text:
\section{Introduction}
Cosmic Microwave Background (CMB) data
are mainly used with the primary aim to infer the
values of the parameters of the cosmological standard model. In addition, CMB data also offer  a
unique opportunity to study the large-scale temperature asymmetries far beyond the size typically accessible with other tools toward nearby
astronomical systems (see, e.g., \citealt{rg,depaolis2011}). Here we continue the use of CMB data to map the dark haloes of nearby galaxies, the latter often studied in most details on other bands or via other methods.
Indeed, in the recent past we have analyzed {\it Planck} data toward four nearby galaxies with the main aim of testing if microwave data show a substantial temperature asymmetry of one side with respect to the other about  the rotation axis of the galactic disks. We have considered, in particular: M31 galaxy and its halo (\citealt{depaolis2014}) the active radio galaxy Centaurus A (Cen A)
that is considered the closest AGN
(\citealt{depaolis2015}),  M82, the largest galaxy in
the M81 Group in the Ursa Major constellation (\citealt{gurzadyan2015}), the M 33 galaxy where we found a substantial temperature asymmetry with respect to its minor axis projected onto the sky plane which extends up to about $3 ^0$ from the galactic center and correlates well with the HI velocity field at 21 cm, at least within about $0.5 ^0$ (\citealt{depaolis2016}).
We emphasize that the
very fact that the detected temperature asymmetries are always   almost frequency independent is a
strong indication of an effect due to the galaxy rotation and remark the importance of
the methodology proposed which, in spite of its simplicity,
may allow one to consistently estimate the galaxy dynamical
mass contained within a certain galactocentric distance. We have also shown that, in general, our method, can be applied to nearby nearly edge-on spirals and may be used to trace the halo bulk dynamics on rather large scales in a model-independent way.
The present paper is a follow-up of the previous paper on the M82 galaxy (\citealt{gurzadyan2015}) where a  substantial North-South and East-West
temperature asymmetry was found, extending up to about $1 ^0$ from the M82 center. The main conclusion about the origin of the temperature asymmetry (almost frequency-independent) was its link with a
Doppler-induced effect regarding the line-of-sight dynamics on the real halo scale - invisible in other bands - the ejections from the
galactic center or the tidal interaction of M82 with the M81 galaxy. Here, we consider again this issue from the point of view of the M81 galaxy, one of the last objects in the Local Group which can be studied by available {\it Planck} data.

\section{Planck data analysis and results toward M81}
M81, also known as Bode's galaxy (or NGC 3031), at J2000 coordinates R.A.: 09$^h$ 55$^m$ 33.1730$^s$, Dec: +69$^0$ 3$^{\prime}$ 55.061$^{\prime\prime}$ (Galactic Longitude $l=142.0918406^0$,  Galactic Latitude $b=40.9001409^0$) is a SA(s)ab type galaxy at a distance of  $3.6\pm 0.2$ Mpc from us (see, e.g., \citealt{Gerke2011}). Following the procedures described in the previous papers, we have used  the publicly released {\it Planck} 2015 data\footnote{From the {\it Planck} Legacy Archive, http://pla.esac.esa.int.} \cite{planck2015a} in the
bands at 70 GHz of the Low Frequency Instrument (LFI),
and in the bands at 100 GHz, 143 GHz and 217 GHz of the High Frequency
Instrument (HFI).
{We have also used the foreground-corrected SMICA band (indicated as SmicaH in Figs. \ref{fig2} and \ref{fig3}) which should display the lowest contamination by the galactic foregrounds.
We notice here that {\it Planck}'s resolution is
$13.2^{\prime}$, $9.6^{\prime}$,  $7.1^{\prime}$ and $5^{\prime}$ in terms of FHWM at 70, 100, 143 and 217 GHz
bands, respectively, and frequency maps \citep{planck2015b}  are provided in CMB temperature at resolution
corresponding to Nside=2048 in HEALPix scheme \citep{gorski2005}. }
\begin{figure}[h!]
 \centering
  \includegraphics[width=0.46\textwidth]{Fig1.pdf}
 \caption{The {\it Planck} field toward the M81  galaxy in the 143 GHz band. The pixel color gives the temperature excess  in $\mu$K with respect to the mean CMB temperature. The optical extension of the M81 galaxy is indicated by the inner ellipse with apparent dimensions   of  $26.9^{\prime}$ and $14.1^{\prime}$, respectively. The four quadrants A1, A2, A3 and A4 are used in the analysis. The thin dashed black line marks the Galactic latitude $b=40^0$ North. We note that the M82 galaxy is clearly visible as the red spot in the A1 region at about $38^{\prime}$ from the center of M81.} \label{fig1}
 \end{figure}
 \begin{figure}[h!]
 \centering
  \includegraphics[width=0.46\textwidth]{Fig2a.pdf}
    \includegraphics[width=0.46\textwidth]{Fig2b.pdf}
 \caption{Upper panel: the temperature asymmetry toward M81 in $\mu$K (with the standard errors) of the A1+A2
region (indicated as A12) with respect to the A3+A4  region (A34)  in the five considered {\it Planck} bands (see text for details) within three radial distances of  $30^{\prime}$ ($R0.50$), $60^{\prime}$ ($R1.00$) and $90^{\prime}$ ($R1.50$). Bottom panel: the same for the 360 control fields with the same geometry (shown in Fig. \ref{fig1}) equally spaced at one degree distance to each other in Galactic longitude and at the same latitude as M81.}
 \label{fig2}
 \end{figure}
   \begin{figure}[h!]
 \centering
  \includegraphics[width=0.46\textwidth]{Fig3a.pdf}
    \includegraphics[width=0.46\textwidth]{Fig3b.pdf}
 \caption{Upper panel: the temperature asymmetry toward M81 in $\mu$K (with the standard errors) of the A1+A4 region (A14) with respect to the A2+A3 region (A23) in the five considered {\it Planck} bands. Bottom panel: the same for the 360 control fields.}
 \label{fig3}
 \end{figure}
{To study in the simplest way the CMB data toward the M81 galaxy, the {\it Planck} field of the region of  interest  (we give in Fig. \ref{fig1} the map in the case of the 143 GHz band) has been divided into  four quadrants A1, A2, A3 and A4. As detailed in the histograms in Figs. \ref{fig2} and \ref{fig3} we have considered the temperature asymmetry in three radial regions about the M81 center within $0.5^0$, $1^0$ and  $1.5^0$ (indicated as  R0.50, R1.00 and R1.50, respectively).
In Fig. \ref{fig1}  the optical extension of the M81 galaxy is  shown, as indicated by the inner ellipse. In Fig. \ref{fig2} we give  the temperature asymmetry toward M81 in
$\mu$K (with the standard errors) of the A1+A2 region (A12) with respect to the A3+A4 region (A34) in the five considered {\it Planck} bands within the three radial distances. In the bottom panel we give  the same for the 360 control fields with the same geometry (shown in Fig. \ref{fig1}) equally spaced at one degree distance to each other in Galactic longitude and at the same latitude as M81.
As one can see from Fig. \ref{fig2} and as expected by considering the rotation direction of the M81 disk about its rotation axis, The A12 region always appears hotter than the  A34 region by  $32-44~\mu$K within $0.5^0$, by   $28-40~\mu$K within $1^0$ and by about  $50-70~\mu$K within $1.5^0$. }
Note that the control fields show a much smaller temperature asymmetry of only a few $\mu$K and that the detected effect is practically the same in any of the five {\it Planck}'s bands.
{We also note that the size of the virial radius of the M81 galaxy (equivalent to the $R_{200}$ radius where the galaxy density is about 200 times larger than the critical density) can be estimated to be about $3.6^0$ \citep{chiboucas2009}.
}
Although from the geometry and the direction rotation of the M81 disk, the expected temperature asymmetry was along the A12/A34 axis, we have also considered  the temperature asymmetry toward M81 in the A14 region with respect to the A23 region and found an even more consistent asymmetry  in all  the considered  {\it Planck} bands. This resembles what was already found towards the companion galaxy M82  \citep{gurzadyan2015}. In the present case the temperature asymmetry amounts to $40-80~\mu$K as implied by a prolate M81 halo rotation, while the control fields always show an asymmetry consistent to zero (see Fig. \ref{fig3}). As far as the
foreground-corrected SMICA band is concerned, the temperature asymmetry  is negligible within $0.5^0$ (although one has to consider that the pixel number in this region is very low) and increases to large values within $1^0$ and $1.5^0$.  Also SMICA data  show a clear and more consistent  A14/A23 temperature asymmetry  with values comparable with those in the other bands within R1.00 and R1.50, although within $0.5^0$ there may be a non-negligible foreground contamination in the other {\it Planck}'s bands.

\section{Discussion}
  {Similar to the case of the other galaxies of the Local Group considered previously and in particular toward M82, we found a consistent North-South and East-West temperature asymmetry also toward the M81 galaxy, that reaches values up to  about $80~\mu$K within $1.5^0$ in all considered {\it Planck} bands.  We believe that the most plausible explanation relies in a Doppler-induced effect due to the spin of the M81 halo, possibly along an axis tilted (up to about $90^0$) with respect to the rotation axis of the M81 disk. In this case the temperature asymmetry can be estimated from the equation  $ \Delta T/T=2v\sin i S\langle\tau\rangle /c$, accordingly  to the model first discussed in \cite{depaolis1995}, where, $v$ is the M81 rotation velocity, $i\simeq 58^0$ is the M81 disk inclination angle, $S$ is the gas or dust filling factor, and $\langle\tau\rangle$ is the averaged optical depth within a given {\it Planck} band.  In order to account for the detected temperature anisotropy, however, the M81 halo should be filled by a relatively large  amount of gas (likely in the form of cold gas clouds), as in the models proposed, e.g., by \cite{pfenniger1994,dpijr1995,gs}.
  A viable explanation of the detected effect could  be, in principle, also the rotational kinematic Sunyaev-Zel'dovich (rkSZ) effect, which is known to be active on galaxy cluster scales \citep{cooray2002,chluba2002,manolopoulou2017}.
 Naturally, to be active, the rkSZ effect does require the presence of a conspicuous population of high energy electrons in the rotating M81 halo.}
  By rescaling to the  free electron density, temperature and size of the galaxy halos, the effect appears to be smaller by at least one order of magnitude (with respect to that in galaxy clusters), giving expected temperature asymmetries below a few $\mu$K. {However, it might be that the temperature asymmetry derives from  a multi component effect due to the presence of cold gas clouds together with a halo of hot and also, possibly, a warm component. Also other effects, such as the case that M81 is an interacting system with a rather recent merging event may induce an increase of the temperature asymmetry, as outlined in the following discussion.

Many galaxies belong to  multiple systems and constitute dynamically linked objects which are affected by their mutual gravitational interaction. An example of such systems  is indeed constituted by the M81, M82  and NGC 3077 galaxies  (the first two objects interacted about 200 Myrs ago,  see e.g. \citealt{makarova2002,sun2005,heithausen2012,oehm2017}). The radio images of the system, obtained for example by the VLA radio telescopes (see \citealt{yun1994}), shows with strong evidence that the HI gas is not only associated with each galaxy but is also present around the galaxies and in the intergalactic space. Clear gaseous filaments are visible  among the three major objects of the galaxy group (which is likely composed by at least six objects: M81, M82, NGC 3077, Holmberg IX, Arp's Loop and NGC 2976 (see, e.g., \citealt{bremnes1998,karachentsev2002,makarova2002}).
This gas and the associated dust could give some  contribution to the detected temperature asymmetry toward the M81 halo. In this respect we note that an enhancement of the emission arising from the rkSZ effect is expected to occur in recent merging events of rich  galaxy clusters giving temperature asymmetries up to $146~\mu$K \citep{chluba2002}. In the present case, the previous merging event between the M81 and M82 galaxies might have generated large scale turbulence and bulk motion with an increase of the free electron density and temperature which may amplify the rkSZ effect producing possibly temperature asymmetries up to $10-20~\mu$K (although a realistic estimate of the effect would require a detailed hydrodynamic modeling of the past merging event). \footnote{We also mention that presence of baryonic jets from an ultraluminous supersoft X-ray source discovered in the M81 galaxy, called ULS-1 (see \cite{liu2015} and references therein), may lead to an unexpected growth of the high-energy electrons and hot plasma populations in the M81 halo.}

Moreover,  there could  also be some contribution to the detected temperature asymmetry from high-latitude gas clouds in our Galaxy along the line of sight towards M81. In this respect we note that M81 is at about $40.9^0$ North of the Galactic disk, where contamination from the Milky Way is expected to be low.
However, interpretation of astronomical observations is often hampered by the lack of direct distance information. Indeed, it is often not  easy to judge whether objects on the same line of sight are physically related or not. Since the discovery of the Arp's Loop \citep{arp1965} the nature of the interstellar clouds in this region has been debated; in particular whether they are related  to the tidal arms around the galaxy triplet \citep{sun2005,demello2008} or to Galactic foreground cirrus \citep{sollima2010,davies2010}.
Already \cite{sandage1976} presented evidence showing that we are observing the M81 triplet through wide spread Galactic foreground cirrus clouds and \citealt{devries1987} built large-scale HI, CO and dust maps which showed Galactic cirrus emission towards the M81 region with $N_H\simeq 1-2\times10^{20}$ cm$^{-2}$. The technique used to distinguish between the emission from extragalactic or Galactic gas and dust relies on spectral measurements and on the identification of the  line of sight velocities which are expected to be different in each case. Unfortunately, in the case of the M81 Group, this technique appears hardly applicable since the radial velocities of  extragalactic and Galactic  clouds share a similar LSR (local standard of rest) velocity range \citep{heithausen2012}.
Several small-area molecular clouds (SAMS), i.e. tiny molecular clouds in  a region where the shielding of the interstellar radiation field is too low (so that these clouds cannot survive for a long time) have been detected by \cite{heithausen2002} toward the M81 Group. More recently, data from the SPIRE instrument onboard {\it Herschel} ESA space observatory and MIPS onboard of {\it Spitzer} allowed the identification of several dust clouds north of  the M81 galaxy with a total hydrogen column density in the range $1.5-5\times10^{20}$ cm$^{-2}$ and dust temperatures between 13 and 17 K \citep{heithausen2012}. However, since there  is no obvious difference  among the individual clouds there was no way to distinguish between Galactic or extragalactic origin although it is likely that some of the IR emission both towards M81 and NGC 3077 is of Galactic origin.  Temperature asymmetry studies in {\it Planck} data may be indicative of the bulk dynamics in the observed region provided that other Local (Galactic) contamination in the data is identified and subtracted. This is not always possible, as in the case of the M81 Group, and therefore it would be important to identify and study other examples of dust clouds where their origin, either Galactic or extragalactic, is not clear.\footnote{One such example might be provided by the interacting system toward NGC 4435/4438 \citep{cortese2010} where the SAMS found appear more consistent with Galactic cirrus clouds than with extragalactic molecular complexes.} Incidentally, the region A1 within R0.50 has been studied by \cite{barker2009}, who found evidence for the presence of an extended structural component beyond the M81 optical disk, with a much flatter surface brightness profile, which  might  contain $\simeq 10-15\%$ of the M81 total V-band luminosity. However the lack of both a similar analysis in the other quadrants (and at larger distances from the M81 center) and the study of the gas/dust component associated to this evolved stellar population, hamper understanding whether  this component may explain the observed temperature asymmetry toward the M81 halo.

Although  the physical origin  of the detected  temperature asymmetry is not clearly identified at present, it appears obvious that the CMB asymmetry method is tracing the M81 halo and intergalactic bridges, not directly revealed in other bands and via conventional methods, based on stellar population studies, ISM, etc. In order to assess this issue, high-resolution and extended (up to $\simeq 1.5 ^0$) observations to infer the distribution of the cold, warm and hot gas components  appears to be an urgent task to be performed. In this respect we emphasize that, in addition to radio observations at 21 cm to map the HI component of the gas (integrated also by other techniques to study small-scale cold structures as done, e.g.,  through interstellar scintillations by \citealt{habibi2011}) and to the X-ray band diffuse emission to infer the amount and distribution of the hot gas component, investigation of the warm gas component with the methodology employed, e.g.,  in \cite{nicastro2016} is extremely important.
Given the serious quantitative disagreement between the microwave temperature asymmetry amplitude revealed for M81 and several other  nearby galaxies and the rkSZ contributions there, the latter's alternative may be  more exotic  halo models (see, e.g., \citealt{Lo2017,Ok2017,Pi2017,van2017,GK}), a dilemma to be solved by future studies.

In conclusion, the available {\it Planck} data, by now, enabled one to trace, by this method, the haloes in the nearby edge-on spirals previously analyzed, while higher resolution data can be efficient for the studies of galaxies also outside the Local Group.
This is particularly important in view of the next generation CMB experiments, such as LiteBird \citep{LiteBird}, CMB-S4 \citep{CMB-S4}, CORE \citep{CORE}, DeepSpace\footnote{See the DeepSpace website at http://deep-space.nbi.ku.dk.}, PIXIE \citep{PIXIE},  and Polarbear \citep{Polarbear}, which will attempt even more precise measurements of the CMB than available so far. Many of these experiments are designed to cover mainly the frequency range around 100 GHz where the relative intensity of the CMB is known to be highest and where one of the most dominant foreground components is dust emission (see, e.g.,
 \citealt{liu2017}). Understanding the properties of dust emission and distinguishing between Galactic foregrounds and extragalactic emission is an important premise for the optimized use of the next generation CMB experiments.

\begin{acknowledgements}
We acknowledge the helpful comments by the referee and the use of {\it Planck} data in the Legacy Archive for
Microwave Background Data Analysis (LAMBDA) and HEALPix
\citep{gorski2005} package. FDP, AAN, and GI acknowledge the support by the INFN
projects TAsP and EUCLID. PJ acknowledges support from the
Swiss National Science Foundation.
AQ is grateful for hospitality to the DST Centre of Excellence in Mathematical \& Statistical Sciences of the University of the Witwatersrand, Johannesburg.
\end{acknowledgements}


text:
\section{Introduction}
\label{sec:intro}
\input{sections/intro}

\section{ATLAS detector}
\label{sec:detector}
\input{sections/detector}

\section{Data and simulation}
\label{sec:datamc}
\input{sections/datamc}

\section{Selection criteria}
\label{sec:selection}
\input{sections/selection}

\section{Uncertainties and background estimation}
\label{sec:background}
\input{sections/background}

\section{Result and interpretations}
\label{sec:result}
\input{sections/result}

\FloatBarrier

\section{Conclusion}
\label{sec:conclusion}
\input{sections/conclusion}

\section*{Acknowledgements}
\input{acknowledgements/Acknowledgements}

\printbibliography

\newpage \input{authorlist/atlas_authlist}

\end{document}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
\label{sec:introduction}

One-dimensional systems are used frequently in quantum mechanics
to approximate real situations because they are relatively easy
to treat mathematically, allowing to get a deeper insight on the
physics involved.
They are useful in a number of applications in contemporary
physics and their simplicity has made them valuable as academic
and research tools.
For instance, it turns up in the description of resonant
tunneling  diode devices, disordered one-dimensional lattices,
realistic one-dimensional solid-state system such as quantum
wells, junctions and superlattices
\cite{PRL.1998.81.5796,PRB.1999.60.10569},
and time analysis of one-dimensional tunneling processes
\cite{RMP.1989.61.917}.
In particular, one-di\-men\-sion\-al rectangular potentials can be
used to model isolated transitions, observed in semiconductors,
from a bound state within a quantum well to a bound state at an
energy greater than the barrier height \cite{Nature.1992.358.565}.
Limiting cases of rectangular potentials can be used to describe
point interactions \cite{Book.1999.Flugge}, as well as
regularized singular interactions \cite{PR.1958.112.1137}.
Also, we should point out that the use of square-barrier and
square-well potentials as simple models for more realistic
physical problems has a long history in the theory of
heterostructures in solid-state physics \cite{Book.1991.Weisbuch}.

However, the propagator or its Fourier transform, the Green's
function, for rectangular potentials, although these are very
simple systems, are difficult to obtain and involve lengthy and
tedious calculations.
Indeed, the exact Green's function and propagators for step
potential and square barrier was obtained only in the late
1980s until the early 1990s
\cite{ZN.1985.40a.379,PRA.1992.46.1233,PRA.1993.47.2562,
PRA.1993.48.2567,PRA.1995.51.2654}.
Moreover, it is well known that semiclassical approach, i.e.,
the Van Vleck-Gutzwiller formula \cite{Book.1990.Gutzwiller}, or
the WKB approach \cite{Book.1998.Merzbacher}, give poor results
when applied to the class of rectangular potentials due to their
discontinuities.

In the present work, we show how to obtain the exact Green's
function
\cite{JPA.1998.31.2975,JPA.2001.34.5041,JPA.2003.36.227},
given as a general sum over paths, in a very simple way avoiding
complicated calculations, for arbitrary rectangular potentials
enlarging the knowledge of exact solutions for Green's functions
and also generalizing and resuming results in the literature.
The procedure outlined here can be thought of as exact version
of the approximation in \cite{AJP.1983.51.897} and provides simple
and direct way to construct recurrence relations for quantum
amplitudes for one-dimensional scattering discussed in
\cite{PRA.1994.49.3310}.
Specifically, we focus on Green's function for single quantum
wells to avoid extra and unnecessary mathematical complications,
but the method is general and could be applied to multiple
quantum wells as well \cite{JPA.2003.36.227}.
From the Green's functions obtained, we  thus describe how
to extract the bound states eigenenergies and eigenfunctions.
Also, we discuss an application of the Green's function approach
to extract information of quasi-bound states in rectangular
potentials.
We should observe that the Green's function approach used in the
present work, has been successfully used in the calculation of
exact Green's functions for segmented potentials
\cite{JPA.1998.31.2975}, in calculation of asymptotic Green's
functions \cite{JPA.2001.34.5041}, in the determination and
discussion of bound states in multiple quantum wells
\cite{JPA.2003.36.227}.
Also, in the calculation of exact Green's function for quantum
graphs \cite{JPA.2003.36.545}, general point interactions
\cite{PRA.2002.66.062712}, and scattering quantum walks
\cite{PRA.2011.84.42343,PRA.2012.86.042309}.

This paper is organized as follows.
In Sec. \ref{sec:gfdp}, we give a briefly review on the
definition and properties of the Green's functions.
In Sec. \ref{sec:gf}, the Green's functions approach for the
rectangular potential is presented.
In Sec. \ref{sec:awp}, the construction of the exact Green's
functions for a general rectangular asymmetric well potential is
presented and how the sought eigenenergies and eigenfunctions are
extracted.
In Sec. \ref{sec:swp}, the case of the square well potential is
discussed and the definite parity eigenfunctions are determined.
In Sec. \ref{sec:iwp} the case of the infinite well potential is
discussed.
In Sec. \ref{sec:qbs} a Green's function approach to extract
information from system with quasi-bound states is presented.
Finally, Sec. \ref{sec:conclusion} contains our conclusions.

\section{Green's function: Definition and properties}
\label{sec:gfdp}

The Green's functions are an important tool in quantum mechanics
for calculating eigenenergies, eigenfunctions and density
of states \cite{Book.2006.Economou}.
It can be defined as solutions of the inhomogeneous differential
equation
\begin{equation}
  \label{eq:defgf}
  [E-\hat{H}(x_{f})]G(x_{f},x_{i};E)=
  \delta(x_{f}-x_{i}),
\end{equation}
subject to certain boundary conditions.
$\hat{H}(x)$ is the Hamiltonian operator
\begin{equation}
  \label{eq:Hamiltonian}
  \hat{H}(x)=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V(x).
\end{equation}

The Green's function $G(x_{f},x_{i};E)$ for a quantum system can be
interpreted as a probability amplitude for a particle that initially at
the point $x_{i}$ moves to the point $x_{f}$ for a fixed
energy \cite{Book.2005.Schulman}
\begin{equation}
G(x_{f},x_{i};E)=\langle x_{f}|\hat{G}(E)|x_{i} \rangle,
\end{equation}
where
$\hat{G}(E)\equiv \lim_{\eta \to 0^{+}} [E-\hat{H}(x)+i\eta]^{-1}$
is the resolvent operator \cite{Book.2006.Kleinert}.

Suppose we have a complete set of normalized energy eigenstates
$\psi_{n}(x)$, $n=0,1,...$ with
\begin{equation}
\hat{H}(x)\psi_{n}=E_{n} \psi_{n},
\end{equation}
so that the solution of Eq. \eqref{eq:defgf} can be written as
\begin{equation}
  \label{eq:segf}
  G(x_{f},x_{i};E)=
  \sum_{n}\frac{\psi_{n}(x_{f})\psi_{n}^{*}(x_ {i})}{E - E_n}.
\end{equation}
For a discrete spectrum, we thus identify the poles of
Green's function with bound states eigenenergies and the
residues at these points give a tensorial product of the bound
states eigenfunctions.
The  continuous part of the spectrum corresponds to a branch
cut of $G(x_{f},x_{i};E)$ \cite{JMP.1992.33.643,Book.2006.Kleinert}.
The following limit
\begin{equation}
\label{eq:limg}
\lim_{E \to E_n}(E-E_n)G(x_f,x_i;E)=\psi_n{(x_f)}\; \psi_n^{*}{(x_i)}
\end{equation}
is used to extract the discrete bound states from the
Green's function.

There are basically three methods for calculating the Green's
functions \cite{Book.2006.Economou}:
solving the differential equation in \eqref{eq:defgf};
summing up the spectral representation in \eqref{eq:segf};
or performing the Feynman path integral expansion for the
Green's function \cite{Book.2010.Feynman}.
In this paper, we present a Feynman-like procedure.
As we are going to see, the approach requires one to determine
the quantum amplitudes of each individual potential, but in
general, this is a much easier task than to calculate numerically
propagation along the whole $V(x)$.

\section{The Green's function approach for rectangular
  potentials}
\label{sec:gf}

\begin{figure}[t*]
    \centering
    \includegraphics*[width=0.85\columnwidth]{fig1.eps}
    \caption{A rectangular potential with three points of
      discontinuities.
      The potential can be considered as a sum of the three
      potential steps.}
    \label{fig:fig1}
\end{figure}

According to Refs.
\cite{JPA.1998.31.2975,JPA.2001.34.5041,JPA.2003.36.227}
the Green's function $G(x_{f},x_{i};E)$ (in what follow, we will
denote it by $G_{f,i}$ for short) for a general 1D
potential is obtained by  writing the potential $V(x)$ as a sum of
$n$ individual potentials $u_{j}$, where each $u_{j}$ vanish
asymptotically.
In the present work, we focus our attention on rectangular
potentials.
As depicted in Fig. \ref{fig:fig1}, a rectangular
potential can be treated as the sum of $n$ step potentials.
The potential function for a rectangular potential can be
written as
\begin{equation}
  V(x)=\left\{
    \begin{array}{llll}
      u_1 & \text{for} & x < x_1,
      & \text{(Region $1$)},\\
      u_j & \text{for} & x_j\leq x < x_{j+1}=x_{j}+\ell_{j},
      & \text{(Region $j$)},\\
      u_n & \text{for} & x \geq x_{n},
      & \text{(Region $n$)},
    \end{array}
  \right.
  \label{eq:potpcp}
\end{equation}
where $u_{j}$, $\ell_j$ and $x_j$ are the height of the potential,
the length of the region $j$ and the localization of the $j$th
discontinuous point, respectively.
The wave number in each different region is
$k_j=\sqrt{2m(E-u_j)/\hbar}=i\kappa_j$.
In what follow, we do not distinguish the cases whether $k_{j}$
is a real or a purely imaginary number ($i\kappa_{j}$), except otherwise
mentioned.

The Green's function for this system, for a fixed energy $E$ and
the end points $x_{i}$ and $x_{f}$, is given by
\begin{equation}
  \label{eq:fgsg}
  G_{f,i}=\frac{m}{i\hbar^2\sqrt{k_{f}k_{i}}}
  \sum_{\rm{sp}}
  W_{\rm{sp}}
  \exp{[\frac{i}{\hbar} S_{\rm{sp}}(x_{f},x_{i};E)]},
\end{equation}
where $W_{\rm sp}$ is the amplitude (or weight) and
$S_{\rm sp}$ is the classical action.
The above sum is performed over all scattering paths (sp)
starting in $x_{i}$ and ending in $x_{f}$.
For each sp, the classical action is obtained from the
propagation over action of potential $u_{j}$,
$S_{\rm sp}(x_b,x_a;E)=\hbar \int_{x_a}^{x_b}k_{j} dx$.
A few comments concerning the Eq. \eqref{eq:fgsg} are necessary.
The expression in Eq. \eqref{eq:fgsg} is obtained from a recursive
procedure, i.e., $G_{f,i}$ for $n$ potentials are derived from
$G_{f,i}$ for $n-1$ potentials.
The $W_{\rm sp}$ are related to local quantum effects,
so they depend on quantum amplitudes ($R_{j}^{(\pm)}$ and
$T_{j}^{(\pm)}$) of individual $u_{j}$
(cf. \ref{sec:appendixa}).
In fact, for the present case of rectangular potentials,
$R_{j}^{(\pm)}$ and $T_{j}^{(\pm)}$ are the usual reflection
and transmission amplitudes of individual $u_{j}$
\cite{PRA.1994.49.3310} (up to a phase
\cite{JPA.1998.31.2975}).
(The superscript $(+)$ ($(-)$) and subscript $j$ denote the
physical quantities of a quantum particle incident
from left (right) at the point $x_{j}$, respectively.)
In the general case of smooth potentials, they are related, but
we need to calculate classical actions for each potential.
In this later case, the interested reader can consult the
Refs. \cite{JPA.2001.34.5041,JPA.2003.36.227} for derivations.

\section{Asymmetric well potential}
\label{sec:awp}

\begin{figure}[t*]
    \centering
    \includegraphics*[width=0.85\columnwidth]{fig2.eps}
    \caption{An asymmetric well potential written as a sum of a
      left and right potential steps.}
    \label{fig:fig2}
\end{figure}

In this section we will consider the case of a quantum particle
confined in the asymmetric well potential defined by
\begin{equation}
  \label{eq:aswp}
  V(x)=
  \left\{
  \begin{array}{llll}
    u_{1} & \text{for} & x < x_{1},
    & \text{(Region 1)},\\
    u_{2} & \text{for} & x_{1} \leq x<x_{2}=x_{1}+\ell_{2},
    &\text{(Region 2)},\\
    u_{3} & \text{for} & x \geq x_{2},
    &\text{(Region 3)},
  \end{array}
  \right.
\end{equation}
and depicted in Fig. \ref{fig:fig2}.
We seek the Green's function for the situation where
$E<u_{3}$.
The eigenvalues will be obtained from the poles of the Green's
function and the eigenfunctions from their respective residues
(cf. Sec. \ref{sec:gfdp}).
We can write nine different Green's functions for the problem,
but for our purposes, we analyze only three relevant
possibilities, namely:
(1) $x_{1}<x_{i}<x_{2}$ and $x_{f}<x_{1}$;
(2) $x_{1}<x_{i},x_{f}<x_{2}$; and
(3) $x_{1}<x_{i}<x_{2}$ and $x_{f}>x_{2}$.
The others six Green's functions can be obtained from symmetry
considerations of the three above cases.

First, we want to exemplify how to use the Eq. \eqref{eq:fgsg}
and how the geometric series appear.
So, let us consider the possibility (2).
To obtain $G_{f,i}^{(2)}$ in a closed form, we need to sum up
all the possibles scattering paths for a quantum particle
starting from $x_{i}$ and arriving at $x_f$, but it can always
be done because the sum in \eqref{eq:fgsg} forms a geometric
series. Without loss of generality, let us set $x_{f}>x_{i}$.

Thus, in Fig. \ref{fig:fig3} is depicted five examples of
scattering paths. The first path in Fig. \ref{fig:fig3}(a) is
the direct propagation from $x_i$ to $x_f$, which contributes
with $e^{i k_{2} (x_f-x_i)}$ for $G_{f,i}^{(2)}$.
In Fig. \ref{fig:fig3}(b), a quantum particle leaves the point
$x_i$, goes to the point $x_{2}$, where it hit the potential and
is reflected, and so, goes to left to arrives at the point $x_f$.
This path contributes with
$e^{i k_{2}(x_{2}-x_{i})}R_{2}^{(+)}e^{-ik_{2}(x_{f}-x_{2})}$,
where $R_{2}^{(+)}$ is the reflection amplitude for a quantum
particle incident from the left of the potential $u_{3}$ at the
point $x_2$.
Following the same reasoning, the contributions for the other
examples in Fig. \ref{fig:fig3} are:
\begin{align*}
\text{(c) } {} &
e^{-ik_{2}(x_{1}-x_{i})}R_{1}^{(-)}e^{ik_{2}(x_{f}-x_{1})};
\nonumber \\
\text{(d) } {} &
e^{ik_{2}(x_{2}-x_{i})}R_{2}^{(+)}e^{ik_{2}\ell_{2}}
R_{1}^{(-)}e^{ik_{2}(x_f-x_{1})};
\nonumber \\
\text{(e) } {} &
e^{-ik_{2}(x_{1}-x_{i})}R_{1}^{(-)}e^{ik_{2}\ell_{2}}
R_{2}^{(+)}e^{-ik_{2}(x_{f}-x_{2})}.
\end{align*}

\begin{figure}[t*]
    \centering
    \includegraphics*[width=\columnwidth]{fig3.eps}
    \caption{In (a)-(e) schematic examples of scattering paths
      and in (f) the families of paths $P_{1}$ and $P_{2}$.}
    \label{fig:fig3}
\end{figure}

The Green's function is then given by the sum of all
contributions of $n$ multiple reflections of all possibles
scattering paths and can be written as ($\ell_{2}=x_{2}-x_{1}$)
\begin{align}
  G_{f,i}^{(2)}  =
  {} & \frac{m}{ik_{2}\hbar^2}
  \Big\{e^{ik_{2}(x_f-x_i)}
  \nonumber \\
  {} & +\sum_{n=0}^{\infty}
  \left(R_{1}^{(-)}R_{2}^{(+)} e^{2  i k_{2} \ell_{2}}\right)^{n}
  \Big[e^{-ik_{2}(x_{1}-x_{i})}R_{1}^{(-)}
   \nonumber \\
  {} & \times
  \left(e^{ik_{2}(x_{f}-x_{1})}+
  R_{2}^{(+)}e^{ik_{2}\ell_{2}}e^{-ik_{2}(x_{f}-x_{2})}\right)
  \nonumber \\
  {} & + e^{ik_{2}(x_{2}-x_{i})}R_{2}^{(+)}
  \nonumber \\
  {} & \times
  \left(e^{-ik_{2}(x_{f}-x_{2})}+
    R_{1}^{(-)}e^{ik_{2}\ell_{2}}e^{ik_{2}(x_{f}-x_{1})}\right)\Big]\Big\}.
\label{eq:gexefinal}
\end{align}
In general, the reflections (transmissions) amplitudes have the
property  $|R_{j}^{(\pm)}|^{2}\leq 1$ ($|T_{j}^{(\pm)}|^{2}\leq 1$),
in such way as the above sum always converge.
In fact, it is a geometric series.
So, after a straightforward algebra the final form for the
Green's function is
\begin{align}
  G_{f,i}^{(2)} =
  {} &\frac{m}{i\hbar^2 k_{2} f_{\rm aw}}
  \left(
    e^{i k_{2} (x_{f}-x_{1})}+
    R_{2}^{(+)}e^{i k_{2} \ell_{2}}
    e^{i k_{2} (x_{2}-x_{f})}
  \right)
  \nonumber \\
  {} & \times
  \left(
    e^{-ik_{2}(x_{i}-x_{1})}+
    R_{1}^{(-)}e^{i k_{2} \ell_{2}}
    e^{-ik_{2}(x_{2}-x_{i})}
  \right) ,
  \label{eq:gaswpcase2}
\end{align}
where $f_{\rm aw}=1-R_{1}^{(-)}R_{2}^{(+)}e^{2 i k_{2} \ell_{2}}$.

The method utilized above consists in sum up all scattering
paths, but this could be very tedious and cumbersome.
However, this can be done by a simple diagrammatic
classification in families of paths, which is a practical way to
identify and perform the geometric series mentioned above.
Consider again the confined particle in the asymmetric well
potential.
The particle starting from $x_{i}$ may
(a) go to the right arriving $x_{f}$;
or (b) go to the left, hit the potential $u_{1}$ at $x_{1}$
being reflected.
There is an infinite family of paths for the particle at
$x_{1}$, we call this family $P_{1}$;
and (c) the particle go to the right, hit the potential $u_{3}$
at $x_{2}$ being reflected.
Like before, there is infinite family of paths for the particle
at $x_{2}$, we call this family $P_{2}$.
These two infinite family of paths are schematically depicted in
Fig. \ref{fig:fig3}(f).
Thus, by using the above prescription, the Green's function can be
written as
\begin{align}
  G_{f,i}^{(2)}
  ={} &\frac{m}{i\hbar^2 k_{2}}
  \left(e^{i k_{2} (x_{f}-x_{i})}\right.
  \nonumber \\
  {} & +
  \left. e^{i k_{2} (x_{2}-x_{i})}R_{2}^{(+)}P_{2} +
  e^{-i k_{2} (x_{1}-x_{i})}R_{1}^{(-)}P_{1}\right).
\label{eq:gsw}
\end{align}
The family $P_{1}$ ($P_{2}$) posses two contributions:
(a) go to the right (left), arriving at $x_f$;
and (b) go to the right (left), hit the potential
$u_{3}$ ($u_{1}$) at $x_{2}$ ($x_{1}$) being reflected followed
by the family $P_{2}$ ($P_{1}$).
Thus,
\begin{subequations}
  \label{eq:Ps}
  \begin{align}
    P_{1}={} & e^{i k_{2} (x_{f}-x_{1})} +
    e^{i k_{2} \ell_{2}}R_{2}^{(+)}P_{2},
    \label{eq:Pls} \\
    P_{2}={} & e^{i k_{2} (x_{f}-x_{2})} +
    e^{i k_{2} \ell_{2}}R_{1}^{(-)}P_{1}.
    \label{eq:Prs}
  \end{align}
\end{subequations}
Solving for $P_{1}$ and $P_{2}$ one obtains
\begin{subequations}
  \label{eq:Psol}
  \begin{align}
    P_{1}={} & \frac{1}{f_{\rm aw}}\left(e^{ i k_{2} (x_{f}-x_{1})}+
      R_{2}^{(+)}e^{i k_{2} \ell_{2}} e^{i k_{2} (x_{2}-x_{f})}\right),
    \label{eq:Pl}\\
    P_{2}={} & \frac{1}{f_{\rm aw}}\left(e^{ i k_{2} (x_{2}-x_{f})}
      + R_{1}^{(-)}e^{i k_{2} \ell_{2}}e^{i k_{2}(x_{f}-x_{1})}\right).
    \label{eq:Pr}
  \end{align}
\end{subequations}
By substitution of Eq. \eqref{eq:Psol} into
Eq. \eqref{eq:gsw}, one obtains the Green's function in
Eq. \eqref{eq:gaswpcase2}.
The other two cases can be obtained in a similar way.
For instance, for case (1) we have
\begin{align}
  \label{eq:gaswpcase1}
  G_{f,i}^{(1)}=
  {} &
  \frac{m}{i\hbar^2 \sqrt{k_{1} k_{2}} f_{\rm aw}}
  \left(
    T_{1}^{(-)} e^{-i k_{1} (x_f-x_{1})}
  \right)
  \nonumber \\
  {} &
  \times
  \left(
    e^{-i k_{2} (x_i-x_{1})} + R_{1}^{(-)} e^{i k_{2} \ell_{2}}
    e^{-i k_{2} (x_{2}-x_i)}
  \right),
\end{align}
and for case (3), we obtain
\begin{align}
  \label{eq:aswpcase3}
  G_{f,i}^{(3)}=
  {} &
  \frac{m}{i\hbar^2 \sqrt{k_{3} k_{2}} f_{\rm aw}}
  \left(
    T_{2}^{(+)} e^{i k_{3}(x_f-x_{2})}
  \right)
  \nonumber \\
  {} & \times
  \left(
    e^{-i k_{2} (x_i-x_{1})} + R_{1}^{(-)}
    e^{ i k_{2} \ell_{2}}e^{-i k_{2} (x_{2}-x_i)}
  \right).
\end{align}

\subsection{Calculation of the bound states}

The bound states are calculated from the residues of the Green's
functions in Eqs. \eqref{eq:gaswpcase2}, \eqref{eq:gaswpcase1}
and \eqref{eq:aswpcase3}.
Its poles $E_{n}=\hbar^{2} k_{n}^{2}/2m$ are all contained in
the term $1/f_{\rm aw}$.
They are calculated from $f_{\rm aw}=0$, which leads to the
following transcendental equation
\begin{equation}
  \label{eq:aswpoles}
  \left(\frac{k_{2}^{(n)}-k_{1}^{(n)}}{k_{1}^{(n)}+k_{2}^{(n)}}\right)
  \left(\frac{k_{2}^{(n)}-k_{3}^{(n)}}{k_{2}^{(n)}+k_{3}^{(n)}}\right)
  e^{2 i k_{2}^{(n)} \ell_{2}}=1,
\end{equation}
where $k_{j}^{(n)}=\sqrt{2m(E_{n}-u_{j})}/\hbar$.
This result agree with the one found by the solution of the
Schr\"{o}dinger equation \cite{Book.1981.Landau}.
Using the formula
\begin{equation}
  \label{eq:limpolo}
  \lim_{E \to E_{n}}\frac{E-E_{n}}{f_{\rm aw}}=
  \frac{\hbar^{2}}{2m}
  \lim_{k \to k_{n}}\frac{k^{2}-k_{n}^{2}}{f_{\rm aw}}=
  \frac{\hbar^{2}}{m} \frac{k_{n}}{f_{\rm aw}^{'(n)}},
\end{equation}
where $f_{\rm aw}^{'(n)}=(d f_{\rm aw}/dk)|_{k=k_{n}}$, we obtain for
the residues of the Green's function in Eqs. \eqref{eq:gaswpcase2},
\eqref{eq:gaswpcase1} and Eq. \eqref{eq:aswpcase3}
\begin{align*}
  \makebox[0.5cm][l]{$\displaystyle
  \psi_{n}^{(2)}(x_f) [\psi_{n}^{(2)}(x_i)]^{*}=
  \frac{\hbar^{2}}{2m}
  \lim_{k \to k_{n}}(k^{2}-k_{n}^{2})G_{f,i}^{(2)}$} &
\nonumber \\
  = {} &\frac{1}{i f_{\rm aw}^{'(n)}}
  \left[
    e^{i k_{2}^{(n)} (x_{f}-x_{1})}+
    \frac{k_{2}^{(n)}-k_{3}^{(n)}}{k_{2}^{(n)}+k_{3}^{(n)}}\;
    e^{i k_{2}^{(n)} (\ell_{2} +x_{2}-x_{f})}
  \right]\nonumber \\
 {} & \times
  \left[
    e^{-ik_{2}^{(n)}(x_{i}-x_{1})}+
    \frac{k_{2}^{(n)}-k_{1}^{(n)}}{k_{1}^{(n)}+k_{2}^{(n)}}\;
    e^{i k_{2}^{(n)}(\ell_{2}-x_{2}+x_{i})}
  \right],
\end{align*}
\begin{align*}
  \makebox[0.6cm][l]{$\displaystyle
  \psi_{n}^{(1)}(x_f) [\psi_{n}^{(2)}(x_i)]^{*}=
  \frac{\hbar^{2}}{2m}\lim_{k \to
    k_{n}}(k^{2}-k_{n}^{2})G_{f,i}^{(2)}$} & \nonumber \\
  = {} & \frac{1}{i f_{\rm aw}^{'(n)}}
  \left[
      \sqrt{\frac{k_{1}^{(n)}}{k_{2}^{(n)}}}
      \frac{2k_{2}^{(n)}}{k_{1}^{(n)}+k_{2}^{(n)}}
    e^{-i k_{1}^{(n)} (x_f-x_{1})}
  \right] \nonumber \\
  {} & \times
  \left[
    e^{-i k_{2}^{(n)} (x_i-x_{1})} +
    \frac{k_{2}^{(n)}-k_{1}^{(n)}}{k_{1}^{(n)}+k_{2}^{(n)}}\;
    e^{i k_{2}^{(n)} (\ell_{2}-x_{2}+x_i)}
  \right],
\end{align*}
\begin{align*}
  \makebox[0.7cm][l]{$\displaystyle\psi_{n}^{(3)}(x_f)
    [\psi_{n}^{(2)}(x_i)]^{*}=
  \frac{\hbar^{2}}{2m}\lim_{k \to
    k_{n}}(k^{2}-k_{n}^{2})G_{f,i}^{(3)}$} & \nonumber \\
  = {} & \frac{1}{i f_{\rm aw}^{'(n)}}
  \left[
      \sqrt{\frac{k_{3}^{(n)}}{k_{2}^{(n)}}}
      \frac{2 k_{2}^{(n)}}{k_{2}^{(n)}+k_{3}^{(n)}}\;
    e^{i k_{3}^{(n)}(x_f-x_{2})}
  \right]\nonumber \\
  {} & \times
  \left[
    e^{-i k_{2}^{(n)} (x_i-x_{1})} +
    \frac{k_{2}^{(n)}-k_{1}^{(n)}}{k_{1}^{(n)}+k_{2}^{(n)}}\;
    e^{i k_{2}^{(n)} (\ell_{2}-x_{2}+x_i)}
  \right].
\end{align*}
The correct normalized eigenfunctions corresponding to the bound
states of the system are thus given by:\\
Region 1: $x<x_{1}$
\begin{equation}
  \psi_{n}^{(1)}(x)=
    \sqrt{\frac{1}{i f_{\rm aw}^{'(n)}}
      \frac{k_{1}^{(n)}}{k_{2}^{(n)}}}
        \frac{2k_{2}^{(n)}}{k_{1}^{(n)}+k_{2}^{(n)}}\;
      e^{-i k_{1}^{(n)} (x-x_{1})},
\end{equation}
Region 2: $x_{1} < x < x_{2}$
\begin{align}
  \psi_{n}^{(2)}(x)=
  {} &
  \sqrt{\frac{1}{i f_{\rm aw}^{'(n)}}}
  \Big(
    e^{i k_{2}^{(n)} (x-x_{1})}
    \nonumber \\
   {} & +
     +\frac{k_{2}^{(n)}-k_{3}^{(n)}}{k_{2}^{(n)}+k_{3}^{(n)}}\;
  e^{i k_{2}^{(n)} (\ell_{2} +x_{2}-x)}
  \Big),
\end{align}
Region 3: $x>x_{2}$
\begin{equation}
  \psi_{n}^{(3)}(x)=
  \sqrt{\frac{1}{i f_{\rm aw}^{'(n)}}
    \frac{k_{3}^{(n)}}{k_{2}^{(n)}}}
  \frac{2 k_{2}^{(n)}}{k_{2}^{(n)}+k_{3}^{(n)}}\;
  e^{i k_{3}^{(n)}(x-x_{2})}.
\end{equation}
It should be observed that $k_{2}^{(n)}$
($k_{1}^{(n)}$ and $k_{3}^{(n)}$) is (are) real (imaginary) number(s),
and consequently $\psi_{n}^{(2)}(x)$ ($\psi_{n}^{(1)}(x)$ and
$\psi_{n}^{(3)}(x)$) is (are) an oscillatory complex (decreasing
real) exponential function(s) of $x$.
Also, the Green's function used here yields the correct
normalization constant which often involves a difficult integral
in the other methods.
In what follow, we apply the results obtained in this section in
other well-known rectangular potentials.

\section{Square well potential}
\label{sec:swp}

The Green's function for a square well potential
(depicted in Fig. \ref{fig:fig4})
can be obtained from the results of the previous section by
setting $u_1=u_3=u_0$ and $u_{2}=0$, thus $k_1=k_3=k_0$ and
$k_{2}=k=\sqrt{2mE}/\hbar$, also $x_1=-a$ and $x_{2}=a$.
From Eqs. \eqref{eq:gaswpcase2}, \eqref{eq:gaswpcase1} and
\eqref{eq:aswpcase3} we thus find
\begin{align}
  \label{eq:swpcase1}
  G_{f,i}^{(1)}=
  {} &
  \frac{m}{i\hbar^2 \sqrt{k_{0} k} f_{\rm sw}}
  \left(
    T_{(-a)}^{(-)} e^{-i k_{0} (x_f+a)}
  \right)
  \nonumber \\
  {} &
  \times
  \left(
    e^{-i k (x_i+a)} + R_{(-a)}^{(-)} e^{2 i k a}
    e^{-i k (a-x_i)}
  \right),
\end{align}
\begin{align}
  \label{eq:sqwcase2}
  G_{f,i}^{(2)}=
  {} &
  \frac{m}{i\hbar^2 k f_{\rm sw}}
  \left(
    e^{i k (x_f+a)} + R_{(a)}^{(+)} e^{2 i k a} e^{i k (a-x_f)}
  \right)
  \nonumber \\
  {} & \times
  \left(
    e^{-i k (x_i+a)} + R_{(-a)}^{(-)} e^{2 i k a}e^{-i k (a-x_i)}
  \right),
\end{align}
\begin{align}
  \label{eq:swpcase3}
  G_{f,i}^{(3)}=
  {} & \frac{m}{i\hbar^2 \sqrt{k_{0} k} f_{\rm sw}}
  \left(
    T_{(a)}^{(+)} e^{i k_{0}(x_f-a)}
  \right)
  \nonumber \\
  {} & \times
  \left(
    e^{-i k (x_i+a)} + R_{(-a)}^{(-)} e^{ 2 i k a}e^{-i k (a-x_i)}
  \right),
\end{align}
where $f_{\rm sw}=1-R_{(-a)}^{(-)}R_{(a)}^{(+)} e^{4ika}$.

\begin{figure}[t*]
    \centering
    \includegraphics*[width=0.85\columnwidth]{fig4.eps}
    \caption{A square well potential.}
    \label{fig:fig4}
\end{figure}

\subsection{Calculation of the bound states}

The eigenenergies are calculated from the condition
$f_{\rm sw}=0$.
By considering the symmetry of the potential, we can infer that
the reflections amplitudes are equal (up to a phase), i.e.,
$R_{(-a)}^{(-)}=R_{(a)}^{(+)}=(k-k_{0})/(k+k_{0})$,
leading to the following transcendental equation
\begin{equation}
  \label{eq:poloswp}
\left(\frac{k_{n}-k_{0}^{(n)}}{k_{n}+k_{0}^{(n)}}\right)^{2}
e^{4 i k_{n} a}=1.
\end{equation}
Using Eq. \eqref{eq:limpolo} the residues are determined and
leads to the following bound state eigenfunctions:\\
Region 1: $x<-a$
\begin{equation}
  \psi_{n}^{(1)}(x)=
    \sqrt{\frac{1}{i f_{\rm sw}^{'(n)}}
      \frac{k_{0}^{(n)}}{k_{n}}}
        \frac{2k_{n}}{k_{0}^{(n)}+k_{n}}\;
      e^{-i k_{0}^{(n)} (x+a)},
\end{equation}
Region 2: $-a < x < a$
\begin{align}
  \psi_{n}^{(2)}(x)=
  {} &
  \sqrt{\frac{1}{i f_{\rm sw}^{'(n)}}}
  \Big(
    e^{i k_{n} (x+a)}
    \nonumber \\
   {} &
     +\frac{k_{n}-k_{0}^{(n)}}{k_{n}+k_{0}^{(n)}}\;
     e^{2i k_{n} a}e^{ik_{n}(a-x)}
  \Big),
\end{align}
Region 3: $x>a$
\begin{equation}
  \psi_{n}^{(3)}(x)=
  \sqrt{\frac{1}{i f_{\rm sw}^{'(n)}}
    \frac{k_{0}^{(n)}}{k_{n}}}
  \frac{2 k_{n}}{k_{n}+k_{0}^{(n)}}\;
  e^{i k_{0}^{(n)}(x-a)}.
\end{equation}

From Eq. \eqref{eq:poloswp}, we can see that there are two
equations describing the eigenenergies.
These eigenenergies are determined by solving
\begin{align*}
(k_{n}-k_{0}^{(n)})e^{2 i k_{n} a}& = +(k_{n}+k_{0}^{(n)}), &
\text{(even parity)}, \\
(k_{n}-k_{0}^{(n)})e^{2 i k_{n} a}& = -(k_{n}+k_{0}^{(n)}), &
\text{(odd parity)},
\end{align*}
and the respective eigenfunctions are
\begin{equation}
  \psi_{n}^{(2)}(x)=2\sqrt{\frac{1}{i f_{\rm sw}^{'(n)}}}
  \begin{cases}
    e^{i k_{n} a} \cos{[k_{n} x]},   & \text{(even parity)}, \\
    i e^{i k_{n} a} \sin{[k_{n} x]}, & \text{(odd parity)},
  \end{cases}
\end{equation}
which are the well-known even and odd solutions for the square
well potential \cite{Book.2011.Sakurai}.

\section{Infinite well potential}
\label{sec:iwp}

In this section, we consider the infinite well potential.
Similarly as in previous sections, the Green's function for the
infinite well potential can be obtained by letting
$u_{1}=u_{3}=\infty$ and $x_1=0$ and $x_2=L$, so that the
potential function is given by
\begin{equation}
  V(x)=
  \begin{cases}
    0  & \text{for } 0 < x < L, \\
    \infty & \text{otherwise}.
  \end{cases}
\end{equation}
As the potentials are infinite in $x_1$ and $x_2$, the
reflection amplitude assume the value
$R_{(0)}^{(-)}=R_{(L)}^{(+)}=-1$
and a vanishing transmission amplitude.
Thus, the Green's function in Regions 1 and 3 vanishes, so
that $\psi_{n}^{(1)}(x)= \psi_{n}^{(3)}(x)=0$.
The Green's function in Region 2 take the form
\begin{align}
  \label{eq:giswp}
  G_{f,i}^{(2)}=
  {} &
  \frac{m}{i\hbar^2 k f_{\rm iw}}
  \left(
    e^{i k x_f}-e^{i k L}e^{i k (L-x_f)}
  \right)
  \nonumber \\
  {} & \times
  \left(
    e^{-i k x_i}-e^{i k L}e^{-i k (L-x_i)}
  \right),
\end{align}
with $f_{\rm iw}=1-e^{2 i k L}$.
After a straightforward algebra, the Eq. \eqref{eq:giswp} can be
written as
\begin{equation}
  G_{f,i}^{(2)}=
  \frac{2m}{\hbar^2 k \sin{[k L]}}
  \sin{[k(x_{f}-L)]\sin{[k x_{i}]}}.
\end{equation}
This result is indeed the exact one obtained through the
spectral expansion of the Green's function in
Eq. \eqref{eq:segf} (cf. \ref{sec:appendixb}).

\subsection{Calculation of the bound states}

The poles are all contained in $\sin{[kL]}$, and eigenenergies of
the  bound states are thus obtained from
$\sin{[k_{n}L]}=0$, i.e.,
\begin{equation}
k_{n}=\frac{n\pi}{L}, \quad n=1,2,3,\ldots.
\end{equation}
From the residues of the Green's function, the correctly
normalized eigenfunctions corresponding for the bound states are
thus given by
\begin{equation}
\psi_{n}^{(2)}(x)= \sqrt{\frac{2}{L}}
  \sin{\left(\frac{n \pi x}{L}\right)}.
\end{equation}
Indeed, these are the exact results for the bound states for the
infinite well potential.

\section{Application on quasi-bound states}
\label{sec:qbs}

In this section, we apply the Green's function approach of
previous sections to extract information of systems with
quasi-bound states.
A quasi-bound state occurs when a particle move inside a system
for a considerable period of time, leaving it when a fairly long
time interval $\tau$ has elapsed \cite{Book.1981.Landau}, where
$\tau$ is called lifetime of the quasi-bound state.
The concept of quasi-bound states is a fundamental one, and has
been applied in all areas of physics.
They have been used to calculate tunneling ionization rates
\cite{PRL.1998.81.2663}, to understand the phenomenon of
diffraction in time \cite{PRA.2011.83.043608}, to describe the
decay of cold atoms in quasi-one-dimensional traps
\cite{EPL.2006.74.965}, and are directly relevant to recent
condensed-matter experiments \cite{PRL.2005.95.066801}.

\begin{figure}[t*]
    \centering
    \includegraphics*[width=0.9\columnwidth]{fig5.eps}
    \caption{A rectangular potential able to support quasi-bound
      states as discussed in the text.}
    \label{fig:fig5}
\end{figure}

Let us consider the potential depicted in Fig. \ref{fig:fig5}.
Suppose the potentials $u_1$ and $u_3$ were infinitely high.
It would then be possible for particles to be trapped inside the
region for $x_1<x<x_2$, i.e., the system  would have genuine
bound states, with well defined energy $E>0$.
They are genuine bound states in the sense that they are
eigenstates of the Hamiltonian with an infinite lifetime.
From the Heisenberg uncertainty principle,
$\Delta E \Delta t \approx \hbar$, if the energy has null
uncertainty its state's lifetime is infinite.

In the situation of a finite barrier as in Fig. \ref{fig:fig5}
(this is rough rectangular approximation for the effective
potential in a central force problem  $V(r)$ plus the
centrifugal barrier $(\hbar^{2}/2m) [l(l+1)/r^{2}]$
\cite{Book.2011.Sakurai}),
the particle can be trapped, but it cannot be trapped forever,
even if $E<u_{3}$, as a consequence of the tunnel effect.
The energy spectrum of these particle will be quasi-discrete, and
it consists of a series of broadened levels, whose width in
represented by $\Gamma=\hbar/\tau$ \cite{Book.1998.Merzbacher},
and the energy values are called quasi-energies.
In the scattering of particles by such potential, the situation
becomes very interesting when the incident energy is close to
the quasi-energy
\begin{equation}
E^{(\rm inc)}\approx E^{(\rm qb)}.
\end{equation}
In this energy interval, the module square of the transmission
amplitude exhibits pronounced peaks, and this is called resonant
scattering \cite{Book.2011.Sakurai}.
In Fig. \ref{fig:fig6} it is depicted a typical transmission probability
as a function of incident energy for a scattering of a potential which
supports quasi-bound states.

\begin{figure}[t*]
    \centering
    \includegraphics*[width=\columnwidth]{fig6.eps}
    \caption{Typical behavior for the transmission coefficient
      for a potential which has two quasi-bound states with
      energies $E_{1}^{(\rm qb)}$ and $E_{2}^{(\rm qb)}$ and widths
      $\Gamma_1$ and $\Gamma_2$, respectively.}
    \label{fig:fig6}
\end{figure}

Now, let us consider the case of a finite square barrier at $x_2$
with an infinite barrier at $x_{1}$.
In this situation, the system can also has quasi-bound states,
due the tunneling through the right square barrier.
The scattering eigenfunction for a particle incident from the
right is given by
\begin{equation}
  \psi (x) \approx
  \frac{1}{\sqrt{2\pi}}(e^{-ik_{4} x}+R_{\rm pot}^{(-)} e^{i k_{4}x}),
\end{equation}
where $R_{\rm pot}^{(-)}$ is the reflection amplitude of whole
potential.
By analogy with the previous case, we would try to extract the
information from quasi-bound states from the reflection
coefficient $R_{\rm pot}^{(-)}$.
Unfortunately, due to the potential to be infinite at left, the
reflection coefficient has the value $|R_{\rm pot}^{(-)}|^{2}=1$ for
all range of energies.
Thus, we cannot extract information of quasi-bound states for
this kind of potential by the above method.
So, we propose a Green's function approach to extract
information of quasi-bound states for such kind of potential,
as we explain below.
Following the same steps described in the Section \ref{sec:awp},
the Green's function for $x_{i}>x_{3}$ and $x_{1}<x_{f}<x_{2}$
is readily obtained and is written as
\begin{align}
  \label{eq:gqbs}
  G_{f,i}=
   {} &
  \frac{m}{i\hbar^2 \sqrt{k_{2} k_{4}}}
  \frac{T_{b}^{(-)}}{f_{\rm qb}} e^{i k_{4} (x_{i}-x_{3})}
  \nonumber \\
  {} & \times
  \left(
    e^{ik_{2}(x_{2}-x_{f})}+R_{1}^{(-)}
    e^{ik_{2}(x_{f}+x_{2}-2x_{1})}
  \right),
\end{align}
with ${f}_{\rm  qb}=1-R_1^{(-)}R_{b}^{(+)}e^{ik_{2}\ell_{2}}$.
$R_{b}^{(+)}$ and $T_{b}^{(-)}$ are the reflection and transmission
amplitudes for the potential barrier given, respectively, by
\begin{equation}
  \label{eq:refb}
  R_{b}^{(+)}=
  R_{2}^{(+)}+
  \frac
  {T_{2}^{(-)}T_{2}^{(+)}R_{3}^{(+)}e^{2ik_{3}\ell_{3}}}
  {1-R_{2}^{(-)}R_{3}^{(+)}e^{2ik_{3}\ell_{3}}},
\end{equation}
\begin{equation}
  \label{eq:transb}
  T_{b}^{(-)}=
  \frac
  {T_{2}^{(-)}T_{3}^{(-)}e^{ik_{3}\ell_{3}}}
  {1-R_{2}^{(-)}R_{3}^{(+)}e^{2ik_{3}\ell_{3}}}.
\end{equation}

\begin{figure}[t*]
    \centering
    \includegraphics*[width=\columnwidth]{fig7.eps}
    \caption{Behavior of $|A|^{2}=|T_{b}^{(-)}/\bar{f}_{qb}|^{2}$ as
      function of the energy displaying the presence of
      quasi-bound states.
      The parameters are $u_{1}=\infty$, $u_2=u_{4}=0$,
      $u_{3}=0.23$ eV, $\ell_{3}=80$ \AA {} and $\ell_{2}=10.05
      \ell_{3}$.
      In the inset is shown the transmission (solid line) and
      reflection (dashed line) coefficients for the potential
      barrier.
    }
    \label{fig:fig7}
\end{figure}

For an infinite barrier at $x_{1}$, we have $R_{1}^{(-)}=-1$,
and the Eq. \eqref{eq:gqbs} simplifies to
\begin{equation}
  \label{eq:gqbss}
  G_{f,i}=
  \frac{2m}{i^{2}\hbar^2 \sqrt{k_{2} k_{4}}}
  \frac{T_{b}^{(-)}}{\bar{f}_{\rm qb}} e^{i k_{4}(x_{i}-x_{3})}
  \sin[k_{2}x_{f}],
\end{equation}
where $\bar{f}_{\rm  qb}=1+R_{b}^{(+)}e^{ik_{2}\ell_{2}}$.
From the interpretation of the Green's function as a probability
amplitude (cf. Section \ref{sec:gfdp}), we can thus interpret
the term
\begin{equation}
 A=\frac{T_{b}^{(-)}}{\bar{f}_{\rm qb}},
\end{equation}
in Eq. \eqref{eq:gqbss} as such amplitude.
If the potential has quasi-bound states, an incident wave with
energy close to the quasi-energy, will have a high probability
of tunneling, entering in the trapping region.
Consequently, a graph of $|A|^{2}$ as a function of the energy,
will have peaks at each energy value close to
$E^{(\rm qb)}$.
So, we can extract information of quasi-energies and its
respective widths from $A$.
It is worthwhile to observe that the amplitude $A$ is not
normalized, but this is not a problem, because we are only
interested in the position of quasi-energies and the width of
quasi-states.

For a numerical example, in the Fig. \ref{fig:fig7} is
shown  a graph of $|A|^{2}$ as function of energy for the
potential in Fig. \ref{fig:fig5}.
For the parameters we choose typical values for heterostructures
in GaAs \cite{JPCM.1989.1.9027}.
The particle's mass is $m=0.07 m_{e}$, where $m_e$ is the
electron mass, $u_{1}=\infty$, $u_2=u_{4}=0$, $u_{3}=0.23$ eV,
$\ell_{3}=80$ \AA {} and $\ell_{2}=10.05 \ell_{3}$.
As we can see, it is evident in the graph the existence of
quasi-bound states.

The approach presented here for rectangular potentials, can be
generalized for smooth potentials, but in this case it is
necessary the calculation of the classical action for the
quantum particle under the action of the potential.
Specifically, we write the amplitude as
\begin{equation}
A=\frac{T_{b}}{f},
\end{equation}
where $T_{b}$ is the generalized transmission amplitude for the
smooth barrier between $x_{2}$ and $x_{3}$,
\begin{equation}
f=1-R R_{\infty} \exp{[\frac{i}{\hbar}S(x_{2},x_{3};E)]},
\end{equation}
where $S(x_{2},x_{3},k)$ is the classical action, $R$ is the
generalized reflection amplitude for the barrier, and
$R_{\infty}$ is the generalized reflection amplitude for the
infinity  barrier at $x_{1}$.
Since $|R_{\infty}|^{2}=1$, we can write
$R_{\infty}=\exp{[-i\phi(E)]}$, in such way that
\begin{equation}
f=1-R \exp{[\frac{i}{\hbar}S(x_{2},x_{3};E)-i\phi(E)]}.
\end{equation}
All those generalized amplitudes above are obtained by the procedure
outlined in \cite{JPA.2001.34.5041,JPA.2003.36.227}.

\section{Conclusion}
\label{sec:conclusion}

In this work, the exact Green's functions for rectangular single
wells are obtained in a rather general way and by a simple
method.
Our results are the exact ones and, although of the simplicity
of the systems considered, Green's functions for such system are
not so easy to obtain by standard procedures (for example,
solving the inhomogeneous differential equation in
Eq. \eqref{eq:defgf}).
The procedures allows one to discuss complete arbitrary
rectangular single wells and barriers, generalizing and resuming
results in the literature.
For instance, by withdrawing of the potential step at $x_{1}$ by
setting $R_{1}^{(-)}=0$, from Eq. \eqref{eq:gaswpcase2} the Green's
function for square barrier of Ref.
\cite{PRA.1993.48.2567,PRA.1995.51.2654}
is obtained.
The method can be applied for general potentials, including those
multidimensional with radial symmetry, but in this case is necessary
the calculation of the classical action for the particle under the
action of the potential \cite{JPA.2001.34.5041,JPA.2003.36.227}.

From the poles and residues of the Green's function the bound
state eigenenergies and eigenfunctions were obtained with the
correct normalization constant.
The determination of the later often involves a difficult
integral in the other methods.

Finally, we also have discussed an application of the Green's
function approach to extract information from quasi-bound states
in systems which standard analysis of the quantum amplitudes
are not possible.
The method could be generalized for smooth potentials and applied to the
well-known alpha decaying and determination of the dwell times and will
be subject of a future work.

\section*{Acknowledgments}

The author would like to thank E. O. Silva, C. F. Woellner and
J. A. O. Freire, for critical reading the manuscript and helpful
discussions.
This work was partially supported by the Fun\-da\-\c{c}\~{a}o
Arauc\'{a}ria (Brazil) under grant number No. 205/2013.