# Formal Aspects of Quantum Decay

D. F. Ramírez Jiménez<EMAIL_ADDRESS>N. G. Kelkar
<EMAIL_ADDRESS>Departamento de Física, Universidad de los Andes,
Cra.1E No.18A-10, Bogotá, D.C., Colombia

###### Abstract

The Fock-Krylov formalism for the calculation of survival probabilities of
unstable states is revisited paying particular attention to the mathematical
constraints on the density of states, the Fourier transform of which gives the
survival amplitude. We show that it is not possible to construct a density of
states corresponding to a purely exponential survival amplitude. he survival
probability $P(t)$ and the autocorrelation function of the density of states
are shown to form a pair of cosine Fourier transforms. This result is a
particular case of the Wiener Khinchin theorem and forces $P(t)$ to be an even
function of time which in turn forces the density of states to contain a form
factor which vanishes at large energies. Subtle features of the transition
regions from the non-exponential to the exponential at small times and the
exponential to the power law decay at large times are discussed by expressing
$P(t)$ as a function of the number of oscillations, $n$, performed by it. The
transition at short times is shown to occur when the survival probability has
completed one oscillation. The number of oscillations depend on the properties
of the resonant state and a complete description of the evolution of the
unstable state is provided by determining the limits on the number of
oscillations in each region.

##  I Introduction

Spontaneous decay is an inherently quantum mechanical process. The likelihood
for the existence of a decaying (unstable) state at a given point of time is
known as the survival probability and must be calculated within the framework
of quantum mechanics. The most intriguing fact that followed from the
theoretical studies is that the survival probability cannot at all times
follow the exponential decay law Khalfin  mostly observed in the laboratory.
The decay law is expected to be quadratic at small times sudarshan ; levitan ;
ghirardi1979 ; boyanov  and a power law at large times Fonda1 ; ournonexpo1 ;
ournonexpo2 ; TorronMuga . The theoretical claims led experimental nuclear and
particle physicists to perform experiments (see expoexpt  and references
therein) with nuclei such as 222Rn, 60Co and 56Mn with half-lives ranging from
hours to days. In spite of performing observations for several half-lives only
an exponential decay law was measured at all times. The unique experiment
where the non-exponential behaviour at large times was confirmed involved the
measurement of the luminescence decays of many dissolved organic materials
after pulsed laser excitation largetexpt . Experimental evidence for short
time non-exponential decay was found in a quantum tunneling experiment
shorttexpt  where ultra-cold sodium atoms were trapped in an accelerating
periodic optical potential created by a standing wave of light. On the
theoretical side, the decay law has been investigated using various different
formalisms in literature (see ourpapercriticaltimes  for a comparison of
approaches). Of great interest is the calculation of the critical time for the
transition from the quadratic to the exponential at small times and the
exponential to the power law at large times. Predictions of the critical times
are useful in deciding the feasibility of experimentally observing a non-
exponential decay ourpapercriticaltimes .

One of the most commonly used formalism for the calculation of survival
probabilities of unstable states is the method introduced by Fock and Krylov
(FK) Fock1 . In this method, the survival amplitude (modulus-squared of which
gives the survival probability) is evaluated as a Fourier transform of the
density of states (DOS) in the resonance. Thus, the DOS is indeed the crucial
quantity required in the calculation and must satisfy certain conditions
Fonda1  for the correct physical behaviour of the survival probability. The
DOS can in principle be constructed using the poles and residues of the
resonances, in a model independent way as was shown in ourpaperSmatrix . An
essential feature of the DOS is the existence of a threshold factor which
ensures the correct power law at large times. In ourpapercriticaltimes , the
present authors obtained the expressions for the DOS using formalisms other
than the FK. In the present work, we revisit the calculation of the survival
probability with the FK framework to discover some subtle features of the
survival probability and constraints on the density of states. One of the main
observations is the fact that the survival probability can only be an even
function of time. The result has consequences for the standard determination
of the critical transition time at small times by expanding an exponential in
all powers of $t$ sudarshan ; levitan . In fact, the result we obtain is a
particular case of the Wiener Khinchin theorem which tells us that the
survival probability and the autocorrelation function (constructed from the
density of states) are cosine Fourier transforms of each other. Following this
result, we investigate the behaviour of the survival probability at small
times with realistic examples of resonances from nuclear physics. Constructing
a functional form of the DOS with the desired physical features we notice that
the absence of an energy dependent form factor (which is usually included in a
model dependent way) in the DOS can lead to unphysical results. Finally, the
theoretical results obtained are demonstrated in a more visual way by applying
the expressions obtained to realistic resonances.

We shall begin by very briefly introducing the Fock-Krylov (FK) method which
can in principle be used to describe the decay of any resonant state, whose
density of states as a function of energy is known. The formalism has been a
common tool for investigating the behaviour of survival probabilities in
literature. This includes the 1958 paper of Khalfin Khalfin , sudarshan  on
the Zeno effect, levitan  on the short time behaviour of $P(t)$, a review
Fonda1  and more recently Ref. urbanowski  where the authors discuss the “true
face” of quantum decay. There exist approaches in literature which do not use
the FK method but rely on a potential based formalism which is useful in
studying tunneling decays (see GC2021  and references therein) and some
interesting aspects of the latter giacosa ; koide  (we refer the reader to
anasto  for a pedagogical review).

In the next section, using the FK formalism, we show that it is not possible
to find a density of states (DOS) corresponding to a purely exponential decay.
Section III derives the relation between the survival probability and the
autocorrelation function of the DOS which leads to a particular case of the
Wiener-Khinchin theorem. Section IV derives the expression for the survival
probability of a system at small times. Here we emphasize that a consequence
of the particular form of the Wiener-Khinchin theorem derived earlier is the
evenness of the survival probability. A functional form of the density of
states is considered in Section V and the dependence of the results on the
choice of the form factor appearing in the density of states is discussed in
Section VI. Expressing the survival probability $P(t)$ as a function of the
number of oscillations performed $n$, we discover interesting features of
$P(n)$ with one of them being that the transition time from the small time
non-exponential to the intermediate time exponential decay law happens at a
time when the survival probability has completed one oscillation. These
results are compared with those obtained from other approaches in literature.
In section VII, we present results for the critical times and behaviour of the
transition regions from the non-exponential to the exponential at small times
and the exponential to the power law behaviour at large times.

##  II Density of States Associated with the Exponential Component of the
Survival Probability

The Fock-Krylov (FK) method has been widely used in literature for the
analysis of unstable states urbanowski ; mnowak ; giraldi ; giraldi1 ; Dijk ;
Boyanovsky . We refer the reader to ourpapercriticaltimes  for details of the
derivation and begin here with the survival amplitude given by,

$A(t)=\int_{E_{min}}^{\infty}{dE\,\rho(E)e^{-iEt}}\,,$ (1)

where $\rho(E)$ is the density of states (DOS) in the resonance and is a real
positive function. In the FK method, one constructs $\rho(E)$ by rewriting the
initial state in terms of the energy eigenstates of the decay products (see
Eqs (4) - (10) in ourpapercriticaltimes ). The commonly used Breit-Wigner
distribution is an example of such a DOS. Though the FK method with an energy
dependent DOS has been extensively used in literature (Khalfin ; sudarshan  to
quote a few), there exist other approaches for the evaluation of survival
probabilities. We refer the reader to ourpapercriticaltimes  for a comparsion
between different approaches such as the Green’s function method, Jost
function based formalism and the Fock-Krylov formalism. In
ourpapercriticaltimes , the survival amplitude within the Green’s function
approach was rewritten in a form similar to that of the Fock-Krylov method to
obtain,

$A(t)=\int_{0}^{\infty}{\biggl{[}\frac{1}{2\pi
i}\,\sum_{n}{C_{n}(k_{n})\bar{C}_{n}(k_{n})\frac{\sqrt{E}}{k_{n}(k_{n}^{2}-E)}}\biggr{]}\,e^{-itE}\,dE},$

where the sum over $n$ is over all poles. Given the form of the above
amplitude, one may identify the quantity in the square brackets as a density
of states as in Eq. (1). The DOS standardly used in the FK method is different
from the quantity in the square brackets. The authors in ourpapercriticaltimes
showed that restricting the sum to only fourth quadrant poles and further
considering one isolated resonance, the above density (referred to as
$\rho^{GC}$ there) is a sum of two terms (see Eqs (69) to (72) in
ourpapercriticaltimes ). The first term is an energy derivative of the phase
shift and the second term is negligible for narrow resonances. Thus, the Fock-
Krylov and Green’s function method GCPhysScripta  agree for isolated narrow
resonances.

The survival amplitude allows us to compute the so-called the survival
probability $P(t)$, which measures the probability that the state of a system
is in its initial state at a time, $t>0$:

$P(t)=\bigl{|}A(t)\bigr{|}^{2}.$ (2)

As a natural manifestation of the normalization condition, we have, $A(0)=1$
and $P(0)=1$. It is known that the survival probability is split in three
well-defined regions: the small time region where $P(t)$ follows a quadratic
law, the intermediate time region where $P(t)$ is dominantly exponential and
the large time region where $P(t)$ is dominated by a power law Fonda1 ;
Khalfin ; Nakazato1 ; Nakazato2 . Since some systems have long intermediate
exponential regions (for instance, nuclear decays), it would be suitable to
describe those decays such that the density of states gives an exponential
survival probability only. However, in the following, we shall show that such
a density of states does not exist. We start from the complex integral

$\oint_{C}{\rho(z)e^{-itz}\,dz},$ (3)

where $C$ is the contour of integration shown in Fig. 1.

Figure 1: Contour of integration for the survival amplitude

The exponential function (or sum of exponential ones) is obtained when the
density of states has simple poles in the fourth quadrant. Since $\rho(z)$
must be a positive real function for positive real $z$, its poles have to be
complex conjugate pairs on the right semiplane, i.e., $\rho(z)$ has simple
poles at

$z=z_{s}\equiv\sigma_{s}-i\omega_{s}\operatorname{sgn}{(s)}/2,$ (4)

where $s=\pm 1,\pm 2,\dotsc$, $\sigma_{s}>0$, and $\omega_{s}>0$ for all $s$.
For $s>0$ the poles are in the fourth quadrant, and for $s<0$ they are in the
first quadrant. Let also $R(z_{s})$ be the residue of $\rho(z)$ at the pole
$z=z_{s}$. As a result of these conventions, we get the following relations:

$\displaystyle z_{s<0}$ $\displaystyle=z_{s>0}^{*},$ (5) $\displaystyle
R(z_{s<0})$ $\displaystyle=R^{*}(z_{s>0}).$ (6)

In the appendix A, we show from the integral (3) that

$A(t)=-2\pi{i}\sum_{s>0}{e^{-iz_{s}t}R(z_{s})}-\int_{-i\infty}^{0}{\rho(z)e^{-izt}\,dz}.$
(A5)

The integral (135) lets us easily separate the exponential and the non-
exponential component of the survival amplitude. Calling these respective
components, $A_{e}(t)$ and $A_{ne}(t)$, from the integral (135) we get:

$\displaystyle A_{e}(t)$
$\displaystyle=-2\pi{i}\sum_{s>0}{e^{-iz_{s}t}R(z_{s})},$ (7) $\displaystyle
A_{ne}(t)$
$\displaystyle=-\int_{-i\infty}^{0}{\rho(z)e^{-izt}\,dz}=-i\int_{0}^{\,\infty}{\rho(-iy)e^{-ty}\,dy}.$
(8)

If the survival amplitude were purely exponential, Eq. (8) would be zero, and
therefore the density of states would vanish along the negative imaginary
axis. Since an analytic function in a region is zero when the function
vanishes along a continuous curve in this region Morse , we deduce that
$\rho(z)=0$ for all points of a domain on the $z$ plane which contains some
segment of the negative imaginary axis. Since this domain is arbitrary, the
density of states would be zero over the complex $z$ plane and then, the
survival amplitude would be zero for all $t$. However, this is imposible
because the density of states has to be different from zero over the real
positive axis and has to have simple poles for $\text{Re\,}{z}>0$. In addition
to that, the survival amplitude could not be zero for all $t$ because the
normalization condition would no longer be satisfied (apart from such an
amplitude losing the meaning of a survival amplitude). Hence, there does not
exist a density of states such that the survival amplitude (taken as a Fourier
transform of this density of states) and the corresponding survival
probability are purely exponential.

A few words to put this result in context with the well-known seminal result
of Khalfin Khalfin  are due. Using the Paley-Wiener theorem, Khalfin showed
that a survival amplitude given as the Fourier transform of a semi-finite
density is not attainable for arbitrary functions $M(t)$, where the survival
amplitude is given by $A(t)=M(t)\exp(\,i\phi t)$. The result obtained in this
section is in some sense a reverse statement of what Khalfin states. We show
that assuming a purely exponential survival probability, $|A(t)|^{2}$, it is
not possible to find a density of states, the Fourier transform of which gives
the survival amplitude as in our Eq. (1). The result is relevant since most
observed decays are exponential and it would be desirable to look for such a
density of states. Apart from this, it has consequences for the calculation of
the autocorrelation function which we shall see later in Section III. It is
worth noting that the above result is based on a simple argument of analytic
continuation and does not rely on the Paley-Wiener theorem.

Eq. (135) lets us descompose the survival probability, $P(t)$, in a suitable
way. Therefore, from the definition of $P(t)$ and Eq. (135), we can write

$P(t)=|A(t)|^{2}=|A_{e}(t)|^{2}+|A_{ne}(t)|^{2}+2\,\text{Re\,}{\bigl{[}A_{e}^{*}(t)A_{ne}(t)\bigr{]}},$
(9)

If we do not focus on the non-exponential behaviour at very small times, we
can see that the survival probability is split into three terms: the first one
comes from the exponential component of $A(t)$, the second one comes from the
non-exponential component of $A(t)$ (at large times) and the last one is
nothing but the interference between the components of $A(t)$. These terms are
denoted respectively as $P_{e}(t)$, $P_{ne}(t)$ and $P_{i}(t)$ and are called
(in the same order) the exponential, non-exponential and interference survival
probability. From the Eqs (7) and (8), those terms are given by:

$\displaystyle P_{e}(t)$
$\displaystyle\equiv|A_{e}(t)|^{2}=4\pi^{2}\biggl{|}\sum_{s>0}{e^{-iz_{s}t}R(z_{s})}\biggr{|}^{2},$
(10) $\displaystyle P_{ne}(t)$
$\displaystyle\equiv|A_{ne}(t)|^{2}=\Biggl{|}\int_{-i\infty}^{0}{\rho(z)e^{-izt}\,dz}\Biggr{|}^{2},$
(11) $\displaystyle P_{i}(t)$ $\displaystyle\equiv
2\,\text{Re\,}{\bigl{[}A_{e}^{*}(t)A_{ne}(t)\bigr{]}}=-4\pi\sum_{s>0}{\text{Re\,}{\Biggl{[}iR^{*}(z_{s})\int_{-i\infty}^{0}{\rho(z)e^{-i(z-z_{s}^{*})t}\,dz}\Biggr{]}}}.$
(12)

This particular descomposition of the survival probability is useful for
analyzing the transition of the decay law from the exponential to the power
law at large times. The analysis of the latter transition using this splitting
can be seen in ourpapercriticaltimes . In section VII we shall study the small
time transition region. Though the above division of regions is in general
valid, there exist exceptions where the decay law is non-exponential at all
times. This happens in the case of broad resonances, of which the $\sigma$
meson is a good example ournonexpo2 . Other examples can be found in GCPRA73 ;
GCPRA93 .

##  III Survival Probability in terms of the autocorrelation of the density of
states

Although it is easier to compute the survival probability simply as the
modulus squared of the survival amplitude, it would be desirable to get an
expression involving some function of the density of states. We shall show
that this function is the autocorrelation function of the density of states.
Furthermore, the expression will be a particular case of the Wiener-Khinchin
theorem111For details about this theorem, see Allen  and Brandwood .. Using
(2),

$P(t)=A(t)A^{*}(t)=\int_{E_{\text{min}}}^{\infty}{\int_{E_{\text{min}}}^{\infty}{\rho(E)\rho(E^{\prime})e^{i(E^{\prime}-E)t}\,dEdE^{\prime}}}.$
(13)

Making the change of variables $E=x$ and $E^{\prime}-E=y$, the new region of
integration $S$ is given by $S=\bigl{\\{}(x,y)\in\mathbb{R}^{2}:x\geq
E_{\text{min}}$ ,$x+y\geq{E_{\text{min}}}\bigr{\\}}$. Integrating first over
$x$ and then over $y$, the integral (13) is split in two integrals:

$P(t)=\int_{0}^{\infty}{\,dy\,e^{\,iyt}\int_{E_{\text{min}}}^{\infty}{\,dx\,\rho(x)\rho(x+y)}}+\int_{-\infty}^{0}{\,dy\,e^{\,iyt}\int_{E_{\text{min}}-y}^{\infty}{\,dx\,\rho(x)\rho(x+y)}}.$
(14)

Making in the second double integral another change of variables, i.e.,
$y^{\prime}=-y$ and $x^{\prime}=x+y$; the region of integration is transformed
to one given by $x^{\prime}\geq E_{\text{min}}$ and $y^{\prime}\geq 0$. Thus,

$P(t)=\int_{0}^{\infty}{\,dy\,e^{\,iyt}\int_{E_{\text{min}}}^{\infty}{\,dx\,\rho(x)\rho(x+y)}}+\int_{0}^{\infty}{\,dy^{\prime}\,e^{-iy^{\prime}t}\int_{E_{\text{min}}}^{\infty}{\,dx^{\prime}\,\rho(x^{\prime}+y^{\prime})\rho(x^{\prime})}}.$
(15)

Now, the integrals can be added. Therefore,

$P(t)=\int_{0}^{\infty}{\,dy\,2\cos{yt}\int_{E_{\text{min}}}^{\infty}{\,dx\,\rho(x)\rho(x+y)}}.$
(16)

The survival probability is then the cosine Fourier transform of the function

${\cal R}(y)=2\int_{E_{\text{min}}}^{\infty}{\,dx\,\rho(x)\rho(x+y)}.$ (17)

${\cal R}(y)$ is the autocorrelation function of the density of states. In
order to obtain the autocorrelation function as an inverse cosine Fourier
transform of $P(t)$, we multiply Eq. (16) by $\cos{y^{\prime}t}$ and then we
integrate with respect to $t$ between $t=0$ and $t=\infty$ so that

$\displaystyle\int_{0}^{\,\infty}{P(t)\cos{y^{\prime}t}\,dt}$
$\displaystyle=\int_{0}^{\infty}{\,dy\biggl{[}\int_{0}^{\,\infty}{\cos{yt}\cos{y^{\prime}t}\,dt}\biggr{]}\cdot
2\int_{E_{\text{min}}}^{\infty}{\,dx\,\rho(x)\rho(x+y)}}$
$\displaystyle=\int_{0}^{\infty}{\,dy\,\frac{\pi}{2}\Bigl{[}\delta{\bigl{(}y-y^{\prime}\bigr{)}}+\delta{\bigl{(}y+y^{\prime}\bigr{)}}\Bigr{]}\cdot
2\int_{E_{\text{min}}}^{\infty}{\,dx\,\rho(x)\rho(x+y)}}$ $\displaystyle=$
$\displaystyle\pi\int_{E_{\text{min}}}^{\infty}{\,dx\,\rho(x)\rho(x+y)}=\frac{\pi}{2}{\cal
R}(y).$ (18)

To summarize, the survival probability and the autocorrelation function of the
density of states are a pair of cosine Fourier transforms given by

$\displaystyle P(t)$ $\displaystyle=\int_{0}^{\,\infty}{{\cal
R}(y)\cos{yt}\,dy},$ (19) $\displaystyle{\cal R}(y)$
$\displaystyle=\frac{2}{\pi}\int_{0}^{\,\infty}{P(t)\cos{yt}\,dt}.$ (20)

A word of caution about the lower limit of integration of the autocorrelation
function of the density of states is in order here. Without loss of
generality, we take its value as $E_{min}=0$. This is equivalent to shifting
the origin of the energy scale at the threshold of the system and is
translated in a change of variable in the integral (17) such that

${\cal R}(y)=2\int_{0}^{\infty}{\,dx\,\rho(x)\rho(y+x)}.$ (21)

Eqs. (19) and (20) still remain valid.

Although the aim of this work is to study the characteristic features of the
survival probability of an unstable quantum system, some applications of the
autocorrelation function are worth mentioning. In the case of a system with a
discrete spectrum, it is simple to compute the autocorrelation function by
taking the Fourier transform of the survival probability. Let
$\bigl{\\{}\bigl{|}n\bigr{\rangle}\bigr{\\}}$ be a discrete eigenstate of a
system with Hamiltonian $H$ and initial state $\bigl{|}i\bigr{\rangle}$222In
order to avoid any confusion related to the notation for the initial state
used at the beginning of this article, this state is denoted as
$\bigl{|}i\bigr{\rangle}$ only in this section., and let
$\bigl{\\{}E_{n}\bigr{\\}}$ be their corresponding eigenenergies:
$H\bigl{|}n\bigr{\rangle}=E_{n}\bigl{|}n\bigr{\rangle}$, where
$n=0,1,2,\dotsc$ and the discrete energies are labeled such that
$E_{0}<E_{1}<E_{2}\cdots$. The survival amplitude and probability are then
given as

$\displaystyle A(t)$
$\displaystyle=\bigl{\langle}i\bigr{|}e^{-iHt}\bigl{|}i\bigr{\rangle}=\sum_{n=0}^{\infty}{\bigl{|}\bigl{\langle}i|n\bigr{\rangle}\bigr{|}^{2}e^{-iE_{n}t}},$
(22) $\displaystyle P(t)$
$\displaystyle=|A(t)|^{2}=\sum_{n=0}^{\infty}{\sum_{m=0}^{\infty}{{\bigl{|}\bigl{\langle}i|n\bigr{\rangle}\bigr{|}^{2}\,\bigl{|}\bigl{\langle}i|m\bigr{\rangle}\bigr{|}^{2}\,e^{-i(E_{n}-E_{m})t}}}}$
$\displaystyle=\sum_{n=0}^{\infty}{\bigl{|}\bigl{\langle}i|n\bigr{\rangle}\bigr{|}^{2}}+2\sum_{n=1}^{\infty}{\sum_{m=0}^{n-1}{\bigl{|}\bigl{\langle}i|n\bigr{\rangle}\bigr{|}^{2}\,\bigl{|}\bigl{\langle}i|m\bigr{\rangle}\bigr{|}^{2}\,\cos{(E_{n}-E_{m})t}}}.$
(23)

The corresponding autocorrelation function can be computed by inspection: Eq.
(23) is obtained from Eq. (19) by multiplying the first sum on the right hand
side of (23) by $\delta(y)$ and substituting $\cos{(E_{n}-E_{m})t}$ by
$\delta{\bigl{[}y-(E_{n}-E_{m})t\bigr{]}}$. Therefore,

${\cal
R}(y)=\delta{(y)}\sum_{n=0}^{\infty}{\bigl{|}\bigl{\langle}i|n\bigr{\rangle}\bigr{|}^{2}}+2\sum_{n=1}^{\infty}{\sum_{m=0}^{n-1}{\bigl{|}\bigl{\langle}i|n\bigr{\rangle}\bigr{|}^{2}\,\bigl{|}\bigl{\langle}i|m\bigr{\rangle}\bigr{|}^{2}\,\delta{\bigl{[}y-(E_{n}-E_{m})t\bigr{]}}}}.$
(24)

Since the autocorrelation function is a convolution-like function, the authors
in caos1 ; caos2 ; caos3  use Eq. (24) together with random matrix theory for
finding signatures of classical chaos in quantum systems. They do so by
comparing numerical results with experiment.

We pointed out in Section II about the impossibility of deriving a density of
states corresponding to a purely exponential survival probability. This
implies that we would not be able to use the autocorrelation function of the
density of states given by Eq. (21) if we wanted to identify the exponential
contribution of the survival probability. Even if we showed that it is
impossible to find a density of states corresponding to a purely exponential
survival probability, one can always find a density of states which leads to a
survival probability with an exponential component in addition to the non-
exponential ones as given in Eqs (7) and (10). In spite of having found a
functional form for the density of states, the general case of the exponential
component of a sum consisting of several poles is difficult to work with. Let
us therefore consider the simpler case of a narrow isolated resonance with a
long intermediate region of exponential decay. In this case, P(t) can be
approximated roughly by its exponential component. Using $z_{s}$ as given by
(4), $A_{e}(t)$ and $P_{e}(t)$ from (7) and (10) can be written as

$\displaystyle A_{e}(t)$ $\displaystyle=-2\pi{i}{e^{-iz_{s}t}R(z_{s})},$ (25)
$\displaystyle P_{e}(t)$
$\displaystyle=4\pi^{2}|R(z_{s})|^{2}e^{-2\text{Im\,}{z_{s}}t}=4\pi^{2}|R(z_{s})|^{2}e^{-{\omega_{s}}t},$
(26)

and Eq. (20) gives us the approximate autocorrelation function:

${\cal R}(y)=\frac{2}{\pi}\int_{0}^{\,\infty}{P(t)\cos{yt}\,dt}\approx
8\pi\omega_{s}\,{\frac{|R(z_{s})|^{2}}{y^{2}+\omega_{s}^{2}}}.$ (27)

If we wish to take other resonances into account, we can improve the
approximation of the autocorrelation function. From Eq. (10):

$P(t)\approx
P_{e}(t)=4\pi^{2}\sum_{s>0}{\sum_{s^{\prime}>0}{R(z_{s})R^{*}(z_{s^{\prime}})\,e^{-i(z_{s}-z_{s^{\prime}}^{*})t}}}.$
(28)

Using Eq. (20), the autocorrelation function is approximately equal to:

${\cal
R}(y)=\frac{2}{\pi}\text{Re\,}{\int_{0}^{\,\infty}{P(t)e^{-iyt}\,dt}}\approx
8\pi\text{Im\,}{\sum_{s>0}{\sum_{s^{\prime}>0}{\frac{R(z_{s})R^{*}(z_{s^{\prime}})}{y+z_{s}-z_{s^{\prime}}^{*}}}}}\\\
=8\pi\sum_{s>0}{|R(z_{s})|^{2}\,\frac{\omega_{s}}{y^{2}+\omega_{s}^{2}}}+16\pi\text{Im\,}{\sum_{s>0}{\sum_{s^{\prime}>s}{\frac{R(z_{s})R^{*}(z_{s^{\prime}})}{y+\sigma_{s}-\sigma_{s^{\prime}}-i\bigl{(}\omega_{s}+\omega_{s^{\prime}}\bigr{)}/2}}}}.$
(29)

##  IV Survival probability of a system for small times

A consequence of Eq. (19) is the evenness of $P(t)$. This property can be
checked directly from the definition of the survival probability:

$P(t)=A(t)A^{*}(t)=\bigl{\langle}0\bigr{|}e^{-iHt}\bigl{|}0\bigr{\rangle}\bigl{\langle}0\bigr{|}e^{-iHt}\bigl{|}0\bigr{\rangle}^{*}=\bigl{\langle}0\bigr{|}e^{-iHt}\bigl{|}0\bigr{\rangle}\bigl{\langle}0\bigr{|}e^{iHt}\bigl{|}0\bigr{\rangle}.$
(30)

Expanding the temporal evolution operators in a Taylor series:

$\displaystyle P(t)$
$\displaystyle=\Biggl{[}\sum_{n=0}^{\infty}{\frac{(-i)^{n}}{n!}\bigl{\langle}0\bigr{|}H^{n}\bigl{|}0\bigr{\rangle}t^{n}}\Biggr{]}\Biggl{[}\sum_{m=0}^{\infty}{\frac{i^{m}}{m!}\bigl{\langle}0\bigr{|}H^{m}\bigl{|}0\bigr{\rangle}t^{m}}\Biggr{]}$
$\displaystyle=\sum_{n=0}^{\infty}{\frac{(it)^{n}}{n!}\sum_{m=0}^{n}{(-1)^{m}\binom{n}{m}\bigl{\langle}0\bigr{|}H^{m}\bigl{|}0\bigr{\rangle}\bigl{\langle}0\bigr{|}H^{n-m}\bigl{|}0\bigr{\rangle}}}$
$\displaystyle=\sum_{n=0}^{\infty}{\frac{i^{n}}{n!}\,p_{n}(H)t^{n}},$ (31)

where $p_{n}(H)$ is given by:

$p_{n}(H)=\sum_{m=0}^{n}{(-1)^{m}\binom{n}{m}\bigl{\langle}0\bigr{|}H^{m}\bigl{|}0\bigr{\rangle}\bigl{\langle}0\bigr{|}H^{n-m}\bigl{|}0\bigr{\rangle}}.$
(32)

The sum given by (32) has the form $\sum_{m=0}^{n}{(-1)^{m}A_{m}A_{n-m}}$,
with $A_{m}$ given by

$A_{m}=\sqrt{\dbinom{n}{m}}\bigl{\langle}0\bigr{|}H^{m}\bigl{|}0\bigr{\rangle}.$
(33)

Since the above sum is zero for $n$ odd, $p_{n}(H)$ is zero for $n$ odd. Thus,

$P(t)=1+\sum_{q=1}^{\infty}{\frac{(-1)^{q}}{(2q)!}p_{2q}(H)\,t^{2q}}.$ (34)

Summarizing the above, we can say that not only is the survival probability an
even function, but also its Taylor expansion around $t=0$ contains even powers
of $t$ only. As a consequence, $P^{\prime}(0)=0$ and

$P^{\prime\prime}(0)=-p_{2}(H)=-2\bigl{\langle}(\Delta
H)^{2}\bigr{\rangle}_{0}<0,$ (35)

where $\bigl{\langle}(\Delta
H)^{2}\bigr{\rangle}_{0}=\bigl{\langle}H^{2}\bigr{\rangle}_{0}-\bigl{\langle}H\bigr{\rangle}_{0}^{2}$.
Furthermore, Eq. (34) implies that the survival probability must follow a
quadratic law for small times, meaning,

$P(t)=1-t^{2}\bigl{\langle}(\Delta H)^{2}\bigr{\rangle}_{0}+O(t^{4}).$ (36)

Another implication of the above result is that it is not possible to have a
purely exponential survival probability because the Taylor expansion of the
exponential survival probability around $t=0$ has both even and odd powers of
$t$. We can see this fact if we expand the Eq. (7) in Taylor series:

$A_{e}(t)=-2\pi
i\sum_{s>0}{R(z_{s})\sum_{n=0}^{\infty}{\frac{(-iz_{s}t)^{n}}{n!}}}=\sum_{n=0}^{\infty}{\frac{(-it)^{n}}{n!}\sum_{s>0}{(-2\pi
i)z_{s}^{n}R(z_{s})}}=\sum_{n=0}^{\infty}{\frac{(-it)^{n}}{n!}\,B_{n}},$ (37)

where $B_{n}=-2\pi i\sum_{s>0}{z_{s}^{n}R(z_{s})}$. Therefore, the
corresponding Taylor expansion for $P_{e}(t)$ will be:

$P_{e}(t)=\bigl{|}A_{e}(t)\bigr{|}^{2}=\sum_{n=0}^{\infty}{\frac{(it)^{n}}{n!}\sum_{m=0}^{n}{(-1)^{m}}\binom{n}{m}B_{m}B_{n-m}^{*}}=\bigl{|}B_{0}\bigr{|}^{2}-2t\,\text{Im\,}{\bigl{(}B_{0}B_{1}^{*}\bigr{)}}-t^{2}\text{Re\,}{\bigl{(}B_{0}B_{2}^{*}-|B_{1}|^{2}\bigr{)}}+\cdots.$
(38)

Eq. (38) lets us interpret the role of the non-exponential and interference
terms of the survival probability. Adding the Taylor expansion of both
functions to the expansion of $P_{e}(t)$, the normalization condition and the
evenness of the survival probability should be ensured. We shall see that this
actually happens. Since the Taylor expansion of the non-exponential and
interference terms of the survival probability depend on the Taylor expansion
of the non-exponentical survival amplitude, we can expand
$A(t)=\bigl{\langle}0\bigr{|}e^{-iHt}\bigl{|}0\bigr{\rangle}$ and use Eq.
(37). Thus, we get the required expansion, i.e.,

$A_{ne}(t)=A(t)-A_{e}(t)=\sum_{n=0}^{\infty}{\frac{(-it)^{n}}{n!}\bigl{(}\bigl{\langle}H^{n}\bigr{\rangle}_{0}-B_{n}\bigr{)}}.$
(39)

Therefore, the Taylor expansion of the non-exponential survival probability is

$P_{ne}(t)=\bigl{|}A_{ne}(t)\bigr{|}^{2}=\sum_{n=0}^{\infty}{\frac{(it)^{n}}{n!}\sum_{m=0}^{n}{(-1)^{m}\binom{n}{m}\bigl{(}\bigl{\langle}H^{m}\bigr{\rangle}_{0}-B_{m}\bigr{)}\bigl{(}\bigl{\langle}H^{n-m}\bigr{\rangle}_{0}-B_{n-m}^{*}\bigr{)}}}\\\
=\bigl{|}1-B_{0}\bigr{|}^{2}-2t\,\text{Im\,}{\Bigl{[}\bigl{(}1-B_{0}\bigr{)}\bigl{(}\bigl{\langle}H\bigr{\rangle}_{0}-B_{1}^{*}\bigr{)}\Bigr{]}}-t^{2}\,\text{Re\,}{\Bigl{[}\bigl{(}1-B_{0}\bigr{)}\bigl{(}\bigl{\langle}H^{2}\bigr{\rangle}_{0}-B_{2}^{*}\bigr{)}-\bigl{|}\bigl{\langle}H\bigr{\rangle}_{0}-B_{1}\bigr{|}^{2}\Bigr{]}}+\cdots.$
(40)

In addition, the Taylor expansion of the interference part of the survival
probability is

$P_{i}(t)=2\text{Re\,}{\bigl{[}A_{e}(t)A_{ne}^{*}(t)\bigr{]}}=2\text{Re\,}{\sum_{n=0}^{\infty}{\frac{(it)^{n}}{n!}\sum_{m=0}^{n}{(-1)^{m}\binom{n}{m}B_{m}\bigl{(}\bigl{\langle}H^{n-m}\bigr{\rangle}_{0}-B_{n-m}^{*}\bigr{)}}}}\\\
=2\text{Re\,}{\bigl{[}B_{0}(1-B_{0}^{*})\bigr{]}}-2t\,\text{Im\,}{\Bigl{[}B_{0}\bigl{(}\bigl{\langle}H\bigr{\rangle}_{0}-B_{1}^{*}\bigr{)}-B_{1}(1-B_{0}^{*})\Bigr{]}}\\\
-{t^{2}}\,\text{Re\,}{\Bigl{[}B_{2}-2\text{Re\,}{\bigl{(}B_{0}B_{2}^{*}\bigr{)}}+B_{0}\bigl{\langle}H^{2}\bigr{\rangle}_{0}-2B_{1}\bigl{(}\bigl{\langle}H\bigr{\rangle}_{0}-B_{1}^{*}\bigr{)}\Bigr{]}}+\cdots.$
(41)

It is gratifying to find that adding the Eqs. (38), (40) and (41), we obtain
after some algebra the Taylor expansion (34). Note that the expansions (40)
and (41) are given in terms of the poles and residues of the DOS and the
expectation values of the integer powers of the Hamiltonian at the initial
state.

Finally, owing to the condition that the coefficients $p_{n}\bigl{(}H\bigr{)}$
have to be finite, some restrictions must be taken into account when the Fock-
Krylov formalism is used. The coefficients $p_{n}\bigl{(}H\bigr{)}$ are finite
if the expectation value of $H^{n}$ at the initial state exists for all
$n=0,1,2,\dotsc$.

$\bigl{\langle}H^{n}\bigr{\rangle}_{0}=\bigl{\langle}0\bigr{|}H^{n}\bigl{|}0\bigr{\rangle}=\bigl{\langle}0\bigr{|}H^{n}\int_{0}^{\infty}{dE\,\int{db\,\bigl{|}E,b\bigr{\rangle}\bigl{\langle}E,b\,|0\bigr{\rangle}}}=\int_{0}^{\,\infty}{E^{n}\rho(E)\,dE}<\infty,\quad
n=0,1,\dotsc$ (42)

The conditions (42) claim that the density of states is such that these
integrals must converge for each value of $n$, and hence we infer that the
convergence of the integrals will be possible if there exists a real and
positive function $g(E)$ such that it is part of the density of states. This
function is indeed the form factor introduced often in literature. Since there
are no analytic methods for obtaining the form factor, it is common to see
phenomenological procedures in the literature (see for instance Fock1 ; Fonda1
; ourpaperSmatrix ; ournonexpo1 ; ournonexpo2 ; Brzeski ). We note that the
existence of the form factor is a consequence of the evenness of the survival
probability.

A word of caution regarding the moments of $H$ is in order before ending this
section. Assuming a general short time dependence of the form, $A\sim
1+bt^{c}$, where $b$ and $c$ are finite constants and requiring the finiteness
of the moments of $H$, it was noted in muga  that the derivatives

$\displaystyle\frac{dA}{dt}\biggr{|}_{t=0}$
$\displaystyle=-\frac{i}{\hbar}\bigl{\langle}\Psi\bigr{|}H\bigl{|}\Psi\bigr{\rangle}=bct^{c-1}\biggr{|}_{t=0},$
$\displaystyle\frac{d^{2}A}{dt^{2}}\biggr{|}_{t=0}$
$\displaystyle=-\frac{i}{\hbar^{2}}\bigl{\langle}\Psi\bigr{|}H^{2}\bigl{|}\Psi\bigr{\rangle}=bc(c-1)t^{c-1}\biggr{|}_{t=0},$

rule out the possibility of the short time $t^{1/2}$ behaviour (sometimes
found in literature such as sudarshan  and garciaisolated ) of $A(t)$ since it
implies an infinite time derivative of $A$ at $t=0$. For the derivatives to be
finite, $c\geq 1$. We also note that studies of the short time behaviour
within a quantum field theoretic (QFT) approach as in maiani1 ; maiani2  find
that the energy uncertainty which depends on the first two moments of $H$ is
infinite. In a more recent QFT based calculation, however, the author
introduces a cut-off parameter $\Lambda$ such that there is no divergence for
times $t\leq 1/\Lambda$. This time scale determines the renormalization of the
bare state and formation of the quasiparticle state. In connection with the
anti-Zeno effect, the author also determines the energy uncertainty at a given
time. A discussion of the finiteness of the moments and the short time
behaviour within solvable models can be found in cordero , where the authors
found that the expansion of the survival probability in terms of resonant
states predicts the possibility of a $t^{3/2}$ short time behavior, which
follows from the fact that in general the energy moments of the Hamiltonian
may diverge. The authors considered the expression,
$P(t)=1-(t/\tau^{*})^{\theta}$, with parameters, $\theta$ and $\tau^{*}$ to
adjust the short time behaviour of calculations using experiment. Two sets,
$\theta$ = 2, $\tau^{*}$ = 12.55 $\mu s$ and $\theta$ = 3/2 and $\tau^{*}$ =
23.15 $\mu s$ were found to agree with the data at small times shorttexpt .

##  V Density of States for Continuum Spectra

All the results obtained so far in this work indicate that knowing the poles
and residues of the density of states and its form factor as well is necessary
if we wish to construct $\rho(E)$ through the Mittag-Leffler theorem. The
latter affirms that a meromorphic function can be constructed by knowing its
poles and residues. Since the calculation of the survival amplitude involves a
Fourier transform which must be performed by going over to the complex energy
plane, we must consider the behaviour of the density of states in the complex
energy plane. Theoretical evidence suggests that the large time behavior of
the survival probability follows a power law that comes from a branch point of
$\rho(E)$ at $E=0$ such that $|\operatorname{Arg}{E}|<\pi$. This can be
established by supposing that the density of states has an asymptotic
expansion around $E=0$ in the form

$\rho(E)\sim E^{\nu}\sum_{n=0}^{\infty}{\beta_{n}E^{n}},\quad E\to 0^{+}.$
(43)

Substituting Eq. (43) in Eq. (135) and applying the Watson’s lemma Ablowitz ,
we find that the survival amplitude and probability have, for larges times,
the asymptotic expansions

$\displaystyle A(t)$
$\displaystyle\sim\frac{1}{i}\sum_{n=0}^{\infty}{(-i)^{\nu+n}\beta_{n}\,\frac{\Gamma\bigl{(}\nu+n+1\bigr{)}}{t^{\nu+n+1}}},$
(44) $\displaystyle P(t)$
$\displaystyle=|\beta_{0}|^{2}\frac{\bigl{|}\Gamma\bigl{(}\nu+1\bigr{)}\bigr{|}^{2}}{t^{2\nu+2}}+O\bigl{(}t^{-2\nu-3}\bigr{)}.$
(45)

It is known from scattering theory that at large times $P(t)\propto
t^{-(2l+3)}$ Fonda1  for a resonance in the $l^{th}$ partial wave. This would
imply $\nu=l+1/2$ which as we will see later is consistent with conditions
imposed on $\nu$. In Section III we noted that the evenness of $P(t)$ follows
from the Wiener-Khinchin theorem and indeed the small time behaviour was
consistent with this requirement. However, the large time $P(t)\propto
t^{-(2l+3)}$ cannot be an even function. Such a strange behavior is however
common in many asymptotic expansions of even functions. As a first example,
consider the function $f(z)=(z^{6}+z^{2}+1)^{-1/2}$, which, for large values
of $|z|$, behaves as $f(z)\sim z^{-3}$. A second example is provided by the
Bessel function of the first kind of order $2n$, $J_{2n}(x)$ whose asymptotic
expression for large $x$ is
$\sqrt{\frac{2}{\pi\,x}}\cos{(x-\frac{\pi}{4}-n\pi)}$ Lebedev . Finally, the
integral

$g(u)=\int_{u^{2}}^{\infty}{\frac{e^{\,it}}{t^{1/2}}\,dt}$

has the asymptotic form $g(u)\sim ie^{\,iu^{2}}/u$ for large $u$ copson . It
is like these functions forget how they were raised in $t=0$ and change
completely as $t\to\infty$. We must note that in spite of the above, there is
in general no contradiction with the result following from the Wiener-Khinchin
theorem since it is the total survival probability (and not just a non-
exponential part) which must be an even function.

Eq. (45) shows that the nature of the branch point is determined by the
exponent of the power law and this feature must be included if we want to
obtain an expression for the density of states. In summary, the density of
states can be built if we know (i) its poles and corresponding residues, (ii)
its form factor and (iii) the threshold factor $\nu$ which also defines the
exponent $2\nu+2$ of the survival probability for large times.
Since the density of states can be decomposed into the product of $E^{\nu}$,
with $\nu>0$, an analytic form factor $g(E)$ with $g(0)\neq 0$, and a
meromorphic function whose poles are the same as those of the density of
states, the deduction of a generic expression for the density of states starts
from the function

$F(z)=\frac{z^{-\nu}}{g(z)}\,\frac{\rho(z)}{z-E},$ (46)

which is the meromorphic component of the density of states, is analytic at
the origin and has simple poles at $z=z_{s}$ and $z=E$. Let $C_{N}$ be the
circle $|z|=R_{N}$ such that $|z_{s}|<R_{N}$ for $s=1,2,\dotsc N$ and contains
the pole $z=E$. From the residue theorem:

$\frac{1}{2\pi
i}\oint_{C_{N}}{\frac{z^{-\nu}}{g(z)}\,\frac{\rho(z)}{z-E}\,dz}=\frac{E^{-\nu}}{g(E)}\,\rho(E)-\sum_{|s|\leq
N}{\frac{1}{z_{s}^{\nu}g(z_{s})}\,\frac{R(z_{s})}{E-z_{s}}}.$ (47)

If $z^{-\nu}\rho(z)/g(z)=O\bigl{(}|z|^{-\delta}\bigr{)}$ for $|z|\to\infty$
and $\delta>0$, the integral vanishes when $N$ tends to infinity. Hence,

$\rho(E)=E^{\nu}\,g(E)\sum_{s}{\biggl{[}-\frac{R(z_{s})}{g(z_{s})}\biggr{]}\,\frac{1}{z_{s}^{\nu}}\,\frac{1}{z_{s}-E}}.$
(48)

Since the density of states has to be a real function, the form factor must
have the following property in the complex $z$-plane:
$g(z_{s<0})=g^{*}(z_{s>0})$. Using this property with Eqs (5) and (6), we can
write an alternative form for $\rho(E)$:

$\rho(E)=\frac{1}{2}\,E^{\nu}\,g(E)\,{\sum_{s}{\frac{\gamma(z_{s})}{z_{s}^{\nu}}\,\frac{1}{z_{s}-E}}}=E^{\nu}\,g(E)\,\text{Re\,}{\sum_{s>0}{\frac{\gamma(z_{s})}{z_{s}^{\nu}}\,\frac{1}{z_{s}-E}}},$
(49)

where $\gamma(z_{s})$ is defined as

$\gamma(z_{s})=-2\,\frac{R(z_{s})}{g(z_{s})},$ (50)

and due to Eq. (5) and the property of the form factor, it satisfies
$\gamma(z_{s<0})=\gamma^{*}(z_{s>0})$. Since the density of states satisfies
the normalization condition, we have

$\int_{0}^{\,\infty}{\rho(E)\,dE}=\text{Re\,}{\sum_{s>0}{\frac{\gamma(z_{s})}{z_{s}^{\nu}}\,\int_{0}^{\,\infty}{\frac{E^{\nu}}{z_{s}-E}\,g(E)\,dE}}}=1.$
(51)

The density of states is such that the integrals (42) will be finite, i.e.,

$\text{Re\,}{\sum_{s>0}{\frac{\gamma(z_{s})}{z_{s}^{\nu}}\,\int_{0}^{\,\infty}{\frac{E^{\nu+n}}{z_{s}-E}\,g(E)\,dE}}}<\infty,\quad
n=1,2,\cdots.$ (52)

Since the non-exponential survival probability given by Eq. (8) is
proportional to the Laplace transform of the density of states on the negative
imaginary axis, by the existence theorem of the Laplace transform, if there
exist real constants $M$ and $\alpha$ such that $\rho(-iy)$ has exponential
order $\alpha$, i.e., $\bigl{|}\rho(-iy)\bigr{|}\leq Me^{\,\alpha y}$, the
integral converges for $t>\alpha$ Schiff . From Eq. (49):

$\frac{1}{2}\bigl{|}y\bigr{|}^{\nu}\,\bigl{|}g(-iy)\bigr{|}\sum_{s}{\biggl{|}\frac{\gamma(z_{s})}{z_{s}^{\nu}}\biggr{|}\,\frac{1}{\bigl{|}z_{s}+iy\bigr{|}}}\leq
Me^{\,\alpha y}.$ (53)

Since $t\geq 0$, the integral (8) must be convergent for $t\geq 0$, and this
means that $\alpha=0$. Hence, the above inequality becomes:

$\bigl{|}y\bigr{|}^{\nu}\,\bigl{|}g(-iy)\bigr{|}\sum_{s}{\biggl{|}\frac{\gamma(z_{s})}{z_{s}^{\nu}}\biggr{|}\,\frac{1}{\bigl{|}z_{s}+iy\bigr{|}}}\leq
M.$ (54)

There, the factor $\tfrac{1}{2}$ was absorbed into the constant $M$. A curious
consequence of the condition (54) is that the form factor cannot be a Gaussian
function, because if $g(E)=e^{-aE^{2}}$, where $a>0$, the left hand of Eq.
(54) would be not bounded over the negative imaginary axis. Finally, notice
that the form factor could be obtained by solving the integral equation (51)
subject to the conditions (52) and (54).

In many of the physical examples of unstable states such as radioactive
nuclei, the poles are such that their imaginary parts are much less that their
real parts, i.e., $\text{Im\,}{z_{s}}\ll\text{Re\,}{z_{s}}$. These poles are
referred to as narrow resonances. In such cases, the coefficients
$\gamma(z_{s})$ do not depend on the residues of the density of states and is
a constant equal to $i$ ourpapercriticaltimes ; ourpaperSmatrix . Thus, for
narrow resonances,

$\gamma(z_{s})=-2\,\frac{R(z_{s})}{g(z_{s})}=-i,\quad\text{Im\,}{z_{s}}\ll\text{Re\,}{z_{s}},$
(55)

or

$\frac{R(z_{s})}{g(z_{s})}=\frac{i}{2},\quad\text{Im\,}{z_{s}}\ll\text{Re\,}{z_{s}}.$
(56)

Using Eq. (55), Eq. (49) transforms to

$\rho(E)=E^{\nu}\,g(E)\,\text{Im\,}{\sum_{s>0}{\frac{1}{z_{s}^{\nu}}\,\frac{1}{z_{s}-E}}},\quad\text{Im\,}{z_{s}}\ll\text{Re\,}{z_{s}}.$
(57)

##  VI Specific form factors

We shall now discuss the advantages or disadvantages of choosing some
particular form of the form factor in the density of states to obtain the
survival amplitude at small times. Our two choices are the exponential form
factor (commonly used for particle and nuclear resonances) and a constant
which is used in a potential description of unstable states.

###  VI.1 Exponential form factor

This is one popular choice in the literature because it allows closed
expressions to be obtained easily. In this case the form factor $g(E)$ takes
the form:

$g(E)=e^{-bE},\quad b>0.$ (58)

Even if there is no physical reason for choosing this particular form, it
satisfies the requirements imposed on the form factor, i.e., it ensures the
convergence of the integrals (42) and the condition (54) will be satisfied if
$0<\nu\leq 1$. Substituting Eq. (58) in Eq. (49), the density of states is
given by

$\rho(E)=\rho(E)=\frac{1}{2}\,E^{\nu}\,e^{-bE}\,{\sum_{s}{\frac{\gamma(z_{s})}{z_{s}^{\nu}}\,\frac{1}{z_{s}-E}}}=E^{\nu}\,e^{-bE}\,\text{Re\,}{\sum_{s>0}{\frac{\gamma(z_{s})}{z_{s}^{\nu}}\,\frac{1}{z_{s}-E}}}.$
(59)

We shall first calculate the expectation values of the powers of the
Hamiltonian at the initial state by substituting Eq. (49) in Eqs (42):

$\bigl{\langle}H^{n}\bigr{\rangle}_{0}=\int_{0}^{\,\infty}{E^{n}\rho(E)\,dE}=\text{Re\,}{\sum_{s>0}{\frac{\gamma(z_{s})}{z_{s}^{\nu}}\,\int_{0}^{\,\infty}{\frac{E^{\nu+n}}{E-z_{s}}\,e^{-bE}\,dE}}}.$
(60)

From the identity Erderly

$\int_{0}^{\,\infty}{\frac{x^{\nu}}{x+\sigma}\,e^{-sx}\,dx}=\Gamma\bigl{(}\nu+1\bigr{)}\,e^{\,\sigma
s}\,\sigma^{\nu}\,\Gamma\bigl{(}-\nu,\sigma
s\bigr{)},\quad\text{Re\,}{s}>0,\text{Re\,}{\nu}>-1,|\operatorname{Arg}{\sigma}|<\pi,$
(61)

where $\Gamma(\alpha,z)$ is the incomplete gamma function333Actually, there
are two differents incomplete gamma functions: the lower and upper ones
Lebedev . Here, we are using the latter one, which is defined as
$\Gamma(\alpha,z)=\int_{z}^{\infty}{t^{\alpha-1}e^{-t}\,dt},\quad\text{Re\,}{\alpha}>0.$
, we obtain:

$\bigl{\langle}H^{n}\bigr{\rangle}_{0}=\int_{0}^{\,\infty}{E^{n}\rho(E)\,dE}=(-1)^{n+1}\Gamma\bigl{(}1+\nu+n\bigr{)}\text{Re\,}{\sum_{s>0}{e^{\,i\nu\pi}{z_{s}^{n}\,\gamma(z_{s})}\,e^{-bz_{s}}\,\Gamma\bigl{(}-\nu-n,-bz_{s}\bigr{)}}}.$
(62)

Since $\bigl{\langle}H^{0}\bigr{\rangle}_{0}=1$, the residues of the density
of states satisfy

$-\Gamma\bigl{(}1+\nu\bigr{)}\text{Re\,}{\sum_{s>0}{e^{\,i\nu\pi}{\,\gamma(z_{s})}\,e^{-bz_{s}}\,\Gamma\bigl{(}-\nu,-bz_{s}\bigr{)}}}=1.$
(63)

Eq. (62) must incorporate Eq. (63) in order to ensure the normalization
condition. Thus,

$\bigl{\langle}H^{n}\bigr{\rangle}_{0}=(-1)^{n}\frac{\Gamma\bigl{(}1+\nu+n\bigr{)}}{\Gamma\bigl{(}1+\nu\bigr{)}}\,\frac{\text{Re\,}{\sum_{s>0}{e^{\,i\nu\pi}{z_{s}^{n}\,\gamma(z_{s})}\,e^{-bz_{s}}\,\Gamma\bigl{(}-\nu-n,-bz_{s}\bigr{)}}}}{\text{Re\,}{\sum_{s>0}{e^{\,i\nu\pi}{\,\gamma(z_{s})}\,e^{-bz_{s}}\,\Gamma\bigl{(}-\nu,-bz_{s}\bigr{)}}}}.$
(64)

In the case of narrow resonances, Eq. (55) gives us an expression for the
residues of the density of states. Therefore, Eq. (64) takes the following
form:

$\bigl{\langle}H^{n}\bigr{\rangle}_{0}=(-1)^{n}\frac{\Gamma\bigl{(}1+\nu+n\bigr{)}}{\Gamma\bigl{(}1+\nu\bigr{)}}\,\frac{\text{Im\,}{\sum_{s>0}{e^{\,i\nu\pi}{z_{s}^{n}\,e^{-bz_{s}}}\,\Gamma\bigl{(}-\nu-n,-bz_{s}\bigr{)}}}}{\text{Im\,}{\sum_{s>0}{e^{\,i\nu\pi}{\,e^{-bz_{s}}}\,\Gamma\bigl{(}-\nu,-bz_{s}\bigr{)}}}},\quad\text{Im\,}{z_{s}}\ll\text{Re\,}{z_{s}}.$
(65)

On the other hand, we can calculate the survival amplitude using the identity
(61):

$A(t)=\int_{0}^{\,\infty}{\rho(E)e^{-iEt}\,dE}=\frac{{\sum_{s}{e^{\,i\nu\pi\operatorname{sgn}{(s)}}{\gamma(z_{s})}\,e^{-pz_{s}}\,\Gamma\bigl{(}-\nu,-pz_{s}\bigr{)}}}}{{\sum_{s}{e^{\,i\nu\pi\operatorname{sgn}(s)}{\,\gamma(z_{s})}\,e^{-bz_{s}}\,\Gamma\bigl{(}-\nu,-bz_{s}\bigr{)}}}},$
(66)

where $p=b+it$. Even though the Eqs (64) and (65) allow us to compute the
coefficients of the Taylor expansion, we have to choose a suitable value for
the parameter $b$. It must be chosen such that the inequality (35) is
satisfied, i.e., $\bigl{\langle}(\Delta H)^{2}\bigr{\rangle}_{0}$ will be
positive. Since establishing the range of the values of $b$ for which
$\bigl{\langle}(\Delta H)^{2}\bigr{\rangle}_{0}>0$ analytically is almost
impossible, it is better to achieve this by employing numerical and graphical
methods. In order to ilustrate this procedure, the following example is worked
out for one isolated, narrow resonance. Introducing the parameters $x_{s}$,
the normalized pole $\xi_{s}$, and the dimensionless variable for time,
$\tau$, as

$\displaystyle x_{s}$
$\displaystyle\equiv\frac{\text{Im\,}{z_{s}}}{\text{Re\,}{z_{s}}},$ (67)
$\displaystyle\xi_{s}$
$\displaystyle\equiv\frac{z_{s}}{\text{Re\,}{z_{s}}}=1-ix_{s}\operatorname{sgn}{(s)},$
(68) $\displaystyle\tau$ $\displaystyle\equiv 2t\text{Im\,}{z_{s}},$ (69)

the terms $bz_{s}$ and $pz_{s}$ in Eqs (64), (65) and (66) have to be replaced
by

$\displaystyle
bz_{s}=\bigl{(}b\text{Re\,}{z_{s}}\bigr{)}\xi_{s}=b_{s}\xi_{s},$ (70)
$\displaystyle
pz_{s}=(b+it)z_{s}=\biggl{(}b_{s}+i\frac{\tau}{2x_{s}}\biggr{)}\xi_{s},$ (71)

where $b_{s}=b\text{Re\,}{z_{s}}$. The parameter $x_{s}\ll 1$ indicates that a
pole represents a narrow resonance, and it is related to the oscillation
frequency $f_{s}$ of the survival probability as well, with $f_{s}$ given by
ourpapercriticaltimes ; winter : $f_{s}=\frac{1}{4\pi x_{s}}$. This frequency
allows us to measure the time of the decay in terms of the number of
oscillations $n$ that the survival probability has performed:

$n=\frac{\tau}{4\pi x_{s}},$ (72)

which is suitable for the description of the survival probability at small
times. The choice of a suitable value of $b$ (or $b_{s}$) depends on the sign
of $\bigl{\langle}(\Delta H)^{2}\bigr{\rangle}_{0}$. For $x_{s}=0.1$ and
$\nu=1/2$, $\bigl{\langle}(\Delta H)^{2}\bigr{\rangle}_{0}$ is negative if
$3.51\times 10^{-4}<b_{s}<0.55$. If we choose a value of $b_{s}$ such that
$\bigl{\langle}(\Delta H)^{2}\bigr{\rangle}_{0}<0$, we would expect the
survival probability to take values greater than one. We illustrate this
feature in Fig. 2 by computing $P(t)$ with $b_{s}=0.1$.

Figure 2: Survival probability for a system described by an isolated resonance
such that $x_{s}=0.1$, $\nu=1/2$, and $b_{s}=0.1$ as a function of the number
of oscillations $n$, and the dimensionless variable for time $\tau$ given by
the Eq. (69). The dashed line represents the exponential component of the
survival probability.

In this case, we can see that the survival probability is greater than one
approximately at the first one-fourth of the oscillation. Since the survival
probability is convex at $t=0$, the form factor with the value of $b_{s}$
chosen forces $P(t)$ to increase until a maximum. This behavior makes no
physical sense.

This situation enables us to pick $b_{s}$ such that $\bigl{\langle}(\Delta
H)^{2}\bigr{\rangle}_{0}>0$ for two intervals. Moreover, the larger the value
of $b_{s}$, the shorter the critical time that characterizes the transition
from the exponential to the power law behavior. Such a sensitivity of the
critical time to the value of $b_{s}$ is however seen only for large values of
$b_{s}$. If $b_{s}\ll 1$ we do not face the problem of a variable critical
time since the effect of the form factor over $P(t)$ is weak. However, the
convergence of the survival probability depends on the condition
$\text{Re\,}{p}>0$, and then, a singularity is located at $p=0$. Hence, we
should expect that the survival probability experiences strong variations when
both $t$ and $b_{s}$ are close to zero. In Fig. 3 we plot the survival
probability for $x_{s}=0.1$ and a relatively large value of $b_{s}$, i.e.,
$b_{s}=1$. Note that this value of $b_{s}$ ensures a survival probability less
than one for $n\geq 0$.

Figure 3: Survival probability for a system described by an isolated resonance
(with $x_{s}=0.1$, $\nu=1/2$, and $b_{s}=1$) as a function of the number of
oscillations $n$, and the dimensionless variable for time $\tau$ given by the
Eq. (69). The dashed line represents the exponential component of the survival
probability.

Fig. 4 shows the survival probability for $x_{s}=0.1$ and $b_{s}=10^{-4}$. As
expected, the survival probability is less than one for $n\geq 0$, its
critical time (for the transition from the exponential to the power law) has
changed with respect to the previous example, i.e., it happens around 15 to 20
oscillations and near $t=0$, the survival probability decreases from 1 to
approximately $e^{-2}$ in less than two thousandth of a period of oscillation
of $P(t)$.

Figure 4: Survival probability for a system described by an isolated resonance
($x_{s}=0.1$, $\nu=1/2$, and $b_{s}=10^{-4}$) as a function of the number of
oscillations $n$, and the dimensionless variable for time $\tau$ given by the
Eq. (69). The upper plot shows $P(n)$ for 30 oscillations, and the lower one
shows $P(n)$ for one oscillation. The dashed line represents the exponential
component of the survival probability.

Knowing how to choose a suitable value of the parameter $b$ ( or $b_{s}$), we
shall now show what would happen with the survival probability for small times
for a realistic case, namely, the decay of ${}^{8}\text{Be}(0^{+})$ into two
alpha particles for S-waves and assuming an exponential form factor. In this
case, the real part of the pole is, $\text{Re\,}{z_{s}}=92$ keV and the
imaginary part is $5.6/2=2.8$ eV. Thus, $x_{s}=3\times 10^{-5}$, and
$\nu=1/2$. The survival amplitude oscillates with a period of $4.6\times
10^{-4}$ mean lifetimes. Here, the sign of $\bigl{\langle}(\Delta
H)^{2}\bigr{\rangle}_{0}$ is negative if $4.8\times 10^{-11}<b_{s}<0.5$. Since
a tiny value of $b_{s}$ does not describe the survival probability near $t=0$
because of the singularity at $p=b+it=0$, the appropiate value of $b_{s}$
should satisfy $b_{s}>0.5$. Hence, we choose $b_{s}=1$, or
$b=b_{s}/\text{Re\,}{z_{s}}=10.83(\text{MeV})^{-1}$. For this value of
$b_{s}$, the survival probability can be seen in Fig. 5.

Figure 5: Survival probability for the decay of ${}^{8}\text{Be}(0^{+})$ into
two alpha particles with $b_{s}=1$ as a function of the number of oscillations
$n$, and the dimensionless variable for time $\tau$ given by the Eq. (69). The
dashed line represents the exponential component of the survival probability.

Let us look at the prominent features of this decay for the parameters chosen
above. The survival probability $P(t)$ is less than one for the value of
$b_{s}$ that we chose, $P(t)$ reaches the large time region after 150000
oscillations approximately (around 70 mean lifetimes) and the intermediate
region starts after the survival amplitude has completed one oscillation (in
$4.6\times 10^{-4}$ mean lifetimes).

Both examples point to the fact that the exponential behavior at small times
is reached approximately after the survival amplitude has completed one
oscillation. Although this could seem to be a coincidence, we shall show in
section VII that the intermediate region indeed starts approximately after the
first oscillation is completed by the survival probability for narrow isolated
resonances.

###  VI.2 Constant form factor

Even though a constant factor is the easiest and simplest choice, we shall
show that this option implies some contradictions. The form factor $g(E)$
takes the form:

$g(E)=1,$ (73)

and the density of states is given by

$\rho(E)=E^{\nu}\sum_{s}{\frac{R(z_{s})}{z_{s}^{\nu}}\,\frac{1}{E-z_{s}}}.$
(74)

From the normalization condition, the density of states must satisfy

$\int_{0}^{\,\infty}{\rho(E)\,dE}=\sum_{s}{\frac{R(z_{s})}{z_{s}^{\nu}}\,\int_{0}^{\,\infty}{\frac{E^{\nu}}{E-z_{n}}\,dE}}=1$
(75)

From a mathematical point of view, the integral in Eq. (75) converges if
$-1<\nu<0$, and we have $\nu>0$ because the density of states must be analytic
at $E=0$. Assuming that $\nu$ takes values in the range $0<\nu<1$, we can
write Eq. (75) as

$\int_{0}^{\,\infty}{\rho(E)\,dE}=1=\sum_{s}{\frac{R(z_{s})}{z_{s}^{\nu}}}\int_{0}^{\,\infty}{E^{\nu-1}\,dE}+\sum_{s}{\frac{R(z_{s})}{z_{s}^{\nu-1}}\,\int_{0}^{\,\infty}{\frac{E^{\nu-1}}{E-z_{n}}\,dE}},$
(76)

and we can see that the integral is not finite because of the integral in the
first term of the right side of Eq. (76) unless

$\sum_{s}{\frac{R(z_{s})}{z_{s}^{\nu}}}=0,\quad 0<\nu<1.$ (77)

From the Eqs (76) and (77), we have

$\int_{0}^{\,\infty}{\rho(E)\,dE}=1=\sum_{s}{\frac{R(z_{s})}{z_{s}^{\nu-1}}\,\int_{0}^{\,\infty}{\frac{E^{\nu-1}}{E-z_{s}}\,dE}}=-\frac{\pi}{\sin{\pi\nu}}\sum_{s}{R(z_{s})\,e^{\,i\pi\nu\operatorname{sgn}{(s)}}}.$
(78)

In conclusion, if $0<\nu<1$ and because of the normalization condition, the
residues and poles of the density of states have to satisfy the conditions

$\displaystyle\sum_{s}{R(z_{s})\,e^{\,i\pi\nu\operatorname{sgn}{(s)}}}$
$\displaystyle=-\frac{\sin{\pi\nu}}{\pi},$ (79)
$\displaystyle\sum_{s}{\frac{R(z_{s})}{z_{s}^{\nu}}}$ $\displaystyle=0.$ (80)

Although the conditions (79) and (80) seem arbitrary, they appear naturally in
systems under the influence of a central potential of finite range whose decay
is calculated by either resonant states or Jost functions
ourpapercriticaltimes . Moreover, substituting $\rho(E)$ given by Eq. (75)
into Eq. (42), we get the expectation values of the integer powers of the
Hamiltonian:

$\displaystyle\int_{0}^{\,\infty}{E^{m}\rho(E)\,dE}$
$\displaystyle=\sum_{s}{\frac{R(z_{s})}{z_{s}^{\nu}}\int_{0}^{\,\infty}{E^{\nu}\frac{E^{m}}{E-z_{s}}\,dE}}$
$\displaystyle=\sum_{s}{\frac{R(z_{s})}{z_{s}^{\nu}}\int_{0}^{\,\infty}{E^{\nu-1}\Biggl{[}\sum_{p=0}^{m}{z_{s}^{p}E^{m-p}}+\frac{z_{s}^{m+1}}{E-z_{s}}\Biggr{]}\,dE}}$
$\displaystyle=\sum_{p=0}^{m}{\Biggl{(}\int_{0}^{\,\infty}{E^{m+\nu-p-1}\,dE}\Biggr{)}\Biggl{(}\sum_{s}{z_{s}^{p}\,\frac{R(z_{s})}{z_{s}^{\nu}}}\Biggr{)}}+\sum_{s}{z_{s}^{m+1}\frac{R(z_{s})}{z_{s}^{\nu}}\int_{0}^{\,\infty}{\frac{E^{\nu-1}}{E-z_{s}}\,dE}}.$
(81)

Eq. (81) will be finite if

$\sum_{s}{z_{s}^{p}\frac{R(z_{s})}{z_{s}^{\nu}}}=0,\quad p=0,1,\dotsc,m.$ (82)

Since this condition should be valid for all positive $m$, it can be written
as:

$\text{Re\,}{\sum_{s>0}{z_{s}^{p}\frac{R(z_{s})}{z_{s}^{\nu}}}}=0,\quad
p=0,1,\dotsc,m,\dotsc$ (83)

The expectation value of $H^{n}$ at the initial state is therefore given by,

$\bigl{\langle}H^{m}\bigr{\rangle}_{0}=\int_{0}^{\,\infty}{E^{m}\rho(E)\,dE}=\sum_{s}{z_{s}^{m+1-\nu}R(z_{s})\int_{0}^{\,\infty}{\frac{E^{\nu-1}}{E-z_{s}}\,dE}}\\\
=-\frac{\pi}{\sin{\pi\nu}}\sum_{s}{z_{s}^{m}R(z_{s})\,e^{\,i\pi\nu\operatorname{sgn}{(s)}}},\quad
m=0,1,2,\dotsc$ (84)

It is worth to point out that it is possible to ensure the convergence of the
integrals (42) by imposing conditions over the poles and residues of the
density of states given by Eqs (82) and (84). For $p=0$, we get the condition
(80) which was obtained by the normalization condition. As a result, the
survival amplitude can be obtained from Eq. (135) and the identity (61). After
a short algebra, we get,

$A(t)=-2\pi
i\sum_{s>0}{R(z_{s})e^{-iz_{s}t}}-\frac{\pi}{\sin{\pi\nu}}e^{-i\pi\nu}\sum_{s}{R(z_{s})e^{-itz_{s}}\frac{\Gamma\bigl{(}-\nu,-iz_{s}t\bigr{)}}{\Gamma\bigl{(}-\nu\bigr{)}}},\quad
0<\nu<1.$ (85)

There are some advantages to studying the decay of a system using a density of
states with a constant form factor. Firstly, it lets us separate the
exponential behavior from both the survival amplitude and probability.
Secondly, it lets us deal with isolated resonances because the conditions (79)
and (80) give us an expression for the residue in terms of its associated pole
only (see Eq. (88) below). Lastly, the descriptions of the survival
probability in the intermediate and large time regions are simpler and allow
us to study in detail the transition from the former to the latter region in a
suitable way. However, using this formalism for the description of $P(t)$ for
small times for one isolated resonance gives us a result that disagrees with
Eq. (34). Let us assume that the isolated resonance has the pole,
$z_{r}=\sigma_{r}-i\omega_{r}/2$, associated with it. Thus, its residue is
obtained from the conditions (63) and (64) to be

$\displaystyle\frac{R(z_{r})}{z_{r}^{\nu}}+\frac{R^{*}(z_{r})}{{z_{r}^{*}}^{\nu}}=0,$
(86) $\displaystyle
R(z_{r})e^{\,i\pi\nu}+R^{*}(z_{r})e^{-i\pi\nu}=-\frac{\sin{\pi\nu}}{\pi}.$
(87)

Solving Eqs (86) and (87), the residue is given by,

$R(z_{r})=-\frac{1}{2\pi
i}\,\frac{z_{r}^{\nu}\sin{\pi\nu}}{\text{Im\,}{\bigl{(}z_{r}^{\nu}\,e^{\,i\pi\nu}\bigr{)}}}.$
(88)

Since $R(z_{r})$ has to satisfy the conditions (66), we get,

$\text{Re\,}{\sum_{s>0}{z_{s}^{p-\nu}R(z_{s})}}=\text{Re\,}{\Biggl{[}z_{r}^{p-\nu}\,\frac{\sin{\pi\nu}}{2\pi
i}\,\frac{z_{r}^{\nu}}{\text{Im\,}{\bigl{(}z_{r}^{\nu}\,e^{\,i\pi\nu}\bigr{)}}}\Biggr{]}}=\frac{\sin{\pi\nu}}{2\pi\text{Im\,}{\bigl{(}z_{r}^{\nu}\,e^{\,i\pi\nu}\bigr{)}}}\text{Im\,}{\bigl{(}z_{r}^{p}\bigr{)}}=0,\text{
for all }p=0,1,2,\dotsc$ (89)

This equation can be written as,
$\sin{\bigl{[}p\operatorname{Arg}{z_{r}}\bigr{]}}=0,\text{ for all
}p=0,1,2,\dotsc$. The only choice that satisfies this equation for all values
of $p$ and for the range of values that $\text{Re\,}{z_{r}}$ and
$\text{Im\,}{z_{r}}$ can take is: $\operatorname{Arg}{z_{r}}=0$, a result that
implies $\text{Im\,}{z_{r}}=0$ and hence a resonance without a width. If one
would still insist to continue with the description of such a resonance and
examine the residue, one would find,

$\lim_{\text{Im\,}{z_{r}}\to 0}{R(z_{r})}=-\frac{1}{2\pi i},$ (90)

and the density of states would be given by

$\rho(E)=\lim_{\text{Im\,}{z_{r}}\to
0}{\biggl{(}-\frac{1}{\pi}\biggr{)}\text{Im\,}{\frac{\bigl{(}E/z_{r}\bigr{)}^{\nu}}{E-\text{Re\,}{z_{r}}+i\text{Im\,}{z_{r}}}}}.$
(91)

From the Plemelj-Dirac formula444The Plemelj-Dirac formula is:
$\lim_{\epsilon\to 0}{\frac{1}{x^{\prime}-x\mp
i\epsilon}}=P\,\frac{1}{x^{\prime}-x}\pm i\pi\delta(x^{\prime}-x),$ where $P$
is the principal value Wyld .

$\displaystyle\rho(E)$
$\displaystyle=-\frac{1}{\pi}\biggl{(}\frac{E}{\text{Re\,}{z_{r}}}\biggr{)}^{\nu}\lim_{\text{Im\,}{z_{r}}\to
0}{\text{Im\,}{\biggl{[}P\,\frac{1}{E-\text{Re\,}{z_{r}}}-i\pi\delta(E-\text{Re\,}{z_{r}})\biggr{]}}}$
$\displaystyle=\biggl{(}\frac{E}{\text{Re\,}{z_{r}}}\biggr{)}^{\nu}\delta(E-\text{Re\,}{z_{r}})=\delta(E-\text{Re\,}{z_{r}}),$
(92)

the survival amplitude reduces to

$A(t)=\int_{0}^{\,\infty}{\delta(E-\text{Re\,}{z_{r}})\,e^{-iEt}\,dt}=e^{-it\text{Re\,}{z_{r}}},$
(93)

and the decay probability will be equal to one, implying no decay. This means
that it is not possible to obtain the survival probability for small times
starting with the energy density of a system which is an isolated resonance.

##  VII Transition Regions and Critical Times

We mentioned before that the survival probability has three well-defined
regions: the small times region where $P(t)$ is dominantly quadratic, the
intermediate region where $P(t)$ is approximately exponential and the large
time region where $P(t)$ displays a power law behaviour. However, there is not
really a sharp separation between the three regions but rather an oscillatory
transition region from the quadratic to the exponential and the exponential to
the power law. Associated with these regions are the critical times, which
indicate when the transition starts and ends or when the transition is
happening. In this section, we shall explore these aspects.

###  VII.1 Critical Time and the Transition Region from Small to Intermediate
Times

In order to find the critical time of transition, $t_{cs}$, from the quadratic
to the exponential region we begin by approximating $P(t)$ given by (36), at
small times, as

$P(t)\approx 1-\bigl{\langle}(\Delta H)^{2}\bigr{\rangle}_{0}t^{2}.$ (94)

Focussing on the intermediate region, we note that from Eqs (7) and (28), we
know the exponential component of both the survival amplitude and probability.
In order to simplify the notation in what follows, we recall that
$\bar{R}(z_{s})=-2\pi i{R}(z_{s})$; and rewriting Eqs (7) and (28) and
expressing the poles in terms of their real and imaginary parts, we get,

$\displaystyle A_{e}(t)$
$\displaystyle=\sum_{s}{\bar{R}(z_{s})e^{-i\sigma_{s}t}\,e^{\,\omega_{s}t/2}}$
(95) $\displaystyle P_{e}(t)$
$\displaystyle=|A_{e}(t)|^{2}=\sum_{s,s^{\prime}}{\bar{R}(z_{s})\bar{R}^{*}(z_{s^{\prime}})e^{-i(\sigma_{s}-\sigma_{s^{\prime}})t}e^{-(\omega_{s}+\omega_{s^{\prime}})t/2}}$
$\displaystyle=\sum_{s}{|\bar{R}(z_{s})|^{2}\,e^{-\omega_{s}t}}+\sum_{s\neq
s^{\prime}}{\bar{R}(z_{s})\bar{R}^{*}(z_{s^{\prime}})e^{-i(\sigma_{s}-\sigma_{s^{\prime}})t}e^{-(\omega_{s}+\omega_{s^{\prime}})t/2}}.$
(96)

In order to study the transition, the simplest choice is to approximate
$P_{e}(t)$ by taking the slowest decreasing term which is associated with the
fourth-quadrant pole of the density of states having the smallest absolute
value of the imaginary part. We call this the decay’s dominant pole,
$z_{d}=\sigma_{d}-i\omega_{d}/2$, so that $\omega_{d}$ must satisfy the
following:

$\omega_{d}=-2\min_{s}{\bigl{\\{}\bigl{|}\text{Im\,}{z_{s}}\bigr{|}\bigr{\\}}}.$
(97)

The intermediate time survival probability can now be written approximately
as,

$P(t)\approx|\bar{R}(z_{d})|^{2}\,e^{-\omega_{d}t}.$ (98)

The transition time $t=t_{cs}$ could be defined as the time for which the
probabilities given by Eqs (94) and (98) are equal. Thus, $t_{cs}$ satisfies
the equation

$1-\bigl{\langle}(\Delta
H)^{2}\bigr{\rangle}_{0}t_{cs}^{2}=|\bar{R}(z_{d})|^{2}\,e^{-\omega_{d}t_{cs}}.$
(99)

It is convenient to define the dimensionless quantity,
$\tau_{cs}=\omega_{d}t_{cs}$, which, after some algebraic manipulations allows
us to write Eq. (99) as,

$f(\tau_{cs})=\biggl{[}1-\biggl{(}\frac{\tau_{cs}}{\alpha}\biggr{)}^{2}\biggr{]}e^{\,\tau_{cs}}=|\bar{R}(z_{d})|^{2},$
(100)

where we have introduced the parameter $\alpha$ defined as,

$\alpha=\frac{\omega_{d}}{\sqrt{\bigl{\langle}(\Delta
H)^{2}\bigr{\rangle}_{0}}}.$ (101)

The problem is thus reduced to finding the values of $\tau_{cs}$ for which
$f(\tau_{cs})$ is equal to $|\bar{R}(z_{d})|^{2}$. The next step is to study
under what conditions will Eq. (100) have a solution. Therefore, we need to
study some properties of this function in the interval
$0\leq\tau_{cs}\leq\alpha$ where the survival probability for small times is
positive. One can easily check these properties to be: (i) For $\tau_{cs}=0$,
$f(0)=1$ for all values of $\alpha$ and $\tau_{cs}=\alpha$, $f(\alpha)=0$.
(ii) Since $f^{\prime}(0)=1$, the straight line which is tangent to
$f(\tau_{cs})$ at $\tau_{0}$ has a slope of 1 and is independent of the value
taken by $\alpha$.
(iii) $f(\tau_{cs})$ has a maximum in the interval $(0,\alpha)$ given by
$\tau_{+}=\sqrt{1+\alpha^{2}}-1$, and the value of the function at that point
is $f(\tau_{+})=f_{+}=2\tau_{+}e^{\,\tau_{+}}/\alpha^{2}$. it is concave. But,
for $\alpha\geq\sqrt{2}$, it has an inflection point at
$\tau_{cs}\equiv\tau_{i}=\sqrt{\alpha^{2}+2}-2$ and it is convex for
$\tau_{cs}<\tau_{i}$ and concave for $\tau_{cs}>\tau_{i}$. The above features
of $f(\tau_{cs})$ in the interval $0\leq\tau_{cs}\leq\alpha$ are represented
graphically in the figure 6.

Figure 6: Sketch of the function $f(\tau_{cs})$.

The maximum of $f(\tau_{cs})$ allows us to deduce a criterion for knowing how
many solutions Eq. (100) can have: (i) If
$\bigl{|}\bar{R}(z_{d})\bigr{|}^{2}>f_{+}$, there is no solution. (ii) If
$f_{+}=\bigl{|}\bar{R}(z_{d})\bigr{|}^{2}$, there is one solution. (iii) If
$1<\bigl{|}\bar{R}(z_{d})\bigr{|}^{2}<f_{+}$, there are two solutions. (iv) If
$\bigl{|}\bar{R}(z_{d})\bigr{|}^{2}<f_{+}<1$, there is one solution. These
cases are represented graphically in Fig. 6. The primary conclusion about this
definition of the transition time is that there exist different solutions that
depend on the modulus squared of the residue of the density of states
evaluated at the dominant pole. Unlike the definitions adopted for the
transition time from the exponential to the power law behavior, there is no
physical criterion for choosing one particular solution in this case. In other
words, this definition of the critical time for the transition from the small
to the intermediate time region does not provide a unique solution and it is
not possible to choose one of them from physical arguments.

At the end of section VI.1, we conjectured based on the examples developed
there that the intermediate time region had been reached once the survival
probability completes its first oscillation. This would mean that the modulus
of the non-exponential survival amplitude at small times would be smaller than
the modulus of the exponential survival amplitude once the survival
probability completes its first oscillation, and this fact may give us a
pointer to define the critical time. Furthermore, this pointer can be easily
studied owing to the fact that the frequency of oscillations of both the
survival probability and the survival amplitude are the same (which shall be
shown in the section VII.3), which permits us to introduce the number of
oscillations of the survival probability in order to analyze the temporal
behavior of the exponential and non-exponential survival amplitudes. Let us
then write $A_{ne}(t)$ given by Eq. (8) and use the expression (49) for the
density of states for one isolated resonance as,

$A_{ne}(t)=\frac{1}{2}\sum_{s=\pm
d}{\frac{\gamma(z_{s})}{(iz_{s})^{\nu}}\int_{0}^{\,\infty}{\frac{y^{\nu}}{iz_{s}-y}\,g(-iy)\,e^{-ty}\,dy}}.$
(102)

Using the notation introduced by Eqs (67), (68) and (69) in section VI.1 for
writing the time in terms of the number of oscillations and making the change
of variable $y\to y\text{Re\,}{z_{s}}$, we get,

$A_{ne}(n)=\frac{1}{2}\sum_{s=\pm
d}{\frac{\gamma(z_{s})}{(i\xi_{s})^{\nu}}\int_{0}^{\,\infty}{\frac{y^{\nu}}{i\xi_{s}-y}\,g(-iy\text{Re\,}{z_{s}})\,e^{-(2\pi
n)y}\,dy}}.$ (103)

Considering the case of large $n$ and writing the asymptotic expansion of
$A_{ne}(n)$ using Watson’s lemma, we get,

$A_{ne}(n)\sim(-i)^{\nu+1}\,g(0)\,\text{Re\,}{\biggl{(}\frac{\gamma(z_{d})}{\xi_{d}^{\nu+1}}\biggr{)}}\,\frac{\Gamma\bigl{(}\nu+1\bigr{)}}{\bigl{(}2\pi
n\bigr{)}^{\nu+1}},\quad n\to\infty$ (104)

Note that for $n=1$, $2\pi n$ which is around 6.28 can be considered large and
thus we could say that the above equation is valid for $n\geq 1$. Although
obtaining an analytic estimation of the error in using the above expression
for all $n\geq 1$ is dificult, numerical tests validate this approximation.
From Eqs. (7), (67), (68) and (69), the exponential survival amplitude can be
written as,

$A_{e}(n)=-2\pi iR(z_{d})\,\exp{\bigl{(}-2\pi i\xi_{d}\,n\bigr{)}}.$ (105)

Hence,

${\cfrac{A_{ne}(n)}{A_{e}(n)}}=\frac{1}{C(x_{d},\nu)}\,\frac{e^{2\pi
x_{d}n}}{n^{\nu+1}}\,e^{\,i2\pi n},\quad n\geq 1,$ (106)

where $C(x_{d},\nu)$ is defined as:

$C(x_{d},\nu)=\frac{(2\pi)^{\nu+2}e^{\,i\pi\nu/2}R(z_{d})}{g(0){\Gamma\bigl{(}\nu+1\bigr{)}}\,\text{Re\,}{\bigl{(}{\gamma(z_{d})}/{\xi_{d}^{\nu+1}}\bigr{)}}}.$
(107)

If $n\sim 1$, $e^{2\pi x_{d}n}\ll n^{\nu+1}$ and $|A_{ne}(n)|\ll|A_{e}(n)|$.
This observation gets better for narrow resonances where $x_{s}\ll 1$. It
allows us to establish approximately when the small time region terminates.
Therefore, we define the critical time for the transition from the quadratic
small time behaviour of the decay law to the intermediate exponential one as
the time for which the survival amplitude has completed its first oscillation.
From Eq. (72), the critical time in the dimensionless units defined before is
given by,

$\tau_{cs}=4\pi x_{d}.$ (108)

On the other hand, we shall now provide expressions for describing not only
the small time survival probability, but also the transition from this region
to the exponential regime. We start by computing the ratio
${{A_{ne}(n)}/{A_{e}(n)}}$ once again from the Eqs. (103) and (105), i.e.,

$\displaystyle\frac{A_{ne}(n)}{A_{e}(n)}$ $\displaystyle=\frac{e^{2\pi
in}}{2\pi i}\sum_{s=\pm
d}{\frac{\gamma(z_{s})}{\gamma(z_{d})}{\frac{1}{(i\xi_{s})^{\nu}}}\int_{0}^{\,\infty}{e^{\,2\pi
x_{d}n}\,\frac{g\bigl{(}-iy\text{Re\,}{z_{s}}\bigr{)}}{g(z_{d})}\,{\frac{y^{\nu}\,e^{-(2\pi
n)y}}{i\xi_{s}-y}}\,\,dy}}$ $\displaystyle=\frac{e^{2\pi in}}{2\pi
i}\int_{0}^{\,\infty}{e^{\,2\pi
x_{d}n}\,\frac{g\bigl{(}-iy\text{Re\,}{z_{d}}\bigr{)}}{g(z_{d})}\,{y^{\nu}\,e^{-(2\pi
n)y}}\Biggl{[}{\frac{1}{(i\xi_{d})^{\nu}}}\,\frac{1}{i\xi_{d}-y}+\frac{\gamma(z_{d}^{*})}{\gamma(z_{d})}\,{\frac{1}{(i\xi_{d}^{*})^{\nu}}}\,\frac{1}{i\xi_{d}^{*}-y}\Biggr{]}\,dy}.$
(109)

Since we are dealing with narrow resonances ($x_{d}\ll 1$) and since $n<1$,
$\gamma(z_{d})\approx-i$, $\exp{\bigl{(}\,2\pi x_{d}n\bigr{)}}\approx 1$, and
$\xi_{d}\approx e^{-ix_{d}}$. Hence, Eq. (109) becomes:

$\frac{A_{ne}(n)}{A_{e}(n)}\approx\frac{e^{2\pi in}}{2\pi
i}\,\int_{0}^{\,\infty}{\frac{g\bigl{(}-iy\text{Re\,}{z_{d}}\bigr{)}}{g(z_{d})}\,(-iy)^{\nu}\,e^{-(2\pi
n)y}\,\Biggl{[}{\frac{e^{-i\nu x_{d}}}{y-ie^{\,ix_{d}}}-\frac{e^{\,i\nu
x_{d}}}{y-ie^{-ix_{d}}}}\Biggr{]}\,dy}.$ (110)

Without the knowledge of the specific form of the form factor, we cannot make
more approximations. However, as $n$ goes from 0 to 1, we could expect that
the integral decreses because of the exponential factor $e^{-2\pi ny}$.
Calling this integral $N(n)$, i.e.,

$N(n)=\frac{1}{2\pi
i}\int_{0}^{\,\infty}{\frac{g\bigl{(}-iy\text{Re\,}{z_{d}}\bigr{)}}{g(z_{d})}\,(-iy)^{\nu}\,e^{-(2\pi
n)y}\,\Biggl{[}{\frac{e^{-i\nu x_{d}}}{y-ie^{\,ix_{d}}}-\frac{e^{\,i\nu
x_{d}}}{y-ie^{-ix_{d}}}}\Biggr{]}\,dy},$ (111)

we have:

$\frac{A_{ne}(n)}{A_{e}(n)}\approx e^{2\pi in}N(n),\quad x_{s}\ll 1,0<n<1.$
(112)

Both the survival probability and the transition from small to intermediate
times can be described through Eq. (112). Furthermore, the survival
probability at small times will be given by

$P(t)=\bigl{|}A_{e}(n)+A_{ne}(n)\bigr{|}^{2}=4\pi^{2}|R(z_{d})|^{2}e^{-4\pi
x_{d}n}\bigl{|}1+e^{2\pi in}N(n)\bigr{|}^{2},$ (113)

expression that shows the oscillatory nature of the transition. An example of
how the transition from small to intermediate time happens can be seen in the
figure 7 for $x_{d}=0.01$ and an exponential form factor with $b_{d}=1$. We
can see that the survival probability for small times given by the Eq. (111)
(dashed line) agrees with the survival probability calculated from the eqs.
(103) and (105) (solid line). We also see that the transition ends near $n=1$
by comparing with the exponential survival probability (dotted line).

Figure 7: Survival probability for small times for a system with $x_{d}=0.01$,
and an exponential form factor with $b_{s}=1$ as a function of the number of
oscillations $n$, and the dimensionless variable for time $\tau$ given by the
Eq. (69). The solid line represents the complete survival probability, the
dashed line represents the survival probability given by the Eq. (113), and
the dotted line represent the exponential component of the survival
probability.

In the section VII.3, we shall use the asymptotic expansion of $A_{ne}(n)$
once again and we shall deduce the conditions on the number of oscillations,
$n$, for the intermediate exponential region and its transition to the power
law at large times. Apart from the above observations, we can also see that
since the survival probability is convex downward at $n=0$ and it becomes
convex upward near $n=1/2$, the quadratic law will be valid approximately in
the first half-cycle of the survival probability, and the transition from the
small to the intermediate time region occurs in the second half-cycle.

###  VII.2 Comparison with other approaches

Investigations of the small time behaviour of the survival probability have a
long history with the interest being particularly enhanced by the possibility
of the so-called quantum Zeno effect sudarshan . Here we shall compare the
results obtained in the present work with some of those in literature for the
behaviour of the survival probability at short times and its transition to the
exponential behaviour at intermediate times. Before performing such a
comparison, we must emphasize that most of the comparisons in literature do
not take into account the fact that the survival probability $P(t)$ must be an
even function of $t$. For example, in sudarshan , on the basis of models, the
authors propose that as $t\to 0$,

$P(t)\rightarrow 1-\frac{\alpha}{\beta}t^{\beta},\,\,\,\beta\neq 1.$ (114)

The critical time is typically determined by comparing the expansion of the
exponential, $e^{-\gamma t}$ at small times, namely, $\exp(-\gamma
t)=1\,-\,\gamma t\,+\,\frac{1}{2}\gamma^{2}t^{2}+\dots$ with the quadratic
behaviour, $P(t)=1-(\Delta H)^{2}t^{2}$. Performing such a comparison,
Ghirardi et al., obtained an expression ghirardi1979  for the critical
transition time from the quadratic to the exponential decay law as,

$t_{cs}=\frac{2\gamma}{2(\Delta E)^{2}+\gamma^{2}}$ (115)

where $\Delta E$ is the uncertainty of the Hamiltonian evaluated at the
initial state and $\gamma$ is the width of the resonance. The authors in
ghirardi1979  initially obtained a result based on an inequality deduced by
Fleming fleming  and showed that the above result coincided with the first one
in the case of $\Delta E\gg\gamma$. We shall briefly describe the derivation
of this result and rewrite it in order to compare it with the result of the
present work.

Fleming derived an inequality which provided a lower bound for the survival
probability such that,

$|A(t)|\geq\cos{(t\Delta E)},\quad 0\leq t\leq\frac{\pi}{2\Delta E}\,.$ (116)

Assuming that the density of states is described as a narrow resonance and
satisfies the Breit-Wigner form near the resonance, the critical time
$\tau^{G}_{cs}$ was defined in ghirardi1979  as the intersection of the
functions $\cos^{2}{D\tau}$ and $e^{-\tau}$. Hence, considering

$\cos^{2}{D\tau^{G}_{cs}}=e^{-\tau^{G}_{cs}},$ (117)

where $D=\Delta E/\gamma$, the approximate solution to Eq. (117) given in the
paper is:

$\tau^{G}_{cs}=\frac{1}{D^{2}}.$ (118)

Translating Eq. (118) to our notation: $\gamma=\omega_{d}$, and $\Delta
E=\sqrt{\bigl{\langle}(\Delta H)^{2}\bigr{\rangle}_{0}}$,

$\tau^{G}_{cs}=\biggl{(}\frac{\gamma}{\Delta
E}\biggr{)}^{2}=\frac{\omega_{d}^{2}}{\bigl{\langle}(\Delta
H)^{2}\bigr{\rangle}_{0}}.$ (119)

The right side of the above equation is nothing but the square of the
parameter $\alpha$ introduced earlier. Thus,

$\tau^{G}_{cs}=\frac{\omega_{d}^{2}}{\bigl{\langle}(\Delta
H)^{2}\bigr{\rangle}_{0}}=\alpha^{2}.$ (120)

In the Fock-Krylov language, the uncertainty of the energy depends on the
density of states and hence depends on the form factor too. Therefore, the
critical time derived in ghirardi1979  is in principle form factor dependent.
Similarly, $\tau_{cs}$ defined by (100) is also dependent on the form factor
used. Our definition (108) of the critical time, i.e., $\tau_{cs}=4\pi x_{d}$,
however, depends only on the poles of the resonance and is hence independent
of the form factor in the density of states.

If we rewrite this critical time in terms of $x_{d}=\omega_{d}/2\sigma_{d}$,
we get,

$\tau^{G}_{cs}=\frac{4x_{d}^{2}}{\bigl{\langle}(\Delta
H)^{2}\bigr{\rangle}_{0}/\sigma_{d}^{2}},$ (121)

and $\bigl{\langle}(\Delta H)^{2}\bigr{\rangle}_{0}\neq 0$ because of the
energy-time uncertainty. We can deduce that $\tau_{cs}\to 0$ when $x_{d}\to
0$. Finally, the critical time measured in terms of the number of oscillations
is:

$n^{G}_{cs}=\frac{\tau^{G}_{cs}}{4\pi
x_{d}}=\frac{x_{d}/\pi}{\bigl{\langle}(\Delta
H)^{2}\bigr{\rangle}_{0}/\sigma_{d}^{2}}.$ (122)

| Critical times | Number of oscillations
---|---|---
$x_{d}$ | $\tau^{G}_{cs}$ |  $\tau_{cs}$ (100) |  $\tau_{cs}$ (108) | $n_{cs}^{G}$ |  $n_{cs}=\tau_{cs}\eqref{e7.7}/(4\pi x_{d}$)
$10^{-12}$ | $4.824\times 10^{-12}$ | $1.988\times 10^{-6}$ | $1.257\times 10^{-11}$ | $0.384$ | $1.582\times 10^{5}$
$10^{-11}$ | $4.825\times 10^{-11}$ | $6.287\times 10^{-6}$ | $1.257\times 10^{-10}$ | $0.384$ | $5.003\times 10^{4}$
$10^{-10}$ | $4.825\times 10^{-10}$ | $1.988\times 10^{-5}$ | $1.257\times 10^{-9}$ | $0.384$ | $1.582\times 10^{4}$
$10^{-9}$ | $4.825\times 10^{-9}$ | $6.287\times 10^{-5}$ | $1.257\times 10^{-8}$ | $0.384$ | $5.003\times 10^{3}$
$10^{-8}$ | $4.825\times 10^{-8}$ | $1.988\times 10^{-4}$ | $1.257\times 10^{-7}$ | $0.384$ | $1.582\times 10^{3}$
$10^{-7}$ | $4.825\times 10^{-7}$ | $6.287\times 10^{-4}$ | $1.257\times 10^{-6}$ | $0.384$ | $5.003\times 10^{2}$
$10^{-6}$ | $4.825\times 10^{-6}$ | $1.989\times 10^{-3}$ | $1.257\times 10^{-5}$ | $0.384$ | $1.582\times 10^{2}$
$10^{-5}$ | $4.825\times 10^{-5}$ | $6.291\times 10^{-3}$ | $1.257\times 10^{-4}$ | $0.384$ | $50.065$
$10^{-4}$ | $4.826\times 10^{-4}$ | $1.993\times 10^{-2}$ | $1.257\times 10^{-3}$ | $0.384$ | $15.858$
$10^{-3}$ | $4.840\times 10^{-3}$ | $6.339\times 10^{-2}$ | $1.257\times 10^{-2}$ | $0.385$ | $5.0446$
$10^{-2}$ | $4.975\times 10^{-2}$ | $0.206$ | $0.1257$ | $0.396$ | $1.639$
$10^{-1}$ | $0.639$ | $0.765$ | $1.257$ | $0.509$ | $0.609$
Table 1: Comparison of the critical transition times, $\tau_{cs}$, from the
non-exponential to the exponential decay law at small times within the
approach of Ref. ghirardi1979  and that of the present work (Eqs (100) and
(108)). $n_{cs}^{G}$ (122) are the number of oscillations performed by the
survival probability before reaching the exponential region. The last column
gives the number $n_{cs}$ corresponding to the solution of (100). The number
corresponding to $\tau_{cs}$ (108) is always $n_{cs}=1$.

Critical times of Ref. ghirardi1979  and our Eq. (100) for different values of
$x_{d}=\omega_{d}/(2\sigma_{d})$, using an exponential form factor with
$b_{s}=2$ in the density of states for an isolated narrow resonance are
compared with our form factor independent definition (108) in Table 1. From
Table 1, we can infer that the critical times of Ref. ghirardi1979  are one
order of magnitude less than the critical times given by (108) whereas several
orders of magnitude smaller than those of (100). The number of oscillations
seem to be constant as long as $x_{d}$ is small but start increasing beyond
$x_{d}$ = 0.1. Since the result of Ref. ghirardi1979  is expected to be valid
for narrow resonances, a comparison with the above results beyond $x_{d}$ =
0.1 may not be appropriate. Considering only the cases below $x_{d}$ =0.1, one
can draw a general conclusion that the survival probability completes about
half an oscillation (i.e., $n_{cs}^{G}\approx 0.4$) before going over to the
dominant exponential decay law in contrast to $n_{cs}=1$ of (108). However,
the value of $n_{cs}^{G}$ is form factor dependent and increases with
decreasing value of $b_{s}$. For the lowest allowed value of $b_{s}=0.65$ for
example, $n_{cs}^{G}\approx 0.9$. The results given in the third and sixth
column, arising from Eq. (100) are grossly different from those of (121) and
(108). One should be cautious while drawing conclusions from the results of
(100) since (i) values of $\tau_{cs}$ obtained using Eq. (100) are strongly
form factor dependent (for example, there is no solution for an exponential
form factor with $b_{s}=1$ even if it falls in the allowed range of values of
$b_{s}$ and gives reasonable results for $\tau_{cs}^{G}$) (ii) the solution of
(100) is not always unique and there are cases as mentioned above when there
exists no solution at all. As a consequence, (108) which depends solely on the
pole value of the resonance, should be considered as the reliable estimate of
$\tau_{cs}$ of the present work.

A brief comment about the increase in the number of oscillations as well as
the existence of a bigger exponential region in $P(t)$ with decreasing $x_{d}$
is in order here. These features depend on the coupling constant in the decay
(see pascual  for a demonstration with a nice model). Explicit examples of
strong and weak decays confirm the same bogda . Taking the ratio $x_{d}$ to
the extremes, a large value of $x_{d}$ implies no exponential decay at all for
the resonance (see for example ournonexpo2  for the case of the broad $\sigma$
resonance) and a tiny $x_{d}$ implies an exponential decay for a large region
of $t$. Though one cannot use the idea of a coupling constant for tunneling
decays in nuclear physics such as that of 8Be to two 4He nuclei, one can see
in Fig. 5 that with $x_{d}$ = 3 $\times$ 10-5, the survival probability
displays an extremely large exponential region and a huge number of
oscillations as expected.

To complete the comparison, we finally note that Fleming fleming  defined the
lifetime of the resonance as $\tau=\int_{0}^{\infty}P(t)dt$ which is the value
of the autocorrelation function ${\cal R}(y)$ of the present work (see Eq.
(21)) at $y=0$.

###  VII.3 Critical time and the transition region between intermediate and
large times

The intermediate time region is characterized by its dominant exponential
nature and in the same way, the large time region is ruled by a strong power
law. One must however take into account the oscillatory nature of the survival
probability too. The oscillatory nature has given rise to a lot of debate in
literature ourpapercriticaltimes ; winter ; GCoscillate ; FondaGhirardi1972 .
Our aim in this section is to obtain expressions that incorporate the
oscillatory feature into the description of these regions for systems
described by one isolated resonance.

In order to do this, we will use the ideas and formalism described in
ourpapercriticaltimes  for studying the transition from the intermediate to
the large time region. In ref. ourpapercriticaltimes , the survival amplitude
given by Eq. (9) is written as the product of the sum of the exponential and
non-exponential survival probabilities, and a modulating function $I(t)$:

$P(t)=\bigl{[}P_{e}(t)+P_{ne}(t)\bigr{]}I(t),$ (123)

where

$I(t)=1+\cfrac{2\,\text{Re\,}{\Biggl{[}\cfrac{A_{e}(t)}{A_{ne}(t)}\Biggr{]}}}{1+\Biggl{|}\cfrac{A_{e}(t)}{A_{ne}(t)}\Biggr{|}^{2}}=1+\cfrac{2\,\cos{\Biggl{[}\operatorname{Arg}{\cfrac{A_{e}(t)}{A_{ne}(t)}}\Biggr{]}}}{\Biggl{|}\cfrac{A_{e}(t)}{A_{ne}(t)}\Biggr{|}+\Biggl{|}\cfrac{A_{ne}(t)}{A_{e}(t)}\Biggr{|}}.$
(124)

Defining the function $I(t)$ seems to be the appropriate path for
incorporating the oscillatory component in the intermediate and large time
regions. To begin with, notice that the oscillation of $P(t)$ comes from the
term

$\cos{\Biggl{[}\operatorname{Arg}{\cfrac{A_{e}(t)}{A_{ne}(t)}}\Biggr{]}},$

and, from the Eqs. (7) and (8), that the oscillation of the last terms comes
from the exponential component of the survival amplitude, mainly, from the
term $e^{-iz_{d}t}=e^{-i\tau/2x_{d}}e^{-\tau/2}$. Thus, the survival amplitude
oscillates with a frequency equal to

$f_{d}=\dfrac{1/2x_{d}}{2\pi}=\frac{1}{4\pi x_{d}},$

and since

$\cos{\Biggl{[}\operatorname{Arg}{\cfrac{A_{e}(t)}{A_{ne}(t)}}\Biggr{]}}\propto\cos{\biggl{(}\frac{1}{2x_{d}}\tau+\delta\biggr{)}},$

where $\delta$ is a constant, $I(t)$ oscillates at the same frequency as well,
and so does the survival probability. On the other hand, substituting Eq.
(106) in Eq. (124), we get,

$I(n)=1+\frac{2\cos{\bigl{[}2\pi
n-\operatorname{Arg}{C(x_{d},\nu)}\bigr{]}}}{|C(x_{d},\nu)|n^{\nu+1}\exp{\bigl{(}-2\pi
x_{d}\,n\bigr{)}}+|C(x_{d},\nu)|^{-1}n^{-\nu-1}\exp{\bigl{(}2\pi
x_{d}\,n\bigr{)}}},\quad n>1,$ (125)

In Fig. 8 we show a plot of $I(n)$ as a function of $n$ with $x_{d}=0.1$,
$\nu=0.5$, and using an exponential form factor with $b_{d}=1$.

Figure 8: Modulating function given by Eq. (125) as a function of the number
of oscillations $n$. Here, $x_{d}=0.1$, $\nu=0.5$ and an exponential form
factor with $b_{d}=1$ is used.

For the intermediate time region, the term
$|C(x_{d},\nu)|^{-1}n^{-\nu-1}\exp{\bigl{(}2\pi x_{d}\,n\bigr{)}}$ can be
neglegible with respect to the term $|C(x_{d},\nu)|n^{\nu+1}\exp{\bigl{(}-2\pi
x_{d}\,n\bigr{)}}$ and $P_{e}(n)\gg P_{ne}(n)$, and, On the other hand, the
large time region satisfies the opposite conditions555A demostration of this
property can be seen in the appendix B.. From Eqs (123) and (125), the
survival probability for the former region is given by

$P(n)=4\pi^{2}|R(z_{s})|^{2}\exp{\bigl{(}-4\pi
x_{d}\,n\bigr{)}}\,\Biggl{\\{}1+\frac{2}{|C(x_{d},\nu)|}\,\frac{\exp{\bigl{(}2\pi
x_{d}\,n\bigr{)}}}{n^{\nu+1}}\cos{\bigl{[}2\pi
n-\operatorname{Arg}{C(x_{d},\nu)}\bigr{]}}\Biggr{\\}},$ (126)

and the survival probability for the latter region is given by:

$P(n)=\frac{4\pi^{2}|R(z_{s})C(x_{d},\nu)|^{2}}{n^{2\nu+2}}\Biggl{\\{}1+{2}{|C(x_{d},\nu)|}\,\frac{n^{\nu+1}}{\exp{\bigl{(}2\pi
x_{d}\,n\bigr{)}}}\cos{\bigl{[}2\pi
n-\operatorname{Arg}{C(x_{d},\nu)}\bigr{]}}\Biggr{\\}}.$ (127)

Although Eqs (126) and (127) incorporate the oscillatory nature of the
survival probability, we do not have characteristic times that allow us to say
when the intermediate or large time region starts or ends, or when the
transition is happening. The first step for establishing these times is to
define the critical time (in terms of the number of oscillations) $n_{cl}$ of
the transition from the intermediate to large times. This is given by the
intersection of the exponential and non-exponential survival probability
ourpapercriticaltimes ; bogda ; garciaisolated :

$\bigl{|}A_{e}(n_{cl})\bigr{|}^{2}=\bigl{|}A_{ne}(n_{cl})\bigr{|}^{2}\quad\therefore\quad
n_{lc}^{2\nu+2}\exp{\bigl{(}-4\pi
x_{d}\,n_{lc}\bigr{)}}=\frac{1}{|C(x_{d},\nu)|^{2}}.$ (128)

This equation has two solutions and the critical time is defined as the
largest one. Another property of this time is related with the nature of the
resonance, i.e., the narrower the resonance is, the larger is the critical
time666Our aim is not to discuss the details or demostrate the properties of
the critical time for the transition from the intermediate to the large time
region. In ourpapercriticaltimes , the authors present a complete analysis of
this for the case $\nu=1/2$, and these properties are similar for the range of
values that $\nu$ can take here, so that the generalization is
straightfoward.. The next step is to analyze the critical points of the
function

$m(n)=\frac{2}{|C(x_{d},\nu)|n^{\nu+1}\exp{\bigl{(}-2\pi
x_{d}\,n\bigr{)}}+|C(x_{d},\nu)|^{-1}n^{-\nu-1}\exp{\bigl{(}2\pi
x_{d}\,n\bigr{)}}}.$ (129)

It is simple to verify that, for $n>1$, $m(n)$ has a maximum at $n=n_{cl}$,
and has a minimum at $n=\frac{\nu+1}{2\pi x_{d}}$, or in dimensionless units,
$\tau=2(\nu+1)$. These two times let us define where the intermediate, the
transition and the large time regions are. The first interval
$1<n<\frac{\nu+1}{2\pi x_{d}}$ corresponds to the intermediate time region.
The second interval $\frac{\nu+1}{2\pi x_{d}}<n<2n_{cl}-\frac{\nu+1}{2\pi
x_{d}}$ corresponds to the transition region: it is defined such that the
critical time is the middle point of the interval. Finally, the interval
$n>2n_{cl}-\frac{\nu+1}{2\pi x_{d}}$ corresponds to the large time region.
Summarizing:

  1. i)

The intermediate time region is defined in the interval $1<n<\frac{\nu+1}{2\pi
x_{d}}$, and the survival probability is given by Eq. (126).

  2. ii)

The transition region from the intermediate to the large time region is
defined in the interval $\frac{\nu+1}{2\pi x_{d}}<n<2n_{cl}-\frac{\nu+1}{2\pi
x_{d}}$, and the survival probability is given by

$P(n)=I(n)\Biggl{[}4\pi^{2}|R(z_{s})|^{2}\exp{\bigl{(}-4\pi
x_{d}\,n\bigr{)}}+\frac{4\pi^{2}|R(z_{s})C(x_{d},\nu)|^{2}}{n^{2\nu+2}}\Biggr{]},$
(130)

where $I(n)$ must be given by Eq. (125).

  3. iii)

The large time region is defined in the interval $n>2n_{cl}-\frac{\nu+1}{2\pi
x_{d}}$, and the survival probability is given by Eq. (127).

In Fig. 9, we show a plot of $P(n)$ as a function of $n$ with $x_{d}=0.1$,
$\nu=0.5$ and using an exponential form factor with $b_{d}=1$. The diferent
regions have been identified by different linestyles.

Figure 9: Survival probability with parameters $x_{d}=0.1$, $\nu=0.5$, and an
exponential form factor with $b_{s}=1$ as a function of the number of
oscillations $n$, and the dimensionless variable for time $\tau$ given by the
Eq. (69). The dotted line is the small time region, the dash-dotted line is
the intermediate time region, the solid line is the transition from the
intermediate to large time region, and the dashed line is large time region.

##  VIII Summary and Conclusions

The time evolution of an unstable state is investigated by studying the
properties and behaviour of the survival probability, $P(t)$, of a decaying
state at all times. It is well known that the quantum mechanical description
of $P(t)$ leads to five distinct regions, namely, the quadratic form at small
times, the dominant exponential decay at intermediate times, the power law at
large times and the two transition regions between the exponential and non-
exponential decay laws at small and large times. Even though the topic as such
has been studied in quite detail over decades, digging deeper into the
behaviour of $P(t)$ in the five regions allowed us to find some subtle
features as well as restrictions in the form of $P(t)$ and the input density
of states (DOS) used to calculate $P(t)$. Working within the commonly used
Fock-Krylov formalism and taking into account the conditions derived, each of
the above mentioned five regions of the survival probability are studied
mathematically as well as numerically with physical examples. Some of the main
observations and conclusions are listed below:

  1. i)

In the FK formalism, the survival amplitude, $A(t)$ is written as a Fourier
transform of the DOS. Since the non-exponential decay regions are hard to
observe and the exponential decay law dominates the observations in the real
world, it would be useful to construct a DOS leading to a purely exponential
decay. We show that the latter is not possible.

  2. ii)

The survival probability and the autocorrelation function of the density of
states are shown to be a pair of cosine Fourier transforms. This is a
particular case of the Wiener-Khinchin theorem.

  3. iii)

A consequence of the previous result is the evenness of $P(t)$.

  4. iv)

A functional form of the DOS which depends on the pole values, a threshold
factor and an energy dependent form factor is provided. Consideration of
particular form factors with physical examples of resonances leads to the
observations that (i) a constant form factor leads to unphysical values of the
energy uncertainty, (ii) the mathematical condition (54) derived in this work
does not allow the usage of a Gaussian form factor and (iii) the commonly used
exponential form factor can be used with restrictions on the parameter in the
exponential.

  5. v)

Transition regions from the small time quadratic law to the exponential decay
law and from the exponential to the large time power law are studied using the
constructed DOS and the mathematical conditions derived in this work.
Expressions for the critical times of transition are provided and compared
with existing literature.

  6. vi)

An interesting novel feature introduced in this work is the description of the
survival amplitude and hence survival probability in terms of the number of
oscillations performed. The small time quadratic behaviour is found to go over
to the exponential decay when the survival probability has completed one
oscillation. Analytical expressions for the number of oscillations performed
in each of the regions of $P(t)$ are given in terms of the pole values and the
threshold factor in the DOS.

An insight into the mathematical construction of the survival probability
alongwith the different constraints and implications of the conditions derived
for the behaviour of $P(t)$ over the entire region of its evolution from small
to large times is thus provided in the present work within the Fock-Krylov
framework. Since the time evolution of an unstable state is often studied in
literature by focussing on a particular region of $P(t)$ or even a particular
aspect of its behaviour, such a complete mathematical evaluation should prove
useful for the focussed studies in future.

##  Appendix A Descomposition of the survival amplitude as a sum of
exponential functions and an integral

Consider the complex integral

$\oint_{C}{\rho(z)e^{-ixt}\,dz},$ (131)

where $C$ is the contour of integration shown in Fig. 1. Since $\rho(z)$
satisfies the Jordan’s lemma, using the residue theorem we get,

$\int_{AO}+\int_{OB}=2\pi i\sum_{s}{e^{-iz_{s}t}R(z_{s})},$ (132)

However, for $AO$, $z=x$. Therefore,

$\int_{AO}=-\int_{0}^{\,\infty}{\rho(x)e^{-ixt}\,dx}=-A(t),$ (133)

and for $OB$,

$\int_{OB}=-\int_{-i\infty}^{0}{\rho(z)e^{-izt}\,dz}.$ (134)

The survival amplitude can be written as:

$A(t)=-2\pi
i\sum_{s}{e^{-iz_{s}t}R(z_{s})}-\int_{-i\infty}^{0}{\rho(z)e^{-izt}\,dz},$
(135)

##  Appendix B Ratio of the exponential and non-exponential survival amplitude

Our aim is to show that $|A_{e}(n)|\gg|A_{ne}(n)|$ in the exponential region
and $|A_{e}(n)|\ll|A_{ne}(n)|$ for large times. Let $\eta(n)$ be the function

$\eta(n)=\biggl{|}\cfrac{A_{e}(n)}{A_{ne}(n)}\biggr{|}=|C|n^{\nu+1}e^{-2\pi
x_{d}n}.$

Taking the derivative with respect to $n$ and calculating the possible
critical points:

$\eta^{\prime}(n)=\eta(n)\biggl{(}\frac{\nu+1}{n}-2\pi
x_{d}\biggr{)}=0,\quad\therefore\quad n=\frac{\nu+1}{2\pi x_{d}}.$

It is easy to show that this critical point is a maximum. Since $\eta(0)=0$
and $\eta(\infty)=0$, this function starts to rise up until a maximum, and
then starts falling down.

These properties allow us to study the function $1/\eta(n)$, i.e.,
$\biggl{|}\dfrac{A_{ne}(n)}{A_{e}(n)}\biggr{|}$. In this case, when $n$
increases from zero, $1/\eta(n)$ falls down until $n=(\nu+1)/(2\pi x_{d})$,
and then it starts to rise up indefinitely. There should be some values of $n$
such that $\eta(n)=\dfrac{1}{\eta(n)}$. The values of $n$ where this condition
is valid are nothing but the critical points for the transition from the
exponential to the power law. Since we are interested in narrow resonances, we
know that the second solution of Eq. (128) goes to zero when $x_{d}$ goes to
zero too. This implies that even for $n=1$ $\eta(n)\gg\dfrac{1}{\eta(n)}$. In
conclusion, $\eta(n)>\dfrac{1}{\eta(n)}$ for $1<n<n_{cl}$, and
$\eta(n)<\dfrac{1}{\eta(n)}$ for $n>n_{cl}$, where $n_{cl}$ is the critical
point for large time transition.

In Fig. 10, we plot $\eta$ (solid line) and $1/\eta$ (dashed line) for
$x_{d}=0.1$ and $b_{s}=1$.

Figure 10: Ratio of the exponential and non-exponential survival amplitudes,
$\eta$ and its inverse as a function of the number of oscillations $n$ of the
survival probability.

Finally, the figure 11 plots $1+m(n)$, where $m(n)$ is given by Eq. (129) and
written in terms of $\eta(n)$ as $m(n)=2/(\eta(n)+\eta(n)^{-1})$ where $m(n)$
(black line) has been computed using Eq. (129) and using the approximations
deduced for $\eta$. The dashed line is when $m(n)\approx 2/\eta$ in the
exponential region and the dotted line is when $m(n)\approx 2\eta$ in the
power law region.

Figure 11: Modulating function as a function of the number of oscillations $n$
of the survival probability.

##  Appendix C Effect of other poles on $P_{e}(t)$

An interesting and typical case is a system whose survival probability can be
written approximately as a sum of the exponential terms only. After certain
time, this sum should reduce to one term associated to the dominant isolated
resonance. In order to see how this reduction law happens, we write Eq. (96)
such that the dominant pole is explicit, i.e.,

$P_{e}(t)=|\bar{R}(z_{d})|^{2}e^{-\omega_{d}t}\Biggl{\\{}1+2\sum_{s>d}{\text{Re\,}{\biggl{[}\frac{\bar{R}(z_{s})}{\bar{R}(z_{d})}e^{-i(\sigma_{s}-\sigma_{d})t}\biggr{]}}\exp{\biggl{(}-\frac{\omega_{s}-\omega_{d}}{2}\,t\biggr{)}}}\Biggr{\\}}\\\
+\sum_{s^{\prime}\neq
d}{|\bar{R}(z_{s}^{\prime})|^{2}e^{-\omega_{s^{\prime}}t}\Biggl{\\{}1+2\sum_{s>s^{\prime}}{\text{Re\,}{\biggl{[}\frac{\bar{R}(z_{s})}{\bar{R}(z_{s}^{\prime})}e^{-i(\sigma_{s}-\sigma_{s}^{\prime})t}\biggr{]}}\exp{\biggl{(}-\frac{\omega_{s}-\omega_{s}^{\prime}}{2}\,t\biggr{)}}}\Biggr{\\}}},$
(136)

and we can approximate the exponential survival amplitude by taking all the
terms associated to the dominant pole:

$P_{e}(t)\approx|\bar{R}(z_{d})|^{2}e^{-\omega_{d}t}\Biggl{\\{}1+2\sum_{s>d}{\text{Re\,}{\biggl{[}\frac{\bar{R}(z_{s})}{\bar{R}(z_{d})}e^{-i(\sigma_{s}-\sigma_{d})t}\biggr{]}}\exp{\biggl{(}-\frac{\omega_{s}-\omega_{d}}{2}\,t\biggr{)}}}\Biggr{\\}}=|\bar{R}(z_{d})|^{2}e^{-\omega_{d}t}M(t),$
(137)

where $M(t)$ is defined as the function

$M(t)={1+2\sum_{s>d}{\text{Re\,}{\biggl{[}\frac{\bar{R}(z_{s})}{\bar{R}(z_{d})}e^{-i(\sigma_{s}-\sigma_{d})t}\biggr{]}}\exp{\biggl{(}-\frac{\omega_{s}-\omega_{d}}{2}\,t\biggr{)}}}}.$
(138)

The modulating function $M(t)$ (which should not be confused with the
modulating function defined by Eq. (124)) contains information about how the
reduction occurs. As is expected, this process does not depend on the dominant
pole only but it also requires the other poles of the density of states for
its description. If we use a “mean field approach” in the sense of
approximating the sum in Eq. (138) by taking the nearest pole to the dominant
one, the modulating function takes the form:

$M(t)\approx{1+2{\text{Re\,}{\biggl{[}\frac{\bar{R}(z_{d+1})}{\bar{R}(z_{d})}e^{-i(\sigma_{d+1}-\sigma_{d})t}\biggr{]}}\exp{\biggl{(}-\frac{\omega_{d+1}-\omega_{d}}{2}\,t\biggr{)}}}},$
(139)

or expressing the time in dimensionless (lifetime) units of the dominant
resonance, i.e., $\tau=\omega_{d}t$, we have,
$P_{e}(\tau)=|\bar{R}(z_{d})|^{2}e^{-\tau}M(\tau),$, where,

$M(\tau)=1+2\text{Re\,}{\Biggl{[}\frac{\bar{R}(z_{d+1})}{\bar{R}(z_{d})}\exp{\biggl{(}-i\frac{\sigma_{d+1}-\sigma_{d}}{\omega_{d}}\,\tau\biggr{)}}\Biggr{]}}\exp{\Biggl{[}-\biggl{(}\frac{\omega_{d+1}}{\omega_{d}}-1\biggr{)}\frac{\tau}{2}\Biggr{]}}.$
(140)

From Eq. (140), we infer that the survival probability experiment an
oscillation with a frequency
$\omega_{t}=(\sigma_{d+1}-\sigma_{d})/\omega_{d}$, and the survival amplitude
goes to the exponential term associated to the dominant pole at a rate of
$2(\omega_{d+1}/\omega_{d}-1)^{-1}$ per lifetime unit (see Fig. 12).

Figure 12: Sketch of the transition of $P_{e}(\tau)$ to the exponential law of
the dominant pole.

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