An Introduction to Scattering Theory
Lecture notes for the $\bm{1}^{\bf st}$ International Summer School on
Advanced Quantum Mechanics
(Prague, September 09-19, 2019)
Milan Šindelka
Laser Plasma Department, Institute of Plasma Physics of the Czech Academy of
Sciences,
U Slovanky 2525/1a, 18200 Prague 8, Czech Republic
<EMAIL_ADDRESS>
()

The purpose of these lectures is to give an accessible and self contained
introduction to quantum scattering theory in one dimension. Part A defines the
theoretical playground, and develops basic concepts of scattering theory in
the time domain (Asymptotic Condition, in- and out- states, scattering
operator $\hat{S}$). The aim of Part B is then to build up, in a step-by-step
fashion, the time independent scattering theory in energy domain. This amounts
to introduce the Lippmann-Schwinger equation for the stationary scattering
states (denoted as $|\psi_{E(\pm 1)}^{\pm}\rangle$), to discuss fundamental
properties of $|\psi_{E(\pm 1)}^{\pm}\rangle$, and subsequently to construct
$\hat{S}$ and $\hat{T}$ operators in terms of $|\psi_{E(\pm 1)}^{\pm}\rangle$.
Physical contents of the $\hat{S}$ and $\hat{T}$ operators is then illuminated
by deriving explicit formulas for the probability of transmission/reflection
of our quantum particle through/from the interaction region of the potential.
An illustrative numerical example is given, which also highlights an existence
of scattering resonances. Finally, Part C elaborates the nonhermitian
scattering theory (Siegert pseudostate formalism), which offers an extremely
powerful tool suitable for clear cut understanding of the resonance phenomena.

Lecture #1

A. Basic theoretical setup, scattering theory in time domain

Based upon Refs. [1, 2].

  * •

A.1 Our playground
Nonrelativistic quantum mechanics
of a single particle moving in 1D along a coordinate $x\in(-\infty,+\infty)$.
The Hamiltonian is assumed to take the form

$\hat{\sf H}\;=\;\hat{\sf H}_{0}\;+\;\hat{\sf V}\hskip 14.22636pt;$ (1)

with $\hat{\sf H}_{0}$ being the kinetic energy operator and $\hat{\sf V}$ the
potential energy.
Explicitly one has

$\hat{\sf H}_{0}\;=\;\frac{\hat{\sf p}^{2}}{2\,m}\hskip 14.22636pt,\hskip
14.22636pt\hat{\sf V}\;=\;V(\hat{\sf x})\hskip 14.22636pt;$ (2)

where as usual $[\hat{\sf x},\hat{\sf p}]=i\hbar\,\hat{\sf 1}$.
The potential function $V(x)$ is assumed to be finite ranged, i.e.,

$V(x)\;=\;0\hskip 14.22636pt{\rm whenever}\hskip 14.22636ptx\leq a\hskip
14.22636pt{\rm or}\hskip 14.22636ptx\geq b\hskip 14.22636pt;$ (3)

where $a$ and $b$ are given parameters ($a<b$).
Dynamical time evolution of the state vector $|\psi(t)\rangle$ is governed by
the Schrödinger equation

$i\hbar\,\partial_{t}\,|\psi(t)\rangle\;=\;\hat{\sf H}\,|\psi(t)\rangle\hskip
14.22636pt.$ (4)

Recall that the Schrödinger equation (4) implies probability conservation.
This is expressed mathematically by the continuity equation which is derived
in Appendix I.
An elementary treatment of the eigenvalue problem of $\hat{\sf H}$ is given in
Appendix II.

  * •

A.2 Time evolution operators
Let $\hat{\sf U}(t)$ be the full evolution operator, defined through an
initial value problem

$i\hbar\;\partial_{t}\,\hat{\sf U}(t)\;=\;\hat{\sf H}\,\hat{\sf U}(t)\hskip
14.22636pt,\hskip 14.22636pt\hat{\sf U}(0)\;=\;\hat{\sf 1}\hskip 14.22636pt.$
(5)

Clearly, one has

$\hat{\sf U}(t)\;=\;e^{-(i/\hbar)\hat{\sf H}t}\hskip 14.22636pt.$ (6)

Dynamical state of our system is then expressible in the form

$|\psi(t)\rangle\;=\;\hat{\sf U}(t)\,|\psi\rangle\hskip 14.22636pt;$ (7)

of course with $|\psi\rangle=|\psi(0)\rangle$.
Let $\hat{\sf U}_{0}(t)$ be the free evolution operator, corresponding to a
situation when the potential is absent.
This entity is defined through an initial value problem

$i\hbar\;\partial_{t}\,\hat{\sf U}_{0}(t)\;=\;\hat{\sf H}_{0}\,\hat{\sf
U}_{0}(t)\hskip 14.22636pt,\hskip 14.22636pt\hat{\sf U}_{0}(0)\;=\;\hat{\sf
1}\hskip 14.22636pt.$ (8)

Clearly, one has

$\hat{\sf U}_{0}(t)\;=\;e^{-(i/\hbar)\hat{\sf H}_{0}t}\hskip 14.22636pt.$ (9)

Freely evolving quantum state vectors are then expressible as

$|\phi(t)\rangle\;=\;\hat{\sf U}_{0}(t)\,|\phi\rangle\hskip 14.22636pt;$ (10)

of course with $|\phi\rangle=|\phi(0)\rangle$.

Lecture #1

  * •

A.3 The asymptotic condition: A qualitative look
Suppose that we have prepared an initial quantum state $|\psi\rangle$ at
$t=0$.
We may view $|\psi\rangle$ as a square integrable wavepacket
$\psi(0,x)=\langle x|\psi\rangle$.
Assume that this wavepacket does not populate any bound states of the studied
problem.
According to the central assumption of the scattering theory,
our wavepacket behaves for $t\neq 0$ as follows:
As $t$ increases from $0$ to $+\infty$, the wavepacket $\psi(t,x)=\langle
x|\psi(t)\rangle$
tends to leave the interaction region (= region of $x\in(a,b)$ with nonzero
potential)
and becomes thus freely evolving at very large positive times.
$[\,$Note that some portion of $\psi(0,x)$ would generally escape to the right
($x\to+\infty$)
$[\,$and some portion to the left ($x\to-\infty$).$\,]$
Similarly, as $t$ decreases from $0$ to $-\infty$, the wavepacket
$\psi(t,x)=\langle x|\psi(t)\rangle$
tends to leave the interaction region and becomes thus freely evolving at very
large negative times.
What we just described is the so called  Asymptotic Condition (AC).
It has crucial importance for all our subsequent elaborations.

  * •

A.4 Formal statement of the Asymptotic Condition
Let us formulate now the AC in a more formal mathematical language.
The AC says that, for each prescribed physical state
$|\psi\rangle=|\psi(0)\rangle$
(which does not populate any eventual bound states of the problem),
there exists an unique incoming state $|\phi_{\rm in}\rangle$ such that

$\hat{\sf U}(t)\,|\psi\rangle\;\to\;\hat{\sf U}_{0}(t)\,|\phi_{\rm
in}\rangle\hskip 14.22636pt{\rm as}\hskip 14.22636ptt\to-\infty\hskip
14.22636pt.$ (11)

Similarly, one assumes that there exists an unique outgoing state $|\phi_{\rm
out}\rangle$ such that

$\hat{\sf U}(t)\,|\psi\rangle\;\to\;\hat{\sf U}_{0}(t)\,|\phi_{\rm
out}\rangle\hskip 14.22636pt{\rm as}\hskip 14.22636ptt\to+\infty\hskip
14.22636pt.$ (12)

Written equivalently,
the AC asserts that the in-state $|\phi_{\rm in}\rangle$ and the out-state
$|\phi_{\rm out}\rangle$ are related to $|\psi\rangle$ as follows:

$|\psi\rangle\;=\;\lim_{t\to-\infty}\,\hat{\sf U}^{\dagger}(t)\,\hat{\sf
U}_{0}(t)\,|\phi_{\rm in}\rangle\hskip 14.22636pt;$ (13)
$|\psi\rangle\;=\;\lim_{t\to+\infty}\,\hat{\sf U}^{\dagger}(t)\,\hat{\sf
U}_{0}(t)\,|\phi_{\rm out}\rangle\hskip 14.22636pt.$ (14)

Alternatively, one may state that

$|\phi_{\rm in}\rangle\;=\;\lim_{t\to-\infty}\,\hat{\sf
U}_{0}^{\dagger}(t)\,\hat{\sf U}(t)\,|\psi\rangle\hskip 14.22636pt;$ (15)
$|\phi_{\rm out}\rangle\;=\;\lim_{t\to+\infty}\,\hat{\sf
U}_{0}^{\dagger}(t)\,\hat{\sf U}(t)\,|\psi\rangle\hskip 14.22636pt.$ (16)

So far, we have just declared the AC to hold without proving it.
A simple and explicit  proof of the AC is worked out in Appendix III.
It is based upon the stationary phase method.

Lecture #1

  * •

A.5 The Scattering Operator: Definition
It should be obvious by now
that there exists a well defined unitary mapping of the in-states into the
out-states.
Indeed, any given dynamical in-state $\hat{\sf U}_{0}(t)\,|\phi_{\rm
in}\rangle$ $[t\to-\infty]$
is converted by an interaction (occurring at finite times)
into some uniquely determined dynamical out-state $\hat{\sf
U}_{0}(t)\,|\phi_{\rm out}\rangle$ $[t\to+\infty]$.
Such that

$|\phi_{\rm out}\rangle\;=\;\lim_{T\to+\infty}\,\hat{\sf
U}_{0}^{\dagger}(+T)\;\hat{\sf U}(+T)\;\hat{\sf U}^{\dagger}(-T)\;\hat{\sf
U}_{0}(-T)\;|\phi_{\rm in}\rangle\hskip 14.22636pt.$ (17)

We shall formally write

$|\phi_{\rm out}\rangle\;=\;\hat{S}\,|\phi_{\rm in}\rangle\hskip 14.22636pt;$
(18)

and call

$\hat{S}\;=\;\lim_{T\to+\infty}\,\hat{\sf U}_{0}^{\dagger}(+T)\;\hat{\sf
U}(+T)\;\hat{\sf U}^{\dagger}(-T)\;\hat{\sf U}_{0}(-T)$ (19)

as the  Scattering Operator.
The just defined scattering operator $\hat{S}$ takes an input
(= an incoming in-state wavepacket set up to collide with our potential)
and converts it into an appropriate output
(= the outgoing out-state wavepacket departing away from the interaction
region to $x\to\pm\infty$).
Thus the knowledge of $\hat{S}$ enables us to determine
an overall output of the studied scattering process from any given input.
This is why  $\hat{S}$ contains all the information about scattering.
Remark: Since we are interested in scattering,
we have tacitly restricted the domain of $\hat{S}=(\ref{hat-S-def})$
just to the scattering sector of our quantum state space.
In other words, the above established definition (18)-(19) of $\hat{S}$
is valid only for such vectors $|\phi_{\rm in}\rangle$ which do not populate
eventual bound states of the problem.

  * •

A.6 The Scattering Operator: Physical meaning, matrix elements
Suppose that in the infinite past (before the scattering takes place)
our system has been prepared in a given freely evolving quantum state
$\hat{\sf U}_{0}(t)\,|\phi_{\rm in}\rangle$.
We wish to look into the infinite future (after the scattering has taken
place)
and determine the probability of finding our system in some freely evolving
quantum state $\hat{\sf U}_{0}(t)\,|\chi_{\rm out}\rangle$.
Clearly, the sought probability equals to

$\wp\;=\;\Bigl{|}{\cal S}\Bigr{|}^{2}\hskip 14.22636pt;$ (20)

where

${\cal S}\;=\;\langle\chi_{\rm out}|\,\hat{\sf U}_{0}^{\dagger}(t)\;\hat{\sf
U}_{0}(t)\,|\phi_{\rm out}\rangle\;=\;\langle\chi_{\rm out}|\phi_{\rm
out}\rangle\;=\;\langle\chi_{\rm out}|\hat{S}|\phi_{\rm in}\rangle\hskip
14.22636pt.$ (21)

Redisplayed once again in final form, one has

$\wp_{\rm out/in}\;=\;\Bigl{|}\,\langle\chi_{\rm out}|\hat{S}|\phi_{\rm
in}\rangle\,\Bigr{|}^{2}\hskip 14.22636pt.$ (22)

This formula gives  physical meaning to the scattering matrix elements.

Lecture #1

B. Passage from the time dependent picture of scattering
B. into the time independent scattering theory

Based upon Refs. [1, 2].

  * •

B.1 Motivation
The above described time dependent picture of scattering (based upon
wavepackets)
is physically intuitive yet still rather formal.
It does not lend itself well to any further theoretical investigation of
$\hat{S}$.
Yet we wish to analyze the properties of $\hat{S}$,
in order to gain deep physics insights (and encounter surprises)
and thus better understand the nature of scattering processes.
Further progress is facilitated via switching from the time domain into the
energy domain.

  * •

B.2 Mathematical developments #1
Our forthcoming task is to relate $|\psi\rangle$ and $|\phi_{\rm in}\rangle$,
$|\phi_{\rm out}\rangle$ more explicitly.
It is a simple matter to verify an identity

$\hat{\sf U}^{\dagger}(t)\,\hat{\sf U}_{0}(t)\,|\phi_{\rm
in}\rangle\;=\;|\phi_{\rm in}\rangle\;-\;\int_{t}^{0}\left\\{\,\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf U}_{0}(t^{\prime})\,|\phi_{\rm
in}\rangle\,\right\\}^{\bm{\cdot}}{\rm d}t^{\prime}\hskip 14.22636pt;$ (23)

here the upper dot superscript denotes time differentiation. Subsequently one
has

$|\psi\rangle\;=\;|\phi_{\rm
in}\rangle\;-\;\int_{-\infty}^{0}\left\\{\,\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf
U}_{0}(t^{\prime})\,\right\\}^{\bm{\cdot}}|\phi_{\rm in}\rangle\;{\rm
d}t^{\prime}\hskip 14.22636pt.$ (24)

Similarly,

$\hat{\sf U}^{\dagger}(t)\,\hat{\sf U}_{0}(t)\,|\phi_{\rm
out}\rangle\;=\;|\phi_{\rm out}\rangle\;+\;\int_{0}^{t}\left\\{\,\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf U}_{0}(t^{\prime})\,|\phi_{\rm
out}\rangle\,\right\\}^{\bm{\cdot}}{\rm d}t^{\prime}\hskip 14.22636pt;$ (25)

and hence

$|\psi\rangle\;=\;|\phi_{\rm
out}\rangle\;+\;\int_{0}^{+\infty}\left\\{\,\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf
U}_{0}(t^{\prime})\,\right\\}^{\bm{\cdot}}|\phi_{\rm out}\rangle\;{\rm
d}t^{\prime}\hskip 14.22636pt.$ (26)

Direct calculation yields the required time derivative,

$\displaystyle\left\\{\,\hat{\sf U}^{\dagger}(t^{\prime})\,\hat{\sf
U}_{0}(t^{\prime})\,\right\\}^{\bm{\cdot}}$ $\displaystyle=$
$\displaystyle\hat{\sf U}^{\dagger\bm{\cdot}}(t^{\prime})\,\hat{\sf
U}_{0}(t^{\prime})\;+\;\hat{\sf U}^{\dagger}(t^{\prime})\,\hat{\sf
U}_{0}^{\bm{\cdot}}(t^{\prime})\;=$ (27) $\displaystyle=$
$\displaystyle\frac{i}{\hbar}\;\hat{\sf
U}^{\dagger}(t^{\prime})\,\left(\hat{\sf H}\,-\,\hat{\sf
H}_{0}\right)\,\hat{\sf U}_{0}(t^{\prime})\;=\;\frac{i}{\hbar}\;\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf V}\,\hat{\sf U}_{0}(t^{\prime})\hskip
14.22636pt.$

Correspondingly, one obtains

$|\psi\rangle\;=\;|\phi_{\rm
in}\rangle\;-\;\frac{i}{\hbar}\,\int_{-\infty}^{0}\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf V}\,\hat{\sf U}_{0}(t^{\prime})\,|\phi_{\rm
in}\rangle\;{\rm d}t^{\prime}\hskip 14.22636pt.$ (28)
$|\psi\rangle\;=\;|\phi_{\rm
out}\rangle\;+\;\frac{i}{\hbar}\,\int_{0}^{+\infty}\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf V}\,\hat{\sf U}_{0}(t^{\prime})\,|\phi_{\rm
out}\rangle\;{\rm d}t^{\prime}\hskip 14.22636pt.$ (29)

This is the sought more explicit relationship between $|\psi\rangle$ and
$|\phi_{\rm in}\rangle$, $|\phi_{\rm out}\rangle$.
Formulas (28)-(29) include integration over time and still do not offer much
insight.
Hence we need to pursue additional developments as detailed below.

Lecture #1

  * •

B.2 Mathematical developments #2
Further elaborations based upon equations (28)-(29) are possible.
An eigenproblem of $\hat{\sf H}_{0}$ can be formally written as follows. One
has

$\hat{\sf H}_{0}\,|\phi_{E\eta}\rangle\;=\;E\,|\phi_{E\eta}\rangle\hskip
14.22636pt;\hskip 14.22636pt[\,E>0\,,\,\eta\in\\{-1,+1\\}\,]$ (30)

with the usual orthonormality and closure relations

$\langle\phi_{E\eta}|\phi_{E^{\prime}\eta^{\prime}}\rangle\;=\;\delta(E-E^{\prime})\;\delta_{\eta\eta^{\prime}}\hskip
14.22636pt,\hskip 14.22636pt\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\,|\phi_{E\eta}\rangle\langle\phi_{E\eta}|\;=\;\hat{\sf
1}\hskip 14.22636pt.$ (31)

It is an elementary task (a simple exercise for you) to verify that
an explicit solution of the eigenproblem (30) satisfying also relations (31)
is given by the prescription

$\langle
x|\phi_{E\eta}\rangle\;=\;\phi_{E\eta}(x)\;=\;\sqrt{\frac{m}{2\,\pi\,\hbar^{2}K}}\;\,e^{+i\eta
Kx}\hskip 14.22636pt;$ (32)

where $\hbar K=\sqrt{2\,m\,E}$.
Clearly, vector $|\phi_{\rm in}\rangle$ can be uniquely expanded in the
$|\phi_{E\eta}\rangle$ basis, such that

$|\phi_{\rm in}\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;c_{E\eta}\,|\phi_{E\eta}\rangle\hskip 14.22636pt,\hskip
14.22636ptc_{E\eta}\;=\;\langle\phi_{E\eta}|\phi_{\rm in}\rangle\hskip
14.22636pt.$ (33)

Converting thus (28) into

$\displaystyle|\psi\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;c_{E\eta}\,|\phi_{E\eta}\rangle\;-\;\frac{i}{\hbar}\,\int_{-\infty}^{0}{\rm
d}t^{\prime}\;\,\int_{0}^{+\infty}{\rm d}E\,\sum_{\eta}\;c_{E\eta}\;\,\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf V}\,\hat{\sf
U}_{0}(t^{\prime})\,|\phi_{E\eta}\rangle\;=$ (34) $\displaystyle=$
$\displaystyle\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;c_{E\eta}\,|\phi_{E\eta}\rangle\;-\;\frac{i}{\hbar}\,\int_{-\infty}^{0}{\rm
d}t^{\prime}\;\,\int_{0}^{+\infty}{\rm d}E\,\sum_{\eta}\;c_{E\eta}\;\,\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf
V}\,e^{-(i/\hbar)Et^{\prime}}\,|\phi_{E\eta}\rangle\hskip 7.11317pt.\hskip
7.11317pt$

An additional refinement can be implemented here,
based upon including an extra factor $e^{+\varepsilon t^{\prime}/\hbar}$ with
$\varepsilon\to+0$. Such that

$\displaystyle|\psi\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;c_{E\eta}\,|\phi_{E\eta}\rangle\;-\;\frac{i}{\hbar}\,\lim_{\varepsilon\to+0}\,\int_{-\infty}^{0}{\rm
d}t^{\prime}\;\,\int_{0}^{+\infty}{\rm d}E\,\sum_{\eta}\;c_{E\eta}\;\,\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf
V}\,e^{-(i/\hbar)(E+i\varepsilon)t^{\prime}}\,|\phi_{E\eta}\rangle\hskip
7.11317pt.\hskip 7.11317pt$ (35)

An importance of inserting $e^{+\varepsilon t^{\prime}/\hbar}$ becomes evident
later on
(see equation (42) below and the accompanying discussion).
For now it is enough to recognize that the presence of $e^{+\varepsilon
t^{\prime}/\hbar}$
does not affect the value of the associated integral in (35) when the limit of
$\varepsilon\to+0$ is taken.

Lecture #1

Similarly we shall write

$|\phi_{\rm out}\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;C_{E\eta}\,|\phi_{E\eta}\rangle\hskip 14.22636pt,\hskip
14.22636ptC_{E\eta}\;=\;\langle\phi_{E\eta}|\phi_{\rm out}\rangle\hskip
14.22636pt;$ (36)

and

$\displaystyle|\psi\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;C_{E\eta}\,|\phi_{E\eta}\rangle\;+\;\frac{i}{\hbar}\,\lim_{\varepsilon\to+0}\,\int_{0}^{+\infty}{\rm
d}t^{\prime}\;\,\int_{0}^{+\infty}{\rm d}E\,\sum_{\eta}\;C_{E\eta}\;\,\hat{\sf
U}^{\dagger}(t^{\prime})\,\hat{\sf
V}\,e^{-(i/\hbar)(E-i\varepsilon)t^{\prime}}\,|\phi_{E\eta}\rangle\hskip
7.11317pt.\hskip 14.22636pt$ (37)

Let us look again now at (35) and (37).
Assuming that the integrations $\int{\rm d}t^{\prime}$,
$\int_{0}^{+\infty}{\rm d}E\,\sum_{\eta}$ are interchangeable, one may write

$|\psi\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;c_{E\eta}\,|\psi_{E\eta}^{+}\rangle\hskip 14.22636pt;$ (38)

where by definition

$|\psi_{E\eta}^{+}\rangle\;=\;|\phi_{E\eta}\rangle\;-\;\frac{i}{\hbar}\,\lim_{\varepsilon\to+0}\,\int_{-\infty}^{0}\hat{\sf
U}^{\dagger}(t^{\prime})\,e^{-(i/\hbar)(E+i\varepsilon)t^{\prime}}\,\hat{\sf
V}\,|\phi_{E\eta}\rangle\;{\rm d}t^{\prime}\hskip 14.22636pt.$ (39)

Similarly one has

$|\psi\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;C_{E\eta}\,|\psi_{E\eta}^{-}\rangle\hskip 14.22636pt;$ (40)

where by definition

$|\psi_{E\eta}^{-}\rangle\;=\;|\phi_{E\eta}\rangle\;+\;\frac{i}{\hbar}\,\lim_{\varepsilon\to+0}\,\int_{0}^{+\infty}\hat{\sf
U}^{\dagger}(t^{\prime})\,e^{-(i/\hbar)(E-i\varepsilon)t^{\prime}}\,\hat{\sf
V}\,|\phi_{E\eta}\rangle\;{\rm d}t^{\prime}\hskip 14.22636pt.$ (41)

In summary, elaborations of this paragraph have enabled us
to replace the formulas (28) and (29) by equivalent relations (38)-(39) and
(40)-(41).
$[\,$Recall also (33), (36) in this context.$\,]$
This is a substantial progress as we shall see shortly.

Lecture #2

  * •

B.3 The explicit Lippmann-Schwinger equation
Take (39) and plug in (6). An integration over $t^{\prime}$ can then be
performed explicitly. Indeed, one has

$\displaystyle|\psi_{E\eta}^{+}\rangle$ $\displaystyle=$
$\displaystyle|\phi_{E\eta}\rangle\;-\;\frac{i}{\hbar}\,\lim_{\varepsilon\to+0}\,\int_{-\infty}^{0}e^{-(i/\hbar)(E-\hat{\sf
H}+i\varepsilon)t^{\prime}}\,\hat{\sf V}\,|\phi_{E\eta}\rangle\;{\rm
d}t^{\prime}\;=$ (42) $\displaystyle=$
$\displaystyle|\phi_{E\eta}\rangle\;+\;\lim_{\varepsilon\to+0}\,\frac{1}{E-\hat{\sf
H}+i\varepsilon}\,\hat{\sf V}\,|\phi_{E\eta}\rangle\hskip 14.22636pt.$

Showing that $|\psi_{E\eta}^{+}\rangle$ is obtainable from
$|\phi_{E\eta}\rangle$ without any reference to the time variable.
Moreover, it is evident that the presence of $i\varepsilon$ in the denominator
of (42) is essential,
since it eliminates the singularity of an ill-defined entity $(E-\hat{\sf
H})^{-1}$.
$[\,$Recall that $\hat{\sf H}$ has a positive definite continuous spectrum,
$[\,$hence $(E-\hat{\sf H})^{-1}$ includes division by zero.$\,]$
The quantity

$\lim_{\varepsilon\to+0}\,\frac{1}{E-\hat{\sf
H}+i\varepsilon}\;=\;\frac{1}{E-\hat{\sf H}+i\,0_{+}}$ (43)

represents a well defined operator.
$[\,$Well definedness of the formula (43) cannot be established just by an
immediate inspection.
$[\,$A detailed self contained analysis of an entity (43) is supplemented in
Appendix IV.$\,]$
It is the so called  retarded Green operator corresponding to $\hat{\sf H}$.
Completely analogous manipulations can be performed also in the case of (41),
we leave them to the reader. One would encounter there the so called
advanced Green operator $(E-\hat{\sf H}-i\,0_{+})^{-1}$.
Stated shortly, it turns out that relations (39) and (41) can be redisplayed
in the final time independent form

${\color[rgb]{1,0,0}|\psi_{E\eta}^{\pm}\rangle\;=\;|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf
H}\pm i\,0_{+}}\;\hat{\sf V}\,|\phi_{E\eta}\rangle\hskip 14.22636pt.}$ (44)

This is the so called  Lippmann-Schwinger equation (LSE, explicit version).
Most importantly,
LSE represents the fundamental starting point of the time independent
scattering theory.

  * •

B.4 The Lippmann-Schwinger equation: Elaborations
Assume a finite $\varepsilon>0$ and define for convenience

$|\psi_{E\eta}^{\pm,\varepsilon}\rangle\;=\;|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf
H}\pm i\,\varepsilon}\;\hat{\sf V}\,|\phi_{E\eta}\rangle\hskip 14.22636pt;$
(45)

such that
$|\psi_{E\eta}^{\pm}\rangle=\lim_{\varepsilon\to+0}|\psi_{E\eta}^{\pm,\varepsilon}\rangle$.
Multiply (45) by $(E-\hat{\sf H}\pm i\,\varepsilon)$ to get

$(E-\hat{\sf H}\pm
i\,\varepsilon)\,|\psi_{E\eta}^{\pm,\varepsilon}\rangle\;=\;(E-\hat{\sf H}\pm
i\,\varepsilon)\,|\phi_{E\eta}\rangle\;+\;\hat{\sf
V}\,|\phi_{E\eta}\rangle\hskip 14.22636pt.$ (46)

Recalling (1) one may simplify the r.h.s. and write down an interesting
formula

$(E-\hat{\sf H}\pm
i\,\varepsilon)\,|\psi_{E\eta}^{\pm,\varepsilon}\rangle\;=\;(E-\hat{\sf
H}_{0}\pm i\,\varepsilon)\,|\phi_{E\eta}\rangle\hskip 14.22636pt.$ (47)

Equation (47) leads towards two important consequences:

Lecture #2

    * 1)

Firstly, one may take (47) and rearrange it into

$(E-\hat{\sf H}_{0}\pm
i\,\varepsilon)\,|\psi_{E\eta}^{\pm,\varepsilon}\rangle\;=\;(E-\hat{\sf
H}_{0}\pm i\,\varepsilon)\,|\phi_{E\eta}\rangle\;+\;\hat{\sf
V}\,|\psi_{E\eta}^{\pm,\varepsilon}\rangle\hskip 14.22636pt;$ (48)

and further into

$|\psi_{E\eta}^{\pm,\varepsilon}\rangle\;=\;|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf
H}_{0}\pm i\,\varepsilon}\,\hat{\sf
V}\,|\psi_{E\eta}^{\pm,\varepsilon}\rangle\hskip 14.22636pt.$ (49)

For $\varepsilon\to+0$ one obtains immediately the so called  implicit
Lippmann-Schwinger equation,

${\color[rgb]{1,0,0}|\psi_{E\eta}^{\pm}\rangle\;=\;|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf
H}_{0}\pm i\,0_{+}}\,\hat{\sf V}\,|\psi_{E\eta}^{\pm}\rangle\hskip
14.22636pt.}$ (50)

Note formal similarity between the explicit LSE (44) and the implicit LSE
(50).
The implicit LSE (50) contains the free Green operator corresponding to
$\hat{\sf H}_{0}$.
$[\,$See Appendix IV for more details regarding this free Green operator.$\,]$
Also, the r.h.s. of (50) contains $|\psi_{E\eta}^{\pm}\rangle$ rather than
$|\phi_{E\eta}\rangle$.
While (44) can be regarded as an explicit definition of
$|\psi_{E\eta}^{\pm}\rangle$,
equation (50) does determine $|\psi_{E\eta}^{\pm}\rangle$ only implicitly.
Yet the implicit LSE (50) possesses great theoretical importance, as we shall
see later on.
For now let us point out the following:
cra The explicit LSE (44) contains the full Green operator $(E-\hat{\sf H}\pm
i\,0_{+})^{-1}$.
cra It can be thus hard to handle (44) numerically when one wishes to
calculate $|\psi_{E\eta}^{\pm}\rangle$.
cra On the other hand, the implicit LSE (50)
cra contains just the free Green operator $(E-\hat{\sf H}_{0}\pm
i\,0_{+})^{-1}$,
cra which is much easier to handle as documented in Appendix IV.
cra Importantly, the solution of (50) can be performed in an iterative
fashion.
cra Such that one starts from (50) and writes

$|\psi_{E\eta}^{\pm}\rangle\;=\;|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf
H}_{0}\pm i\,0_{+}}\,\hat{\sf
V}\left(|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf H}_{0}\pm
i\,0_{+}}\,\hat{\sf V}\,|\psi_{E\eta}^{\pm}\rangle\right)\hskip 14.22636pt;$
(51)

cra giving

$\displaystyle\hskip 14.22636pt|\psi_{E\eta}^{\pm}\rangle$ $\displaystyle=$
$\displaystyle|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf H}_{0}\pm
i\,0_{+}}\,\hat{\sf V}\,|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf H}_{0}\pm
i\,0_{+}}\,\hat{\sf V}\,\frac{1}{E-\hat{\sf H}_{0}\pm i\,0_{+}}\,\hat{\sf
V}\,|\psi_{E\eta}^{\pm}\rangle\;.\hskip 7.11317pt\hskip 14.22636pt$ (52)

cra By proceeding forward in the same spirit one gets

$\displaystyle|\psi_{E\eta}^{\pm}\rangle$ $\displaystyle=$
$\displaystyle|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf H}_{0}\pm
i\,0_{+}}\,\hat{\sf V}\,|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf H}_{0}\pm
i\,0_{+}}\,\hat{\sf V}\,\frac{1}{E-\hat{\sf H}_{0}\pm i\,0_{+}}\,\hat{\sf
V}\,|\psi_{E\eta}^{\pm}\rangle\;=$ (53) $\displaystyle=$
$\displaystyle|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf H}_{0}\pm
i\,0_{+}}\,\hat{\sf V}\,|\phi_{E\eta}\rangle\;+\;\frac{1}{E-\hat{\sf H}_{0}\pm
i\,0_{+}}\,\hat{\sf V}\,\frac{1}{E-\hat{\sf H}_{0}\pm i\,0_{+}}\,\hat{\sf
V}\,|\phi_{E\eta}\rangle$ $\displaystyle+$ $\displaystyle\frac{1}{E-\hat{\sf
H}_{0}\pm i\,0_{+}}\,\hat{\sf V}\,\frac{1}{E-\hat{\sf H}_{0}\pm
i\,0_{+}}\,\hat{\sf V}\,\frac{1}{E-\hat{\sf H}_{0}\pm i\,0_{+}}\,\hat{\sf
V}\,|\psi_{E\eta}^{\pm}\rangle\hskip 14.22636pt;$

cra and so on. This is the so called Born series expansion for the LSE.
cra Note that successive terms of (53) contain successively higher powers of
$\hat{\sf V}$.
cra Born series corresponds thus to the perturbation theory in $\hat{\sf V}$.
cra The Born series approach is therefore most convenient
cra when one encounters scattering by weak potentials.

Lecture #2

    * 2)

Secondly, one may simply perform the limit of $\varepsilon\to+0$ in (47),
and take advantage of the property (30). This reveals an important insight of
having

${\color[rgb]{1,0,0}\hat{\sf
H}\,|\psi_{E\eta}^{\pm}\rangle\;=\;E\,|\psi_{E\eta}^{\pm}\rangle\hskip
14.22636pt.}$ (54)

Hence  $|\psi_{E\eta}^{\pm}\rangle$ is an eigenvector of the full Hamiltonian
$\hat{\sf H}$,
with the same eigenvalue $E$ as appropriate for the eigenvector
$|\phi_{E\eta}\rangle$ of $\hat{\sf H}_{0}$.
Vectors $|\psi_{E\eta}^{\pm}\rangle$, $|\phi_{E\eta}\rangle$ and the
eigenvalue $E$ are mutually interconnected
through the LSEs (44) and (50).

  * •

B.5 Stationary scattering states $\psi_{E\eta}^{\pm}(x)$
Let us define $\psi_{E\eta}^{\pm}(x)=\langle x|\psi_{E\eta}^{\pm}\rangle$, and
call:
$\psi_{E\eta}^{+}(x)$ = the retarded stationary scattering wavefunctions;
$\psi_{E\eta}^{-}(x)$ = the advanced stationary scattering wavefunctions.
Formula (54) shows that the entities $\psi_{E\eta}^{\pm}(x)$ solve the time
independent Schrödinger equation

${\color[rgb]{1,0,0}\left\\{\,-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,+\,V(x)\,\right\\}\psi_{E\eta}^{\pm}(x)\;=\;E\,\psi_{E\eta}^{\pm}(x)\hskip
14.22636pt.}$ (55)

As we know from elementary quantum mechanics,
each continuum energy level $E>0$ is twice degenerate.
Thus $\psi_{E\eta}^{\pm}(x)$ is not determined uniquely just by the
Schrödinger equation (55).
An unambiguous specification of $\psi_{E\eta}^{\pm}(x)$ requires two
additional pieces of information:
Appropriate boundary conditions and an adequate normalization factor.
These two additional characteristics of $\psi_{E\eta}^{\pm}(x)$ are found by
returning to the implicit LSE (50).
Namely, the position representation of equation (50) takes the form

$\psi_{E\eta}^{\pm}(x)\;=\;\phi_{E\eta}(x)\;+\;\int_{a}^{b}{\rm d}y\;\,\langle
x|\,\frac{1}{E-\hat{\sf H}_{0}\pm
i\,0_{+}}\,|y\rangle\;V(y)\;\psi_{E\eta}^{\pm}(y)\hskip 14.22636pt;$ (56)

where $\phi_{E\eta}(x)=(\ref{phi-E-eta-
def})=\sqrt{\frac{m}{2\,\pi\,\hbar^{2}K}}\;\,e^{+i\eta Kx}$ and $\hbar
K=\sqrt{2\,m\,E}$.
The free Green operator $(E-\hat{\sf H}_{0}\pm i\,0_{+})^{-1}$ entering into
(56)
has been analyzed in much detail within Appendix IV,
and the associated equations (276) and (289) supply an explicit result

$\langle x|\,\frac{1}{E-\hat{\sf H}_{0}\pm i\,0_{+}}\,|y\rangle\;=\;(\mp
i)\,\frac{m}{\hbar^{2}K}\;e^{\pm iK|x-y|}\hskip 14.22636pt.$ (57)

If so, then the implicit LSE (56) can be redisplayed as

$\psi_{E\eta}^{\pm}(x)\;=\;\sqrt{\frac{m}{2\,\pi\,\hbar^{2}K}}\;\,e^{+i\eta
Kx}\;\mp\;\frac{i\,m}{\hbar^{2}K}\,\int_{a}^{b}{\rm d}y\;\,e^{\pm
iK|x-y|}\;V(y)\;\psi_{E\eta}^{\pm}(y)\hskip 14.22636pt.$ (58)

Lecture #2

To simplify our subsequent notations, let us set conveniently

${\color[rgb]{1,0,0}\psi_{E\eta}^{\pm}(x)\;=\;\sqrt{\frac{m}{2\,\pi\,\hbar^{2}K}}\;\,\tilde{\psi}_{E\eta}^{\pm}(x)\hskip
14.22636pt;}$ (59)

this converts the LSE (58) into

$\tilde{\psi}_{E\eta}^{\pm}(x)\;=\;e^{+i\eta
Kx}\;\mp\;\frac{i\,m}{\hbar^{2}K}\,\int_{a}^{b}{\rm d}y\;\,e^{\pm
iK|x-y|}\;V(y)\;\tilde{\psi}_{E\eta}^{\pm}(y)\hskip 14.22636pt.$ (60)

Most importantly,
equation (60) enables us to understand now the qualitative behavior of
$\tilde{\psi}_{E\eta}^{\pm}(x)$
in the left asymptotic region of $x\leq a$ and in the right asymptotic region
of $x\geq b$.
Indeed, a brief inspection of formula (60) reveals that, inevitably,

$\displaystyle{\color[rgb]{1,0,0}\tilde{\psi}_{E(+1)}^{+}(x>b)}$
$\displaystyle{\color[rgb]{1,0,0}=}$ $\displaystyle{\color[rgb]{1,0,0}\hskip
50.3615ptT_{E(+1)}^{+}\;e^{+iKx}\hskip 14.22636pt;}$ (61)
$\displaystyle{\color[rgb]{1,0,0}\tilde{\psi}_{E(+1)}^{+}(x<a)}$
$\displaystyle{\color[rgb]{1,0,0}=}$
$\displaystyle{\color[rgb]{1,0,0}e^{+iKx}\;+\;R_{E(+1)}^{+}\;e^{-iKx}\hskip
14.22636pt;}$ (62)
$\displaystyle{\color[rgb]{1,0,0}\tilde{\psi}_{E(-1)}^{+}(x>b)}$
$\displaystyle{\color[rgb]{1,0,0}=}$
$\displaystyle{\color[rgb]{1,0,0}e^{-iKx}\;+\;R_{E(-1)}^{+}\;e^{+iKx}\hskip
14.22636pt;}$ (63)
$\displaystyle{\color[rgb]{1,0,0}\tilde{\psi}_{E(-1)}^{+}(x<a)}$
$\displaystyle{\color[rgb]{1,0,0}=}$ $\displaystyle{\color[rgb]{1,0,0}\hskip
50.3615ptT_{E(-1)}^{+}\;e^{-iKx}\hskip 14.22636pt.}$ (64)

Here the $T_{E(\pm 1)}^{+}$ and $R_{E(\pm 1)}^{+}$ are some as yet unknown
coefficients, given formally by prescriptions

$\displaystyle T_{E(+1)}^{+}$ $\displaystyle=$ $\displaystyle
1\;-\;\frac{i\,m}{\hbar^{2}K}\,\int_{a}^{b}{\rm
d}y\;\,e^{-iKy}\;V(y)\;\tilde{\psi}_{E(+1)}^{+}(y)\hskip 14.22636pt;$ (65)
$\displaystyle R_{E(+1)}^{+}$ $\displaystyle=$ $\displaystyle\hskip
11.38092pt-\;\frac{i\,m}{\hbar^{2}K}\,\int_{a}^{b}{\rm
d}y\;\,e^{+iKy}\;V(y)\;\tilde{\psi}_{E(+1)}^{+}(y)\hskip 14.22636pt;$ (66)

and analogously for $T_{E(-1)}^{+}$, $R_{E(-1)}^{+}$.
$[\,$Note that the r.h.s. of equations (65)-(66) contains
$\tilde{\psi}_{E\eta}^{\pm}(y)$,
$[\,$hence we cannot use (65)-(66) to directly access $T_{E(+1)}^{+}$ and
$R_{E(+1)}^{+}$ prior to knowing $\tilde{\psi}_{E\eta}^{\pm}(y)$.$\,]$
Similarly one finds that

$\displaystyle\tilde{\psi}_{E(+1)}^{-}(x>b)$ $\displaystyle=$ $\displaystyle
e^{+iKx}\;+\;R_{E(+1)}^{-}\;e^{-iKx}\hskip 14.22636pt;$ (67)
$\displaystyle\tilde{\psi}_{E(+1)}^{-}(x<a)$ $\displaystyle=$
$\displaystyle\hskip 50.3615ptT_{E(+1)}^{-}\;e^{+iKx}\hskip 14.22636pt;$ (68)
$\displaystyle\tilde{\psi}_{E(-1)}^{-}(x>b)$ $\displaystyle=$
$\displaystyle\hskip 50.3615ptT_{E(-1)}^{-}\;e^{-iKx}\hskip 14.22636pt;$ (69)
$\displaystyle\tilde{\psi}_{E(-1)}^{-}(x<a)$ $\displaystyle=$ $\displaystyle
e^{-iKx}\;+\;R_{E(-1)}^{-}\;e^{+iKx}\hskip 14.22636pt;$ (70)

where again the $T_{E(\pm 1)}^{-}$ and $R_{E(\pm 1)}^{-}$ are some as yet
unknown coefficients.
Equations (61)-(64) and (67)-(70) combined with formula (59) supply
appropriate boundary conditions and an adequate normalization factor
for the continuum wavefunctions $\psi_{E\eta}^{\pm}(x)$ solving the time
independent Schrödinger equation (55).
The stationary scattering states $\psi_{E\eta}^{\pm}(x)$ are thus uniquely
characterized now.

Lecture #2

Additional remarks to be taken into account:
Firstly, in our later elaborations we shall assign direct physical meaning to
$T_{E(\pm 1)}^{+}$ and $R_{E(\pm 1)}^{+}$,
see equations (108)-(109), (113)-(114) and the accompanying discussion.
Secondly, moment of reflection validates insightful properties

$\psi_{E(-1)}^{-}(x)\;=\;\left(\psi_{E(+1)}^{+}(x)\right)^{*}\hskip
14.22636pt,\hskip
14.22636pt\psi_{E(-1)}^{+}(x)\;=\;\left(\psi_{E(+1)}^{-}(x)\right)^{*}\hskip
14.22636pt.$ (71)

In other words, the advanced scattering wavefunctions $\psi_{E\eta}^{-}(x)$
are just complex conjugates of their retarded counterparts
$\psi_{E\eta}^{+}(x)$.
Thirdly, the retarded stationary scattering wavefunctions
$\psi_{E\eta}^{+}(x)$
are analyzed once again in a self contained fashion within Appendix II.
In particular, their specific boundary conditions (61)-(64) are rederived
$[\,$see equations (204)-(207)$\,]$,
two fundamental properties of the coefficients $T_{E(\pm 1)}^{+}$ and
$R_{E(\pm 1)}^{+}$ are demonstrated to hold
$[\,$see equations (208)-(209)$\,]$, and the corresponding
orthonormality & closure relations are established $[\,$see equations (227),
(229)$\,]$.
Also the associated Wronskian is examined,
this leads to an additional insight $[\,$see equations (231)-(232)$\,]$.
Fourthly, in Appendix V we describe an extremely simple numerical algorithm
for calculating the stationary scattering wavefunction $\psi_{E(+1)}^{+}(x)$.
This algorithm is based on solving the boundary value problem (55), (61)-(62)
on a discrete $x$-grid using an elementary finite difference method.
An analogous approach can of course be employed also in the case of
$\psi_{E(-1)}^{+}(x)$ and $\psi_{E(\pm 1)}^{-}(x)$.

Lecture #3

  * •

B.6 Determination of the scattering operator $\bm{\hat{}}{\bm{S}}$: Part #1
Let us return now to the scattering matrix element (21), $\langle\chi_{\rm
out}|\hat{S}|\phi_{\rm in}\rangle$.
The given prescribed in-wavepacket $|\phi_{\rm in}\rangle$ can be expanded as
done above in (33).
The associated $t=0$ physical state $|\psi\rangle=(\ref{psi-from-phi-in})$
takes then the form (38), with $|\psi_{E\eta}^{+}\rangle$ related to
$|\phi_{E\eta}\rangle$ via the LSE (44) or (50).
An arbitrarily chosen out-wavepacket $|\chi_{\rm out}\rangle$ can be expanded
similarly as above in (36), that is,

$|\chi_{\rm out}\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;b_{E\eta}\,|\phi_{E\eta}\rangle\hskip 14.22636pt,\hskip
14.22636ptb_{E\eta}\;=\;\langle\phi_{E\eta}|\chi_{\rm out}\rangle\hskip
14.22636pt.$ (72)

The associated $t=0$ physical state

$|\varphi\rangle\;=\;\lim_{t\to+\infty}\,\hat{\sf U}^{\dagger}(t)\,\hat{\sf
U}_{0}(t)\,|\chi_{\rm out}\rangle$ (73)

$[\,$cf. relation (14)$\,]$ takes then the form analogous to (40), namely,

$|\varphi\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;b_{E\eta}\,|\psi_{E\eta}^{-}\rangle\hskip 14.22636pt;$ (74)

with $|\psi_{E\eta}^{-}\rangle$ related to $|\phi_{E\eta}\rangle$ via the LSE
(44) or (50).
Our sought scattering matrix element can now be evaluated as follows:

$\displaystyle\langle\chi_{\rm out}|\hat{S}|\phi_{\rm in}\rangle$
$\displaystyle=$ $\displaystyle\lim_{T\to+\infty}\,\langle\chi_{\rm
out}|\,\hat{\sf U}_{0}^{\dagger}(+T)\;\hat{\sf U}(+T)\;\hat{\sf
U}^{\dagger}(-T)\;\hat{\sf U}_{0}(-T)\,|\phi_{\rm in}\rangle\;=$ (75)
$\displaystyle=$ $\displaystyle\lim_{T\to+\infty}\,\left(\langle\chi_{\rm
out}|\,\hat{\sf U}_{0}^{\dagger}(+T)\;\hat{\sf U}(+T)\right)\left(\hat{\sf
U}^{\dagger}(-T)\;\hat{\sf U}_{0}(-T)\,|\phi_{\rm
in}\rangle\right)\;=\;\langle\varphi|\psi\rangle\;=$ $\displaystyle=$
$\displaystyle\int_{0}^{+\infty}{\rm d}E\,\sum_{\eta}\,\int_{0}^{+\infty}{\rm
d}E^{\prime}\,\sum_{\eta^{\prime}}\;\,b_{E\eta}^{*}\;c_{E^{\prime}\eta^{\prime}}\;\langle\psi_{E\eta}^{-}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\hskip
14.22636pt.$

We have used along the way the definition (19) of $\hat{S}$ together with
relations (13), (73).
Now it is meaningful to introduce the overlap entities

${\color[rgb]{1,0,0}S_{(E\eta)(E^{\prime}\eta^{\prime})}\;=\;\langle\psi_{E\eta}^{-}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\hskip
14.22636pt.}$ (76)

Clearly, knowledge of $S_{(E\eta)(E^{\prime}\eta^{\prime})}$
enables us to evaluate $\langle\chi_{\rm out}|\hat{S}|\phi_{\rm in}\rangle$
for any chosen $|\phi_{\rm in}\rangle$, $|\chi_{\rm out}\rangle$ via (75).
Hence the fundamental scattering operator $\hat{S}$ is completely determined
by $S_{(E\eta)(E^{\prime}\eta^{\prime})}$.
In addition to the overlaps
$S_{(E\eta)(E^{\prime}\eta^{\prime})}=(\ref{S-matel-basic})$
we also conveniently introduce similarly looking entities

${\color[rgb]{1,0,0}Q_{(E\eta)(E^{\prime}\eta^{\prime})}^{\pm}\;=\;\langle\psi_{E\eta}^{\pm}|\psi_{E^{\prime}\eta^{\prime}}^{\pm}\rangle\hskip
14.22636pt.}$ (77)

Quantities (76)-(77) can be analyzed solely using the time independent
scattering formalism,
and especially using the LSE (44), (50).
Such an analysis is pursued in the next two paragraphs.

Lecture #3

  * •

B.7 Mathematical interlude #1
Let us evaluate explicitly the overlap entity
$Q_{(E\eta)(E^{\prime}\eta^{\prime})}^{+}=\langle\psi_{E\eta}^{+}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle$.
One begins with substituting for $\langle\psi_{E\eta}^{+}|$ from the explicit
LSE (44), to get

$\displaystyle Q_{(E\eta)(E^{\prime}\eta^{\prime})}^{+}$ $\displaystyle=$
$\displaystyle\langle\phi_{E\eta}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;+\;\langle\phi_{E\eta}|\,\hat{\sf
V}\,\frac{1}{E-\hat{\sf
H}-i\,\varepsilon}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;=$ (78)
$\displaystyle=$
$\displaystyle\langle\phi_{E\eta}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;+\;\langle\phi_{E\eta}|\,\hat{\sf
V}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;\frac{1}{E-E^{\prime}-i\,\varepsilon}\hskip
14.22636pt.$

We have exploited here the property (54), i.e., $\hat{\sf
H}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\,=\,E^{\prime}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle$.
The limit of $\varepsilon\to+0$ is of course implicitly understood in (78).
Our next step is to substitute for
$|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle$ from the implicit LSE (50)
while taking the same value of $\varepsilon\to+0$. This yields

$\displaystyle Q_{(E\eta)(E^{\prime}\eta^{\prime})}^{+}$ $\displaystyle=$
$\displaystyle\langle\phi_{E\eta}|\phi_{E^{\prime}\eta^{\prime}}\rangle\;+\;\langle\phi_{E\eta}|\,\frac{1}{E^{\prime}-\hat{\sf
H}_{0}+i\varepsilon}\,\hat{\sf
V}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;+\;\langle\phi_{E\eta}|\,\hat{\sf
V}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;\frac{1}{E-E^{\prime}-i\,\varepsilon}\;=$
(79) $\displaystyle=$
$\displaystyle\delta(E-E^{\prime})\;\delta_{\eta\eta^{\prime}}\;+\;\langle\phi_{E\eta}|\,\hat{\sf
V}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\,\left\\{\frac{1}{E^{\prime}-E+i\,\varepsilon}\,+\,\frac{1}{E-E^{\prime}-i\varepsilon}\right\\}\hskip
14.22636pt.$

We have exploited here the property (30), i.e., $\hat{\sf
H}_{0}\,|\phi_{E\eta}\rangle\,=\,E\,|\phi_{E\eta}\rangle$.
Also the orthonormality relations (31) have been incorporated.
Importantly, the $\\{\bm{\cdots}\\}$ term of (79) equals to zero, therefore
$Q_{(E\eta)(E^{\prime}\eta^{\prime})}^{+}=\delta(E-E^{\prime})\,\delta_{\eta\eta^{\prime}}$.
The same outcome would result also for
$Q_{(E\eta)(E^{\prime}\eta^{\prime})}^{-}$. Thus, altogether, we may conclude
that

${\color[rgb]{1,0,0}\langle\psi_{E\eta}^{\pm}|\psi_{E^{\prime}\eta^{\prime}}^{\pm}\rangle\;=\;\delta(E-E^{\prime})\;\delta_{\eta\eta^{\prime}}\hskip
14.22636pt.}$ (80)

The just derived formula (80) reveals that the solutions
$|\psi_{E\eta}^{\pm}\rangle$ of the LSE (44) or (50)
satisfy orthonormality relations analogous to those possessed by
$|\phi_{E\eta}\rangle$ and listed in (31).
The corresponding eigenvalue equations (30) and (54) are worth recalling in
this context.
In passing we note that retarded version of the orthonormality relations (80)
has been derived by elementary means also in Appendix II, see equation (227).

Lecture #3

  * •

B.8 Mathematical interlude #2
Next, let us evaluate explicitly the fundamental entity
$S_{(E\eta)(E^{\prime}\eta^{\prime})}=(\ref{S-matel-basic})$ relevant for
scattering.
We shall proceed analogously as in the previous paragraph. Namely,
one begins with substituting for $\langle\psi_{E\eta}^{-}|$ from the explicit
LSE (44), to get

$\displaystyle S_{(E\eta)(E^{\prime}\eta^{\prime})}$ $\displaystyle=$
$\displaystyle\langle\phi_{E\eta}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;+\;\langle\phi_{E\eta}|\,\hat{\sf
V}\,\frac{1}{E-\hat{\sf
H}+i\,\varepsilon}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;=$ (81)
$\displaystyle=$
$\displaystyle\langle\phi_{E\eta}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;+\;\langle\phi_{E\eta}|\,\hat{\sf
V}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;\frac{1}{E-E^{\prime}+i\,\varepsilon}\hskip
14.22636pt.$

We have exploited here the property (54), i.e., $\hat{\sf
H}\,|\psi_{E^{\prime}\eta^{\prime}}^{-}\rangle\,=\,E^{\prime}\,|\psi_{E^{\prime}\eta^{\prime}}^{-}\rangle$.
The limit of $\varepsilon\to+0$ is of course implicitly understood in (81).
Our next step is to substitute for
$|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle$ from the implicit LSE (50)
while taking the same value of $\varepsilon\to+0$. This yields

$\displaystyle S_{(E\eta)(E^{\prime}\eta^{\prime})}$ $\displaystyle=$
$\displaystyle\langle\phi_{E\eta}|\phi_{E^{\prime}\eta^{\prime}}\rangle\;+\;\langle\phi_{E\eta}|\,\frac{1}{E^{\prime}-\hat{\sf
H}_{0}+i\varepsilon}\,\hat{\sf
V}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;+\;\langle\phi_{E\eta}|\,\hat{\sf
V}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;\frac{1}{E-E^{\prime}+i\,\varepsilon}\;=$
(82) $\displaystyle=$
$\displaystyle\delta(E-E^{\prime})\;\delta_{\eta\eta^{\prime}}\;+\;\langle\phi_{E\eta}|\,\hat{\sf
V}\,|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\,\left\\{\frac{1}{E^{\prime}-E+i\,\varepsilon}\,+\,\frac{1}{E-E^{\prime}+i\varepsilon}\right\\}\hskip
14.22636pt.$

We have exploited here the property (30), i.e., $\hat{\sf
H}_{0}\,|\phi_{E\eta}\rangle\,=\,E\,|\phi_{E\eta}\rangle$.
Also the orthonormality relations (31) have been incorporated.
Importantly, the $\\{\bm{\cdots}\\}$ term of (82) does not vanish, but equals
to

$\\{\bm{\cdots}\\}\;=\;-\,2\,i\,\pi\;\,\frac{1}{\pi}\,\frac{\varepsilon}{(E-E^{\prime})^{2}+\varepsilon^{2}}\;=\;-\,2\,i\,\pi\;\,\delta_{\varepsilon}(E-E^{\prime})\hskip
14.22636pt;$ (83)

with the Lorentzian profile function

$\delta_{\varepsilon}(E-E^{\prime})\;=\;\frac{1}{\pi}\,\frac{\varepsilon}{(E-E^{\prime})^{2}+\varepsilon^{2}}$
(84)

reducing to $\delta(E-E^{\prime})$ in the limit of $\varepsilon\to+0$. If so,
then

$\displaystyle{\color[rgb]{1,0,0}S_{(E\eta)(E^{\prime}\eta^{\prime})}}$
$\displaystyle{\color[rgb]{1,0,0}=}$
$\displaystyle{\color[rgb]{1,0,0}\delta(E-E^{\prime})\;\delta_{\eta\eta^{\prime}}\;-\;2\,i\,\pi\;\,\delta(E-E^{\prime})\;T_{(E\eta)(E\eta^{\prime})}\hskip
14.22636pt;}$ (85)

where by definition

$\displaystyle{\color[rgb]{1,0,0}T_{(E\eta)(E\eta^{\prime})}}$
$\displaystyle{\color[rgb]{1,0,0}=}$
$\displaystyle{\color[rgb]{1,0,0}\langle\phi_{E\eta}|\,\hat{\sf
V}\,|\psi_{E\eta^{\prime}}^{+}\rangle\;=}$ (86)
$\displaystyle{\color[rgb]{1,0,0}=}$
$\displaystyle{\color[rgb]{1,0,0}\langle\phi_{E\eta}|\,\hat{\sf
V}\,+\,\hat{\sf V}\,\frac{1}{E-\hat{\sf H}+i\,0_{+}}\,\hat{\sf
V}\,|\phi_{E\eta^{\prime}}\rangle\hskip 14.22636pt.}$ (87)

In (87) we have again substituted for $|\psi_{E\eta^{\prime}}^{+}\rangle$ from
the explicit LSE (44).

Lecture #3

  * •

B.9 Determination of the scattering operator $\bm{\hat{}}{\bm{S}}$: Part #2
We have already shown that the fundamental scattering operator $\hat{S}$
is completely determined by the matrix elements
$S_{(E\eta)(E^{\prime}\eta^{\prime})}=(\ref{S-matel-basic})=(\ref{S-matel-
take-3})$.
The r.h.s. of equation (85) is proportional to $\delta(E-E^{\prime})$.
This so called on-shell energy factor
indicates that the energy $E$ is conserved in scattering processes.
Quantity (87) represents the so called on-shell $T$-matrix element.
It can be conveniently viewed as

${\color[rgb]{1,0,0}T_{(E\eta)(E\eta^{\prime})}\;=\;\langle\phi_{E\eta}|\,\hat{T}(E)\,|\phi_{E\eta^{\prime}}\rangle\hskip
14.22636pt;}$ (88)

where by definition

${\color[rgb]{1,0,0}\hat{T}(E)\;=\;\hat{\sf V}\;+\;\hat{\sf
V}\,\frac{1}{E-\hat{\sf H}+i\,0_{+}}\,\hat{\sf V}\hskip 14.22636pt.}$ (89)

Prescription (89) introduces the so called  Transition Operator or
$\hat{T}$-operator
which depends parameterically upon the impact energy $E$.
Practical evaluation of $T_{(E\eta)(E\eta^{\prime})}$ for a concrete physical
problem
can be based upon two main strategies:

    * i)

Numerical construction of the retarded stationary scattering states
$|\psi_{E\eta}^{+}\rangle$,
followed by an application of the formula (86).
Recall that $\psi_{E(+1)}^{+}(x)=\langle x|\psi_{E(+1)}^{+}\rangle$ represents
an unique solution of the boundary value problem (55), (61)-(62).
Numerical treatment of the just mentioned boundary value problem is
straightforward,
the corresponding algorithm is described in a self contained fashion in
Appendix V.
Treatment of $\psi_{E(-1)}^{+}(x)=\langle x|\psi_{E(-1)}^{+}\rangle$ is
completely analogous.
The just presented route i) will be pursued forward in our forthcoming
elaborations.

    * ii)

An analytic (perturbative) calculation of $|\psi_{E\eta}^{\pm}\rangle$ based
upon the Born series expansion (53),
again followed by an application of the formula (86). As noted already,
the Born series approach is most convenient/justified
when one encounters scattering by weak potentials.
In fact, there exists even a quicker route leading towards the Born expansion
of $\hat{T}(E)=(\ref{hat-T-E-def})$,
which is based upon the Born series expansion of the retarded Green operator
$(E-\hat{\sf H}+i\,0_{+})^{-1}$.
Such an expansion is derived in Appendix IV, see equation (•).
Combination of (•) and (89) yields the desired perturbative Born series
expansion for $\hat{T}(E)$:

$\displaystyle\hat{T}(E)$ $\displaystyle=$ $\displaystyle\hat{\sf
V}\;+\;\hat{\sf V}\,\frac{1}{E-\hat{\sf H}_{0}+i\,0_{+}}\,\hat{\sf
V}\;+\;\hat{\sf V}\,\frac{1}{E-\hat{\sf H}_{0}+i\,0_{+}}\,\hat{\sf
V}\,\frac{1}{E-\hat{\sf H}_{0}+i\,0_{+}}\,\hat{\sf V}\;+\;\bm{\ldots}\hskip
14.22636pt.$ (90)

The just presented route ii) possesses its important merits,
but we shall not pursue forward this direction of thought in our forthcoming
elaborations.

Lecture #3

  * •

B.10 Determination of the scattering operator $\bm{\hat{}}{\bm{S}}$: Part #3
Let us return to our discussion of $\hat{S}$ which began in subsection B.6.
Suppose we aim at determining $|\phi_{\rm out}\rangle$ for a given prescribed
$|\phi_{\rm in}\rangle$.
One may take $|\phi_{\rm in}\rangle$ and proceed via (33), (44) or (50), (38)
to get the associated $|\psi\rangle$.
One finds that

$|\psi\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;|\psi_{E\eta}^{+}\rangle\,\langle\phi_{E\eta}|\phi_{\rm
in}\rangle\hskip 14.22636pt.$ (91)

Subsequently we wish to apply (40), (44) or (50), (36) and calculate the
sought $|\phi_{\rm out}\rangle$.
Equations relevant for this purpose take the following appearance:

$|\psi\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;|\psi_{E\eta}^{-}\rangle\,\langle\phi_{E\eta}|\phi_{\rm
out}\rangle\hskip 14.22636pt;$ (92) $|\phi_{\rm
out}\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;|\phi_{E\eta}\rangle\,\langle\phi_{E\eta}|\phi_{\rm
out}\rangle\hskip 14.22636pt.$ (93)

Formula (93) requires knowledge of an overlap $\langle\phi_{E\eta}|\phi_{\rm
out}\rangle$.
Yet the same overlap appears also in (92), and can be thus obtained from
$|\psi\rangle$
by exploiting the orthonormality property (80) in combination with (91).
Indeed, one gets

$\langle\phi_{E\eta}|\phi_{\rm
out}\rangle\;=\;\langle\psi_{E\eta}^{-}|\psi\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E^{\prime}\,\sum_{\eta^{\prime}}\;\,\langle\psi_{E\eta}^{-}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\,\langle\phi_{E^{\prime}\eta^{\prime}}|\phi_{\rm
in}\rangle\hskip 14.22636pt.$ (94)

Returning back to (93), we have explicitly

$|\phi_{\rm out}\rangle\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\,\int_{0}^{+\infty}{\rm
d}E^{\prime}\,\sum_{\eta^{\prime}}\;\,|\phi_{E\eta}\rangle\,\langle\psi_{E\eta}^{-}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\,\langle\phi_{E^{\prime}\eta^{\prime}}|\phi_{\rm
in}\rangle\hskip 14.22636pt.$ (95)

The just derived equation (95) enables us
to determine the sought outgoing asymptotic state $|\phi_{\rm out}\rangle$
for any given incoming asymptotic state $|\phi_{\rm in}\rangle$,
provided only that the overlap entity
$\langle\psi_{E\eta}^{-}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle=(\ref{S-matel-
basic})=(\ref{S-matel-take-3})$ has been evaluated.
Thus the map $|\phi_{\rm in}\rangle\mapsto|\phi_{\rm out}\rangle$ is
explicitly constructed now,
solely within the time independent scattering formalism based upon LSE.
Moreover, a comparison between our explicit result (95) and the formal
definition (18) of $\hat{S}$
implies inevitably

${\color[rgb]{1,0,0}\hat{S}\;=\;\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\,\int_{0}^{+\infty}{\rm
d}E^{\prime}\,\sum_{\eta^{\prime}}\;\,|\phi_{E\eta}\rangle\,\langle\psi_{E\eta}^{-}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\,\langle\phi_{E^{\prime}\eta^{\prime}}|\hskip
14.22636pt;}$ (96)

or, in other words,

${\color[rgb]{1,0,0}\langle\phi_{E\eta}|\,\hat{S}\,|\phi_{E^{\prime}\eta^{\prime}}\rangle\;=\;\langle\psi_{E\eta}^{-}|\psi_{E^{\prime}\eta^{\prime}}^{+}\rangle\;=\;S_{(E\eta)(E^{\prime}\eta^{\prime})}\;=\;(\ref{S-matel-
take-3})\hskip 14.22636pt.}$ (97)

Meaning that  the fundamental scattering operator $\hat{S}$ is explicitly and
simply determined
by solutions $|\psi_{E\eta}^{\pm}\rangle$ of the LSE (44) or (50).

Lecture #3

Let us look again at the scattering matrix elements (75).
Equations (97) and (85) enable us to simplify (75) into the final form

$\displaystyle\langle\chi_{\rm out}|\hat{S}|\phi_{\rm in}\rangle$
$\displaystyle=$ $\displaystyle\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\;b_{E\eta}^{*}\;c_{E\eta}\;-\;2\,i\,\pi\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\,\sum_{\eta^{\prime}}\;\,b_{E\eta}^{*}\;c_{E\eta^{\prime}}\;T_{(E\eta)(E\eta^{\prime})}\;=$
(98) $\displaystyle=$ $\displaystyle\langle\chi_{\rm out}|\phi_{\rm
in}\rangle\;-\;2\,i\,\pi\int_{0}^{+\infty}{\rm
d}E\,\sum_{\eta}\,\sum_{\eta^{\prime}}\;\,b_{E\eta}^{*}\;c_{E\eta^{\prime}}\;T_{(E\eta)(E\eta^{\prime})}\hskip
14.22636pt.$

Here we recall that $c_{E\eta}=\langle\phi_{E\eta}|\phi_{\rm in}\rangle$ and
$b_{E\eta}=\langle\phi_{E\eta}|\chi_{\rm out}\rangle$ as stated in (33) and
(72).

Lecture #4

  * •

B.11 Transmission and reflection phenomena: Take #1
Our formal treatment of the scattering operator $\hat{S}$ has been finalized.
Let us explore now in more detail what does $\hat{S}$ actually tell us about
scattering.
Specifically, we shall focus in the two subsequent paragraphs
on  deriving simple expressions for
the probability of transmission/reflection of our quantum particle
through/from the interaction region $(a,b)$ of the nonvanishing potential
$V(x)$.
We start by putting together formulas (85), (95), (97). This yields

$\displaystyle|\phi_{\rm out}\rangle$ $\displaystyle=$
$\displaystyle|\phi_{\rm in}\rangle\;-\;2\,i\,\pi\int_{0}^{\infty}{\rm
d}E\;\sum_{\eta}\sum_{\eta^{\prime}}\;|\phi_{E\eta}\rangle\;\,T_{(E\eta)(E\eta^{\prime})}\;\langle\phi_{E\eta^{\prime}}|\phi_{\rm
in}\rangle\hskip 14.22636pt.$ (99)

The associated on-shell $T$-matrix element $T_{(E\eta)(E\eta^{\prime})}$ is
expressible as a spatial integral
using equations (86) and (32). One has

$T_{(E\eta)(E\eta^{\prime})}\;=\;\sqrt{\frac{m}{2\,\pi\,\hbar^{2}K}}\,\int_{-\infty}^{+\infty}{\rm
d}x\;\,e^{+i\eta Kx}\;V(x)\;\psi_{E\eta^{\prime}}^{+}(x)\hskip 14.22636pt.$
(100)

Hereafter we shall assume
that the chosen incoming wavepacket $|\phi_{\rm in}\rangle$ moves from the
left towards the right,
and is thus populated only by positive momenta. Hence

$\langle\phi_{E(-1)}|\phi_{\rm in}\rangle\;=\;0\hskip 14.22636pt;$ (101)

valid for all $E$. Formula (99) is then reduced into

$\displaystyle|\phi_{\rm out}\rangle$ $\displaystyle=$
$\displaystyle|\phi_{\rm in}\rangle\;-\;2\,i\,\pi\int_{0}^{\infty}{\rm
d}E\;\sum_{\eta}\;\,|\phi_{E\eta}\rangle\;\,T_{(E\eta)(E(+1))}\;\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle\;=$ $\displaystyle=$ $\displaystyle|\phi_{\rm
in}\rangle\;-\;2\,i\,\pi\int_{0}^{\infty}{\rm
d}E\;\,|\phi_{E(+1)}\rangle\;\,T_{(E(+1))(E(+1))}\;\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle$ $\displaystyle\phantom{|\phi_{\rm
in}\rangle}\;-\;2\,i\,\pi\int_{0}^{\infty}{\rm
d}E\;\,|\phi_{E(-1)}\rangle\;\,T_{(E(-1))(E(+1))}\;\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle\hskip 14.22636pt.$

Most importantly, the two $T$-matrix elements entering into (•)
are expressible solely in terms of the coefficients $T_{E(+1)}^{+}$ and
$R_{E(+1)}^{+}$
which we have encountered before when dealing with the boundary conditions for
$\psi_{E(+1)}^{+}(x)$,
see equations (61), (62) above. Namely,
after combining formula (100) with (59), (65)-(66) we find that

$\displaystyle T_{(E(+1))(E(+1))}$ $\displaystyle=$
$\displaystyle\frac{i}{2\,\pi}\;\Bigl{(}\,T_{E(+1)}^{+}-1\,\Bigr{)}\hskip
14.22636pt;$ (103) $\displaystyle T_{(E(-1))(E(+1))}$ $\displaystyle=$
$\displaystyle\frac{i}{2\,\pi}\;R_{E(+1)}^{+}\hskip 50.64586pt.$ (104)

Lecture #4

An insertion of (103)-(104) inside (•) provides in turn a compact outcome

$\displaystyle|\phi_{\rm out}\rangle$ $\displaystyle=$
$\displaystyle|\phi_{\rm in}\rangle\;-\;\int_{0}^{\infty}{\rm
d}E\;\,|\phi_{E(+1)}\rangle\langle\phi_{E(+1)}|\phi_{\rm in}\rangle$
$\displaystyle+$ $\displaystyle\int_{0}^{\infty}{\rm
d}E\;\,|\phi_{E(+1)}\rangle\;\,T_{E(+1)}^{+}\;\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle$ $\displaystyle+$ $\displaystyle\int_{0}^{\infty}{\rm
d}E\;\,|\phi_{E(-1)}\rangle\;\,R_{E(+1)}^{+}\;\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle\hskip 14.22636pt.$

The first line on the r.h.s. of (•) actually vanishes,
as one may verify immediately by recalling (101) and the closure property
(31).
Our outgoing asymptotic state $|\phi_{\rm out}\rangle$ possesses thus a simply
looking final appearance

$\displaystyle|\phi_{\rm out}\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\rm
d}E\;\,|\phi_{E(+1)}\rangle\;\,T_{E(+1)}^{+}\;\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle$ $\displaystyle+$ $\displaystyle\int_{0}^{\infty}{\rm
d}E\;\,|\phi_{E(-1)}\rangle\;\,R_{E(+1)}^{+}\;\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle\hskip 14.22636pt.$

Instead of (•) one can write even more insightfully

${\color[rgb]{1,0,0}|\phi_{\rm out}\rangle\;=\;|\phi_{\rm out}^{\rm
T}\rangle\;+\;|\phi_{\rm out}^{\rm R}\rangle\hskip 14.22636pt;}$ (107)

where by definition

$\displaystyle{\color[rgb]{1,0,0}|\phi_{\rm out}^{\rm T}\rangle}$
$\displaystyle{\color[rgb]{1,0,0}=}$
$\displaystyle{\color[rgb]{1,0,0}\int_{0}^{\infty}{\rm
d}E\;\,|\phi_{E(+1)}\rangle\;\,T_{E(+1)}^{+}\;\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle\hskip 14.22636pt;}$ (108)
$\displaystyle{\color[rgb]{1,0,0}|\phi_{\rm out}^{\rm R}\rangle}$
$\displaystyle{\color[rgb]{1,0,0}=}$
$\displaystyle{\color[rgb]{1,0,0}\int_{0}^{\infty}{\rm
d}E\;\,|\phi_{E(-1)}\rangle\;\,R_{E(+1)}^{+}\;\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle\hskip 14.22636pt.}$ (109)

Clearly, $|\phi_{\rm out}^{\rm T}\rangle$ populates only positive momenta
(much as $|\phi_{\rm in}\rangle$ does),
we recognize thus $|\phi_{\rm out}^{\rm T}\rangle$ as the outgoing transmitted
wavepacket.
On the other hand, $|\phi_{\rm out}^{\rm R}\rangle$ populates only negative
momenta (as opposed to $|\phi_{\rm in}\rangle$),
we recognize thus $|\phi_{\rm out}^{\rm R}\rangle$ as the outgoing reflected
wavepacket.
Our just presented findings justify to name
an entity $T_{E(+1)}^{+}$ as the transmission coefficient, and
an entity $R_{E(+1)}^{+}$ as the reflection coefficient.
Recall that numerical calculation of $T_{E(+1)}^{+}$ and $R_{E(+1)}^{+}$ for a
given potential $V(x)$
is very straightforward. Indeed, for any prescribed energy $E>0$
one solves the boundary value problem (55), (61)-(62) for
$\psi_{E(+1)}^{+}(x)$,
using an algorithm which is described in a self contained fashion in Appendix
V.
Before proceeding further,
let us point out that results equivalent to (107), (108)-(109)
have been derived in Appendix III using an alternative and perhaps more
elementary approach
(based upon the stationary phase method). $[\,$See especially equations (252)
and (261).$\,]$

Lecture #4

  * •

B.12 Transmission and reflection phenomena: Take #2
Quantum mechanical norm conservation
$[\,$which is encoded into unitarity of $\hat{S}$ in the scattering sector of
our quantum state space$\,]$
guarantees that $\langle\phi_{\rm out}|\phi_{\rm out}\rangle=\langle\phi_{\rm
in}|\phi_{\rm in}\rangle=1$. Yet formulas (107), (108)-(109) imply

$1\;=\;\langle\phi_{\rm out}|\phi_{\rm out}\rangle\;=\;\langle\phi_{\rm
out}^{\rm T}|\phi_{\rm out}^{\rm T}\rangle\;+\;\langle\phi_{\rm out}^{\rm
R}|\phi_{\rm out}^{\rm R}\rangle\hskip 14.22636pt;$ (110)

with

$\displaystyle\langle\phi_{\rm out}^{\rm T}|\phi_{\rm out}^{\rm T}\rangle$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}{\rm
d}E\;\,\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}\,\Bigl{|}\,\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle\Bigr{|}^{2}\hskip 14.22636pt;$ (111) $\displaystyle\langle\phi_{\rm
out}^{\rm R}|\phi_{\rm out}^{\rm R}\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\rm
d}E\;\,\Bigl{|}\,R_{E(+1)}^{+}\,\Bigr{|}^{2}\,\Bigl{|}\,\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle\Bigr{|}^{2}\hskip 14.22636pt.$ (112)

Equation (110) assigns an important physical interpretation
to the self overlaps $\langle\phi_{\rm out}^{\rm T}|\phi_{\rm out}^{\rm
T}\rangle$ and $\langle\phi_{\rm out}^{\rm R}|\phi_{\rm out}^{\rm R}\rangle$:
Quantity $\langle\phi_{\rm out}^{\rm T}|\phi_{\rm out}^{\rm
T}\rangle=(\ref{T-probability})$ possesses the meaning of the overall
transmission probability, whereas
quantity $\;\langle\phi_{\rm out}^{\rm R}|\phi_{\rm out}^{\rm
R}\rangle=(\ref{R-probability})$ possesses the meaning of the overall
reflection probability.
An additional simplification becomes possible when the incoming wavepacket
$|\phi_{\rm in}\rangle$
populates only a narrow interval of energies, say $E_{0}-\Delta_{E}/2\leq
E\leq E_{0}+\Delta_{E}/2$, with $\Delta_{E}\to 0$.
Both quantities $T_{E(+1)}^{+}$ and $R_{E(+1)}^{+}$ are continuous functions
of $E$,
and remain thus almost constant inside the mentioned energy interval for small
enough $\Delta_{E}$.
The transmission/reflection probabilities (111)-(112) then boil down into

$\displaystyle\langle\phi_{\rm out}^{\rm T}|\phi_{\rm out}^{\rm T}\rangle$
$\displaystyle=$
$\displaystyle\Bigl{|}\,T_{E_{0}(+1)}^{+}\,\Bigr{|}^{2}\,\int_{0}^{\infty}{\rm
d}E\;\,\Bigl{|}\,\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle\Bigr{|}^{2}\;=\;\Bigl{|}\,T_{E_{0}(+1)}^{+}\,\Bigr{|}^{2}\hskip
14.22636pt;$ (113) $\displaystyle\langle\phi_{\rm out}^{\rm R}|\phi_{\rm
out}^{\rm R}\rangle$ $\displaystyle=$
$\displaystyle\Bigl{|}\,R_{E_{0}(+1)}^{+}\,\Bigr{|}^{2}\,\int_{0}^{\infty}{\rm
d}E\;\,\Bigl{|}\,\langle\phi_{E(+1)}|\phi_{\rm
in}\rangle\Bigr{|}^{2}\,=\;\Bigl{|}\,R_{E_{0}(+1)}^{+}\,\Bigr{|}^{2}\hskip
14.22636pt.$ (114)

We have just proven that entities $\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}$ and
$\Bigl{|}\,R_{E(+1)}^{+}\,\Bigr{|}^{2}$
are interpreted physically as the transmission/reflection probabilities
corresponding to the given impact energy $E$.
Probability conservation is granted via equation (110), which ensures having

${\color[rgb]{1,0,0}\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}\;+\;\Bigl{|}\,R_{E(+1)}^{+}\,\Bigr{|}^{2}\;=\;1\hskip
14.22636pt;}$ (115)

valid for all $E$.
Note that property (115) has been proven also in Appendix II
by alternative and more elementary means, see equation (208).
Note also that Appendix III contains results equivalent to (111)-(112) and
(113)-(114)
which are derived using an alternative and perhaps more elementary approach
(stationary phase method). $[\,$See especially equations (253) and (262).$\,]$
Not only amplitudes but also phases of $T_{E(+1)}^{+}$, $R_{E(+1)}^{+}$ are
physically significant.
Indeed, it can be shown that the values of ${\rm arg}\,T_{E(+1)}^{+}$ and
${\rm arg}\,R_{E(+1)}^{+}$ are related
to the time which the transmitted/reflected particle spends inside the
interaction region $(a,b)$.
$[\,$We shall not delve deeper into these matters within the present
course.$\,]$
All what we said above about $T_{E(+1)}^{+}$ and $R_{E(+1)}^{+}$ applies of
course also to entities $T_{E(-1)}^{+}$, $R_{E(-1)}^{+}$
$[\,$These obviously describe a situation when $|\phi_{\rm in}\rangle$
populates only negative momenta.$\,]$

Lecture #4

  * •

B.13 Transmission and reflection phenomena: An illustrative numerical example
All our theoretical elaborations pursued so far can be now conveniently
illustrated
by a straightforward numerical calculation, performed for a simple model
problem.
Namely, we shall take $m=1$, $\hbar=1$, and consider a potential function of
the form

$V(x)\;=\;\left(0.5\,x^{2}-0.8\right)\,e^{-0.1\,x^{2}}\hskip 14.22636pt;$
(116)

see Ref. [3]. The profile of $V(x)$ is plotted below:

Figure 1:  Our model potential $V(x)=(\ref{V-Jolanta})$.

One can see that $V(x)$ is a symmetric potential consisting of a single well
and two barriers.
A simple numerical algorithm of Appendix V can now be employed
to calculate the corresponding energy dependent transmission probability
$\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}$.
The resulting outcome is depicted below:

Figure 2:  The calculated transmission probability
$\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}$.

Note in passing that $T_{E(+1)}^{+}=T_{E(-1)}^{+}=T_{E}^{+}$ as stated by
equation (231) of Appendix II.

Lecture #4

Fig.2 shows that the obtained profile of
$\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}$ exhibits the following behavior:

    * i)

For $E\to 0$ one encounters $\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}\to 0$. This
agrees with physics intuition.
$[\,$Low energy particles approaching our potential are always reflected
back.$\,]$

    * ii)

For $E\to+\infty$ one encounters $\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}\to 1$.
This agrees with physics intuition, too.
$[\,$High energy particles approaching our potential are always
transmitted.$\,]$

    * iii)

Most importantly, the profile of $\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}$ is
not smooth, but contains sharp peaks.
This phenomenon seems counter-intuitive and hardly explainable
using the theory presented up to now in the present course.
Indeed, one may wonder why there is an almost unit transmission for $E\doteq
0.621$,
but almost zero transmission at $E=0.620$ or $E=0.622$.
Similarly for the peak observed at $E\doteq 1.327$ and eventually for the
higher peaks.

The behavior of $\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}$ described in iii) is
an example of a  resonance phenomenon.
An adequate theoretical explanation of the just touched resonance phenomena
is  outside the scope of the hermitian scattering theory.
Desire of physics theoreticians to deeper understand the resonance phenomena
has motivated an intense research activity, leading to the development of the
so called
nonhermitian scattering theory or  nonhermitian quantum mechanics.
This nonhermitian theory
still remains an active field of research which continues to flourish even
nowadays.
One elegant version of the nonhermitian scattering theory
is developed in the subsequent Part C of our course.

Lecture #4

  * •

B.14 Final summary of the conventional scattering theory in 1D
Our introduction to the standard or orthodox scattering theory in 1D has been
brought to an end.
In Part A we have looked at scattering from the time dependent perspective.
The AC was formulated, leading to entities $|\psi\rangle$, $|\phi_{\rm
in}\rangle$, $|\phi_{\rm out}\rangle$ and ultimately to $\hat{S}$.
We discussed physical meaning of the scattering matrix elements.
Part B was motivated by our desire to switch from the time domain into the
energy domain.
In order to reach such a fundamental goal, we needed to proceed in multiple
steps.
Stated briefly, a more explicit interconnection between $|\psi\rangle$ and
$|\phi_{\rm in}\rangle$ or $|\phi_{\rm out}\rangle$ has been established,
through abstractly looking formulas (28) and (29). Subsequently, formulas (28)
and (29)
were replaced by equivalent yet time independent relations $[\,$(33), (36),
(38), (40), (44), (50)$\,]$.
The LSE (44) and (50) have led to the concept of stationary scattering states
$|\psi_{E(\pm 1)}^{\pm}\rangle$,
which were uniquely characterized by basic equations $[\,$(54), (55), (59),
(61)-(70)$\,]$,
and which were shown to possess the orthonormality property (80).
Proceeding further on, by taking an advantage of the LSE (44) and (50),
we were able to explicitly view the scattering operator $\hat{S}$ (and the
closely related operator $\hat{T}$)
solely in terms of the time independent entities $|\psi_{E(\pm
1)}^{\pm}\rangle$.
See the most important relations $[\,$(85), (86), (87), (88), (89), (96),
(97)$\,]$.
We recall also formula (98) enabling us to calculate the scattering matrix
elements (21),
and the associated (experimentally measurable) transition probabilities (22).
In the next step, we have illuminated the physical contents of $\hat{S}$ and
$\hat{T}$
by studying transmission/reflection of our quantum particle
through/from the interaction region of the potential.
Formulas (108)-(109) lead us towards the concept of the
transmission/reflection coefficient,
while in equations (113)-(114) we have determined
the corresponding transmission/reflection probabilities.
Finally, an illustrative numerical example was supplied,
which has also highlighted an existence of scattering resonances.
Thus, all in all, the general time independent scattering theory in 1D has
been built up,
and Part B can be closed. We hope that Part B has convinced the readers that
the time independent approach to scattering is not only advantageous
computationally,
but leads also to deep physics insights.

Lecture #5

C. Nonhermitian Scattering Theory

Based upon Ref. [4].

  * •

C.1 Motivational considerations
In Part A we have introduced the basic theoretical setup of quantum scattering
theory,
via looking at scattering phenomena from the time dependent perspective.
The AC was formulated, leading to entities $|\psi\rangle$, $|\phi_{\rm
in}\rangle$, $|\phi_{\rm out}\rangle$ and ultimately to $\hat{S}$.
Operator $\hat{S}$ was shown to contain all the information about an overall
scattering outcome.
We discussed physical meaning of the matrix elements of $\hat{S}$.
In Part B we have developed the conventional or orthodox scattering theory,
based upon switching from the time domain (wavepackets)
into the energy domain (continuum eigenstates of the total Hamiltonian
$\hat{\sf H}$).
Central ingredient of this approach was the LSE (44) and (50).
The just mentioned conventional or orthodox formulation is undoubtedly valid,
exact, general,
and used by vast majority of researchers working in scattering theory.
But, remember,  the whole energy domain formalism of Part B
(time independent scattering theory, LSE)
represents merely a tool for constructing $\hat{S}$ and for understanding its
properties.
In physics we are ultimately interested in the mapping $|\psi_{\rm
out}\rangle\,=\,\hat{S}\,|\phi_{\rm in}\rangle$ of equation (18),
not in the stationary eigenstates of $\hat{\sf H}$.
By the way,  the continuum eigenstates of $\hat{\sf H}$ are not square
integrable,
so they do not represent legitimate physical state vectors of quantum
mechanics.
Taking into account these remarks … one may reasonably ask now:
Is there just a single possible way of studying the scattering phenomena ?
$[\,$$\equiv$ the one outlined above in Part B ?$\,]$
Or, are there other possible ways of getting $\hat{S}$
and the associated physical information about scattering ?
If yes, do these alternative ways possess any merits/offer any benefits
compared to the above outlined conventional approach of Part B ?
Stated concisely, an answer to these questions looks as follows.
Nowadays there exist two alternative routes leading towards $\hat{S}$
and to the associated scattering observables, namely:

    * 1)

Wavepacket propagation in the time domain
Meaning that one solves the time dependent Schrödinger equation (4)
numerically for a given incoming state $|\phi_{\rm in}\rangle$ and gets thus
the corresponding outgoing state $|\phi_{\rm out}\rangle$. No need to deal
with the energy domain, or continuum eigenstates of $\hat{\sf H}$, or LSE.
Such a strategy looks somewhat like a ”brute force” approach based upon heavy
computing, but some pragmatically oriented people prefer this style.

Lecture #5

    * 2)

Nonhermitian scattering theory (and this we should study from now on)
Motivated by our desire to better understand resonance phenomena and related
matters.
$[\,$Resonances $\sim$ rather unexpected and often sharp peaks in the
transmission profile $\Bigl{|}\,T_{E(+1)}^{+}\,\Bigr{|}^{2}$.$\,]$
Based upon solving an eigenvalue problem

$\left(\,-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,+\,V(x)\,\right)\varphi_{j}(x)\;=\;E_{j}\,\varphi_{j}(x)\hskip
14.22636pt;$ (117)

where the eigenvalue $E_{j}$ is now allowed to be complex valued,
and where the pertinent eigenfunction $\varphi_{j}(x)$ satisfies the so called
Siegert boundary conditions
$[\,$see our more detailed discussion below, especially Eqs. (124) and
(126)$\,]$.
Complex valued $E_{j}$’s indicate where the word ”nonhermitian” comes from.
$[\,$Hermitian operators of the standard quantum mechanics have only real
eigenvalues.$\,]$
As we shall see later on, knowledge of $\\{E_{j},\varphi_{j}(x)\\}_{j}$
enables us to build $\hat{S}$
and describe thus adequately all the scattering phenomena.
Moreover, when the nonhermitian formulation of the problem is adopted,
the concept of a resonance becomes sharply defined,
and the role of resonances in scattering processes is then far better
understood.
In addition, the nonhermitian formalism itself is valuable by its own right,
since it allows us to look at the scattering problems from an unconventional
perspective.
This in turn often leads towards a compelling physical interpretation
and facilitates predictions of novel unexpected phenomena
which are difficult to anticipate when working within the conventional
formalism of Part B.

The three above discussed approaches to scattering theory
$[\,$0) conventional or orthodox, 1) wavepacket based, 2) nonhermitian ]
are complementary, each of them possesses its own advantages and
disadvantages.
The real expert should ideally be conversant both with 0) and 1) and 2),
and be able to combine these approaches neatly
while taking into account specific features of a concrete quantum mechanical
problem under study.

Lecture #5

  * •

C.2 Interlude: Bound states in the standard (hermitian) quantum mechanics
Before elaborating more on the nonhermitian scattering theory, let us prepare
the ground a bit,
by recalling the solution of equation (117) for the case of bound states.
$[\,$This is a standard stuff which can be found in any conventional textbook
on quantum mechanics.$\,]$
Let the range $(a,b)$ of our potential $V(x)$
be conveniently changed into $(-a,+a)$ from now onwards $[\,$here $a>0$ is a
prescribed parameter$\,]$.
Suppose that $E_{j}$ and $\varphi_{j}(x)$ is a bound state solution of
equation (117). Then $E_{j}<0$ as we know.
In the right asymptotic region of $x>(+a)$, equation (117) reduces to

$-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,\varphi_{j}(x)\;=\;E_{j}\,\varphi_{j}(x)\hskip
14.22636pt.$ (118)

General solution of (118) takes of course an explicit form

$\varphi_{j}(x)\;=\;{\cal A}_{+}\,e^{+\kappa_{j}x}\;+\;{\cal
A}_{-}\,e^{-\kappa_{j}x}\hskip 14.22636pt;$ (119)

where

$\kappa_{j}\;=\;\sqrt{(-2)\,m\,E_{j}}/\hbar\hskip 14.22636pt;$ (120)

while ${\cal A}_{\pm}$ are as yet arbitrary complex coefficients. Note that
$\kappa_{j}>0$.
Since we are dealing with a bound state,
the coefficient ${\cal A}_{+}$ pertaining to the exploding exponential
$e^{+\kappa_{j}x}$ must vanish, thereby

$\varphi_{j}(x\to+\infty)\;\sim\;e^{-\kappa_{j}x}\hskip 14.22636pt.$ (121)

Similar sequence of arguments applies also for the left asymptotic region of
$x\leq(-a)$.
One encounters again (118), with the general solution $\varphi_{j}(x)={\cal
B}_{+}\,e^{+\kappa_{j}x}+\,{\cal B}_{-}\,e^{-\kappa_{j}x}$ $({\rm
here}\;\,{\cal B}_{\pm}\in{\mathbb{C}})$.
Since we are dealing with a bound state,
the coefficient ${\cal B}_{-}$ pertaining to the exploding exponential
$e^{-\kappa_{j}x}$ must vanish, thereby

$\varphi_{j}(x\to-\infty)\;\sim\;e^{+\kappa_{j}x}\hskip 14.22636pt.$ (122)

Properties (121) and (122) indicate that $\varphi_{j}(x)$ behaves in the
asymptotic region of $x\to\pm\infty$
as a single exponential $e^{\mp\kappa_{j}x}$, not as a superposition of two
exponentials.
This observation will be recalled in the next paragraph,
where it will motivate us to introduce the Siegert boundary conditions.
Before proceeding further, let us make one additional remark regarding the
bound state quantization.
Namely, quantization of the bound state energy levels $E_{j}$ arises due to
imposing
two boundary conditions (121) and (122) on the solutions of the second order
ODE (117). Indeed,
while the ODE (117) possesses always two linearly independent particular
solutions
for any trial $E_{j}$ (even for $E_{j}$ complex),
the two boundary conditions (121) and (122) can simultaneously be met
only for a discrete set of specific eigenenergies.
$[\,$See again any standard quantum mechanics textbook for details.
$[\,$For bound states we require additionally $E_{j}$ to be real and
negative.$\,]$

Lecture #5

  * •

C.3 Nonhermitian Quantum Mechanics: Introducing the Siegert states
In the previous paragraph we have dealt with the bound state solutions of
equation (117).
Let us conveniently introduce the bound state problem once again using a
slightly different language.
Namely, when looking for the bound states of a given short ranged potential
$V(x)$,
we are interested in solutions $E_{j}$ and $\varphi_{j}(x)$ of an eigenvalue
problem

${\color[rgb]{1,0,0}\left(\,-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,+\,V(x)\,\right)\varphi_{j}(x)\;=\;E_{j}\,\varphi_{j}(x)\hskip
14.22636pt;}$ (123)

subjected to the boundary conditions

${\color[rgb]{1,0,0}\varphi_{j}(x\to+\infty)\;\sim\;e^{-ik_{j}x}\hskip
14.22636pt,\hskip 14.22636pt\varphi_{j}(x\to-\infty)\;\sim\;e^{+ik_{j}x}\hskip
14.22636pt;}$ (124)

where by definition

$k_{j}\;=\;\sqrt{2\,m\,E_{j}}/\hbar\hskip 14.22636pt.$ (125)

The associated energy eigenvalue $E_{j}$ is required to be real and negative
for the case of bound states.
Instead of (124) one may write equivalently

${\color[rgb]{1,0,0}\Bigl{(}\partial_{x}\,+\,i\,k_{j}\Bigr{)}\,\varphi_{j}(x)\Bigr{|}_{x=+a}=\;0\hskip
14.22636pt,\hskip
14.22636pt\Bigl{(}\partial_{x}\,-\,i\,k_{j}\Bigr{)}\,\varphi_{j}(x)\Bigr{|}_{x=-a}=\;0\hskip
14.22636pt.}$ (126)

Here, as we recall, symbol $a>0$ stands for the potential range $[\,$such that
$V(x)=0$ for $|x|\geq a$$\,]$.
It is a simple matter to verify that (124) implies (126) and vice versa.
Most importantly, the nonhermitian quantum mechanics
generalizes the notion of a bound state into the so called Siegert state.
Namely, when looking for the Siegert states of a given short ranged potential
$V(x)$,
we are interested in solutions $E_{j}$ and $\varphi_{j}(x)$ of an eigenvalue
problem (123)
subjected to the Siegert boundary conditions (124) or (126). Definition (125)
remains valid,
yet the eigenvalue ${\bm{E}}_{\bm{j}}$ is allowed to be generally complex.
Hence $k_{j}=(\ref{k-j-from-E-j})$ is also generally complex (not just
negative imaginary as for the bound states),
and the exponentials $e^{\mp ik_{j}x}$ appearing in (124) might be exploding!
Bound states are apparently special cases of the Siegert states.
If a given Siegert state solution $\varphi_{j}(x)$ of (123)-(124)/(126)
explodes at $x\to\pm\infty$,
then it is not square integrable and not normalizable in the usual sense.
$[\,$The normalization issue is left unresolved for now, we will come to it
later on in the course.$\,]$
Not only the bound states,
but all the Siegert state solutions $E_{j}$ and $\varphi_{j}(x)$ defined by
equations (123)-(124)/(126)
possess quantized (discrete) energy levels (some of these being complex).
Indeed, quantization of $E_{j}$ arises in this context again due to imposing
two boundary conditions (124)/(126) on the solutions of the second order ODE
(123).
While the ODE (123) possesses
always two linearly independent particular solutions for any trial complex
$E_{j}$,
the two boundary conditions (124)/(126) can simultaneously be met
only for a discrete set of specific eigenenergies.

Lecture #5

  * •

C.4 Looking at the Siegert states once again
So far we have just defined the Siegert states $\varphi_{j}(x)$
as solutions of the mathematical eigenvalue problem (123)-(124)/(126).
Now it is appropriate to ask the following important questions:

    * i)

How can one calculate all these Siegert states in practice (numerically)
for a given potential $V(x)$ ?

    * ii)

How about normalization ? How about the orthonormality and closure relations ?

    * iii)

What are the Siegert states $\varphi_{j}(x)$ actually good for ?
How are they related to the scattering theory, and in particular to quantities
like $\hat{S}$ or $T_{E(+1)}^{+}$
(where $E>0$ is of course real valued and continuous) ?

    * iv)

Can the language of Siegert states provide deeper insights into such phenomena
like e.g. resonance scattering ?

All the just posed questions are addressed very explicitly in the paragraphs
below,
which describe an elegant and clear cut formulation developed
by Oleg Tolstikhin in 1990’s and onwards [4].
Note that the concept of a Siegert state was introduced by Gamow and Siegert
back in 1930’s.
However, it was only Tolstikhin in 1990’s who succeeded
to bridge the gap between the world of Siegert states and the standard quantum
scattering theory.
It was Tolstikhin who sharpened fundamentally the Siegert state formalism
and brought it to practical use.

  * •

C.5 Developing the Siegert state formalism: Take #1
Let us recall equations (123), (126)
and define the Siegert states as solutions of the following eigenvalue
problem:

${\color[rgb]{1,0,0}\left(\,-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,+\,V(x)\,\right)\,\varphi(x)\;=\;E\,\,\varphi(x)\hskip
14.22636pt;}$ (127)
${\color[rgb]{1,0,0}\Bigl{(}\,\partial_{x}\,-\,ik\,\Bigr{)}\,\varphi(x)\,\Bigr{|}_{x=+a}\;=\;0\hskip
14.22636pt;}$ (128)
${\color[rgb]{1,0,0}\Bigl{(}\,\partial_{x}\,+\,ik\,\Bigr{)}\,\varphi(x)\,\Bigr{|}_{x=-a}\;=\;0\hskip
14.22636pt.}$ (129)

Here $k$ stands for an as yet unknown generally complex parameter
which gives rise to the complex energy $E=\frac{\hbar^{2}k^{2}}{2\,m}$.
Let $\\{{\it b}_{\nu}(x)\\}_{\nu}$ be an arbitrary complete orthonormal basis
set on an interval $x\in[-a,+a]$,
such that

$\int_{-a}^{+a}b_{\nu}^{*}(x)\,b_{\nu^{\prime}}(x)\;{\rm
d}x\;=\;\delta_{\nu\nu^{\prime}}\hskip 14.22636pt.$ (130)

Note that the basis functions ${\it b}_{\nu}(x)$ are not constrained to vanish
at the edges $x=\pm a$.
$[\,$Completeness actually requires at least some of the $b_{\nu}$’s to be
nonvanishing at $x=\pm a$.$\,]$
For example one may take

$b_{\nu}(x)\;=\;\sqrt{\frac{2\,\nu+1}{2\,a}}\;P_{\nu}\left(\frac{x}{a}\right)\hskip
14.22636pt,\hskip 14.22636pt\nu\,=\,0,1,2,3\ldots\hskip 14.22636pt;$ (131)

with $P_{\nu}(u)$ being the conventional Legendre polynomials.

Lecture #5

Alternatively one may employ the Fourier basis set

$b_{\nu}(x)\;=\;\frac{e^{+i\,\nu\,\pi\,\frac{x}{a}}}{\sqrt{2\,a}}\hskip
14.22636pt,\hskip 14.22636pt\nu\,\in\,\\{\ldots,-2,-1,0,+1,+2,\ldots\\}\hskip
14.22636pt.$ (132)

The as yet unknown Siegert state wavefunction can be for $x\in[-a,+a]$
expanded as

${\color[rgb]{1,0,0}\varphi(x)\;=\;\sum_{\nu}\,c_{\nu}\,{\it b}_{\nu}(x)\hskip
14.22636pt;}$ (133)

where the coefficients $c_{\nu}$ remain to be determined.
$[\,$Boundary condition (128) implies that
$[\,$$\varphi(x\geq(+a))\,=\,{\cal C}_{+}\,e^{+ikx}$, similarly
$\varphi(x\leq(-a))\,=\,{\cal C}_{-}\,e^{-ikx}$ due to (129).
$[\,$Factors ${\cal C}_{\pm}$ are fixed by demanding continuity of
$\varphi(x)$ at $x=\pm a$.$\,]$
In order to find the as yet unknown expansion coefficients $c_{\nu}$
we multiply (127) by ${\it b}_{\nu}^{*}(x)$ and integrate over $x\in[-a,+a]$.
One gets

$-\,\frac{\hbar^{2}}{2\,m}\,\int_{-a}^{+a}{\it
b}_{\nu}^{*}(x)\,\partial_{xx}\,\varphi(x)\;{\rm
d}x\;\,+\;\,\int_{-a}^{+a}{\it
b}_{\nu}^{*}(x)\,\Bigl{(}V(x)-E\Bigr{)}\,\varphi(x)\;{\rm d}x\;=\;0\hskip
14.22636pt.$ (134)

Subsequently we integrate the first term by parts,
and incorporate the Siegert boundary conditions (128)-(129).
We also insert the basis set expansion (133) afterwards.
This yields an intermediate outcome

$\displaystyle-\,\frac{i\hbar^{2}k}{2\,m}\,\sum_{\nu^{\prime}}\,c_{\nu^{\prime}}\,\Bigl{(}\,{\it
b}_{\nu}^{*}(+a)\,{\it b}_{\nu^{\prime}}(+a)\,+\,{\it b}_{\nu}^{*}(-a)\,{\it
b}_{\nu^{\prime}}(-a)\,\Bigr{)}\;+\;\frac{\hbar^{2}}{2\,m}\;\sum_{\nu^{\prime}}\,c_{\nu^{\prime}}\int_{-a}^{+a}\Bigl{(}\partial_{x}{\it
b}_{\nu}^{*}(x)\Bigr{)}\Bigl{(}\partial_{x}{\it
b}_{\nu^{\prime}}(x)\Bigr{)}\,{\rm d}x$
$\displaystyle+\;\sum_{\nu^{\prime}}\,c_{\nu^{\prime}}\int_{-a}^{+a}{\it
b}_{\nu}^{*}(x)\,V(x)\,{\it b}_{\nu^{\prime}}(x)\;{\rm
d}x\;-\;\frac{\hbar^{2}k^{2}}{2\,m}\;c_{\nu}\;=\;0\hskip 14.22636pt.$ (135)

Next, let us conveniently translate formula (•) into the language of linear
algebra.
For this purpose we shall introduce the following matrix elements:

$\displaystyle{\color[rgb]{1,0,0}H_{\nu\nu^{\prime}}}$
$\displaystyle{\color[rgb]{1,0,0}=}$
$\displaystyle{\color[rgb]{1,0,0}\frac{\hbar^{2}}{2\,m}\,\int_{-a}^{+a}\Bigl{(}\partial_{x}{\it
b}_{\nu}^{*}(x)\Bigr{)}\Bigl{(}\partial_{x}{\it
b}_{\nu^{\prime}}(x)\Bigr{)}\,{\rm d}x\;+\;\int_{-a}^{+a}{\it
b}_{\nu}^{*}(x)\,V(x)\,{\it b}_{\nu^{\prime}}(x)\;{\rm d}x\hskip 14.22636pt;}$
(136) $\displaystyle{\color[rgb]{1,0,0}L_{\nu\nu^{\prime}}}$
$\displaystyle{\color[rgb]{1,0,0}=}$
$\displaystyle{\color[rgb]{1,0,0}\frac{\hbar^{2}}{2\,m}\,\Bigl{(}\,{\it
b}_{\nu}^{*}(+a)\,{\it b}_{\nu^{\prime}}(+a)\,+\,{\it b}_{\nu}^{*}(-a)\,{\it
b}_{\nu^{\prime}}(-a)\,\Bigr{)}\hskip 14.22636pt.}$ (137)

Instead of (•) one may then write equivalently and concisely

${\color[rgb]{1,0,0}\left({\bm{H}}\,-\,i\,k\,{\bm{L}}\,-\,\frac{\hbar^{2}k^{2}}{2\,m}\,{\bm{I}}\right){\bm{c}}\;=\;{\bm{0}}\hskip
14.22636pt;}$ (138)

with obvious notations.

Lecture #5

Suppose that the basis set expansion (133) is truncated to $N$ terms
(where $N$ is arbitrarily large in principle, it is of course restricted in
practical calculations).
The matrix equation (138) is then correspondingly truncated as well.
$[\,$It contains then $N$-by-$N$ matrices ${\bm{H}}$, ${\bm{L}}$, ${\bm{I}}$
and and $N$-dimensional column vector ${\bm{c}}$.$\,]$
Equation (138)$|_{N\,{\rm finite}}$ can now be classified as a quadratic
eigenvalue problem of linear algebra,
to be solved for unknowns $k$ and ${\bm{c}}$. An adequate solution of
(138)$|_{N\,{\rm finite}}$ is performed
by taking advantage of a linearization technique which is described in the
next paragraph.
For the sake of clarity, we point out here that equation (138)$|_{N\,{\rm
finite}}$ or (138)$|_{N\to\infty}$ is implied
by our original Siegert eigenvalue problem (127), (128)-(129). On the other
hand,
it is not obvious whether every solution $\\{k,{\bm{c}}\\}$ of equation
(138)$|_{N\,{\rm finite}}$
corresponds to some approximate Siegert eigenstate (133)
which would satisfy (127), (128)-(129) in the limit of $N\to\infty$.
For this reason we prefer to call all the solutions of (138)$|_{N\,{\rm
finite}}$ as  Siegert pseudostates.

Lecture #6

  * •

C.6 Developing the Siegert state formalism: Take #2
The quadratic eigenvalue problem (138)$|_{N\,{\rm finite}}$ can be redisplayed
in a more standard appearance

${\color[rgb]{1,0,0}\left({\bm{A}}\,+\,\lambda\,{\bm{B}}\,+\,\lambda^{2}\,{\bm{I}}\right){\bm{c}}\;=\;{\bm{0}}\hskip
14.22636pt;}$ (139)

where

${\color[rgb]{1,0,0}{\bm{A}}\;=\;+\,\frac{2\,m}{\hbar^{2}}\;{\bm{H}}\hskip
14.22636pt,\hskip
14.22636pt{\bm{B}}\;=\;-\,\frac{2\,m}{\hbar^{2}}\;{\bm{L}}\hskip 14.22636pt;}$
(140)

and

${\color[rgb]{1,0,0}\lambda\;=\;i\,k\hskip 14.22636pt.}$ (141)

To facilitate further progress and linearize the l.h.s. of (139) with respect
to $\lambda$,
we conveniently introduce a $2N$-component column vector

${\color[rgb]{1,0,0}\left(\matrix{{\bm{c}}\cr\bm{\tilde{}}{\bm{c}}}\right)\;=\;\left(\matrix{{\bm{c}}\cr\lambda\,{\bm{c}}}\right)\hskip
14.22636pt.}$ (142)

Equation (139) is then equivalent to

$\left(\matrix{{\bm{O}}&{\bm{I}}\cr-{\bm{A}}&-{\bm{B}}}\right)\left(\matrix{{\bm{c}}\cr\bm{\tilde{}}{\bm{c}}}\right)\;=\;\lambda\,\left(\matrix{{\bm{c}}\cr\bm{\tilde{}}{\bm{c}}}\right)\hskip
14.22636pt.$ (143)

Indeed, one may easily check that (139) implies (143) and vice versa.
Importantly, the transformed problem (143) acquires doubled matrix dimension
$2N$,
yet it is linear in $\lambda$ as opposed to the $N$-dimensional matrix problem
(139).
$[\,$One may observe that the replacement (139) $\mapsto$ (143) resembles
$[\,$the well known passage from a single second order ODE to two coupled
first order ODEs.$\,]$
Proceeding further, we multiply (143) by matrix

$\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\hskip
14.22636pt;$ (144)

while noting that

$\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)^{-1}\;=\;\left(\matrix{{\bm{0}}&{\bm{I}}\cr{\bm{I}}&-{\bm{B}}}\right)\hskip
14.22636pt.$ (145)

$[\,$Due to (145) the matrix (144) is certainly regular.$\,]$ This converts
equation (143) into an equivalent symmetric form

${\color[rgb]{1,0,0}\left(\matrix{-{\bm{A}}&{\bm{O}}\cr{\bm{O}}&{\bm{I}}}\right)\left(\matrix{{\bm{c}}\cr\bm{\tilde{}}{\bm{c}}}\right)\;=\;\lambda\,\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\left(\matrix{{\bm{c}}\cr\bm{\tilde{}}{\bm{c}}}\right)\hskip
14.22636pt.}$ (146)

Symmetric linear eigenvalue problems (146) are treatable (even numerically)
by standard well established methods of linear algebra.
Hence also equations (139) and (138)$|_{N\,{\rm finite}}$ can be discussed and
solved in this way.

Lecture #6

  * •

C.7 Developing the Siegert state formalism: Take #3
Let us look again at problem (139), and define the corresponding quadratic
matrix polynomial

${\color[rgb]{1,0,0}{\bm{M}}(\lambda)\;=\;{\bm{A}}\,+\,\lambda\,{\bm{B}}\,+\,\lambda^{2}\,{\bm{I}}\hskip
14.22636pt.}$ (147)

Instead of (139) one can write then

${\bm{M}}(\lambda)\;{\bm{c}}\;=\;{\bm{0}}\hskip 14.22636pt.$ (148)

The matrix equation (148) possesses a nontrivial solution ${\bm{c}}$ iff ${\rm
det}[{\bm{M}}(\lambda)]=0$.
But ${\rm det}[{\bm{M}}(\lambda)]$ is a $\lambda$-polynomial of degree $2N$.
Polynomials of degree $2N$ generically have $2N$ distinct roots.
Hereafter we shall assume, for the sake of maximum simplicity,
that this is the case also for our polynomial equation ${\rm
det}[{\bm{M}}(\lambda)]=0$.
We shall also assume, again for the sake of maximum simplicity,
that all the $2N$ roots of ${\rm det}[{\bm{M}}(\lambda)]=0$ are nonzero.
$[\,$Correctness of the just made assumptions can always be checked by
numerical calculation
$[\,$performed a posteriori for each concrete physical system under study and
for concrete $N$.$\,]$
If so, equations (148), (146), (139) possess exactly $2N$ distinct
eigensolutions

${\color[rgb]{1,0,0}\lambda^{(n)}\neq 0\hskip 14.22636pt,\hskip
14.22636pt\left(\matrix{{\bm{c}}^{(n)}\cr\bm{\tilde{}}{\bm{c}}^{(n)}}\right)\hskip
14.22636pt,\hskip 14.22636pt1\leq n\leq 2N\hskip 14.22636pt.}$ (149)

Here the eigenvectors are linearly independent due to (143) and (146).
$[\,$Hence the $2N$ eigenvectors listed in (149) form a basis in the pertinent
linear space ${\mathbb{C}}^{2N}$.$\,]$
Correspondingly, in the finite basis set representation (138)$|_{N\,{\rm
finite}}$
of our original eigenvalue problem (127), (128)-(129)
we have obtained exactly $2N$ distinct linearly independent Siegert
pseudostates (133), that is,

${\color[rgb]{1,0,0}\varphi^{(n)}(x)\;=\;\sum_{\nu}\,c_{\nu}^{(n)}\,{\it
b}_{\nu}(x)\hskip 14.22636pt.}$ (150)

The associated Siegert eigenvalue follows immediately from equation (141). One
has

${\color[rgb]{1,0,0}k^{(n)}\;=\;-i\,\lambda^{(n)}\hskip 14.22636pt,\hskip
14.22636ptE^{(n)}\;=\;\frac{\hbar^{2}\left(k^{(n)}\right)^{2}}{2\,m}\;=\;-\,\frac{\hbar^{2}\left(\lambda^{(n)}\right)^{2}}{2\,m}\hskip
14.22636pt.}$ (151)

We have just finished answering question i) raised in the paragraph C.4.
From now on we shall conveniently adopt the notation $\varphi^{(n)}(x)$
instead of writing $\varphi_{j}(x)$ as we did tentatively in C.3-C.4.
The same applies also for the pertinent eigenvalues $E^{(n)}$ and the related
$k^{(n)}$.

Lecture #6

  * •

C.8 Developing the Siegert state formalism: Take #4
Let us establish now appropriate orthonormality relations
for the eigensolutions (149) of the problem (148), (146), (139).
Take any two distinct eigensolutions (149), such that

$\left(\matrix{-{\bm{A}}&{\bm{O}}\cr{\bm{O}}&{\bm{I}}}\right)\left(\matrix{{\bm{c}}^{(n)}\cr\bm{\tilde{}}{\bm{c}}^{(n)}}\right)\;=\;\lambda^{(n)}\,\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\left(\matrix{{\bm{c}}^{(n)}\cr\bm{\tilde{}}{\bm{c}}^{(n)}}\right)\hskip
14.22636pt;$ (152)
$\left(\matrix{-{\bm{A}}&{\bm{O}}\cr{\bm{O}}&{\bm{I}}}\right)\left(\matrix{{\bm{c}}^{(n^{\prime})}\cr\bm{\tilde{}}{\bm{c}}^{(n^{\prime})}}\right)\;=\;\lambda^{(n^{\prime})}\,\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\left(\matrix{{\bm{c}}^{(n^{\prime})}\cr\bm{\tilde{}}{\bm{c}}^{(n^{\prime})}}\right)\hskip
14.22636pt.$ (153)

Multiply (152) by
$({\bm{c}}^{(n^{\prime})T},\bm{\tilde{}}{\bm{c}}^{(n^{\prime})T})$ and (153)
by $({\bm{c}}^{(n)T},\bm{\tilde{}}{\bm{c}}^{(n)T})$. Subtract the two to get

$(\lambda^{(n)}-\lambda^{(n^{\prime})})\,\left(\matrix{{\bm{c}}^{(n)T}&\bm{\tilde{}}{\bm{c}}^{(n)T}}\right)\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\left(\matrix{{\bm{c}}^{(n^{\prime})}\cr\bm{\tilde{}}{\bm{c}}^{(n^{\prime})}}\right)\;=\;0\hskip
14.22636pt.$ (154)

Formula (154) reveals that for $\lambda^{(n)}\neq\lambda^{(n^{\prime})}$ (and
thus for $n\neq n^{\prime}$)
the eigensolutions (149) are orthogonal with respect to the scalar product

$\left(\matrix{{\bm{c}}^{(n)T}&\bm{\tilde{}}{\bm{c}}^{(n)T}}\right)\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\left(\matrix{{\bm{c}}^{(n^{\prime})}\cr\bm{\tilde{}}{\bm{c}}^{(n^{\prime})}}\right)\hskip
14.22636pt.$ (155)

Note also that for all values of $n$ $(1\leq n\leq 2N)$ we necessarily have

$\left(\matrix{{\bm{c}}^{(n)T}&\bm{\tilde{}}{\bm{c}}^{(n)T}}\right)\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\left(\matrix{{\bm{c}}^{(n)}\cr\bm{\tilde{}}{\bm{c}}^{(n)}}\right)\;\neq\;0\hskip
14.22636pt.$ (156)

Since if the l.h.s. of (156) was zero for some $n$, then the vector
$\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\left(\matrix{{\bm{c}}^{(n)}\cr\bm{\tilde{}}{\bm{c}}^{(n)}}\right)$
would be orthogonal to all the $2N$-dimensional vectors of our linear space
${\mathbb{C}}^{2N}$,
hence
$\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\left(\matrix{{\bm{c}}^{(n)}\cr\bm{\tilde{}}{\bm{c}}^{(n)}}\right)$
would vanish, and thus also
$\left(\matrix{{\bm{c}}^{(n)}\cr\bm{\tilde{}}{\bm{c}}^{(n)}}\right)$ would
vanish $[\,$cf. property (145)$\,]$.
Our above made observations lead us towards imposing the normalization
convention

${\color[rgb]{1,0,0}\left(\matrix{{\bm{c}}^{(n)T}&\bm{\tilde{}}{\bm{c}}^{(n)T}}\right)\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\left(\matrix{{\bm{c}}^{(n^{\prime})}\cr\bm{\tilde{}}{\bm{c}}^{(n^{\prime})}}\right)\;=\;2\,\lambda^{(n)}\;\delta_{nn^{\prime}}\hskip
14.22636pt.}$ (157)

The extra factor $2\,\lambda^{(n)}$ appearing on the r.h.s. of (157) has been
inserted
for the sake mathematical convenience. Recall in this context that
$\lambda^{(n)}\neq 0$ for all $n$.
An equivalent version of (157) is obtained by exploiting (142) and reads as

${\color[rgb]{1,0,0}\Bigl{(}\,\lambda^{(n)}+\lambda^{(n^{\prime})}\Bigr{)}\,{\bm{c}}^{(n)T}{\bm{c}}^{(n^{\prime})}\;+\;{\bm{c}}^{(n)T}{\bm{B}}\,{\bm{c}}^{(n^{\prime})}\;=\;2\,\lambda^{(n)}\;\delta_{nn^{\prime}}\hskip
14.22636pt.}$ (158)

Lecture #6

  * •

C.9 Developing the Siegert state formalism: Take #5
Let us establish now appropriate closure relations
for the eigensolutions (149) of the problem (148), (146), (139). As mentioned
already,
the $2N$ eigenvectors listed in (149) form a basis in the pertinent linear
space ${\mathbb{C}}^{2N}$.
Meaning that every $2N$-dimensional column vector
$\left(\matrix{{\bm{u}}\cr{\bm{v}}}\right)$ can be expressed in the form

$\left(\matrix{{\bm{u}}\cr{\bm{v}}}\right)\;=\;\sum_{n=1}^{2N}\;q_{n}\left(\matrix{{\bm{c}}^{(n)}\cr\bm{\tilde{}}{\bm{c}}^{(n)}}\right)\hskip
14.22636pt;$ (159)

with the coefficients

$q_{n}\;=\;\frac{1}{2\,\lambda^{(n)}}\,\left(\matrix{{\bm{c}}^{(n)T}&\bm{\tilde{}}{\bm{c}}^{(n)T}}\right)\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\,\left(\matrix{{\bm{u}}\cr{\bm{v}}}\right)\hskip
14.22636pt;$ (160)

as being implied by (157). Consider an $2N$-by-$2N$ matrix

${\bm{Q}}\;=\;\sum_{n=1}^{2N}\,\frac{1}{2\,\lambda^{(n)}}\,\left(\matrix{{\bm{c}}^{(n)}\cr\bm{\tilde{}}{\bm{c}}^{(n)}}\right)\left(\matrix{{\bm{c}}^{(n)T}&\bm{\tilde{}}{\bm{c}}^{(n)T}}\right)\hskip
14.22636pt.$ (161)

Clearly,

${\bm{Q}}\,\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)\left(\matrix{{\bm{u}}\cr{\bm{v}}}\right)\;=\;\left(\matrix{{\bm{u}}\cr{\bm{v}}}\right)\hskip
14.22636pt;$ (162)

valid for all $\left(\matrix{{\bm{u}}\cr{\bm{v}}}\right)$. Meaning that

${\bm{Q}}\;=\;\left(\matrix{{\bm{B}}&{\bm{I}}\cr{\bm{I}}&{\bm{O}}}\right)^{-1}\;=\;\left(\matrix{{\bm{O}}&{\bm{I}}\cr{\bm{I}}&-{\bm{B}}}\right)\hskip
14.22636pt;$ (163)

cf. equation (145). From (161) and (163) one obtains using (142)
the following important closure properties:

${\color[rgb]{1,0,0}\sum_{n=1}^{2N}\,\frac{1}{\lambda^{(n)}}\;{\bm{c}}^{(n)}\,{\bm{c}}^{(n)T}\;=\;{\bm{O}}\hskip
14.22636pt;}$ (164)
${\color[rgb]{1,0,0}\sum_{n=1}^{2N}\,{\bm{c}}^{(n)}\,{\bm{c}}^{(n)T}\;=\;2\,{\bm{I}}\hskip
14.22636pt;}$ (165)
${\color[rgb]{1,0,0}\sum_{n=1}^{2N}\,\lambda^{(n)}\;{\bm{c}}^{(n)}\,{\bm{c}}^{(n)T}\;=\;-2\,{\bm{B}}\hskip
14.22636pt.}$ (166)

We have just finished answering question ii) raised in the paragraph C.4.

Lecture #6

  * •

C.10 Developing the Siegert state formalism: Take #6
As the last step of our formal mathematical elaborations
we shall derive spectral representation of the inverse of the quadratic matrix
polynomial (147).
We claim that

${\color[rgb]{1,0,0}{\bm{M}}^{-1}(\lambda)\;=\;\sum_{n=1}^{2N}\,\frac{{\bm{c}}^{(n)}\,{\bm{c}}^{(n)T}}{2\,\lambda^{(n)}(\lambda-\lambda^{(n)})}\hskip
14.22636pt;}$ (167)

valid for all complex $\lambda$ except of course of the roots $\lambda^{(n)}$
of equation ${\rm det}[{\bm{M}}(\lambda)]=0$.
In order to prove (167), we shall multiply both sides by ${\bm{M}}(\lambda)$
from the left,
and see what comes out. One has

${\bm{M}}(\lambda)\;{\bm{M}}^{-1}(\lambda)\;=\;\sum_{n=1}^{2N}\,\frac{{\bm{M}}(\lambda)\,{\bm{c}}^{(n)}\,{\bm{c}}^{(n)T}}{2\,\lambda^{(n)}(\lambda-\lambda^{(n)})}\hskip
14.22636pt.$ (168)

But
${\bm{M}}(\lambda)\,{\bm{c}}^{(n)}\,=\,{\bm{M}}(\lambda_{n})\,{\bm{c}}^{(n)}\,+\,(\lambda-\lambda^{(n)})\,\Bigl{(}{\bm{B}}+(\lambda+\lambda^{(n)}){\bm{I}}\Bigr{)}\,{\bm{c}}^{(n)}$.
Hence

${\bm{M}}(\lambda)\;{\bm{M}}^{-1}(\lambda)\;=\;\sum_{n=1}^{2N}\,\frac{\Bigl{(}{\bm{B}}+(\lambda+\lambda^{(n)}){\bm{I}}\Bigr{)}\,{\bm{c}}^{(n)}\,{\bm{c}}^{(n)T}}{2\,\lambda^{(n)}}\;=\;{\bm{I}}\hskip
14.22636pt;$ (169)

as a direct consequence of (164)-(165). The validity of (167) is thus
established.

  * •

C.11 Siegert state representation of the retarded Green function: Take #1
Having passed through all the necessary mathematical preparations,
we are slowly moving on towards addressing the question iii) raised in
paragraph C.4.
Most importantly,  the gap between
the mathematical world of Siegert states and the physical world of quantum
scattering theory
is bridged via examining the retarded (outgoing) Green function
$G_{E}^{+}(x,y)$
which has been introduced and discussed at length in Appendix IV.
According to equations (274) and (294) from Appendix IV,
an entity $G_{E}^{+}(x,y)$ is defined as an unique solution of the following
boundary value problem:

${\color[rgb]{1,0,0}\left\\{\,-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,+\,V(x)\,-\,E\,\right\\}\,G_{E}^{+}(x,y)\;=\;-\,\delta(x-y)\hskip
14.22636pt;}$ (170)
$G_{E}^{+}(x\,_{<\,\min(-a,y)}^{>\,\max(+a,y)},y)\;\simeq\;e^{\pm ikx}\hskip
14.22636pt.$ (171)

Here $E>0$ stands for the physical impact energy and $k=\sqrt{2\,m\,E}/\hbar$
as usual.
Let us hereafter assume $y\in[-a,+a]$. Requirement (171)
can then be replaced by the Siegert type boundary conditions analogous to
(128)-(129), that is,

${\color[rgb]{1,0,0}\Bigl{(}\,\partial_{x}\,-\,ik\,\Bigr{)}\,G_{E}^{+}(x,y)\,\Bigr{|}_{x=+a}\;=\;0\hskip
14.22636pt;}$ (172)
${\color[rgb]{1,0,0}\Bigl{(}\,\partial_{x}\,+\,ik\,\Bigr{)}\,G_{E}^{+}(x,y)\,\Bigr{|}_{x=-a}\;=\;0\hskip
14.22636pt.}$ (173)

If so, then our boundary value problem (170), (172)-(173) may be for $|y|\leq
a$ analyzed and solved
using the same method as we applied above for the Siegert eigenproblem (127),
(128)-(129).

Lecture #6

Namely, let us restrict $x$ like we did for $y$, such that $x,y\in[-a,+a]$,
and expand $G_{E}^{+}(x,y)$ similarly as done in (133) for the case of
$\varphi(x)$. This amounts to set

$G_{E}^{+}(x,y)\;=\;\sum_{\nu\nu^{\prime}}\,G_{E,\nu\nu^{\prime}}^{+}\,{\it
b}_{\nu}(x)\,{\it b}_{\nu^{\prime}}(y)\hskip 14.22636pt;$ (174)

where the as yet unknown expansion coefficients $G_{E,\nu\nu^{\prime}}^{+}$
remain to be determined.
Much as before in (133),
the summation over $\nu$ is truncated in (174) just to $N$ terms, and
similarly for $\nu^{\prime}$.
In order to find the $N^{2}$ matrix elements $G_{E,\nu\nu^{\prime}}^{+}$, we
multiply (170) by ${\it b}_{\nu}^{*}(x)\,{\it b}_{\nu^{\prime}}^{*}(y)$,
and integrate over both spatial coordinates $x,y\in[-a,+a]$.
This yields, with the help of (130), a preliminary formula

$\displaystyle-\,\frac{\hbar^{2}}{2\,m}\,\int_{-a}^{+a}{\rm
d}x\int_{-a}^{+a}{\rm d}y\;\,{\it b}_{\nu}^{*}(x)\,{\it
b}_{\nu^{\prime}}^{*}(y)\;\partial_{xx}\,G_{E}^{+}(x,y)$
$\displaystyle+\;\int_{-a}^{+a}{\rm d}x\int_{-a}^{+a}{\rm d}y\;\,{\it
b}_{\nu}^{*}(x)\,{\it
b}_{\nu^{\prime}}^{*}(y)\,\Bigl{(}V(x)-E\Bigr{)}\,G_{E}^{+}(x,y)\;=\;-\,\delta_{\nu\nu^{\prime}}\hskip
14.22636pt.$

Subsequently we integrate the first term of (•) by parts,
and incorporate the Siegert boundary conditions (172)-(173).
We also insert the basis set expansion (174) afterwards,
and take again an advantage of (130).
All the mentioned manipulations provide altogether an intermediate outcome

$\displaystyle-\,\frac{i\hbar^{2}k}{2\,m}\,\sum_{\nu^{\prime\prime}}\,G_{E,\nu^{\prime\prime}\nu^{\prime}}^{+}\,\Bigl{(}\,{\it
b}_{\nu}^{*}(+a)\,{\it b}_{\nu^{\prime\prime}}(+a)\,+\,{\it
b}_{\nu}^{*}(-a)\,{\it
b}_{\nu^{\prime\prime}}(-a)\,\Bigr{)}\;+\;\frac{\hbar^{2}}{2\,m}\;\sum_{\nu^{\prime\prime}}\,G_{E,\nu^{\prime\prime}\nu^{\prime}}^{+}\int_{-a}^{+a}\Bigl{(}\partial_{x}{\it
b}_{\nu}^{*}(x)\Bigr{)}\Bigl{(}\partial_{x}{\it
b}_{\nu^{\prime\prime}}(x)\Bigr{)}\,{\rm d}x$
$\displaystyle+\;\sum_{\nu^{\prime\prime}}\,G_{E,\nu^{\prime\prime}\nu^{\prime}}^{+}\int_{-a}^{+a}{\it
b}_{\nu}^{*}(x)\,V(x)\,{\it b}_{\nu^{\prime\prime}}(x)\;{\rm
d}x\;-\;\frac{\hbar^{2}k^{2}}{2\,m}\;G_{E,\nu\nu^{\prime}}^{+}\;=\;-\,\delta_{\nu\nu^{\prime}}\hskip
14.22636pt;$ (176)

which means however nothing else than

${\color[rgb]{1,0,0}{\bm{G}}_{E}^{+}\,\left({\bm{H}}\,-\,i\,k\,{\bm{L}}\,-\,\frac{\hbar^{2}k^{2}}{2\,m}\,{\bm{I}}\right)\;=\;-\;{\bm{I}}\hskip
14.22636pt.}$ (177)

Here ${\bm{G}}_{E}^{+}$ stands of course for an $N$-by-$N$ matrix of elements
$G_{E,\nu\nu^{\prime}}^{+}$.
Matrices ${\bm{H}}$ and ${\bm{L}}$ are defined above in (136)-(137).
Instead of (177) one may write equivalently

${\color[rgb]{1,0,0}{\bm{G}}_{E}^{+}\,\left({\bm{A}}\,+\,\lambda\,{\bm{B}}\,+\,\lambda^{2}\,{\bm{I}}\right)\;=\;-\,\frac{2\,m}{\hbar^{2}}\,{\bm{I}}\hskip
14.22636pt;}$ (178)

with the matrices ${\bm{A}}$ and ${\bm{B}}$ being given by prescription (140),
and $\lambda=ik$ as in (141).
Even more concisely we can state that

${\color[rgb]{1,0,0}{\bm{G}}_{E}^{+}\;{\bm{M}}(\lambda)\;=\;-\,\frac{2\,m}{\hbar^{2}}\,{\bm{I}}\hskip
14.22636pt;}$ (179)

where the matrix polynomial ${\bm{M}}(\lambda)=(\ref{bm-M-lambda-def})$.

Lecture #6

  * •

C.12 Siegert state representation of the retarded Green function: Take #2
Our introduction to nonhermitian scattering theory approaches an epic final.
In this paragraph and in the next one,
all the above pursued technical developments of Part C come to fruition.
Most importantly, equation (179) enables us to conclude that

${\bm{G}}_{E}^{+}\;=\;-\,\frac{2\,m}{\hbar^{2}}\;{\bm{M}}^{-1}(\lambda)\hskip
14.22636pt.$ (180)

Our previously obtained formula (167) implies in turn

${\color[rgb]{1,0,0}{\bm{G}}_{E}^{+}\;=\;\frac{m}{\hbar^{2}}\;\sum_{n=1}^{2N}\,\frac{{\bm{c}}^{(n)}\,{\bm{c}}^{(n)T}}{\lambda^{(n)}(\lambda^{(n)}-\lambda)}\hskip
14.22636pt;}$ (181)

or, in other words,

$G_{E,\nu\nu^{\prime}}^{+}\;=\;\frac{m}{\hbar^{2}}\;\sum_{n=1}^{2N}\,\frac{c_{\nu}^{(n)}\,c_{\nu^{\prime}}^{(n)}}{\lambda^{(n)}(\lambda^{(n)}-\lambda)}\hskip
14.22636pt.$ (182)

Subsequently one may combine (182) with (174) and (150).
This yields a simple, fully explicit, and compelling result

${\color[rgb]{1,0,0}G_{E}^{+}(x,y)\;=\;\frac{m}{\hbar^{2}}\;\sum_{n=1}^{2N}\,\frac{\varphi^{(n)}(x)\;\varphi^{(n)}(y)}{\lambda^{(n)}(\lambda^{(n)}-\lambda)}\;=\;\frac{m}{\hbar^{2}}\;\sum_{n=1}^{2N}\,\frac{\varphi^{(n)}(x)\;\varphi^{(n)}(y)}{k^{(n)}(k-k^{(n)})}\hskip
14.22636pt.\hskip 14.22636pt[\,-a\leq x,y\leq+a\,]}$ (183)

Consistency of the whole mathematical formulation leading to (183)
excludes an existence of singularities in the denominator term
$(k-k^{(n)})^{-1}$.
$[\,$Another remark: Boundary condition (172) implies that
$[\,$$G_{E}^{+}(x\geq(+a),y)={\cal G}_{+}\,e^{+ikx}$, similarly
$G_{E}^{+}(x\leq(-a),y)={\cal G}_{-}\,e^{-ikx}$ due to (173). $(\\#)$
$[\,$This holds of course assuming $|y|\leq a$.
$[\,$Factors ${\cal G}_{\pm}$ are fixed by demanding continuity of
$G_{E}^{+}(x,y)$ at $x=\pm a$.$\,]$
Equation (183) and our comment $(\\#)$ express
the retarded Green function $G_{E}^{+}(x,y)$ of the conventional quantum
scattering theory
solely in terms of the nonhermitian Siegert pseudostates (150) and their
pertinent eigenvalues (151).
$[\,$We tacitly assume $(-a)\leq y\leq(+a)$ and $x\in(-\infty,+\infty)$ in the
just presented statements.$\,]$
We recall from Part B that knowledge of $G_{E}^{+}(x,y)$ for $|y|\leq a$
enables us to straightforwardly determine
the $\hat{T}$-operator and thus also the fundamental scattering operator
$\hat{S}$
$[\,$see equations (85), (88), (89)$\,]$. If so, then
the entire quantum scattering theory can be reformulated
solely in terms of the Siegert pseudostates
$\left\\{\lambda^{(n)},\,\varphi^{(n)}(x)\right\\}_{n=1}^{2N}$
which contain complete physical information about all the scattering
phenomena.
Valid of course as a well defined approximation,
which becomes exact in the complete basis set limit of $N\to\infty$.
As one can see, the fundamental equation (183) of Tolstikhin contains merely
a discrete sum over all the $2N$ calculated Siegert pseudostates.
The stationary scattering states $\psi_{E(\pm 1)}^{+}(x)$ of Part B are not
referenced at all in (183).

Lecture #6

  * •

C.13 Siegert state representation of the transmission coefficient
Let us supply now
a concrete example of an application of the nonhermitian Siegert pseudostate
formalism,
namely, a Siegert based calculation of the transmission coefficient
$T_{E}^{+}=T_{E(\pm 1)}^{+}$.
$[\,$For the sake of clarity we recall in this context equation (231) of
Appendix II.$\,]$
We shall take advantage of the formula (308) which was derived in Appendix IV.
When translated into our notation of Part C, equation (308) looks as follows:

$G_{E}^{+}(-a,+a)\;=\;\frac{(-i)\,m}{\hbar^{2}k}\;\,T_{E}^{+}\;\,e^{+2ika}\hskip
14.22636pt.$ (184)

Combination of (184) and (183) provides immediately
an interesting, important, and explicit result

${\color[rgb]{1,0,0}T_{E}^{+}\;=\;i\;k\;e^{-2ika}\,\sum_{n=1}^{2N}\,\frac{\varphi^{(n)}(-a)\;\varphi^{(n)}(+a)}{k^{(n)}(k-k^{(n)})}\hskip
14.22636pt.}$ (185)

This demonstrates that the transmission coefficient $T_{E}^{+}$ of the
conventional scattering theory
is expressible solely in terms of the Siegert pseudostates.
Formula (185) lends itself well to a numerical test.
$[\,$As a benchmark one may take here our conventional calculation of
$T_{E}^{+}=T_{E(\pm 1)}^{+}$
$[\,$which was performed above in the paragraph B.13.
$[\,$The just sketched program will be implemented in a very explicit fashion
during our Project.$\,]$
It is worthy to look again at equation (185) and examine its physical
contents.
In (185), the transmission coefficient $T_{E}^{+}$ is determined
as a discrete sum over separate contributions of all the corresponding $2N$
Siegert pseudostates.
For a particular physical system under study, and for a prescribed impact
energy $E=\frac{\hbar^{2}k^{2}}{2m}$,
it may sometimes happen that the physical value of $k$
almost coincides with $k^{(n)}$ at some single $n$ (say at $n=1$).
If so, then the $n=1$ term often dominates in the summation (185),
and hence $T_{E}^{+}$ can be approximated just by

$T_{E}^{+}\;\doteq\;i\;k\;e^{-2ika}\;\frac{\varphi^{(1)}(-a)\;\varphi^{(1)}(+a)}{k^{(1)}(k-k^{(1)})}\hskip
14.22636pt.$ (186)

Suppose that $\varphi^{(1)}(x)$ solves the Siegert boundary value problem
(127), (128)-(129).
Then it is not hard to see that, necessarily, $\Re\,k^{(1)}>0$ and
$\Im\,k^{(1)}<0$.
$[\,$Indeed, since $k>0$ is close to $k^{(1)}$, then $\Re\,k^{(1)}>0$ and
$\Im\,k^{(1)}$ is small in magnitude.
$[\,$Furthermore, as we know, $\varphi^{(1)}(x\,^{>}_{<}\pm a)={\cal
C}_{\pm}\,e^{\pm ik^{(1)}x}$.
$[\,$For $\Im\,k^{(1)}>0$ the exponential $e^{\pm ik^{(1)}x}$ would decay with
$x\to\pm\infty$,
$[\,$thus $\varphi^{(1)}(x)$ would be a bound state pertinent to real
$E^{(1)}$ and thus to real $k^{(1)}$, contradiction.
$[\,$For $\Im\,k^{(1)}=0$, $\varphi^{(1)}(x)$ would be an ordinary continuum
eigenstate of $\hat{\sf H}$ having real $E^{(1)}>0$,
$[\,$yet this is inconsistent with the Siegert boundary conditions (128)-(129)
$[\,$due to properties (115), (208).$\,]$
Correspondingly, the associated complex energy eigenvalue
$E^{(1)}=\,\hbar^{2}(k^{(1)})^{2}/(2\,m)$
takes the form $E^{(1)}={\cal E}_{\rm res}^{(1)}-i\,\Gamma_{\rm res}^{(1)}/2$
where both ${\cal E}_{\rm res}^{(1)}>0$ and $\Gamma_{\rm res}^{(1)}>0$.
We call $\varphi^{(1)}(x)$ as a  resonance state.

Lecture #6

Equation (186) can be now further transformed, into

$T_{E}^{+}\;\doteq\;i\;\frac{\hbar^{2}}{2\,m}\left(\frac{k(k+k^{(1)})}{k^{(1)}}\right)e^{-2ika}\;\frac{\varphi^{(1)}(-a)\;\varphi^{(1)}(+a)}{E-E^{(1)}}\;\doteq\;i\;\frac{\hbar^{2}k^{(1)}}{m}\;e^{-2ika}\;\frac{\varphi^{(1)}(-a)\;\varphi^{(1)}(+a)}{E-{\cal
E}_{\rm res}^{(1)}+i\,\Gamma_{\rm res}^{(1)}/2}\hskip 14.22636pt.$ (187)

The just obtained approximation (187) of the transmission coefficient
$T_{E}^{+}=(\ref{Siegert-T-E-+-final})$
is expected to work very accurately in an energy range of $E\approx{\cal
E}_{\rm res}^{(1)}$,
provided that $\Gamma_{\rm res}^{(1)}$ is small in magnitude compared to
${\cal E}_{\rm res}^{(1)}$, and provided that
there are no other resonances $E^{(n)}={\cal E}_{\rm res}^{(n)}-i\,\Gamma_{\rm
res}^{(n)}/2$ located nearby $E^{(1)}={\cal E}_{\rm res}^{(1)}-i\,\Gamma_{\rm
res}^{(1)}/2$.
Assuming that (187) holds reasonably well,
the resulting transmission probability possesses for $E\approx{\cal E}_{\rm
res}^{(1)}$ a Lorentzian form

${\color[rgb]{1,0,0}\Bigl{|}\,T_{E}^{+}\,\Bigr{|}^{2}\;\doteq\;{\cal
Q}\;\frac{\left(\frac{\Gamma_{\rm res}^{(1)}}{2}\right)^{2}}{\left(E-{\cal
E}_{\rm res}^{(1)}\right)^{2}+\left(\frac{\Gamma_{\rm
res}^{(1)}}{2}\right)^{2}}\hskip 14.22636pt;}$ (188)

with a prefactor

${\cal Q}\;=\;\left|\,\frac{\hbar^{2}k^{(1)}}{m}\,\frac{2}{\Gamma_{\rm
res}^{(1)}}\,\varphi^{(1)}(-a)\;\varphi^{(1)}(+a)\,\right|^{2}\hskip
14.22636pt.$ (189)

Formula (188) represents the so called
Breit-Wigner profile of a resonance peak in the transmission probability.
Let us revisit now our illustrative scattering calculation discussed in the
paragraph B.13.
We claim that each of the two sharp peaks of
$\Bigl{|}\,T_{E}^{+}\,\Bigr{|}^{2}$ encountered in Fig.2 of B.13
appears merely due to the  resonance transmission phenomenon,
which is described adequately by the nonhermitian equation (188).
$[\,$Here apparently ${\cal Q}\doteq 1$, an explanation exists but reaches
beyond the scope of our present course.$\,]$
In other words, we anticipate that the two mentioned narrow transmission peaks
of Fig.2
arise due to  presence of isolated resonances in our model potential $V(x)$
$[\,$defined by equation (116) and plotted in Fig.1$\,]$.
The just presented claim will be verified by an explicit numerical calculation
during our computational Project.
In the paragraphs C.12-C.13 we have answered, arguably quite convincingly,
the fundamental questions iii) and iv) raised before in C.4.
Our short introduction to nonhermitian scattering theory can thus be brought
to an end.

Project

D. Mini-project: Transmission phenomena studied using the conventional
scattering theory
and using the nonhermitian Siegert based approach

  * •

The aim of this tutorial mini-project will be to investigate numerically
the problem of scattering of a quantum particle in 1D.
We shall focus on evaluating the transmission probability
$\Bigl{|}\,T_{E}^{+}\,\Bigr{|}^{2}$
as a function of the incident particle energy $E$, using two approaches:

    * a)

The standard hermitian time independent scattering theory of Part B.
See especially the paragraph B.13, which contains also the definition of our
model problem.

    * b)

The nonhermitian scattering theory, based upon the Siegert pseudostate
formalism
as outlined in Part C. See especially the paragraph C.13.

  * •

Our first major goal will be to demonstrate that both theoretical frameworks
a) and b)
provide the same transmission profiles. $[\,$Which should be the same as
plotted in Fig.2.$\,]$
Our second goal will be to highlight fundamental advantage of the nonhermitian
formalism
$[\,$approach b)$\,]$ for the description of resonance scattering.
This would amount to calculate explicitly the corresponding resonance
wavefunctions $\varphi^{(n)}(x)$,
and to identify their role in forming Lorentzian peaks (188) in the obtained
transmission profile.

  * •

Our third (optional) goal might be to proceed further
by considering other suitable 1D potentials, and to try to understand the role
of the so called exceptional points and/or anti-bound states in scattering
phenomena.

Appendix I

Appendix I: Continuity equation and related matters

Based upon Refs. [1, 2].

  * •

Consider the time dependent Schrödinger equation (4) in the position
representation,

$i\hbar\;\partial_{t}\,\psi(t,x)\;=\;-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,\psi(t,x)\;+\;V(x)\,\psi(t,x)\hskip
14.22636pt.$ (190)

Let $\psi(t,x)$ be any particular solution of (190),
it can be a wavepacket but also a spatially delocalized function,
depending upon the prescribed initial condition $\psi(t_{\rm init},x)$.
A combination of formulas

$\displaystyle(+i)\hbar\;\partial_{t}\,\psi^{\phantom{*}}(t,x)$
$\displaystyle=$
$\displaystyle-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,\psi^{\phantom{*}}(t,x)\;+\;V(x)\,\psi^{\phantom{*}}(t,x)$
(191) $\displaystyle(-i)\hbar\;\partial_{t}\,\psi^{*}(t,x)$ $\displaystyle=$
$\displaystyle-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,\psi^{*}(t,x)\;+\;V(x)\,\psi^{*}(t,x)$
(192)

provides the following outcome,

$\displaystyle\partial_{t}\Bigl{(}\psi^{*}(t,x)\,\psi(t,x)\Bigr{)}$
$\displaystyle=$
$\displaystyle\frac{i\,\hbar}{2\,m}\,\Bigl{(}\,\psi^{*}(t,x)\,\partial_{xx}\,\psi(t,x)\,-\,\psi(t,x)\,\partial_{xx}\,\psi^{*}(t,x)\,\Bigr{)}\;=$
(193) $\displaystyle=$
$\displaystyle\frac{i\,\hbar}{2\,m}\,\partial_{x}\Bigl{(}\,\psi^{*}(t,x)\,\partial_{x}\,\psi(t,x)\,-\,\psi(t,x)\,\partial_{x}\,\psi^{*}(t,x)\,\Bigr{)}\hskip
14.22636pt.$

Subsequently one gets

$\partial_{t}\Bigl{(}\psi^{*}(t,x)\,\psi(t,x)\Bigr{)}\;=\;-\,\partial_{x}\,\frac{1}{2}\left\\{\,\psi^{*}(t,x)\left(\frac{\hat{\sf
p}}{m}\,\psi(t,x)\right)\,-\,\psi(t,x)\left(\frac{\hat{\sf
p}}{m}\,\psi(t,x)\right)^{*}\,\right\\}\hskip 14.22636pt.$ (194)

After setting

$\varrho(t,x)\;=\;\psi^{*}(t,x)\,\psi(t,x)\hskip 7.11317pt,\hskip
7.11317ptJ(t,x)\;=\;\frac{1}{2}\left\\{\,\psi^{*}(t,x)\left(\frac{\hat{\sf
p}}{m}\,\psi(t,x)\right)\,-\,\psi(t,x)\left(\frac{\hat{\sf
p}}{m}\,\psi(t,x)\right)^{*}\,\right\\}$ (195)

one converts (194) into the desired continuity equation, namely,

$\partial_{t}\,\varrho(t,x)\;+\;\partial_{x}\,J(t,x)\;=\;0\hskip 14.22636pt.$
(196)

Recall that the continuity equation (196)
expresses mathematically the law of probability conservation.
It is valid for an arbitrary solution $\psi(t,x)$ of the time dependent
Schrödinger equation (190).
In particular, $\psi(t,x)$ does not need to be square integrable or spatially
localized.

Appendix I

  * •

Equation (193) admits an additional and rather important generalization.
Namely, let $\psi_{1}(t,x)$ and $\psi_{2}(t,x)$ be two arbitrary (generally
distinct) particular solutions of (190).
A combination of formulas

$\displaystyle(+i)\hbar\;\partial_{t}\,\psi_{2}^{\phantom{*}}(t,x)$
$\displaystyle=$
$\displaystyle-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,\psi_{2}^{\phantom{*}}(t,x)\;+\;V(x)\,\psi_{2}^{\phantom{*}}(t,x)$
(197) $\displaystyle(-i)\hbar\;\partial_{t}\,\psi_{1}^{*}(t,x)$
$\displaystyle=$
$\displaystyle-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,\psi_{1}^{*}(t,x)\;+\;V(x)\,\psi_{1}^{*}(t,x)$
(198)

provides the following outcome,

$\displaystyle\partial_{t}\Bigl{(}\psi_{1}^{*}(t,x)\,\psi_{2}(t,x)\Bigr{)}$
$\displaystyle=$
$\displaystyle\frac{i\,\hbar}{2\,m}\,\Bigl{(}\,\psi_{1}^{*}(t,x)\,\partial_{xx}\,\psi_{2}(t,x)\,-\,\psi_{2}(t,x)\,\partial_{xx}\,\psi_{1}^{*}(t,x)\,\Bigr{)}\;=$
(199) $\displaystyle=$
$\displaystyle\frac{i\,\hbar}{2\,m}\,\partial_{x}\Bigl{(}\,\psi_{1}^{*}(t,x)\,\partial_{x}\,\psi_{2}(t,x)\,-\,\psi_{2}(t,x)\,\partial_{x}\,\psi_{1}^{*}(t,x)\,\Bigr{)}\hskip
14.22636pt.$

Valid again even if the solutions $\psi_{1,2}(t,x)$ of (190) are not square
integrable or spatially localized.

  * •

Let us revisit equation (199), and assume that both $\psi_{1}(t,x)$ and
$\psi_{2}(t,x)$ are stationary states of $\hat{\sf H}$.
Thus $\psi_{1,2}(t,x)\,=\,\psi_{E}^{(1,2)}(x)\;e^{-(i/\hbar)Et}$ with obvious
notations.
Equation (193) is correspondingly simplified into

$\partial_{x}\Bigl{(}\,\psi_{E}^{(1)*}(x)\,\partial_{x}\,\psi_{E}^{(2)}(x)\,-\,\psi_{E}^{(2)}(x)\,\partial_{x}\,\psi_{E}^{(1)*}(x)\,\Bigr{)}\;=\;0\hskip
14.22636pt.$ (200)

Subsequent integration over $x\in(a,b)$ yields then

$\Bigl{(}\,\psi_{E}^{(1)*}(x)\,\partial_{x}\,\psi_{E}^{(2)}(x)\,-\,\psi_{E}^{(2)}(x)\,\partial_{x}\,\psi_{E}^{(1)*}(x)\,\Bigr{)}\Bigr{|}_{x=a}^{x=b}\;=\;0\hskip
14.22636pt.$ (201)

Formula (201) will be exploited in Appendix II.

Appendix II

Appendix II: One-dimensional stationary Schrödinger equation

Based upon Refs. [1, 2].

  * •

Let us consider an eigenvalue problem of the full Hamiltonian $\hat{\sf
H}=(\ref{hat-sf-H-H0-V})$ in the position representation.
One has

$\left\\{\,-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,+\,V(x)\,\right\\}\psi_{E}(x)\;=\;E\,\psi_{E}(x)\hskip
14.22636pt.$ (202)

Here $\psi_{E}(x)$ is a provisional notation for the eigenfunctions.
In what follows, we shall be concerned only with the continuum eigenstates of
$\hat{\sf H}$,
associated with $E>0$. We omit any discussion of eventual bound states.

  * •

At any given $E>0$,
the problem (202) represents a second order ordinary differential equation
(ODE) for $\psi_{E}(x)$.
Since our potential $V(x)$ vanishes for all $x\geq b$,
it is legitimate to set $\psi_{E}(x\geq b)=e^{+iKx}$ with $\hbar
K=\sqrt{2\,m\,E}$.
Subsequent back propagation of the ODE (202) yields uniquely $\psi_{E}(x)$
also for all $x<b$.
Since $V(x)$ vanishes for all $x\leq a$, the wavefunction $\psi_{E}(x\leq a)$
is necessarily some linear combination of two exponentials $e^{\pm iKx}$.
Thus $\psi_{E}(x\leq a)=A(E)\,e^{+iKx}+B(E)\,e^{-iKx}$,
where $A(E)$ and $B(E)$ are some uniquely determined coefficients.
Let us recall now formula (201) from Appendix I and set
$\psi_{E}^{(1)}(x)=\psi_{E}^{(2)}(x)=\psi_{E}(x)$, such that

$\Bigl{(}\,\psi_{E}^{*}(x)\,\partial_{x}\,\psi_{E}(x)\,-\,\psi_{E}(x)\,\partial_{x}\,\psi_{E}^{*}(x)\,\Bigr{)}\Bigr{|}_{x=a}^{x=b}\;=\;0\hskip
14.22636pt.$ (203)

After plugging into (203) the above discussed simple explicit forms of
$\psi_{E}(x\leq a)$ and $\psi_{E}(x\geq b)$,
one finds that $|A(E)|^{2}=|B(E)|^{2}+1$. Showing that $A(E)$ is certainly
nonzero.
If so, one may redefine $\psi_{E}(x)$ via dividing it by $A(E)$.
Then $\psi_{E}(x\leq a)=e^{+iKx}+R(E)\,e^{-iKx}$ and $\psi_{E}(x\geq
b)=T(E)\,e^{+iKx}$,
where $R(E)=B(E)/A(E)$ and $T(E)=1/A(E)$ are uniquely determined coefficients.
Since $|A(E)|^{2}=|B(E)|^{2}+1$, one inevitably has $|T(E)|^{2}+|R(E)|^{2}=1$.

  * •

Let us summarize the findings of the above paragraph
using a slightly modified notation (updated to match our notation in the main
text).
We have shown that there exists an unique particular solution
$\tilde{\psi}_{E(+1)}^{+}(x)$ of the ODE (202)
which exhibits the following kind of behavior in the asymptotic regions of
$x\leq a$ and $x\geq b$:

$\displaystyle\tilde{\psi}_{E(+1)}^{+}(x\geq b)$ $\displaystyle=$
$\displaystyle\hskip 50.53212ptT_{E(+1)}^{+}\;e^{+iKx}\hskip 14.22636pt;$
(204) $\displaystyle\tilde{\psi}_{E(+1)}^{+}(x\leq a)$ $\displaystyle=$
$\displaystyle e^{+iKx}\;+\;R_{E(+1)}^{+}\;e^{-iKx}\hskip 14.22636pt.$ (205)

Here $\hbar K=\sqrt{2\,m\,E}$ as before, while $T_{E(+1)}^{+}$ and
$R_{E(+1)}^{+}$ are some uniquely determined coefficients
(these carry a lot of physical meaning, as demonstrated in much detail within
the main text).
A completely analogous reasoning reveals that there exists also another
unique particular solution $\tilde{\psi}_{E(-1)}^{+}(x)$ of the ODE (202),
which behaves in the asymptotic regions of $x\leq a$ and $x\geq b$ as follows:

$\displaystyle\tilde{\psi}_{E(-1)}^{+}(x\geq b)$ $\displaystyle=$
$\displaystyle e^{-iKx}\;+\;R_{E(-1)}^{+}\;e^{+iKx}\hskip 14.22636pt;$ (206)
$\displaystyle\tilde{\psi}_{E(-1)}^{+}(x\leq a)$ $\displaystyle=$
$\displaystyle\hskip 51.35701ptT_{E(-1)}^{+}\;e^{-iKx}\hskip 14.22636pt;$
(207)

Appendix II

here again $T_{E(-1)}^{+}$ and $R_{E(-1)}^{+}$ are some uniquely determined
coefficients.
In the main text we have assigned direct physical meaning to $T_{E(\pm
1)}^{+}$ and $R_{E(\pm 1)}^{+}$,
see equations (108)-(109), (113)-(114) and the accompanying discussion.

  * •

Solutions $\tilde{\psi}_{E(+1)}^{+}(x)$ and $\tilde{\psi}_{E(-1)}^{+}(x)$ of
the ODE (202) are linearly independent,
and span thus the two-dimensional eigenspace of $\hat{\sf H}$ assigned to the
given continuum energy level $E$.
The entities $T_{E(\pm 1)}^{+}$ and $R_{E(\pm 1)}^{+}$ possess two important
properties, namely,

$\Bigl{|}\,T_{E(\pm 1)}^{+}\,\Bigr{|}^{2}\;+\;\Bigl{|}\,R_{E(\pm
1)}^{+}\,\Bigr{|}^{2}\;=\;1\hskip 14.22636pt;$ (208)

and

$T_{E(\pm 1)}^{+*}\,R_{E(\mp 1)}^{+}\;+\;R_{E(\pm 1)}^{+*}\,T_{E(\mp
1)}^{+}\;=\;0\hskip 14.22636pt.$ (209)

The normalization property (208) is a direct consequence of the law of
probability conservation.
It can be quicky verified by taking equation (201) and choosing
$\psi_{E}^{(1)}(x)=\psi_{E}^{(2)}(x)=\tilde{\psi}_{E(\pm 1)}^{+}(x)$.
$[\,$Recall that the diagonal version $\psi_{E}^{(1)}(x)=\psi_{E}^{(2)}(x)$ of
equation (201) is indeed closely related
$[\,$to the law of probability conservation for stationary states, see
Appendix I.$\,]$
The orthogonality property (209) follows immediately from equation (201)
after setting $\psi_{E}^{(1)}(x)=\tilde{\psi}_{E(\pm 1)}^{+}(x)$ and
$\psi_{E}^{(2)}(x)=\tilde{\psi}_{E(\mp 1)}^{+}(x)$.

  * •

Proceeding further,
let us establish appropriate orthonormality relations for the eigenfunctions
$\tilde{\psi}_{E(\pm 1)}^{+}(x)$.
To accomplish this goal, we shall set $\eta\in\\{-1,+1\\}$, and examine an
overlap

$I_{(E\eta)(E^{\prime}\eta^{\prime})}^{L}\;=\;\int_{-L}^{+L}\tilde{\psi}_{E\eta}^{+*}(x)\,\tilde{\psi}_{E^{\prime}\eta^{\prime}}^{+}(x)\;{\rm
d}x\hskip 14.22636pt;\hskip 14.22636pt[\,L>0\,]$ (210)

with an intention of taking the box length $L\to+\infty$ later on.
One has

$\displaystyle\left\\{\,-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,+\,V(x)\,\right\\}\;\,\tilde{\psi}_{E\eta}^{+*}(x)$
$\displaystyle=$ $\displaystyle E\;\tilde{\psi}_{E\eta}^{+*}(x)\hskip
14.22636pt;$ (211)
$\displaystyle\left\\{\,-\,\frac{\hbar^{2}}{2\,m}\,\partial_{xx}\,+\,V(x)\,\right\\}\tilde{\psi}_{E^{\prime}\eta^{\prime}}^{+}(x)$
$\displaystyle=$ $\displaystyle
E^{\prime}\,\tilde{\psi}_{E^{\prime}\eta^{\prime}}^{+}(x)\hskip 14.22636pt.$
(212)

Multiply (211) by $\tilde{\psi}_{E^{\prime}\eta^{\prime}}^{+}(x)$, and (212)
by $\tilde{\psi}_{E\eta}^{+*}(x)$.
Subtract (212) from (211), and integrate both sides of the resulting equation
over $x\in(-L,+L)$.
This yields an intermediate outcome

$(E-E^{\prime})\,I_{(E\eta)(E^{\prime}\eta^{\prime})}^{L}\;=\;\frac{\hbar^{2}}{2\,m}\,\int_{-L}^{+L}\Bigl{\\{}\tilde{\psi}_{E\eta}^{+*}(x)\,\partial_{xx}\,\tilde{\psi}_{E^{\prime}\eta^{\prime}}^{+}(x)\,-\,\tilde{\psi}_{E^{\prime}\eta^{\prime}}^{+}(x)\,\partial_{xx}\,\tilde{\psi}_{E\eta}^{+*}(x)\Bigr{\\}}\;{\rm
d}x\hskip 14.22636pt.$ (213)

After integrating by parts one gets

$(E-E^{\prime})\,I_{(E\eta)(E^{\prime}\eta^{\prime})}^{L}\;=\;\frac{\hbar^{2}}{2\,m}\,\left.\Bigl{\\{}\tilde{\psi}_{E\eta}^{+*}(x)\,\partial_{x}\,\tilde{\psi}_{E^{\prime}\eta^{\prime}}^{+}(x)\,-\,\tilde{\psi}_{E^{\prime}\eta^{\prime}}^{+}(x)\,\partial_{x}\,\tilde{\psi}_{E\eta}^{+*}(x)\Bigr{\\}}\right|_{x=-L}^{x=+L}\hskip
14.22636pt.$ (214)

Appendix II

Assume now that the interval $(-L,+L)$ is large enough
as to cover the whole interaction region $x\in(a,b)$.
In that case, the r.h.s. of (214) can be worked out further
using the asymptotic forms (204)-(205), (206)-(207) of the involved
wavefunctions.
Leading to the following expressions:

$\displaystyle\left(\frac{\hbar^{2}K^{2}}{2\,m}-\frac{\hbar^{2}K^{\prime
2}}{2\,m}\right)I_{(E(+1))(E^{\prime}(+1))}^{L}\;=$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2\,m}\,\Biggl{\\{}\,i\,T_{E(+1)}^{+*}\,T_{E^{\prime}(+1)}^{+}\,(K+K^{\prime})\,e^{-i(K-K^{\prime})L}\;+\;i\,R_{E(+1)}^{+*}\,R_{E^{\prime}(+1)}^{+}\,(K+K^{\prime})\,e^{-i(K-K^{\prime})L}\;-\;i\,(K+K^{\prime})\,e^{+i(K-K^{\prime})L}$
$\displaystyle\phantom{\frac{\hbar^{2}}{2\,m}\,\Biggl{\\{}}+\;i\,R_{E(+1)}^{+*}\,(K-K^{\prime})\,e^{-i(K+K^{\prime})L}-\;i\,R_{E^{\prime}(+1)}^{+}\,(K-K^{\prime})\,e^{+i(K+K^{\prime})L}\,\Biggr{\\}}\hskip
14.22636pt;$
$\displaystyle\left(\frac{\hbar^{2}K^{2}}{2\,m}-\frac{\hbar^{2}K^{\prime
2}}{2\,m}\right)I_{(E(+1))(E^{\prime}(-1))}^{L}\;=$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2\,m}\,\Biggl{\\{}\,i\,T_{E(+1)}^{+*}\,R_{E^{\prime}(-1)}^{+}\,(K+K^{\prime})\,e^{-i(K-K^{\prime})L}\;+\;i\,R_{E(+1)}^{+*}\,T_{E^{\prime}(-1)}^{+}\,(K+K^{\prime})\,e^{-i(K-K^{\prime})L}$
$\displaystyle\phantom{\frac{\hbar^{2}}{2\,m}\,\Biggl{\\{}}+\;i\,T_{E(+1)}^{+*}\,(K-K^{\prime})\,e^{-i(K+K^{\prime})L}-\;i\,T_{E^{\prime}(-1)}^{+}\,(K-K^{\prime})\,e^{+i(K+K^{\prime})L}\,\Biggr{\\}}\hskip
14.22636pt.$

Here of course $K^{\prime}=\sqrt{2\,m\,E^{\prime}}$.
Analogous expressions result also for $I_{(E(-1))(E^{\prime}(+1))}^{L}$ and
$I_{(E(-1))(E^{\prime}(-1))}^{L}$,
we do not display them here to save space.
Consider now $I_{(E\eta)(E^{\prime}\eta^{\prime})}^{L}=(\ref{I-overlap-def})$
to be a distributional entity, whose
only physical meaning arises due to integrals ${\cal
I}_{\eta\eta^{\prime}}^{L}[\chi_{1},\chi_{2}]=\int_{0}^{+\infty}{\rm
d}E\int_{0}^{+\infty}{\rm
d}E^{\prime}\,I_{(E\eta)(E^{\prime}\eta^{\prime})}^{L}\,\chi_{1}(E)\,\chi_{2}(E^{\prime})$,
where $\chi_{1,2}(E)$ are continuous test functions of the energy $E>0$ or
equivalently of $K>0$.
$[\,$We also tacitly assume that $\chi_{1,2}(E)$ falls off to zero at
$E\to+\infty$ faster than any power of $K$,
$[\,$this makes ${\cal I}_{\eta\eta^{\prime}}^{L}[\chi_{1},\chi_{2}]$ always
well defined.$\,]$
In the limit of $L\to+\infty$, the exponential $e^{\pm i(K+K^{\prime})L}$
appearing in (•)-(•)
oscillates rapidly with $K$ and $K^{\prime}$ unless both $K$ and $K^{\prime}$
are close to zero.
Meaning that all the terms of (•)-(•) containing $(K-K^{\prime})\,e^{\pm
i(K+K^{\prime})L}$
do not contribute to ${\cal I}_{\eta\eta^{\prime}}^{L}[\chi_{1},\chi_{2}]$,
and can thus be discarded for large $L$.
On the other hand, in the limit of $L\to+\infty$, the exponential $e^{\pm
i(K-K^{\prime})L}$ appearing in (•)-(•)
oscillates rapidly with $K$ and $K^{\prime}$ unless $(K-K^{\prime})$ is close
to zero.
Hence (•)-(•) can be correspondingly simplified as follows:

$\displaystyle+i\,(K-K^{\prime})\;I_{(E(+1))(E^{\prime}(+1))}^{L}$
$\displaystyle=$ $\displaystyle
e^{+i(K-K^{\prime})L}\;-\;\Bigl{|}\,T_{E(+1)}^{+}\Bigr{|}^{2}\,e^{-i(K-K^{\prime})L}\;-\;\Bigl{|}\,R_{E(+1)}^{+}\Bigr{|}^{2}\,e^{-i(K-K^{\prime})L}\hskip
14.22636pt;$ (217)
$\displaystyle-i\,(K-K^{\prime})\;I_{(E(+1))(E^{\prime}(-1))}^{L}$
$\displaystyle=$ $\displaystyle
T_{E(+1)}^{+*}\,R_{E^{\prime}(-1)}^{+}\,e^{-i(K-K^{\prime})L}\;+\;R_{E(+1)}^{+*}\,T_{E^{\prime}(-1)}^{+}\,e^{-i(K-K^{\prime})L}\hskip
44.10185pt.$ (218)

$[\,$Note that we have kept $E^{\prime}$ in (218) for our later convenience,
cf. equation (221) below.$\,]$
Analogous expressions would result also for $I_{(E(-1))(E^{\prime}(+1))}^{L}$
and $I_{(E(-1))(E^{\prime}(-1))}^{L}$.

Appendix II

An additional dramatic simplification is possible due to properties
(208)-(209).
Namely, after exploiting (208), equation (217) boils down into

$I_{(E(+1))(E^{\prime}(+1))}^{L}\;=\;\frac{2\,\sin((K-K^{\prime})L)}{(K-K^{\prime})}\hskip
14.22636pt.$ (219)

For $L\to+\infty$, the familiar textbook formula reveals that

$I_{(E(+1))(E^{\prime}(+1))}^{\infty}\;=\;2\,\pi\;\delta(K-K^{\prime})\hskip
14.22636pt.$ (220)

Furthermore, equation (218) leads to

$\displaystyle-i\,(K-K^{\prime})\;I_{(E(+1))(E^{\prime}(-1))}^{L}$
$\displaystyle=$
$\displaystyle\left(\,T_{E(+1)}^{+*}\,R_{E(-1)}^{+}\,+\,R_{E(+1)}^{+*}\,T_{E(-1)}^{+}\right)e^{-i(K-K^{\prime})L}$
(221) $\displaystyle+$
$\displaystyle\left(\,T_{E(+1)}^{+*}\,\partial_{E}\,R_{E(-1)}^{+}\,+\,R_{E(+1)}^{+*}\,\partial_{E}\,T_{E(-1)}^{+}\right)(E^{\prime}-E)\;e^{-i(K-K^{\prime})L}\hskip
14.22636pt.$

The first term on the r.h.s. of (221) vanishes because of (209), hence

$\displaystyle i\;I_{(E(+1))(E^{\prime}(-1))}^{L}$ $\displaystyle=$
$\displaystyle\left(\,T_{E(+1)}^{+*}\,\partial_{E}\,R_{E(-1)}^{+}\,+\,R_{E(+1)}^{+*}\,\partial_{E}\,T_{E(-1)}^{+}\right)\frac{\hbar^{2}}{2\,m}\;(K^{\prime}+K)\;e^{-i(K-K^{\prime})L}\hskip