vb, Moiseyev:1978aa,Moiseyev:1998aa,Afnan:1991kb,ho_method_1983,Reinhardt:1982aa} enables a description of resonances with, essentially, bound-state techniques. This is achieved by expressing the wave function not as usual along the real $r$ axis, but on a contour rotated into the complex-$r$ plane. This can be achieved by applying the transformation
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
r \to r e^{\ii \phi}
|
||||||
|
\label{eq:r-scaled}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
to Eq.~\eqref{eq:SE_rad}, with some angle $\phi$. The proper choice of $\phi$ in general depends on the position of the resonance one wishes to study. If the state of interest has a complex energy $E$, then it is necessary to ensure that $\phi > {-}\arg(E)/2$. As $E$ is usually not known beforehand, one might repeat the calculation while increasing $\phi$ until a resonance is found.
|
||||||
|
\footnote{One might think of simply setting $\phi=\pi/4$ to accommodate all possible resonances. However, in most cases, large $\phi$ angles lead to potentials not vanishing fast enough along the contour, thereby demanding higher momentum cutoffs, or even potentials becoming completely divergent.}
|
||||||
|
|
||||||
|
With the convention in Eq.~\eqref{eq:r-scaled}, $r$ is still a real parameter but no longer equals to the physical radial coordinate of the system. The overall argument $kr e^{\ii \phi}$ of the Riccati-Hankel function in Eq.~\eqref{eq:pobc} satisfies $\Ip(kr e^{\ii \phi})>0$, and therefore square-integrability of the wave function as a function of $r$ is recovered. An example of such a scaled wave function is illustrated in Fig.~\ref{fig:csm_illustration}.
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\begin{figure}
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=0.8\textwidth]{Chapter-4/csm_illustration.pdf}
|
||||||
|
\caption{Effect on the reduced radial wave function of a typical $S$-wave resonance (Gamow state), due to complex-scaling of the $r$ contour. The solid (dotted) line corresponds to the real (imaginary) parts. After complex-scaling, it asymptotically converges to an exponentially decaying Riccati-Hankel function, $\hat{h}^+_0(pr e^{\ii\phi})=\hat{h}^+_0(\Tilde{p}r)=\exp(\ii \Tilde{p} r)$, where we define $\Tilde{p} = p e^{\ii \phi}$, the \emph{effective} wave number with $\Ip(\Tilde{p})>0$, so that it is normalizable just like bound-state wave functions.}
|
||||||
|
\label{fig:csm_illustration}
|
||||||
|
\end{figure}
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
|
||||||
|
It was shown in Ref.~\cite{Afnan:1991kb} that the scaling of the radial coordinate $r$ is equivalent to a rotation in momentum representation that goes in the opposite (clockwise) direction with the same angle $\phi$. That is, if we consider the wave function of the state as a function of a momentum coordinate $q$, then complex scaling is implemented via
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
q \to q e^{{-}\ii \phi} \,.
|
||||||
|
\label{eq:q-scaled}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
This procedure then makes it possible to alternatively calculate resonance wave functions in momentum space. Note furthermore that scaling in momentum space can also be understood as a rotation of the branch cut in the complex-energy plane by an angle $2\phi$ clockwise, thereby exposing a section of the second Riemann sheet where resonances are located.
|
||||||
|
|
||||||
|
After this transformation, we can absorb the $e^{\ii \phi}$ phase into the wave number $k$ and define the \emph{effective} wave number as $\Tilde{k} = k e^{\ii \phi}$,
|
||||||
|
so that the asymptotic form in Eq.~\eqref{eq:pobc} is preserved as
|
||||||
|
\begin{equation}
|
||||||
|
\psi_{l,k}(r e^{\ii \phi}) \xrightarrow[r \rightarrow \infty]{} N \, \hat{h}^+_l(\Tilde{k} r) \,.
|
||||||
|
\end{equation}
|
||||||
|
The scaling technique can thus be interpreted as mapping resonances from the \nth{4} quadrant in the complex-$k$ plane to the \nth{1} quadrant in the $\Tilde{k}$ plane.
|
||||||
|
For future reference, we note that, at the same time, it will effectively map bound states from the positive imaginary $k$ axis to the \nth{2} quadrant in the complex $\Tilde{k}$ plane.
|
||||||
|
|
||||||
|
\section{Complex scaling in 3D Cartesian coordinates}
|
||||||
|
|
||||||
|
While the method of complex scaling is easier to explain in a partial-wave framework~\citep{Afnan:1991kb,ho_method_1983}, the equivalent 3D Cartesian formulation, which is most appropriate for the finite-volume studies in Sec.\ref{TBA}, needs to be carefully stated. To that end we note that complex scaling of each individual component of $\vec{r}=(x,y,z)$ is equivalent to complex scaling of the radial coordinate $r$,
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
r = \sqrt{x^2+y^2+z^2} \to \sqrt{(x e^{\ii \phi})^2+(y e^{\ii \phi})^2+(z e^{\ii \phi})^2} = r e^{\ii \phi} \,,
|
||||||
|
\label{eq:radial_coordinate_scaling}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
but it leaves the angles $\theta$ and $\varphi$ in spherical coordinates $\vecr = (r,\theta,\phi)$ unaffected:
|
||||||
|
%
|
||||||
|
\begin{subalign}
|
||||||
|
\cos{\theta} &= \frac{z}{r} = \frac{z e^{\ii \phi}}{r e^{\ii \phi}} \,, \\
|
||||||
|
\tan{\varphi} &= \frac{y}{x} = \frac{y e^{\ii \phi}}{x e^{\ii \phi}} \,.
|
||||||
|
\end{subalign}
|
||||||
|
%
|
||||||
|
Therefore, we can apply complex scaling directly to Cartesian coordinates simply via $\vec{r}_i \to \vec{r}_i \ee^{\ii \phi}$. Note that we have implicitly altered usual Euclidean norm in Eq.~\eqref{eq:radial_coordinate_scaling} so it that preserves complex scaling, \viz
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
|\vec{r}| = \sqrt{x^2+y^2+z^2}
|
||||||
|
\mathtext{for} \vec{r}=(x,y,z) \mathtext{where} x,y,z \in \CC \,,
|
||||||
|
\end{equation}
|
||||||
|
instead of,
|
||||||
|
\begin{equation}
|
||||||
|
|\vec{r}| \neq \sqrt{|x|^2+|y|^2+|z|^2} \, .
|
||||||
|
\label{eq:norm-incorrect}
|
||||||
|
\end{equation}
|
||||||
|
This can be justified in the spirit of analytic continuation, because the standard Euclidean norm showing in Eq.~\eqref{eq:norm-incorrect} would not be analytic.
|
||||||
|
|
||||||
|
Finally, letting $\vec{q}$ be the momenta conjugate to $\vec{r}$, we see that the transformation
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
\vec{q} \to \vec{q} \ee^{{-}\ii \phi}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
is necessary to preserve the canonical commutation relation,
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
\left[ \vec{r} \ee^{\ii \phi},\vec{q} \ee^{{-}\ii \phi} \right] = \left[ \vec{r},\vec{q} \right] = \ii \, .
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
This agrees nicely with the claim made in Eq.~\eqref{eq:q-scaled}.
|
||||||
|
|
||||||
|
\subsection{Non-Hermiticity and the c-product}
|
||||||
|
|
||||||
|
In traditional quantum mechanics, one requires the Hamiltonian $H$ to be Hermitian ($H^\dagger=H$) to ensure that the energy spectrum, being a physical observable, is real and that time evolution is strictly unitary, \ie, the norm of quantum states are preserved under the time evolution operator $\ee^{{-}\ii H t / \hbar}$. However, when considering decay, an inherently time-dependent phenomenon, in a time-independent framework such as the complex-scaling method, the Hamiltonian is no longer Hermitian. Instead, in the present case, it becomes complex symmetric~\citep{Moiseyev:1998aa}, \ie,
|
||||||
|
\begin{equation}
|
||||||
|
H^\intercal=H \, .
|
||||||
|
\end{equation}
|
||||||
|
This permits the energy spectrum to include complex eigenvalues, which is precisely what is needed to describe resonances. In fact, the non-Hermiticity and the corresponding non-unitary time evolution of Gamow states are well aligned with the physical interpretation of resonances as metastable systems that ultimately decay.
|
||||||
|
|
||||||
|
Similarly to how non-degenerate eigenvectors of a Hermitian operator are orthogonal under the inner product defined on the Hilbert space, the non-degenerate eigenvectors of a complex symmetric operator are orthogonal under the so-called ``c-product''~\cite{Moiseyev:1978aa,Moiseyev:2011}.
|
||||||
|
For eigenstates $\ket{\psi_1}$ and $\ket{\psi_2}$ with equal angular-momentum quantum numbers $(l,m)$, we define the c-product in coordinate representation as
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
\braket{\psi_1 | \psi_2} = \int \dd r \, \psi_1(r)\psi_2(r) \,,
|
||||||
|
\label{eq:CP-vecr}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
and similarly in momentum space. Note that $\psi_1(r)$ appears without complex conjugation under the integral. This is precisely the c-product introduced in Ref.~\cite{Moiseyev:1978aa} with the notation $(\psi_1|\psi_2)$. In this manuscript, we use the standard notation $\braket{\psi_1 | \psi_2}$ with the implicit
|
||||||
|
understanding that for complex-scaled systems this is meant to denote the c-product.
|
||||||
|
|
||||||
|
Equivalently, one can change the definition of bra states so that no complex conjugation is involved when they are associated with a complex-scaled system. This is so even for bound states calculated with complex scaling. Although the energies of such states remain real, wave functions become complex when defined along the rotated contour and the orthogonality of states with different binding energies is ensured only if no complex conjugation is performed for bras, leading again to the c-product~\cite{Moiseyev:2011}. Ultimately, these concepts can be understood by properly distinguishing bra and ket states as, respectively, left and right
|
||||||
|
eigenvectors of the non-Hermitian complex-scaled Hamiltonian~\cite{Afnan:1991kb}. Even more rigorously, a comprehensive theory for Gamow bras and kets can be developed within the RHS formalism mentioned previously~\cite{delaMadrid:2012aa}. However, in practice we find it convenient and sufficient to employ complex scaling along with the c-product.
|
||||||
|
|
||||||
|
To prove c-orthogonality for complex symmetric $H$, we start by considering two non-degenerate eigenvectors (written as vectors instead of bras or kets for clarity),
|
||||||
|
%
|
||||||
|
\begin{subalign}
|
||||||
|
H \vec{v_1} = E_1 \vec{v_1} \, ,
|
||||||
|
\label{eq:eig_a}\\
|
||||||
|
H \vec{v_2} = E_2 \vec{v_2} \, ,
|
||||||
|
\label{eq:eig_b}
|
||||||
|
\end{subalign}
|
||||||
|
%
|
||||||
|
where $E_1 \neq E_2$. Taking the transpose of \ref{eq:eig_b} gives,
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:eig_trans}
|
||||||
|
\vec{v_2}^\intercal H = E_2 \vec{v_2}^\intercal \, ,
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
since $H^\intercal=H$. Note that complex conjugation was not involved. Now left-multiply \ref{eq:eig_a} by $v_2^\intercal$, right-multiply \ref{eq:eig_trans} by $v_1$, and finally subtract them to get,
|
||||||
|
%
|
||||||
|
\begin{spliteq}
|
||||||
|
\vec{v_2}^\intercal H \vec{v_1} - \vec{v_2}^\intercal H \vec{v_1}
|
||||||
|
& = (E_1-E_2) \vec{v_2}^\intercal \vec{v_1}\\
|
||||||
|
0 & = (E_1-E_2) \vec{v_2}^\intercal \vec{v_1}\\
|
||||||
|
0 & = \vec{v_2}^\intercal \vec{v_1} \, . \qed
|
||||||
|
\end{spliteq}
|
||||||
|
|
||||||
|
\ny{TODO: Self-orthogonal states}
|
||||||
|
|
||||||
|
\section{Purely outgoing boundary conditions}
|
||||||
|
\label{sec:POBC}
|
||||||
|
|
||||||
|
One can calculate Gamow states (and bound states) by solving the Schr\=odinger equation using an differential equation solver, subject to \emph{purely outgoing boundary conditions} (POBC).
|
||||||
|
|
||||||
|
The asymptotic behavior for the regular function is
|
||||||
|
\begin{equation}
|
||||||
|
\phi_p(r) \xrightarrow[r \rightarrow \infty]{} \frac{\ii}{2} \left[ \Jostf_l(p) e^{-\ii pr} - \Jostf_l(-p) e^{\ii pr} \right] \, .
|
||||||
|
\end{equation}
|
||||||
|
Then the first derivative is
|
||||||
|
\begin{equation}
|
||||||
|
\phi'_p(r) \xrightarrow[r \rightarrow \infty]{} \frac{p}{2} \left[ \Jostf_l(p) e^{-\ii pr} + \Jostf_l(-p) e^{\ii pr} \right] \, .
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
Solving for $\Jostf_l(\pm p)$ gives
|
||||||
|
\begin{equation}
|
||||||
|
\left( \frac{\phi'_p}{p} \pm \ii \phi_p \right) e^{\mp \ii pr} \xrightarrow[r \rightarrow \infty]{} \Jostf_l(\pm p) \, .
|
||||||
|
\end{equation}
|
||||||
|
But we know that $\Jostf_l(p) = 0$ for bound states and Gamow states. Therefore, for an integration interval $(0,R]$,
|
||||||
|
the boundary conditions
|
||||||
|
\begin{spliteq}
|
||||||
|
\phi_p(0) &= 0 \\
|
||||||
|
\frac{\phi'_p(R)}{p} + \ii \phi_p(R) &= 0 \\
|
||||||
|
\end{spliteq}
|
||||||
|
would yield the discrete bound state and Gamow state energies $E = \frac{p^2}{2\mu}$.
|
||||||
|
|
||||||
|
\section{Zeldovich regularization}
|
||||||
|
\label{sec:Zeldovich}
|
||||||
|
|
||||||
|
\ny{Only worked out for $S$-wave. Need to extend.}
|
||||||
|
|
||||||
|
Here we discuss Zeldovich regularization, as a method to calculate inner products
|
||||||
|
of configuration space wave functions determined via POBC as discussed in~\ref{sec:POBC}.
|
||||||
|
We will exploit the knowledge of asymptotic forms of wave functions to define a regularized inner-product.
|
||||||
Using this, we can implement CA-EC in configuration space, without the complications
|
Using this, we can implement CA-EC in configuration space, without the complications
|
||||||
of complex-scaled contours. As $\Ip(p)<0$ for resonance states,
|
of complex-scaled contours.
|
||||||
|
|
||||||
|
As $\Ip(p)<0$ for resonance states,
|
||||||
the wave function is ever increasing and not square integrable.
|
the wave function is ever increasing and not square integrable.
|
||||||
They are said to belong in a ``Rigged Hilbert Space''. Inner products and
|
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
|
normalization factors of Gamow states can be calculated by inserting a regularization
|
||||||
|
|
@ -115,3 +310,69 @@ 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
|
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
|
($re^{-2\ii\phi}$) for some arbitrary angle $\phi$. As mentioned above, this method
|
||||||
is very inefficient in the Zeldovich implementation compared to RHA-EC.
|
is very inefficient in the Zeldovich implementation compared to RHA-EC.
|
||||||
|
|
||||||
|
\subsection{Analytic continuation of bound states}
|
||||||
|
\label{sec:analytic_cont}
|
||||||
|
|
||||||
|
In bound-to-resonance extrapolation, we need to work in a complex-scaled
|
||||||
|
basis to accommodate resonances.
|
||||||
|
We need the bound state wave functions expanded on this complex-scaled contour to use as
|
||||||
|
training data. These can be simply obtained by diagonalizing the Hamiltonian, analytically
|
||||||
|
continued onto the complex-scaled basis.
|
||||||
|
|
||||||
|
However, if one already has access to a set of pre-calculated bound state wave
|
||||||
|
functions on the real (un-rotated) contour, they
|
||||||
|
can be analytically continued onto the complex-scaled contour, without having to solve the Schr\"odinger's equation again, using the homogeneous equation,
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
\ket{\psi}=\frac{1}{E-H_0}V\ket{\psi} \, ,
|
||||||
|
\label{eq:homogenous}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
where $G(E)$ is the Green's operator defined as
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
G(E)=\frac{1}{E-H_0} \, .
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
|
||||||
|
Rewriting the above equation as in partial-wave notation gives
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
u(\Tilde{q})=\int_{0}^{\infty} q\,dq\, \frac{1}{E-\frac{q^2}{2\mu}}V(\Tilde{q},q) u(q) \, .
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
Now the reduced radial wave function $u(\Tilde{q})$ can be calculated on the complex-scaled contour in momentum space.
|
||||||
|
|
||||||
|
Equivalently, in configuration space, the Eq.~\eqref{eq:homogenous} reads,
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
\psi(\Tilde{\vec{r}})={-}\frac{\mu}{2 \pi} \int_{0}^{\infty} d^3 \vec{r} \, \frac{e^{\ii p |\Tilde{\vec{r}}-\vec{r}|}}{|\Tilde{\vec{r}}-\vec{r}|} V(\vec{r}) \psi(\vec{r}) \, .
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
However, as we work in a partial-wave basis, we should be able to find a simpler version of this equation that does not involve a 3-dimensional integration over $\vec{r}$.
|
||||||
|
|
||||||
|
To that end, let us start from Eq.~\eqref{eq:homogenous} and insert few complete basis sets of ``radial'' states introduced in Sec.~\ref{sec:partial_wave}.
|
||||||
|
\begin{spliteq}
|
||||||
|
\braket{\Tilde{r},\Tilde{l},\Tilde{m} | \psi} &= \iint dr \, dr' \, \sum_{l,m} \sum_{l',m'} \bra{\Tilde{r},l,m} \frac{1}{E-H_0} \ket{r,l,m} \bra{r,l,m}V\ket{r',l',m'} \braket{r',l',m' | \psi} \\
|
||||||
|
&= \int dr \, \sum_{l,m} \braketmatrix{\Tilde{r},\Tilde{l},\Tilde{m}}{\frac{1}{E-H_0}}{r,l,m} V(r) \braket{r,l,m | \psi}
|
||||||
|
\end{spliteq}
|
||||||
|
|
||||||
|
It is clear that we need an expression for $\braketmatrix{\Tilde{r},\Tilde{l},\Tilde{m}}{\frac{1}{E-H_0}}{r,l,m}$ which is the Green's function expanded in a partial-wave basis. This can be carried out as follows.
|
||||||
|
\begin{multline}
|
||||||
|
\braketmatrix{\Tilde{r},\Tilde{l},\Tilde{m}}{\frac{1}{E-H_0}}{r,l,m} \\
|
||||||
|
= \iint dE' \, dE'' \, \sum_{l',m'} \sum_{l'',m''} \braket{\Tilde{r},\Tilde{l},\Tilde{m} | E',l',m'} \braketmatrix{E',l',m'}{\frac{1}{E-H_0}}{E'',l'',m''} \braket{E'',l'',m'' | r,l,m} \\
|
||||||
|
= \int dE' \, \sum_{l',m'} \braket{\Tilde{r},\Tilde{l},\Tilde{m} | E',l',m'} \frac{1}{E-E'} \braket{E',l',m' | r,l,m} \\
|
||||||
|
= \delta_{\Tilde{l},l} \, \delta_{\Tilde{m},m} \int \frac{p'}{\mu} \, dp' \, \left[ \ii^l \sqrt{\frac{2\mu}{\pi p'}} \hat{j}_l(p' \Tilde{r}) \, \right]^* \frac{2\mu}{p^2-{p'}^2} \left[ \ii^l \sqrt{\frac{2\mu}{\pi p'}} \hat{j}_l(p' r) \, \right] \\
|
||||||
|
= \delta_{\Tilde{l},l} \, \delta_{\Tilde{m},m} \frac{4\mu}{\pi} \int_0^\infty dp' \, \frac{\hat{j}_l(p' \Tilde{r}) \hat{j}_l(p' r)}{p^2-{p'}^2}
|
||||||
|
\end{multline}
|
||||||
|
|
||||||
|
This integral can be evaluated to obtain the partial-wave representation of the Green's function~\citep{Fuda:1973zz},
|
||||||
|
\begin{equation}
|
||||||
|
\braketmatrix{\Tilde{r},\Tilde{l},\Tilde{m}}{\frac{1}{E-H_0}}{r,l,m} = {-} \frac{2 \mu}{p} \, \hat{j}_l(p r_{<}) \, \hat{h}^+_l(p r_{>}) \, \delta_{\Tilde{l},l} \, \delta_{\Tilde{m},m} \, ,
|
||||||
|
\end{equation}
|
||||||
|
where $r_<=\min(r,\Tilde{r})$ and $r_>=\max(r,\Tilde{r})$.
|
||||||
|
Finally, we have,
|
||||||
|
\begin{equation}
|
||||||
|
u(\Tilde{r}) = {-}\frac{2 \mu}{p} \int_0^\infty dr \, \hat{j}_l(p r_{<}) \, \hat{h}^+_l(p r_{>}) V(r) u(r) \, .
|
||||||
|
\end{equation}
|
||||||
|
|
|
||||||
Binary file not shown.
Binary file not shown.
|
|
@ -1,4 +1,264 @@
|
||||||
\chapter{Eigenvector continuation}
|
\chapter{Finite volume}
|
||||||
\label{chap:Eigenvector_continuation}
|
\label{chap:Finite_volume}
|
||||||
|
|
||||||
Introduced in Ref.~\cite{Frame:2017fah}.
|
\section{Simple relative coordinates}
|
||||||
|
|
||||||
|
The coordinate system we choose for finite-volume calculations must satisfy several conditions:
|
||||||
|
|
||||||
|
\begin{enumerate}
|
||||||
|
\item The Hamiltonian in terms of the new coordinates $(\vec{x}_i,\vec{p}_i)$ must be computationally efficient to apply.
|
||||||
|
\item An $n$-body system results in $n - 1$ degrees of freedom when the center-of-mass is subtracted out. This center-of-mass of motion should be isolated as a separate coordinate which can then be dropped.
|
||||||
|
\item Periodic boundary condition, which translates to the quantization condition for the original momenta $\vec{q}_i$ must follow from quantization condition for the new momenta $\vec{p}_i$.
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
These conditions are satisfied by simple relative coordinates introduced in ~\cite{TBA}.
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
\vec{x}_i = \begin{cases}
|
||||||
|
\vec{r}_i - \vec{r}_n & \text{for}\ i<n \,, \\
|
||||||
|
\frac{1}{n} \sum_{j=1}^n \vec{r}_j & \text{for}\ i=n \,.
|
||||||
|
\end{cases}
|
||||||
|
\label{eq:simple_relative_coordinates}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
The last coordinate $\vec{x}_n$ corresponds to the center-of-mass and is therefore ultimately dropped. Nevertheless, we define it here for the sake of completeness, in the sense that, a transformation has to be invertible. To see that this transformation is canonical, we start with an ansatz for the generating function of \nth{2} type \citep{jose_classical_2006}.
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
G_2(\vec{r}_i,\vec{p}_i) = \sum_{i=1}^{n-1} (\vec{r}_i - \vec{r}_n)\vec{p}_i + \frac{\vec{p}_n}{n} \sum_{i=1}^n \vec{r}_i
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
Now, it can be seen that
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
\vec{x}_i=\frac{\partial G_2}{\partial \vec{p}_i}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
recovers the definition given by Eq.~\ref{eq:simple_relative_coordinates}. Then we evaluate
|
||||||
|
%
|
||||||
|
\begin{spliteq}
|
||||||
|
\vec{q}_i &= \frac{\partial G_2}{\partial \vec{r}_i} \\
|
||||||
|
&= \frac{\vec{p}_n}{n} + \begin{cases}
|
||||||
|
\vec{p}_i & \text{for}\ i<n \,, \\
|
||||||
|
-\sum_{j=1}^{n-1} \vec{p}_j & \text{for}\ i=n \,.
|
||||||
|
\end{cases}
|
||||||
|
\end{spliteq}
|
||||||
|
%
|
||||||
|
Setting to zero the center-of-mass momentum $\vec{p}_n=0$, we can see that quantization of $\vec{p}_j$ implies quantization of $\vec{q}_j$, as required for periodicity.
|
||||||
|
|
||||||
|
Finally, noting that $\frac{\partial G_2}{\partial t}=0$, we can calculate the free Hamiltonian, with the center-of-mass energy subtracted out.
|
||||||
|
%
|
||||||
|
\begin{spliteq}
|
||||||
|
H_0 &= \sum_{i=1}^n \frac{\vec{q}_i^2}{2m} - \frac{1}{2nm} \parenth {\sum_{i=1}^n \vec{q}_i}^2 \\
|
||||||
|
&= \frac{1}{2m} \sum_{i=1}^{n-1} \parenth{\frac{\vec{p}_n}{n} + \vec{p}_i}^2 + \frac{1}{2m} \parenth{\frac{\vec{p}_n}{n} - \sum_{i=1}^{n-1} \vec{p}_i}^2
|
||||||
|
- \frac{1}{2nm} \parenth {\sum_{i=1}^n \vec{p}_i}^2 \\
|
||||||
|
&= \frac{1}{m} \sum_{i=1}^{n-1} \sum_{j=1}^{i} \vec{p}_i \cdot \vec{p}_j
|
||||||
|
\label{eq:H0_in_SRC}
|
||||||
|
\end{spliteq}
|
||||||
|
%
|
||||||
|
Evidently, the Hamiltonian doesn't depend on $\vec{p}_n$, eliminating one degree-of-freedom, thereby reducing the basis size. However, this includes some cross terms, unlike, for example, Jacobi coordinates.
|
||||||
|
|
||||||
|
For instance, a 3-body system will have the simple form
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
H_0 = \frac{1}{m} \parenth{\vec{p}_1^2 + \vec{p}_2^2 + \vec{p}_1 \cdot \vec{p}_2 } \, ,
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
enabling computationally efficient strategies for multiplying with a vector without explicitly constructing the whole matrix.
|
||||||
|
|
||||||
|
\section{$n$-body Hamiltonian}
|
||||||
|
|
||||||
|
Using a uniformly discretized mesh containing $N$ lattice points in each direction,
|
||||||
|
\begin{equation}
|
||||||
|
x = \frac{L}{N} k \quad \text{for} \quad k={-}\frac{N}{2}, \cdd \frac{N}{2}+1 \, ,
|
||||||
|
\end{equation}
|
||||||
|
we can construct the $n$-body basis as follows. Let $k_{i,c}$ denote the ``lattice index'' corresponding to the $c$th spatial dimension of the $\vec{x}_i$ coordinate. For arbitrary number of particles $n$ and spatial dimensions $d$, basis states can be labelled as
|
||||||
|
%
|
||||||
|
\begin{spliteq}
|
||||||
|
\ket{s}
|
||||||
|
&= \ket{k_{{1,1}},\cdd k_{{1,d}},\cdd k_{{n-1,1}},\cdd k_{{n-1,d}}} \\
|
||||||
|
&= \bigotimes_{\substack{i=1,\cdot\cdot n-1 \\c=1,\cdot\cdot d}} \ket{k_{i,c}} \,.
|
||||||
|
\label{eq:s}
|
||||||
|
\end{spliteq}
|
||||||
|
%
|
||||||
|
Note that $\ket{s}$ can in addition include discrete quantum numbers such as spin and isospin but they are neglected here for simplicity. This decomposition implies that the momentum operators $p_{i,c}$ are diagonal in $i$ and $c$. Furthermore, the kinetic operators $K_c$ are diagonal in $c$, allowing the free Hamiltonian to be decomposable as,
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
H_0 = K \oplus K \oplus \ldots \oplus K \quad \text{($d$ times)} \,,
|
||||||
|
\label{eq:H_0}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
where $\oplus$ denotes the Kronecker sum~\citep{KroneckerSumWolfram} and $K$ is the 1D kinetic energy operator. Note that $K_c$ in not necessarily diagonal in $i$ due to our choice of simple relative coordinates introducing the cross terms shown in Eq.~\eqref{eq:H0_in_SRC}. For example, for a three-body system in $d=3$ dimensions, this construction amounts to a sparse matrix with entries
|
||||||
|
%
|
||||||
|
\begin{spliteq}
|
||||||
|
&\bra{k_{1,1}, k_{1,2}, k_{1,3}, k_{2,1}, k_{2,2}, k_{2,3}} H_0 \ket{l_{1,1}, l_{1,2}, l_{1,3}, l_{2,1}, l_{2,2}, l_{2,3}} \\
|
||||||
|
&\hspace{2em}= \frac{1}{2\mu} \left\{ \left[(p^2)_{k_{1,1},l_{1,1}} + (p^2)_{k_{2,1},l_{2,1}} + p_{k_{1,1},l_{1,1}} p_{k_{2,1},l_{2,1}}\right]
|
||||||
|
\delta_{k_{1,2},l_{1,2}} \delta_{k_{1,3},l_{1,3}} \delta_{k_{2,2},l_{2,2}} \delta_{k_{2,3},l_{2,3}} \right. \\
|
||||||
|
&\hspace{4em}+ \left[(p^2)_{k_{1,2},l_{1,2}} + (p^2)_{k_{2,2},l_{2,2}} + p_{k_{1,2},l_{1,2}} p_{k_{2,2},l_{2,2}}\right]
|
||||||
|
\delta_{k_{1,1},l_{1,1}} \delta_{k_{1,3},l_{1,3}} \delta_{k_{2,1},l_{2,1}} \delta_{k_{2,3},l_{2,3}} \\
|
||||||
|
& \left. \hspace{4em}+ \left[(p^2)_{k_{1,3},l_{1,3}} + (p^2)_{k_{2,3},l_{2,3}} + p_{k_{1,3},l_{1,3}} p_{k_{2,3},l_{2,3}}\right]
|
||||||
|
\delta_{k_{1,1},l_{1,1}} \delta_{k_{1,2},l_{1,2}} \delta_{k_{2,1},l_{2,1}} \delta_{k_{2,2},l_{2,2}} \right\} \, .
|
||||||
|
\label{eq:d3n3}
|
||||||
|
\end{spliteq}
|
||||||
|
%
|
||||||
|
The matrix representations of the momentum operator $p$ will be discussed later in Sec.~\ref{sec:DVR}. The potential energy matrix is diagonal in the configuration space (assuming local potentials) and therefore takes the form,
|
||||||
|
%
|
||||||
|
\begin{spliteq}
|
||||||
|
&\bra{k_{1,1}, k_{1,2}, k_{1,3}, k_{2,1}, k_{2,2}, k_{2,3}} V \ket{l_{1,1}, l_{1,2}, l_{1,3}, l_{2,1}, l_{2,2}, l_{2,3}} \\
|
||||||
|
&\hspace{2em}= V(x_{1,1}, x_{1,2}, x_{1,3}, x_{2,1}, x_{2,2}, x_{2,3}) \delta_{k_{1,1},l_{1,1}} \delta_{k_{1,2},l_{1,2}} \delta_{k_{1,3},l_{1,3}} \delta_{k_{2,1},l_{2,1}} \delta_{k_{2,2},l_{2,2}} \delta_{k_{2,3},l_{2,3}} \, .
|
||||||
|
\end{spliteq}
|
||||||
|
%
|
||||||
|
This resulting sparsity of the matrix should be exploited for numerical implementations, as the matrix is simply too large be be stored in computer memory for most practical applications. Alternatively, the matrix elements can be calculated on-demand when the multiplying with a vector. This is made possible due to most eigenvalue algorithms such as the implicitly restarted Arnoldi method (IRAM) only requiring the results of matrix-vector multiplications for a set of given vectors~\citep{lehoucq1998arpack}.
|
||||||
|
|
||||||
|
Alternatively, the tensor decomposition of the basis states can be exploited with the fact that modern computers (especially GPUs) are getting more and more efficient at large-scale tensor contractions, due to the rising popularity of machine learning. To this end, let us express the wave function as a rank-$\left[(n-1)\times d\right]$ tensor.
|
||||||
|
\begin{equation}
|
||||||
|
\braket{k_{{1,1}},\cdd k_{{1,d}},\cdd k_{{n-1,1}},\cdd k_{{n-1,d}}|\psi}=\psi_{k_{{1,1}},\cdd k_{{1,d}},\cdd k_{{n-1,1}},\cdd k_{{n-1,d}}}
|
||||||
|
\end{equation}
|
||||||
|
Similarly, the Hamiltonian can be written as,
|
||||||
|
\begin{equation}
|
||||||
|
\braket{k_{{1,1}},\cdd k_{{1,d}},\cdd k_{{n-1,1}},\cdd k_{{n-1,d}}|H|l_{{1,1}},\cdd l_{{1,d}},\cdd l_{{n-1,1}},\cdd l_{{n-1,d}}}=H^{l_{{1,1}},\cdd l_{{1,d}},\cdd l_{{n-1,1}},\cdd l_{{n-1,d}}}_{k_{{1,1}},\cdd k_{{1,d}},\cdd k_{{n-1,1}},\cdd k_{{n-1,d}}} \, .
|
||||||
|
\end{equation}
|
||||||
|
Now the free Hamiltonian derived in Eq.~\eqref{eq:H0_in_SRC} can be applied via,
|
||||||
|
\begin{equation}
|
||||||
|
(H_0 \psi)_{k_{{1,1}},\cdd k_{{1,d}},\cdd k_{{n-1,1}},\cdd k_{{n-1,d}}} =
|
||||||
|
\frac{1}{2\mu} \sum_{c=1}^d \sum_{i=1}^{n-1} \left( p^{l_{i,c}}_{k_{i,c}} \sum_{j=1}^i p^{l_{j,c}}_{k_{j,c}} \psi_{l_{{1,1}},\cdd l_{{1,d}},\cdd l_{{n-1,1}},\cdd l_{{n-1,d}}} \right) \, ,
|
||||||
|
\end{equation}
|
||||||
|
where Einstein summation is implied. The potential energy can be trivially implemented as point-wise multiplication with the diagonal elements of the potential energy tensor $V_{k_{{1,1}},\cdd k_{{1,d}},\cdd k_{{n-1,1}},\cdd k_{{n-1,d}}}^{k_{{1,1}},\cdd k_{{1,d}},\cdd k_{{n-1,1}},\cdd k_{{n-1,d}}}$.
|
||||||
|
\ny{Illustrate as a tensor network.}
|
||||||
|
|
||||||
|
\section{Discrete variable representation}
|
||||||
|
\label{sec:DVR}
|
||||||
|
|
||||||
|
Now we are left with the task of implementing the 1D momentum operator $p={-}\ii\partial$ in discretized configuration space. The simplest approach would be to adopt a finite-difference scheme for approximating the derivative, one of the most basic of which being,
|
||||||
|
\begin{equation}
|
||||||
|
\partial^l_k = \frac{n}{L} (\delta^{l+1}_k - \delta^l_k) \, ,
|
||||||
|
\end{equation}
|
||||||
|
where we have used the tensor notation again for convenience. However, this leads to significant numerical errors, even for a wave function as basic as a plane-wave $\psi(x)=\exp(\ii k x)$, as discussed in \cite{Konig:2020lzo}.
|
||||||
|
|
||||||
|
As pointed out in \cite{Bulgac:2013mz}, calculating the derivative in the \emph{discrete} Fourier basis, leads to much more accurate results. One could employ a fast Fourier transform (FFT) algorithm to efficiently go between representations and apply the kinetic energy operator efficiently in its diagonal basis. Since such an operation is still a linear map in the configuration space, alternatively, we can derive the ``derivative operator'' $\partial$ as a matrix or a rank-2 tensor, which when multiplied gives the same result numerically~\citep{Klos:2018sen}.
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
\partial^l_k = \frac{\pi (-1)^{k-l}}{L}
|
||||||
|
\begin{cases}
|
||||||
|
{-}\ii & \text{if } k=l \\
|
||||||
|
\frac{\exp \left[ {-}\ii\frac{\pi (k-l)}{N} \right]} {\sin{\frac{\pi (k-l)}{N}}} & \text{otherwise} \\
|
||||||
|
\end{cases}
|
||||||
|
\label{eq:daba_matrix}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
The second derivative equivalent of above can also be useful for optimizing the implementation.
|
||||||
|
%
|
||||||
|
\begin{equation}
|
||||||
|
(\partial^2)^l_k = {-} \frac{\pi^2 (-1)^{k-l}}{L^2}
|
||||||
|
\begin{cases}
|
||||||
|
\frac{N^2 + 2}{3} & \text{if } k=l \\
|
||||||
|
\frac{2} {\sin^2{\frac{\pi (k-l)}{N}}} & \text{otherwise} \\
|
||||||
|
\end{cases}
|
||||||
|
\label{eq:daba2_matrix}
|
||||||
|
\end{equation}
|
||||||
|
%
|
||||||
|
|
||||||
|
\section{Resonances as avoided crossings}
|
||||||
|
|
||||||
|
[\ldots]
|
||||||
|
|
||||||
|
\section{Bound-to-resonance extrapolation in finite-volume}
|
||||||
|
\label{sec:B2R_FV}
|
||||||
|
|
||||||
|
As proven in~~\ref{TBA}, it is possible to employ EC to extrapolate from bound states to resonances. But for situations where complex-scaling is not possible, it is worth studying the same approach in finite volume where resonances can be indirectly identified from avoided crossings. That is, we attempt to train with finite volume bound states, which does not contain any avoided crossings, and then extrapolate to the resonant regime to see if resonances manifest as avoided crossings.
|
||||||
|
|
||||||
|
Consider 2 particles in a 3-D periodic box interacting with a potential,
|
||||||
|
\begin{equation}
|
||||||
|
V(c,r)=-(0.5-c)\exp \left( \frac{-r^2}{30} \right) +(0.5+c)\exp \left(\frac{-(r-7)^2}{9} \right) \, ,
|
||||||
|
\label{eq:Yapa_potential}
|
||||||
|
\end{equation}
|
||||||
|
where $c$ is a parameter to be varied. This potential is engineered to have a barrier at large $c$ values that can support a resonance, and a trough at small $c$ values that can support a bound state. The shape of the potential for 3 values of $c$ are shown in Fig.~\ref{fig:potential}.
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=\textwidth]{Chapter-5/potential.pdf}
|
||||||
|
\caption{The shape of the potential given by Eq.~\eqref{eq:Yapa_potential} for 3 values of $p$. As shown, the last curve contains a pronounced barrier, suggesting the possibility to support a resonance.}
|
||||||
|
\label{fig:potential}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
The EC procedure is carried out by training at two points, $c=-0.45$ and $c=-0.40$, and then extrapolating at $c=0.10$, for each $L$ value. The resulting spectra are shown in Fig.~\ref{fig:spectra}. The extrapolated spectrum seems to show an avoided crossing. A zoomed in version of this spectrum is shown in Fig.~\ref{fig:last_spectrum} along with an exact calculation for comparison. As it can be seen, the exact and the extrapolated spectrum seems to agree closely, displaying a clear avoided crossing at the same energy level. This rather remarkable result shows that EC can predict an avoided crossing corresponding to a resonance, even when trained with bound spectra that does not contain at least a hint of such a feature.
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=\textwidth]{Chapter-5/spectra.pdf}
|
||||||
|
\caption{The finite volume spectra for the two $c$ values, $c=-0.45$ and $c=-0.40$, shown on the left and the middle respectively, that are used for training the EC basis. On the right, the extrapolated spectrum is shown which shows a clear avoided crossing, thereby implying the existence of a resonance.}
|
||||||
|
\label{fig:spectra}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=0.7\textwidth]{Chapter-5/last_spectrum.pdf}
|
||||||
|
\caption{The finite volume spectra for $c=0.10$ in Fig.~\ref{fig:spectra} zoomed in, along with an exact calculation of the spectrum as a comparison. Both spectra seems to agree well, as far as the energy level of the avoided crossing due to the resonance is concerned.}
|
||||||
|
\label{fig:last_spectrum}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\section{Volume extrapolation}
|
||||||
|
|
||||||
|
In previous calculations, we still had to deal with repeatedly solving the exact spectrum for a range of $L$ values. It turns out that we can employ EC again to tackle this problem. In this section, we will empirically show how that it is possible to obtain a highly accurate finite-volume spectrum only with few $L$ values, as opposed to mesh of $L$ values dense enough to identify important features such as avoided crossings.
|
||||||
|
|
||||||
|
We lay out the formalism for ``finite-volume eigenvector continuation (FVEC)'', where we extrapolate from periodic boxes with sizes $L_i$, $i=1,\cdd N$ to a target volume $L_*$. This should be distinguished from performing at a fixed single volume $L$ to extrapolate a parametric dependence of the Hamiltonian as we did in Sec.~\ref{sec:B2R_FV}. Specifically, we want to consider states $\ket{\psi_{L_i}}$ at volume $L_i$ (or sets of states $\{\ket{\psi_{L_i}^{(j)}},\,j=1,\cdd N_i\}$) and perform EC using Hamiltonian and norm matrices
|
||||||
|
%
|
||||||
|
\begin{subalign}[eq:H-N-naive]
|
||||||
|
H_{ij} &= \braket{\psi_{L_i} | H_{L_*} | \psi_{L_j}} \,, \\
|
||||||
|
N_{ij} &= \braket{\psi_{L_i} | \psi_{L_j}} \,.
|
||||||
|
\end{subalign}
|
||||||
|
%
|
||||||
|
However, at face value the above definitions appear problematic because the dependence on $L$ does not simply stem from the Hamiltonian; it is inherent in the definition of the Hilbert space. Two states $\ket{\psi_{L_i}}$ and $\ket{\psi_{L_j}}$ are actually vectors in different Hilbert spaces for $i\neq j$, and it is not immediately clear how the matrix elements written down naively in Eqs.~\eqref{eq:H-N-naive} can be well-defined quantities. To resolve this issue, we develop the notion of a vector space that accommodates states with arbitrary periodicities and show how it relates to FVEC calculations.
|
||||||
|
|
||||||
|
\ny{Incomplete}
|
||||||
|
|
||||||
|
\section{Direct resonances in finite volume}
|
||||||
|
|
||||||
|
[\ldots]
|
||||||
|
|
||||||
|
\section{Finite-volume correction}
|
||||||
|
|
||||||
|
The L\"uscher formalism \citep{Luscher:1985dn,Luscher:1986pf,Luscher:1990ux} for bound-states lets us calculate the energy correction $\Delta E = E_L - E_\infty$ for a two-body system simulated in a periodic box of length $L$, compared to the same system in infinite volume. This lets one extrapolate the energy, from some finite $L$ larger than the range of the potential $R$, to infinite volume, $L \to \infty$, which is usually the quantity of interest. We start by constructing the finite-volume Hamiltonian $H_L = H_0 + V_L$ by replicating the interaction potential $V$ (assumed to be local for simplicity) at each cubic image,
|
||||||
|
\begin{equation}
|
||||||
|
V_L(\vec r) = \sum_{\vec n \in \mathbb{Z}^3} V(\vec r + \vec n L) \, .
|
||||||
|
\end{equation}
|
||||||
|
Inspired by this, we construct and approximation for the finite-volume wave function $\psi_L(\vec r)$,
|
||||||
|
\begin{equation}
|
||||||
|
\Tilde{\psi}_L(\vec r) = \sum_{\vec n \in \mathbb{Z}^3} \psi_\infty(\vec r + \vec n L) \, .
|
||||||
|
\end{equation}
|
||||||
|
To justify this approximation, let us see the effect of multiplying the above with $H_L$.
|
||||||
|
\begin{spliteq}
|
||||||
|
H_L \Tilde\psi_L(\vec r)
|
||||||
|
&= H_0 \sum_{\vec n'} \psi_\infty(\vec r + \vec n' L) + \sum_{\vec n'} \sum_{\vec n} V(\vec r + \vec n L) \psi_\infty(\vec r + \vec n' L) \\
|
||||||
|
&= \sum_{\vec n'} \left\{ \left[ H_0 + V(\vec r + \vec n' L) \right] \psi_\infty(\vec r + \vec n' L) + \sum_{\vec n \neq \vec n'} V(\vec r + \vec n L) \psi_\infty(\vec r + \vec n' L) \right\} \\
|
||||||
|
&= E_\infty \sum_{\vec n'} \psi_\infty(\vec r + \vec n' L) + \sum_{\vec n'} \sum_{\vec n \neq \vec n'} V(\vec r + \vec n L) \psi_\infty(\vec r + \vec n' L) \\
|
||||||
|
&= E_\infty \, \Tilde\psi_L(\vec r) + \eta(\vec r) \\
|
||||||
|
\end{spliteq}
|
||||||
|
where we have defined,
|
||||||
|
\begin{spliteq}
|
||||||
|
\eta(\vec r) &= \sum_{\vec n'} \sum_{\vec n \neq \vec n'} V(\vec r + \vec n L) \psi_\infty(\vec r + \vec n' L) \\
|
||||||
|
&= \sum_{\vec n} \sum_{\vec n' \neq 0} V(\vec r + \vec n L) \psi_\infty(\vec r + (\vec n + \vec n') L) \, . \\
|
||||||
|
\end{spliteq}
|
||||||
|
Now, based on the previous discussion in Sec.~\ref{TBA}, we consider the asymptotic form of the wave function for the region $|\vec r| > R$,
|
||||||
|
\begin{equation}
|
||||||
|
\psi_\infty(\vec r) \to \ii^l \gamma \, Y^m_l(\vec {\hat r}) \frac{\hat h^+_l(\ii \kappa r)}{r} = \mathcal O (\ee^{{-}\kappa r}) \, .
|
||||||
|
\end{equation}
|
||||||
|
Now we can see that $\eta(\vec r)$ only has contributions from the asymptotic region of the wave function since $V(r) = 0$ for $r > L/2$. Moreover, these contributions are suppressed exponentially with respect to $\kappa|\vec n'|L$, with the leading contribution being of order $\mathcal O (\ee^{{-}\kappa L})$. Therefore we have proved that $\Tilde\psi_L(\vec r)$ is an \emph{approximate} eigenfunction of $H_L$ with an eigenvalue of $E_\infty$.
|
||||||
|
|
||||||
|
To estimate the finite-volume correction, we first consider the matrix element $\braket{\psi_L|H_L|\Tilde\psi_L}$. Acting $H_L$ towards the left gives,
|
||||||
|
\begin{equation}
|
||||||
|
\braket{\psi_L|H_L|\Tilde\psi_L} = E_L \braket{\psi_L|\Tilde\psi_L} \, ,
|
||||||
|
\end{equation}
|
||||||
|
whereas acting towards the right gives,
|
||||||
|
\begin{equation}
|
||||||
|
\braket{\psi_L|H_L|\Tilde\psi_L} = E_\infty \braket{\psi_L|\Tilde\psi_L} + \braket{\psi_L|\eta} \, ,
|
||||||
|
\end{equation}
|
||||||
|
so that we have,
|
||||||
|
\begin{equation}
|
||||||
|
E_L - E_\infty = \frac{\braket{\psi_L|\eta}}{\braket{\psi_L|\Tilde\psi_L}}\, .
|
||||||
|
\end{equation}
|
||||||
|
It can be shown that this expression reduces to,
|
||||||
|
\begin{equation}
|
||||||
|
\Delta E = E_L - E_\infty = \sum_{|\vec n| = 1} \int \dd^3 r \, \psi^*_\infty(\vec r) V(\vec r) \psi_\infty(\vec r + \vec n L) + \mathcal O (\ee^{{-}\sqrt{2}\kappa r}) \, .
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\ny{Incomplete}
|
||||||
Binary file not shown.
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|
|
@ -1,4 +1,4 @@
|
||||||
@article{Frame:2017fah,
|
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|
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@ -9,3 +9,368 @@
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year = "1971"
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{Balslev:1971vb,
|
||||||
|
author = "Balslev, E. and Combes, J. M.",
|
||||||
|
title = "{Spectral properties of many-body schroedinger operators with dilatation-analytic interactions}",
|
||||||
|
doi = "10.1007/BF01877511",
|
||||||
|
journal = "Commun. Math. Phys.",
|
||||||
|
volume = "22",
|
||||||
|
pages = "280--294",
|
||||||
|
year = "1971"
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{Reinhardt:1982aa,
|
||||||
|
author = {Reinhardt, W. P.},
|
||||||
|
title = {Complex coordinates in the theory of atomic and molecular structure and dynamics},
|
||||||
|
journal = {Annu. Rev. Phys. Chem.},
|
||||||
|
volume = {33},
|
||||||
|
pages = {223},
|
||||||
|
year = {1982},
|
||||||
|
keywords = {},
|
||||||
|
url = {https://dx.doi.org/10.1146/annurev.pc.33.100182.001255}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{ho_method_1983,
|
||||||
|
title = {The method of complex coordinate rotation and its applications to atomic collision processes},
|
||||||
|
volume = {99},
|
||||||
|
issn = {0370-1573},
|
||||||
|
url = {https://www.sciencedirect.com/science/article/pii/0370157383901126},
|
||||||
|
doi = {https://doi.org/10.1016/0370-1573(83)90112-6},
|
||||||
|
number = {1},
|
||||||
|
journal = {Physics Reports},
|
||||||
|
author = {Ho, Y. K.},
|
||||||
|
year = {1983},
|
||||||
|
pages = {1--68},
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{Konig:2019adq,
|
||||||
|
title = {Eigenvector Continuation as an Efficient and Accurate Emulator for Uncertainty Quantification},
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|
author = {K{\"o}nig, S. and Ekstr{\"o}m, A. and Hebeler, K. and Lee, D. and Schwenk, A.},
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year = {2020},
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journal = {Physics Letters B},
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volume = {810},
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pages = {135814},
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issn = {0370-2693},
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doi = {10.1016/j.physletb.2020.135814}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{Moiseyev:1978aa,
|
||||||
|
title = {Resonance Properties of Complex-Rotated Hamiltonians},
|
||||||
|
author = {Moiseyev, N. and Certain, P.R. and Weinhold, F.},
|
||||||
|
year = {1978},
|
||||||
|
volume = {36},
|
||||||
|
number = {6},
|
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|
pages = {1613--1630},
|
||||||
|
issn = {0026-8976},
|
||||||
|
doi = {10.1080/00268977800102631},
|
||||||
|
langid = {english},
|
||||||
|
journal = {Mol. Phys.}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{Moiseyev:1998aa,
|
||||||
|
title = {Quantum Theory of Resonances: Calculating Energies, Widths and Cross-Sections by Complex Scaling},
|
||||||
|
shorttitle = {Quantum Theory of Resonances},
|
||||||
|
author = {Moiseyev, Nimrod},
|
||||||
|
year = {1998},
|
||||||
|
volume = {302},
|
||||||
|
number = {5},
|
||||||
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pages = {212--293},
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|
issn = {0370-1573},
|
||||||
|
doi = {10.1016/S0370-1573(98)00002-7},
|
||||||
|
langid = {english},
|
||||||
|
journal = {Phys. Rept.}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{Afnan:1991kb,
|
||||||
|
author = "Afnan, I. R.",
|
||||||
|
title = "{Resonances in few body systems}",
|
||||||
|
doi = "10.1071/PH910201",
|
||||||
|
journal = "Austral. J. Phys.",
|
||||||
|
volume = "44",
|
||||||
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pages = "201--216",
|
||||||
|
year = "1991"
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||||||
|
}
|
||||||
|
|
||||||
|
@book{Moiseyev:2011,
|
||||||
|
title = {Non-{{Hermitian Quantum Mechanics}}},
|
||||||
|
author = {Moiseyev, Nimrod},
|
||||||
|
year = {2011},
|
||||||
|
publisher = {{Cambridge University Press}},
|
||||||
|
isbn = {978-1-139-49699-5}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{Luscher:1985dn,
|
||||||
|
title = {Volume dependence of the energy spectrum in massive quantum field theories - {I}. {Stable} particle states},
|
||||||
|
volume = {104},
|
||||||
|
issn = {1432-0916},
|
||||||
|
url = {https://doi.org/10.1007/BF01211589},
|
||||||
|
doi = {10.1007/BF01211589},
|
||||||
|
language = {en},
|
||||||
|
number = {2},
|
||||||
|
urldate = {2021-01-25},
|
||||||
|
journal = {Comm. Math. Phys.},
|
||||||
|
author = {Lüscher, M.},
|
||||||
|
month = jun,
|
||||||
|
year = {1986},
|
||||||
|
pages = {177--206},
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{Luscher:1986pf,
|
||||||
|
title = {Volume dependence of the energy spectrum in massive quantum field theories - {II}. {Scattering} states},
|
||||||
|
volume = {105},
|
||||||
|
issn = {1432-0916},
|
||||||
|
url = {https://doi.org/10.1007/BF01211097},
|
||||||
|
doi = {10.1007/BF01211097},
|
||||||
|
language = {en},
|
||||||
|
number = {2},
|
||||||
|
urldate = {2021-01-25},
|
||||||
|
journal = {Comm. Math. Phys.},
|
||||||
|
author = {Lüscher, M.},
|
||||||
|
month = jun,
|
||||||
|
year = {1986},
|
||||||
|
pages = {153--188},
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{Luscher:1990ux,
|
||||||
|
title = {Two-particle states on a torus and their relation to the scattering matrix},
|
||||||
|
volume = {354},
|
||||||
|
issn = {0550-3213},
|
||||||
|
url = {https://www.sciencedirect.com/science/article/pii/0550321391903666},
|
||||||
|
doi = {10.1016/0550-3213(91)90366-6},
|
||||||
|
language = {en},
|
||||||
|
number = {2-3},
|
||||||
|
urldate = {2021-01-25},
|
||||||
|
journal = {Nucl. Phys. B},
|
||||||
|
author = {Lüscher, M.},
|
||||||
|
month = may,
|
||||||
|
year = {1991},
|
||||||
|
note = {Publisher: North-Holland},
|
||||||
|
pages = {531--578},
|
||||||
|
}
|
||||||
|
|
@ -24,7 +24,6 @@
|
||||||
\usepackage{bbm}
|
\usepackage{bbm}
|
||||||
\usepackage{nicefrac}
|
\usepackage{nicefrac}
|
||||||
\usepackage{slashed}
|
\usepackage{slashed}
|
||||||
%\usepackage{hyperref} % cannot compile with this
|
|
||||||
\usepackage{leftidx}
|
\usepackage{leftidx}
|
||||||
\usepackage{environ}
|
\usepackage{environ}
|
||||||
\usepackage{mathtools}
|
\usepackage{mathtools}
|
||||||
|
|
@ -102,7 +101,7 @@
|
||||||
%\memberIV{Member 4 name} % unnecessary if committeesize=3, 4
|
%\memberIV{Member 4 name} % unnecessary if committeesize=3, 4
|
||||||
|
|
||||||
%% Student writing thesis, \student{First Middle}{Last}
|
%% Student writing thesis, \student{First Middle}{Last}
|
||||||
\student{Nuwan Subhashana}{Yapa} % a full middle name
|
\student{Yapa Mudiyanselage Nuwan Subhashana}{Yapa} % a full middle name
|
||||||
|
|
||||||
%% Degree program e.g. Marine, Earth, and Atmospheric Science
|
%% Degree program e.g. Marine, Earth, and Atmospheric Science
|
||||||
\program{Physics}
|
\program{Physics}
|
||||||
|
|
|
||||||
|
|
@ -20,7 +20,7 @@ Abstract text ...
|
||||||
|
|
||||||
%% -------------------------------- Biography ------------------------------- %%
|
%% -------------------------------- Biography ------------------------------- %%
|
||||||
\begin{biography}
|
\begin{biography}
|
||||||
The author was born in Kandy, Sri Lanka. \ldots
|
Also known as Nuwan Yapa, the author was born in Kandy, Sri Lanka. \ldots
|
||||||
\end{biography}
|
\end{biography}
|
||||||
|
|
||||||
%% ----------------------------- Acknowledgements --------------------------- %%
|
%% ----------------------------- Acknowledgements --------------------------- %%
|
||||||
|
|
|
||||||
|
|
@ -246,4 +246,13 @@
|
||||||
|
|
||||||
\newtheorem{lemma}{Lemma}
|
\newtheorem{lemma}{Lemma}
|
||||||
|
|
||||||
|
\newcommand\abssym[1]{\left|#1\right|}
|
||||||
|
\newcommand\normsym[1]{\left\Vert#1\right\Vert}
|
||||||
|
\newcommand\norm[1]{\|#1\|}
|
||||||
|
\newcommand\parenth[1]{\left(#1\right)}
|
||||||
|
|
||||||
|
\newcommand{\Jostf}{\mathcal{F}}
|
||||||
|
|
||||||
\newcommand{\ny}[1]{\textcolor{gray}{NY: #1}}
|
\newcommand{\ny}[1]{\textcolor{gray}{NY: #1}}
|
||||||
|
|
||||||
|
\newcommand{\braketmatrix}[3]{\left \langle #1 \middle| #2 \middle| #3 \right \rangle}
|
||||||
|
|
|
||||||
10
optional.tex
10
optional.tex
|
|
@ -13,11 +13,11 @@
|
||||||
%% the links are not visually distinct from normal text (i.e. no change
|
%% the links are not visually distinct from normal text (i.e. no change
|
||||||
%% in color or extra boxes).
|
%% in color or extra boxes).
|
||||||
\usepackage[
|
\usepackage[
|
||||||
pdfauthor={Change author in optional.tex},
|
pdfauthor={Nuwan Yapa},
|
||||||
pdftitle={Change title in optional.tex},
|
pdftitle={Quantum few-body problem in nuclear theory},
|
||||||
pdfcreator={pdftex},
|
pdfcreator={pdftex},
|
||||||
pdfsubject={NC State ETD Dissertation},
|
pdfsubject={NC State ETD Dissertation},
|
||||||
pdfkeywords={change key words in optional.tex},
|
pdfkeywords={nuclear theory quantum physics},
|
||||||
colorlinks=true,
|
colorlinks=true,
|
||||||
linkcolor=black,
|
linkcolor=black,
|
||||||
citecolor=black,
|
citecolor=black,
|
||||||
|
|
@ -53,8 +53,8 @@
|
||||||
|
|
||||||
|
|
||||||
%% Utopia
|
%% Utopia
|
||||||
\usepackage[T1]{fontenc}
|
%\usepackage[T1]{fontenc}
|
||||||
\usepackage[adobe-utopia]{mathdesign}
|
%\usepackage[adobe-utopia]{mathdesign}
|
||||||
|
|
||||||
%% Palatino
|
%% Palatino
|
||||||
%\usepackage[T1]{fontenc}
|
%\usepackage[T1]{fontenc}
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue