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Gamow states; rigged Hilbert space; resonances; complex delta function; Breit-Wigner amplitude
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\title{\bf The resonance amplitude associated with the Gamow states}
\author{Rafael de la Madrid \\
\small{\it Department of Physics, The Ohio State University at Newark,
Newark, OH 43055} \\
\small{E-mail: \texttt{rafa@mps.ohio-state.edu}}}
\date{}
\maketitle
\begin{abstract}
\noindent The Gamow states describe the quasinormal modes of
quantum systems. It is shown that the resonance amplitude associated with the
Gamow states is given by the complex delta function. It is also shown that
under the near-resonance approximation of neglecting the lower bound of the
energy, such resonance amplitude becomes the Breit-Wigner amplitude. This
result establishes the precise connection between the Gamow states,
Nakanishi's complex delta function and the Breit-Wigner amplitude. In addition,
this result provides another theoretical basis for the
phenomenological fact that the almost-Lorentzian peaks in cross sections
are produced by intermediate, unstable particles.
\end{abstract}
\noindent {\it Keywords}: Gamow states; resonances; rigged Hilbert space;
complex delta function; Breit-Wigner lineshape
\noindent PACS: 03.65.-w, 03.65.Bz; 03.65.Ca; 03.65.Db
\section{Introduction}
\setcounter{equation}{0}
\label{sec:Intro}
Resonances appear in all areas of quantum physics, in both the relativistic
and non-relativistic regimes. Resonances are intrinsic properties of a
quantum system, and they describe the system's preferred ways of
decaying. Experimentally, resonances appear as sharp peaks
in the cross section that resemble the Breit-Wigner (Lorentzian) lineshape.
The Gamow states are the natural wave functions of resonances, and they were
introduced by Gamow in his paper on $\alpha$-decay of radioactive
nuclei~\cite{GAMOW}. Since then, they have been used by a number of authors,
see e.g.~\cite{SIEGERT,PEIERLS,HUMBLET,ZELDOVICH,BERGGREN,GASTON,BERGGREN78,SUDARSHAN,
MONDRAGON83,CURUTCHET,BL,LIND,BERGGREN96,BOLLINI,FERREIRA,BETAN,MICHEL1,AJP02,KAPUSCIK1,MONDRAGON03,
MICHEL2,KAPUSCIK2,MICHEL3,MICHEL4,MICHEL5,URRIES,MICHEL6,TOMIO}. Likewise
the bound
states, the Gamow states are properties of the Hamiltonian, and they are
associated with the natural frequencies of the system. The usefulness
of the Gamow states is attested by the remarkable success of the Gamow Shell
Model~\cite{MICHEL1,MICHEL2,MICHEL3,MICHEL4,MICHEL5,MICHEL6} and similar
nuclear-structure formalisms~\cite{FERREIRA,BETAN}.
Because resonances leave a quasi-Lorentzian fingerprint in the cross section,
and because the Gamow states are the natural wave functions of resonances,
the resonance amplitude associated with the Gamow states must be related
to the Breit-Wigner amplitude. The purpose of this paper is to show that
the resonance amplitude associated with the Gamow states
is proportional to the complex delta function,
$\delta (E-z_{\rm R})$, and that such amplitude can be
approximated in the near-resonance region by the Breit-Wigner amplitude. More
precisely, we will show that the transition amplitude from a resonance
state of energy $z_{\rm R}$ to a scattering state of energy $E\geq 0$,
${\cal A}(z_{\rm R} \to E)$, is given by
\begin{equation}
{\cal A}(z_{\rm R} \to E)= \rmi \sqrt{2\pi}{\cal N}_{\rm R}
\delta (E-z_{\rm R})
\simeq -\frac{{\cal N}_{\rm R}}{\sqrt{2\pi}} \frac{1}{E-z_{\rm R}} \, ,
\quad E\geq 0 \, ,
\label{aeqconsBW}
\end{equation}
where
\begin{equation}
{\cal N}_{\rm R}^2 \equiv \rmi \, {\rm res}[S(E)]_{E=z_{\rm R}} \equiv
\rmi \, {\rm r}_{\rm R} \, ,
\label{normafac}
\end{equation}
$S$ denotes the $S$ matrix, and ${\rm r}_{\rm R}$ denotes the residue
of $S$ at $z_{\rm R}$. In addition, we will see that the lower bound of
the energy (threshold) is the reason why this amplitude is not exactly but
only approximately given by the Breit-Wigner amplitude.
Section~\ref{sec:basics} provides a quick summary of the most important
properties of the Gamow states, along with some basic phenomenological
properties of resonances. The proof of~(\ref{aeqconsBW}) is provided in
Sec.~\ref{sec:proof}. The conclusions are included in
Sec.~\ref{sec:conclusions}.
For the sake of clarity, we shall prove Eq.~(\ref{aeqconsBW}) using the
example of the spherical shell potential for zero angular momentum. However,
as explained in Appendix~\ref{sec:appendix-gener}, the result is valid for
any partial wave and for spherically symmetric potentials that fall off
faster than exponentials. Finally, in Appendix~\ref{sec:appendix-cdf},
we provide a thorough characterization of the complex delta function
and its associated functional, since they have rarely appeared in the
literature.
\section{Basics of resonances and Gamow states}
\setcounter{equation}{0}
\label{sec:basics}
Resonance peaks are characterized by the energy $E_{\rm R}$
at which they occur and by their width $\Gamma _{\rm R}$. The resonance
peak is related to a pole of the $S$ matrix at the complex number
$z_{\rm R}= E_{\rm R} - \rmi \, \Gamma _{\rm R}/2$, because the theoretical
expression of the cross section in terms of the $S$ matrix fits the
experimental cross section in the neighborhood of $E_{\rm R}$, see
Eqs.~(\ref{Smex}) and~(\ref{crossS}) below.
When the peak is too narrow and its width cannot be measured, one measures
the lifetime $\tau _{\rm R}$ of the decaying particle. Decaying systems follow
the exponential decay law, except for short- and long-term deviations.
Although a decaying particle has a finite lifetime, it is otherwise assigned
all the properties that are attributed to stable particles, like
angular momentum, charge, spin and parity. For example, a radioactive nucleus
has a finite lifetime, but otherwise it possesses all the properties of stable
nuclei; in fact, it is included in the periodic table of the elements along
with the stable nuclei. Similarly, most elementary particles are
unstable, and they are listed along with the stable ones in the Particle Data
Table~\cite{PDT} and attributed
values for the mass, spin and width (or lifetime). Thus, stable particles
differ from unstable ones by the value of their width, which is zero
in the case of stable particles and different from zero in the case of
unstable ones. Hence, phenomenologically, unstable particles are not less
fundamental than the stable ones.
A priori, resonances and decaying particles are different entities. A
resonance refers to the energy distribution of the outgoing particles in
a scattering process, and it is characterized by its energy and
width. A decaying state is described in a time-dependent setting by its
energy and lifetime. Yet the difference is quantitative rather than
qualitative, and both concepts are related by
\begin{equation}
\Gamma _{\rm R}=\frac{\hbar}{\tau _{\rm R}} \, ,
\label{lifewidre}
\end{equation}
though in most systems one can measure either $\tau _{\rm R}$ or
$\Gamma _{\rm R}$, but not both.
Theoretically, however, the relation~(\ref{lifewidre}) is usually justified
as an approximation, $\tau _{\rm R}\Gamma _{\rm R} \sim \hbar$, as a kind of
time-energy uncertainty relation. For a long time, it was not possible to
experimentally check whether the relation~(\ref{lifewidre}) is exact or
approximate, since
the lifetime and width could not be measured in the same system. This
changed with the measurements of the width~\cite{OATES} and
lifetime~\cite{VOLZ} of the $3p\ ^2P_{3/2}$ state of Na, which provide a firm
experimental basis that Eq.~(\ref{lifewidre}) holds exactly, not just
approximately. Thus, resonances and decaying systems are two sides of the
same phenomenon.
Although the resonance peaks in the cross section resemble the Lorentzian,
the resonance lineshape does not coincide exactly with the
Lorentzian. Two features of the cross section reveal so. First, the maximum
of the resonance peak never occurs at $E=E_{\rm R}$, whereas the maximum
of the Lorentzian occurs exactly at $E=E_{\rm R}$. And second, the Laurent
expansion of the $S$ matrix around the resonance pole,
\begin{equation}
S(E)=\frac{{\rm r}_{\rm R}}{E-z_{\rm R}}+B(E) \, ,
\label{Smex}
\end{equation}
which produces the Lorentzian peak in the cross section~\cite{NOTE1},
\begin{equation}
\sigma \sim \frac{1}{(E-E_{\rm R})^2+(\Gamma _{\rm R}/2)^2} \, ,
\label{crossS}
\end{equation}
is valid only in the vicinity of the resonance pole. Because~(\ref{Smex})
and~(\ref{crossS}) are valid only in the vicinity of the resonance energy, the
Lorentzian lineshape is just a near-resonance approximation to the
exact resonance lineshape.
Because the Lorentzian does not coincide exactly with the resonance lineshape,
the Breit-Wigner amplitude cannot coincide exactly with the resonance
amplitude. One can reach the same conclusion by using the point of view of
decaying states as follows. The Breit-Wigner amplitude
yields the exponential decay law only when it is defined over the
whole of the energy real line $(-\infty , \infty )$ rather than just over the
scattering spectrum (see e.g.~\cite{FONDA}). Because in quantum mechanics
the scattering spectrum has a lower bound, the Breit-Wigner amplitude would
yield the exponential decay law only if it was defined also at energies that
do not belong to the scattering spectrum. Thus, the Breit-Wigner amplitude is
incompatible with the exponential decay law, and therefore cannot coincide
with the exact resonance/decay amplitude.
Mathematically, the Gamow states are eigenvectors of the Hamiltonian with
a complex eigenvalue $z_{\rm R}=E_{\rm R} -\rmi \, \Gamma _{\rm R}/2$,
\begin{equation}
H|z_{\rm R}\rangle = z_{\rm R} |z_{\rm R}\rangle \, ,
\label{tiSeq}
\end{equation}
and, in the radial position representation, they satisfy a
``purely outgoing boundary condition'' (POBC) at infinity:
\begin{equation}
\langle r|z_{\rm R}\rangle \sim
\rme ^{{\rm i}\sqrt{(2m/\hbar ^2)z_{\rm R}} \, r}
\, , \quad \mbox{as} \ r \to \infty \, .
\label{POBC}
\end{equation}
The time-independent Schr\"odinger equation~(\ref{tiSeq}) subject to the
POBC~(\ref{POBC}) is equivalent to the following integral equation of the
Lippmann-Schwinger type:
\begin{equation}
|z_{\rm R}\rangle =\frac{1}{z_{\rm R}-H_0+\rmi \hskip0.2mm 0}
V |z_{\rm R}\rangle \, ,
\label{inteGam}
\end{equation}
where $H_0$ is the free Hamiltonian and $V$ is the potential. Since
Eq.~(\ref{inteGam}) also yields
the bound states, the Gamow states are a natural generalization to resonances
of the wave functions of bound states.
%(i.e., of normal modes).
The bound and
resonance energies obtained by solving~(\ref{inteGam}) coincide with the
poles of the $S$ matrix.
The time evolution of a Gamow state is given by
\begin{equation}
\rme ^{-\rmi Ht/\hbar}|z_{\rm R}\rangle =
\rme ^{-\rmi E_{\rm R}t/\hbar} \rme ^{-\Gamma _{\rm R}t/(2\hbar)}
|z_{\rm R}\rangle \, ,
\label{tdSeq}
\end{equation}
and therefore the Gamow states abide by the exponential decay law. Because
the eigenvalue of Eq.~(\ref{tiSeq}) is also a pole of the $S$ matrix,
Eq.~(\ref{tdSeq}) implies that Eq.~(\ref{lifewidre}) holds. In this
way, the Gamow states unify the concepts of resonance and decaying
particle, and they provide a ``particle status'' for them.
Furthermore, since one can obtain both the bound and the resonance energies
from Eq.~(\ref{inteGam}), or from the poles of the $S$ matrix, resonances are
qualitatively the same as bound states. The only difference is
quantitative: The Gamow states have a non-zero width (i.e., finite lifetime),
whereas the bound states have a zero width (i.e., infinite lifetime).
An important feature of the Gamow states is that they form a basis that expands
any wave packet $\varphi ^+$, see e.g.~review~\cite{05CJP}. The basis
formed by the Gamow states is not complete though, and one has to add an
additional set of kets to complete the basis. In a system with several
resonances, we have that
\begin{equation}
\varphi ^+(t)=\sum_n \rme ^{-\rmi z_nt/\hbar}
|z_n\rangle \langle z_n|\varphi ^+\rangle
+\int_0^{-\infty}\rmd E \, \rme ^{-\rmi Et/\hbar}
|E^+\rangle \langle ^+E|\varphi ^+ \rangle \, ,
\label{resoexpan}
\end{equation}
where $z_n=E_n-\rmi \, \Gamma _n/2$ denotes the $n$th resonance energy. In this
equation, the sum contains the resonance contribution, whereas the integral
contains the background. For simplicity, we have omitted the
contribution from the bound states. The main virtue of resonance expansions
is to isolate each resonance's contribution to the wave packet.
Resonance expansions allow us to understand the deviations from exponential
decay~\cite{RAIZEN}. In the energy region where one resonance ${\rm R}$ is
dominant, the expansion~(\ref{resoexpan}) can be written as
\begin{equation}
\varphi ^+(t)= \rme ^{-\rmi z_{\rm R}t/\hbar}
|z_{\rm R}\rangle \langle z_{\rm R}|\varphi ^+\rangle +
{\rm background(R)} \, ,
\label{resoexpanR}
\end{equation}
where the term ``background(R)'' contains all contributions not associated
with the resonance ${\rm R}$, including those from other resonances. Because
``background(R)'' will always be nonzero, there will always be deviations
from exponential decay. The magnitude of these deviations depends
on how well we tune the system around the resonance energy: The better
we tune the system around the Gamow state $|z_{\rm R}\rangle$, the smaller
``background(R)'' will be. Note that ``background(R)'' is the analog to
the background $B(E)$ of the expansion~(\ref{Smex}).
\section{Proof}
\setcounter{equation}{0}
\label{sec:proof}
\subsection{Preliminaries}
\label{sec:prelmi}
The proof of~(\ref{aeqconsBW}) presented below is a straightforward application
of the theory of distributions. Rather than working in a general setting, we
will use the example of the spherical shell potential,
\begin{equation}
V({\bf x})= V(r)=\left\{ \begin{array}{ll}
0 &00$ and
$E$ has zero or negative imaginary part. The reason why we use this regulator
is that for complex $z$, the wave functions $\widehat{\psi}^-(z^*)^*$ grow
slower than $\rme ^{|{\rm Im}(\sqrt{2m/\hbar ^2 \, z})|^2}$ in the lower half
plane of the second sheet~\cite{LS2}. More precisely, Proposition~3
in~\cite{LS2} shows that for each $n=1,2,\ldots$ and for each $\beta >0$,
there is a $C>0$ such that in the lower half plane
of the second sheet, $\widehat{\psi}^-(z^*)^*$ is bounded by
\begin{equation}
|\widehat{\psi}^-(z^*)^*| \leq C \,
\frac{1} {|z|^{1/4}|1+z|^{n}} \,
\rme ^{\frac{\, \, |{\rm Im}(\sqrt{2m/\hbar ^2 \, z\,})|^2}{2\beta}} \, .
\label{boundinwhvalp+}
\end{equation}
This estimate implies that $\rme^{-\rmi\alpha z} \widehat{\psi}^-(z^*)^*$
tends to zero in the infinite arc of the lower half of the second sheet,
\begin{equation}
\lim_{z\to \infty} \rme ^{-\rmi z\alpha}\widehat{\psi}^-(z^*)^* = 0 \, ,
\quad \alpha >0 \, .
\label{indifli}
\end{equation}
In its turn, the limit~(\ref{indifli}) enables us to apply Cauchy's theorem
to obtain
\begin{equation}
\widehat{\psi}^-(z_{\rm R}^*)^*=
\lim _{\alpha \to 0}- \frac{1}{2\pi \rmi} \int_{-\infty}^{\infty}\rmd E \,
\rme ^{-\rmi \alpha E} \widehat{\psi}^-(E)^* \frac{1}{E-z_{\rm R}} \, ,
\label{CAUSK}
\end{equation}
where the integral is performed infinitesimally below the real axis of the
second sheet. By
multiplying both sides of~(\ref{CAUSK}) by $\rmi \sqrt{2\pi}{\cal N}_{\rm R}$,
and by recalling~(\ref{profosjd}), we obtain
\begin{equation}
\langle \widehat{\psi}^-|\widehat{z}_{\rm R}^-\rangle =
\lim _{\alpha \to 0} \int_{-\infty}^{\infty}\rmd E \,
\rme ^{-\rmi\alpha E} \psi ^-(E)^* (-1)
\frac{{\cal N}_{\rm R}}{\sqrt{2\pi}} \frac{1}{E-z_{\rm R}} \, .
\label{itenrmind}
\end{equation}
In the bra-ket notation, Eq.~(\ref{itenrmind}) reads as
\begin{equation}
\langle \widehat{\psi}^-|\widehat{z}_{\rm R}\rangle =
\lim _{\alpha \to 0} \int_{-\infty}^{\infty}\rmd E \,
\rme ^{-\rmi\alpha E} \langle \widehat{\psi}^-|\widehat{E}^-\rangle
\langle ^-\widehat{E}|\widehat{z}_{\rm R}\rangle \, .
\label{itenrmindb-k}
\end{equation}
Comparison of~(\ref{itenrmind}) with~(\ref{itenrmindb-k}) yields the
following expression for the amplitude~(\ref{tildeamp}):
\begin{equation}
\widetilde{ {\cal A}}(z_{\rm R} \to E) =
-\frac{{\cal N}_{\rm R}}{\sqrt{2\pi}} \frac{1}{E-z_{\rm R}} \, ,
\ E\in (-\infty , \infty ) \, .
\end{equation}
Thus, if the scattering spectrum was the whole real line, the resonance
amplitude would be exactly the Breit-Wigner amplitude. However, because
the scattering spectrum has a lower bound, the resonance amplitude is not
exactly the Breit-Wigner amplitude. Only when we can neglect the effect
of the threshold, the resonance amplitude coincides with the Breit-Wigner
amplitude:
\begin{equation}
{\cal A}(z_{\rm R} \to E) \simeq \widetilde{ {\cal A}}(z_{\rm R} \to E)
\, ,
\end{equation}
which is the approximation on the right-hand side of~(\ref{aeqconsBW}). In
particular, when the threshold can be ignored, the complex delta function
becomes for all intends and purposes the Breit-Wigner amplitude.
It should be stressed that the amplitude~(\ref{tildeamp}) is not physical,
because in~(\ref{tildeamp}) the energy $E$ runs over the whole real line rather
than over the scattering spectrum. However, such unphysical amplitude helps us
understand what the physical amplitude --the complex delta function-- is, by
allowing us to see how the resonance would decay if the scattering
spectrum was the whole real line.
\subsection{Further remarks}
\label{sec:fR}
Aside from phase space factors, cross sections are determined by the
transition amplitude from an ``in'' to an ``out'' state,
${\cal A}(E_i\to E_f )$. If $|E^{\pm}\rangle$ denote the ``in'' and ``out''
solutions of the Lippmann-Schwinger equation, then
\begin{equation}
{\cal A}(E_i\to E_f ) = \langle ^-E_f|E_i^+\rangle =
S(E_i) \delta (E_f -E_i) \, .
\end{equation}
If we imagine now that instead of an initial state $|E_i^+\rangle$ we had an
unstable particle $|z_{\rm R}\rangle$, the transition (decay) amplitude
${\cal A}(z_{\rm R} \to E_f )$ would be given by~(\ref{aeqconsBW}). Using the
approximate decay amplitude of~(\ref{aeqconsBW}), one obtains the following
approximate decay probability:
\begin{equation}
|{\cal A}(z_{\rm R} \to E_f )|^2 \simeq
\frac{|{\cal N}_{\rm R}|^2}{2\pi}
\frac{1}{(E_f-E_{\rm R})^2+ (\Gamma _{\rm R}/2)^2} \, ;
\label{almoslpro}
\end{equation}
that is, the decay probability of a resonance is given by the Lorentzian
when the effect of the threshold can be ignored.\footnote{A much more
detailed study of the dependence of the cross section (and
expectation values of observables) on the Breit-Wigner
amplitude can be found in e.g.~\cite{SIEGERT,BERGGREN78,BERGGREN96}.} Because
the almost-Lorentzian
decay probability~(\ref{almoslpro}) coincides with the almost-Lorentzian
peaks in cross sections, resonances can be interpreted as intermediate,
unstable particles.
Finally, it is worthwhile to compare the Gamow states with the states
introduced by Kapur and Peierls~\cite{KAPUR}. As mentioned above, the
Gamow states are eigenfunctions of the Hamiltonian that satisfy the
POBC~(\ref{POBC}) at infinity; the wave numbers involved in the
POBC~(\ref{POBC}) are complex and proportional to the square root of the
complex eigenenergies of the Gamow states; such complex eigenenergies are the
same as the poles of the $S$ matrix, and they do not depend on any external
parameter or energy. By contrast, the Kapur-Peierls states are
eigenfunctions of the Hamiltonian that satisfy a POBC at a finite radial
distance $r_0$, where $r_0$ is such that the potential vanishes for $r>r_0$;
the wave numbers involved in the POBC satisfied by the Kapur-Peierls
states are real and proportional to the square root of the real energy
of the incoming particle; the POBC satisfied by the Kapur-Peierls states
makes them
and their associated complex eigenenergies depend on $r_0$ and on the real
energy of the incoming particle; also, the complex eigenenergies of the
Kapur-Peierls states are not the same as the poles of the $S$ matrix. Thus, the
Kapur-Peierls states do not seem to be related to the standard Breit-Wigner
amplitude, because such amplitude does not depend on $r_0$ and its complex
energy does not depend on the energy of the incoming particle.
\section{Conclusions}
\label{sec:conclusions}
Since resonances leave an almost-Lorentzian fingerprint in the cross section,
and since the Gamow states are the wave functions of resonances, the decay
amplitude provided by a Gamow state should be linked to the Breit-Wigner
amplitude. In this paper, we have found that the precise link is given by
Eq.~(\ref{aeqconsBW}), and we have interpreted this result by saying that
the resonance amplitude associated with a Gamow state is exactly given by
the complex delta function, and that the Breit-Wigner amplitude is an
approximation to such resonance amplitude, which approximation is valid when
we can neglect the effect of the threshold. Thus, Eq.~(\ref{aeqconsBW})
establishes the precise relation between the Gamow state, Nakanishi's
complex delta function and the Breit-Wigner amplitude. In addition,
Eq.~(\ref{aeqconsBW}) affords another theoretical argument in favor of
interpreting the almost-Lorentzian peaks in cross sections as intermediate,
unstable particles---resonances are real (as opposed to virtual) particles,
in accordance with resonance phenomenology.
As is well known, the actual resonance lineshape of cross sections can be
very different from a quasi-Lorentzian one, due to the effect of thresholds,
other resonances, or extra channels. The usefulness
of~(\ref{aeqconsBW}) does not lie in predicting the exact shape of the cross
section, but rather in identifying what contribution to the cross section
comes from each pole of the $S$ matrix. In particular, although the
equality in Eq.~(\ref{aeqconsBW}) is always exact, for practical purposes
the approximation in Eq.~(\ref{aeqconsBW}) is useful only for narrow resonances.
When we add Eq.~(\ref{aeqconsBW}) to the other known properties of the Gamow
states, we see that such states have all the necessary properties to describe
resonance/unstable particles:
\begin{itemize}
\item[$\bullet$] They are associated with poles of the $S$ matrix.
\item[$\bullet$]They exhibit the correct phenomenological signatures of
both resonances (almost-Lorentzian lineshape) and unstable particles
(exponential decay), and they provide a firm theoretical basis
for~(\ref{lifewidre}).
\item[$\bullet$] They are basis vectors that isolate each resonance's
contribution to a wave packet.
\end{itemize}
\section*{Acknowledgments}
The author thanks Casey Koeninger for encouragement, and Alfonso Mondrag\'on
for enlightening criticisms. The author thanks Alfonso Mondrag\'on also for
his precise explanation of the relation between the Gamow and the Kapur-Peierls
states.
\appendix
\def\thesection{\Alph{section}}
\section{Generalizations}
\setcounter{equation}{0}
\label{sec:appendix-gener}
Equation~(\ref{aeqconsBW}) is not valid only for the spherical
shell potential~(\ref{potential}) but actually holds for a quite large
class of potentials. The reason can be found in well-known results of
scattering
theory~\cite{TAYLOR,NUSSENZVEIG}. As explained in~\cite{TAYLOR}, page~191,
partial wave analysis is valid whenever the spherically symmetric potential
satisfies the following requirements:
\begin{enumerate}
\item[$\widetilde{\rm I}$.] $V(r)=O(r^{-3-\epsilon})$ as $r\to \infty$.
\item[II.] $V(r)=O(r^{-3/2+\epsilon})$ as $r\to 0$.
\item[III.] $V(r)$ is continuous for $0b$ (with different expressions
for the Jost functions), the general proof goes through exactly the same lines
as the proof for the spherical shell potential. Finally, the argument extends
without difficulty to higher angular momentum.
\def\thesection{\Alph{section}}
\section{The complex delta functional}
\setcounter{equation}{0}
\label{sec:appendix-cdf}
In quantum mechanics, the complex delta function was originally introduced by
Nakanishi~\cite{NAKANISHI} to describe resonances in the Lee
model~\cite{LEE}. In mathematics, the complex delta function was introduced
by Gelfand and Shilov~\cite{GELFAND}. The purpose of this appendix is to
introduce the precise mathematical definition of the complex delta
function and to show that, when the test functions are analytic, such
definition coincides with the one given by Nakanishi.
\subsection{Three definitions of the (linear) complex delta functional}
The complex delta functional has different forms depending on the properties
of the test functions on which it acts. We shall review the three most
important forms, namely when the complex delta functional acts on
analytic functions (this form is used in this paper and and was
introduced in~\cite{GELFAND}), when it acts on meromorphic functions (this is
the form used by Nakanishi~\cite{NAKANISHI}), and when
it acts on non-meromorphic functions (this form was introduced
in~\cite{GELFAND}). When the space of test functions are analytic,
as is our case, these three forms coincide (as they should) and can be written
as in Eq.~(\ref{Cdelta-anti-mt}).
\subsubsection{First definition---the test functions are analytic}
\label{sec:firsdief}
According to page~1 of Volume~I of Ref.~\cite{GELFAND}, a distribution is a
function that associates a complex number with each function belonging to
a vector space:
\begin{equation}
\begin{array}{rcl}
{\rm distribution} :\{ {\rm Space \ of \ functions}\} & \longmapsto &
{\mathbb C}
\\
{\rm function} & \longmapsto & {\rm complex \ number} \, .
\end{array}
\label{distributions}
\end{equation}
The functions in the ``$\{ {\rm Space\ of\ functions}\}$'' are usually called
test
functions. Because a distribution maps functions into complex numbers, they
are usually called functionals. Such functionals can be linear or antilinear.
A more precise definition is the following. If $\Phi$ is a vector space
of test functions endowed with a topology, a linear (antilinear)
distribution $F$ is a function from $\Phi$ to $\mathbb C$
\begin{equation}
\begin{array}{rcl}
F : \Phi & \longmapsto & {\mathbb C} \\
\phi & \longmapsto & F(\phi)
\end{array}
\label{F}
\end{equation}
such that
\begin{itemize}
\item[({\it i})] $F$ is well defined,
\item[({\it ii})] $F$ is linear (antilinear),
\item[({\it iii})] $F$ is continuous.
\end{itemize}
A very important example of distribution is the (linear) Schwartz delta
functional at a real number $E$. Such functional associates
with each test function $\phi$ the value that $\phi$ takes at $E$:
\begin{equation}
\begin{array}{rcl}
\delta _E : \Phi_{\rm Schw} & \longmapsto & {\mathbb C} \\
\phi & \longmapsto & \delta _E(\phi)= \phi (E) \, ,
\end{array}
\label{SDdelta}
\end{equation}
where the test functions of $\Phi _{\rm Schw}$ are infinitely differentiable
and of polynomial falloff. It is
straightforward to show that definition~(\ref{SDdelta}) satisfies the
above requirements~({\it i})-({\it iii}).
The (linear) complex delta functional is defined in a completely
analogous way. As stated by Gelfand and Shilov~\cite[Vol.~2, page~85]{GELFAND},
the point $E$ in Eq.~(\ref{SDdelta}) may be complex in the spaces
of analytic functions. If
$\Phi _{\rm anal}$ denotes a vector space of {\it analytic} functions
at the complex point $z_0$, then the linear complex delta
functional at $z_0$ is defined as a function that associates with each test
function $\phi$ the value that the analytic continuation of $\phi$ takes
at $z_0$:
\begin{equation}
\begin{array}{rcl}
\delta _{z_0} : \Phi _{\rm anal} & \longmapsto & {\mathbb C} \\
\phi & \longmapsto & \delta _{z_0}(\phi)=
\phi (z_0) \, .
\end{array}
\label{Cdelta}
\end{equation}
Two important comments are in order here. First, the test functions of
$\Phi _{\rm anal}$ must be analytic at $z_0$; that is, $z_0$ is not
a singularity (e.g., a pole) of any $\phi$, otherwise
definition~(\ref{Cdelta}) makes no sense. And second, the complex delta
functional is completely specified by Eq.~(\ref{Cdelta}) because the
test functions are analytic at $z_0$, and therefore one does not need to
introduce any contour in the definition of~(\ref{Cdelta}), even though
one could use such a contour, as in Eq.~(\ref{Cdeltameroph}) below.
Definition~(\ref{Cdelta}) actually fulfills the
requirements~({\it i})-({\it iii}).\footnote{In this paper, we omit any
explicit discussion on the continuity requirement~({\it iii}). The
reason is that first, the continuity of the complex delta function is
guaranteed by the results of~\cite{LS2}, and second, continuity is not
essential to our main discussion.} The only property that is conceptually
challenging is~({\it i}). Because we are assuming that the test
functions are analytic at $z_0$, $\phi (z_0)$ exists and is unique, which
grants requirement~({\it i}). Thus, definition~(\ref{Cdelta}) completely,
rigorously and unambiguously defines the complex delta functional.
\subsubsection{Second definition---the test functions are meromorphic}
\label{sec:second-linear}
Many functions are not analytic
but just meromorphic. That is, when we analytically continue them, they have
isolated singularities (``poles'') in the complex plane. At such poles,
definition~(\ref{Cdelta}) makes no sense, and one has to extend it. If
$\Phi _{\rm mero}$ is a vector space of meromorphic functions at $z_0$, the
(linear) complex delta functional at $z_0$ is defined as
\begin{equation}
\begin{array}{rcl}
\delta _{z_0} : \Phi _{\rm mero} & \longmapsto & {\mathbb C} \\
\phi & \longmapsto & \delta _{z_0}(\phi)=
\frac{1}{2\pi \rmi} \oint \rmd z \, \frac{\phi (z)}{z-z_0} \, .
\end{array}
\label{Cdeltameroph}
\end{equation}
One can again check very easily that definition~(\ref{Cdeltameroph})
satisfies requirements~({\it i})-({\it iii}). Note that because in
definition~(\ref{Cdeltameroph}) the test functions are
meromorphic, such definition depends on Cauchy's theorem and on the
contour used.\footnote{The contour used in Eq.~(\ref{Cdeltameroph}) is assumed
to be a circle around $z_0$ such that the test function $\phi$ is analytic
inside such circle except perhaps at $z_0$.}
If we denote by $a_0$ the zeroth term of the Laurent expansion of
$\phi (z)$ around $z_0$, then definition~(\ref{Cdeltameroph}) associates
$a_0$ with each test function $\phi$, since
\begin{equation}
a_0=\frac{1}{2\pi \rmi} \oint \rmd z \, \frac{\phi (z)}{z-z_0} \, .
\end{equation}
Thus, we may write definition~(\ref{Cdeltameroph}) as
\begin{equation}
\begin{array}{rcl}
\delta _{z_0} : \Phi _{\rm mero} & \longmapsto & {\mathbb C} \\
\phi & \longmapsto & \delta _{z_0}(\phi)= a_0 \, .
\end{array}
\label{Cdeltameroph-alter}
\end{equation}
Obviously, both~(\ref{Cdeltameroph}) and~(\ref{Cdeltameroph-alter}) define
the same functional, because both associate the same complex number with
the same function, even though in~(\ref{Cdeltameroph-alter}) no contour
integral has been explicitly used.
Now, when $\phi (z)$ is analytic at $z_0$, $a_0$ is simply $\phi (z_0)$. Thus,
when the test functions are not just meromorphic but also {\it analytic}
at $z_0$, definitions~(\ref{Cdeltameroph}) and~(\ref{Cdeltameroph-alter})
become definition~(\ref{Cdelta}), {\sf because in such case all these
definitions associate each function $\phi$ with one and the
same complex number $\phi (z_0)$}. This is why, when the test functions
$\phi$ are all {\it analytic} at $z_0$, one can define the complex delta
functional by way of Eq.~(\ref{Cdelta}), as Gelfand and Shilov do
in page~85, Vol.~II of~\cite{GELFAND}.
\subsubsection{Third definition---the test functions are not meromorphic}
\label{sec:third-linear}
When the test functions are not meromorphic, definitions~(\ref{Cdelta}),
(\ref{Cdeltameroph}) and~(\ref{Cdeltameroph-alter}) make no sense. One
can still define a complex delta
functional at the origin following the prescription of Gelfand and
Shilov~\cite[Vol.~I, Appendix B]{GELFAND}. When the functions are
meromorphic, such definition of the complex delta functional at
the origin becomes~(\ref{Cdeltameroph}) and~(\ref{Cdeltameroph-alter}).
However, because in this paper we use test functions that are analytic
at the resonance energies, we do not need to use this general definition or
definition~(\ref{Cdeltameroph}), because all these definitions actually
become~(\ref{Cdelta}).
\subsection{Three definitions of the (antilinear) complex delta functional}
\label{sec:anti}
In this paper, we have used antilinear (rather than linear)
functionals. We will therefore briefly explain how one
defines such functionals for the cases considered in the
previous section.
The (antilinear) Schwartz delta functional at a real number $E$
associates with each test function $\phi$, the complex conjugate of the value
that $\phi$ takes at $E$:
\begin{equation}
\begin{array}{rcl}
\widehat{\delta}_E : \Phi_{\rm Schw} & \longmapsto & {\mathbb C} \\
\phi & \longmapsto & \widehat{\delta}_E(\phi)= \phi (E)^* \, .
\end{array}
\label{SDdelta-anti}
\end{equation}
When we write the action of $\widehat{\delta}_E$ as an integral operator, the
kernel of such integral operator is Dirac's delta function:
\begin{equation}
\widehat{\delta}_E(\phi )= \int_0^{\infty}\rmd E' \,
\delta (E'-E) \phi (E')^*
=\phi (E)^* \, .
\label{SDdeltaIO}
\end{equation}
If $\Phi _{\rm anal}$ denotes a vector space of test functions $\phi$ such that
$\phi ^*$ are all {\it analytic} at $z_0$, then
the antilinear complex delta functional at $z_0$ is a
function that associates with each test function $\phi$, the value that
the analytic continuation of $\phi ^*$ takes at $z_0$:
\begin{equation}
\begin{array}{rcl}
\widehat{\delta}_{z_0} : \Phi _{\rm anal} & \longmapsto &
{\mathbb C} \\
\phi & \longmapsto & \widehat{\delta}_{z_0}(\phi)=
\phi (z_0^*)^* \, .
\end{array}
\label{Cdelta-anti}
\end{equation}
When we write the expression for $\widehat{\delta}_{z_0}$ as an integral
operator, the kernel of such integral operator is the complex delta function:
\begin{equation}
\widehat{\delta}_{z_0}(\phi )= \int_0^{\infty}\rmd E' \,
\delta (E'-{z_0}) \phi (E')^* =
\phi (z_0^*)^*\, .
\label{decdf}
\end{equation}
When the test functions are only meromorphic and $z_0$ is one of their poles,
definition~(\ref{Cdelta-anti}) needs to be changed to
\begin{equation}
\begin{array}{rcl}
\widehat{\delta}_{z_0} : \Phi _{\rm mero} & \longmapsto & {\mathbb C} \\
\phi & \longmapsto & \widehat{\delta}_{z_0}(\phi)=
\frac{1}{2\pi \rmi} \oint \rmd z \, \frac{\phi (z^*)^*}{z-z_0} \, .
\end{array}
\label{Cdeltameroph-anti}
\end{equation}
If we denote by $a_0^*$ the zeroth term of the Laurent expansion of
$\phi (z^*)^*$ around $z_0$, then definition~(\ref{Cdeltameroph-anti})
associates $a_0^*$ with each test function $\phi$, and therefore we can write
\begin{equation}
\begin{array}{rcl}
\widehat{\delta}_{z_0} : \Phi _{\rm mero} & \longmapsto & {\mathbb C} \\
\phi & \longmapsto & \widehat{\delta}_{z_0}(\phi)= a_0^* \, .
\end{array}
\label{Cdeltameroph-anti-a8}
\end{equation}
If the functions are not even meromorphic, we need to use the prescription of
Gelfand and Shilov~\cite[Vol.~I, Appendix B]{GELFAND}.
The same conclusions as in the previous section apply to the antilinear
complex delta functional. When $\phi (z^*)^*$ are all {\it analytic}
at $z_0$, $a_0^*$ is simply $\phi (z_0^*)^*$. Thus, when the test functions
are all {\it analytic} at $z_0$, definition~(\ref{Cdeltameroph-anti}) becomes
definition~(\ref{Cdelta-anti}), and we are allowed to use~(\ref{Cdelta-anti}).
\subsection{Nakanishi's definition}
\label{sec:Nakadef}
Nakanishi~\cite{NAKANISHI} uses a slightly different version of the
complex delta function. When he writes $\delta _{\rm N}(\phi )$ as
an integral operator, Nakanishi uses the following expression:
\begin{equation}
\delta _{\rm N}(\phi )= \int_{\gamma} \rmd E \,
\phi (E^*)^* \delta _{\rm N}(E-z_{\rm R}) \, ,
\label{Nakscon1}
\end{equation}
where
\begin{equation}
\delta _{\rm N}(E-z_{\rm R}) = \frac{1}{2\pi \rmi} \left(
\frac{1}{E^{(-)}-z_{\rm R}} - \frac{1}{E^{(+)}-z_{\rm R}}
\right) \, ,
\end{equation}
and where the contour $\gamma$ is such that the integral in
Eq.~(\ref{Nakscon1}) decomposes into two terms. The end points of the
integration paths are the same for the two terms, namely, $0$ and
$+\infty$. The
integration path for the first term, $\frac{1}{E^{(-)}-z_{\rm R}}$, passes
below $z_{\rm R}$, whereas the integration path for the second term,
$\frac{1}{E^{(+)}-z_{\rm R}}$, passes above $z_{\rm R}$. Adding the two
terms we obtain
\begin{equation}
\int_{\gamma}\rmd E \, \phi (E^*)^* \delta _{\rm N}(E-z_{\rm R})=
\frac{1}{2\pi \rmi} \oint \rmd E \, \frac{\phi (E^*)^*}{E-z_{\rm R}} =
\phi (z_{\rm R}^*)^* \, .
\label{Nakscon}
\end{equation}
Thus, the distributional definition~(\ref{Cdelta-anti}) is equivalent
to Nakanishi's definition~(\ref{Nakscon1})-(\ref{Nakscon}), because both
approaches associate the same complex number, $\phi (z_{\rm R}^*)^*$, with
the same test function, $\phi$.
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\end{document}
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