Difference between revisions of "Properties of the Linear Schrodinger Equation"

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{{nonlinear waves course
 
{{nonlinear waves course
 
  | chapter title = Properties of the Linear Schrodinger Equation
 
  | chapter title = Properties of the Linear Schrodinger Equation
  | next chapter = [[Properties of the Linear Schrodinger Equation]]
+
  | next chapter = [[Connection betwen KdV and the Schrodinger Equation]]
 
  | previous chapter = [[Introduction to the Inverse Scattering Transform]]
 
  | previous chapter = [[Introduction to the Inverse Scattering Transform]]
 
}}
 
}}
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incident waves. This is easiest seen through the following examples
 
incident waves. This is easiest seen through the following examples
  
==Example <math>\delta</math> function potential==
+
==Example 1: <math>\delta</math> function potential==
  
We consider here the case when <math>u\left(  x,0\right)  =\delta\left(  x\right)
+
We consider here the case when <math>u\left(  x,0\right)  = u_0 \delta\left(  x\right)
 
.</math> Note that this function can be thought of as the limit as of the potential
 
.</math> Note that this function can be thought of as the limit as of the potential
 
<center><math>
 
<center><math>
 
u\left(  x\right)  =\left\{
 
u\left(  x\right)  =\left\{
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
 
0 & x\notin\left[  -\varepsilon,\varepsilon\right] \\
 
0 & x\notin\left[  -\varepsilon,\varepsilon\right] \\
 
\frac{u_{0}}{2\varepsilon} & x\in\left[  -\varepsilon,\varepsilon\right]
 
\frac{u_{0}}{2\varepsilon} & x\in\left[  -\varepsilon,\varepsilon\right]
Line 30: Line 30:
 
In this case we need to solve
 
In this case we need to solve
 
<center><math>
 
<center><math>
\partial_{x}^{2}w+u_{0}\delta\left( x\right) w=-\lambda x
+
\partial_{x}^{2}w+ u_0\delta(x) w=-\lambda w
 
</math></center>
 
</math></center>
 
We consider the case of <math>\lambda<0</math> and <math>\lambda>0</math> separately. For the first
 
We consider the case of <math>\lambda<0</math> and <math>\lambda>0</math> separately. For the first
Line 37: Line 37:
 
w\left(  x\right)  =\left\{
 
w\left(  x\right)  =\left\{
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
 
ae^{kx}, & x<0\\
 
ae^{kx}, & x<0\\
be^{kx}, & x>0
+
be^{-kx}, & x>0
 
\end{matrix}
 
\end{matrix}
 
\right.
 
\right.
Line 45: Line 45:
 
We have two conditions at <math>x=0,</math> <math>w</math> must be continuous at <math>0</math> and
 
We have two conditions at <math>x=0,</math> <math>w</math> must be continuous at <math>0</math> and
 
<math>\partial_{x}w\left(  0^{+}\right)  -\partial_{x}w\left(  0^{-}\right)
 
<math>\partial_{x}w\left(  0^{+}\right)  -\partial_{x}w\left(  0^{-}\right)
+u_{0}w\left(  0\right)  =0.<math> This gives the condition that </math>a=b</math> and
+
+u_0 w\left(  0\right)  =0.</math> This final condition is obtained by integrating `across' zero as follows
 +
<center><math>\begin{align}
 +
\int_{0^{-}}^{0^{+}} \left(\partial_x^2 w +u_0\delta(x) w + \lambda w \right) \ \mathrm{d}x = 0.
 +
\end{align}
 +
</math></center>
 +
 
 +
This gives the condition that <math>a=b</math> and
 
<math>k=u_{0}/2.</math> We need to normalise the eigenfunctions so that
 
<math>k=u_{0}/2.</math> We need to normalise the eigenfunctions so that
 
<center><math>
 
<center><math>
\int_{-\infty}^{\infty}\left(  w\left(  x\right)  \right)  ^{2}dx=1.
+
\int_{-\infty}^{\infty}\left(  w\left(  x\right)  \right)  ^{2}\mathrm{d}x=1.
 
</math></center>
 
</math></center>
 
Therefore
 
Therefore
 
<center><math>
 
<center><math>
2\int_{0}^{\infty}\left(  ae^{-u_{0}x/2}\right)  ^{2}dx=1
+
2\int_{0}^{\infty}\left(  ae^{-u_{0}x/2}\right)  ^{2}\mathrm{d}x=1
 
</math></center>
 
</math></center>
 
which means that <math>a=\sqrt{u_{0}/2}.</math> Therefore, there is only one discrete
 
which means that <math>a=\sqrt{u_{0}/2}.</math> Therefore, there is only one discrete
Line 59: Line 65:
 
w_{1}\left(  x\right)  =\left\{
 
w_{1}\left(  x\right)  =\left\{
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
 
\sqrt{k_{1}}e^{k_{1}x}, & x<0\\
 
\sqrt{k_{1}}e^{k_{1}x}, & x<0\\
 
\sqrt{k_{1}}e^{-k_{1}x}, & x>0
 
\sqrt{k_{1}}e^{-k_{1}x}, & x>0
Line 69: Line 75:
 
w\left(  x\right)  =\left\{
 
w\left(  x\right)  =\left\{
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
e^{-ikx}+re^{ikx}, & x<0\\
+
\mathrm{e}^{-\mathrm{i}kx}+r\mathrm{e}^{\mathrm{i}kx}, & x<0\\
ae^{-ikx}, & x>0
+
t\mathrm{e}^{-\mathrm{i}kx}, & x>0
 
\end{matrix}
 
\end{matrix}
 
\right.
 
\right.
Line 79: Line 85:
 
+u_{0}w\left(  0\right)  =0.</math> This gives us
 
+u_{0}w\left(  0\right)  =0.</math> This gives us
 
<center><math>\begin{matrix}
 
<center><math>\begin{matrix}
1+r  &  =a\\
+
1+r  &  =t\\
-ika+ik-ikr  &  =-au_{0}
+
-ikt+ik-ikr  &  =-tu_{0}
 
\end{matrix}</math></center>
 
\end{matrix}</math></center>
 
which has solution
 
which has solution
 
<center><math>\begin{matrix}
 
<center><math>\begin{matrix}
 
r  &  =\frac{u_{0}}{2ik-u_{0}}\\
 
r  &  =\frac{u_{0}}{2ik-u_{0}}\\
a &  =\frac{2ik}{2ik-u_{0}}
+
t &  =\frac{2ik}{2ik-u_{0}}
 
\end{matrix}</math></center>
 
\end{matrix}</math></center>
  
 +
==Example 2: Hat Function Potential==
  
===Example: Scattering by a Well===
+
The properties of the eigenfunction is perhaps seem most easily through the
 
 
The properties of the eigenfunction is prehaps seem most easily through the
 
 
following example
 
following example
 
<center><math>
 
<center><math>
 
u\left(  x\right)  =\left\{
 
u\left(  x\right)  =\left\{
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
0 & x\notin\left[  -\varepsilon,\varepsilon\right] \\
+
0 & x\notin\left[  -\varsigma,\varsigma\right] \\
b & x\in\left[  -\varepsilon,\varepsilon\right]
+
b & x\in\left[  -\varsigma,\varsigma\right]
 
\end{matrix}
 
\end{matrix}
 
\right.
 
\right.
Line 104: Line 109:
 
where <math>b>0.</math>
 
where <math>b>0.</math>
  
\paragraph{Case when <math>\lambda>0</math>}
+
===Case when <math>\lambda<0</math>===
  
 
If we solve this equation for the case when <math>\lambda<0,</math> <math>\lambda=-k^{2}</math> we
 
If we solve this equation for the case when <math>\lambda<0,</math> <math>\lambda=-k^{2}</math> we
Line 111: Line 116:
 
w\left(  x\right)  =\left\{
 
w\left(  x\right)  =\left\{
 
\begin{matrix}
 
\begin{matrix}
[c]
+
a_{1}e^{kx}, & x<-\varsigma,\\
a_{1}e^{kx}, & x<-\varepsilon\\
+
b_{1}\cos\kappa x+b_{2}\sin\kappa x, & -\varsigma< x <\varsigma,\\
b_{1}\cos\kappa x+b_{2}\sin\kappa x & -\varepsilon<x<\varepsilon\\
+
a_{2}e^{-kx}, & x>\varsigma,
a_{2}e^{-kx} & x>\varepsilon
 
 
\end{matrix}
 
\end{matrix}
 
\right.
 
\right.
Line 120: Line 124:
 
where <math>\kappa=\sqrt{b-k^{2}}</math> which means that <math>0\leq k\leq\sqrt{b}</math> (there is
 
where <math>\kappa=\sqrt{b-k^{2}}</math> which means that <math>0\leq k\leq\sqrt{b}</math> (there is
 
no solution for <math>k>\sqrt{b}).</math> We then match <math>w</math> and its derivative at
 
no solution for <math>k>\sqrt{b}).</math> We then match <math>w</math> and its derivative at
<math>x=\pm\varepsilon</math> to solve for <math>a</math> and <math>b</math>. This leads to two system of
+
<math>x=\pm\varsigma</math> to solve for <math>a</math> and <math>b</math>. This leads to two system of
equation, one for the even (<math>a_{1}=a_{2}</math> and <math>b_{2}=0</math> ) and one for the odd
+
equations, one for the even (<math>a_{1}=a_{2}</math> and <math>b_{2}=0</math> ) and one for the odd
 
solutions (<math>a_{1}=-a_{2}</math> and <math>b_{1}=0)</math>. The solution for the even solutions
 
solutions (<math>a_{1}=-a_{2}</math> and <math>b_{1}=0)</math>. The solution for the even solutions
 
is
 
is
 +
<center><math>
 +
w\left(  x\right)  =\left\{
 +
\begin{matrix}
 +
a_{1}e^{kx}, & x<-\varsigma,\\
 +
b_{1}\cos\kappa x, & -\varsigma< x <\varsigma,\\
 +
a_{1}e^{-kx}, & x>\varsigma,
 +
\end{matrix}
 +
\right.
 +
</math></center>
 +
If we impose the condition that the function and its derivative are continuous at
 +
<math>x=\pm\varsigma</math> we obtain the following equation
 
<center><math>
 
<center><math>
 
\left(
 
\left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
e^{-k\varepsilon} & -\cos\kappa\varepsilon\\
+
e^{-k\varsigma} & -\cos\kappa\varsigma\\
ke^{-k\varepsilon} & -\kappa\sin\kappa\varepsilon
+
ke^{-k\varsigma} & -\kappa\sin\kappa\varsigma
 
\end{matrix}
 
\end{matrix}
 
\right)  \left(
 
\right)  \left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
 
a_{1}\\
 
a_{1}\\
 
b_{1}
 
b_{1}
Line 139: Line 154:
 
\right)  =\left(
 
\right)  =\left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
 
0\\
 
0\\
 
0
 
0
Line 149: Line 164:
 
\det\left(
 
\det\left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
e^{-k\varepsilon} & -\cos\kappa\varepsilon\\
+
e^{-k\varsigma} & -\cos\kappa\varsigma\\
ke^{-k\varepsilon} & -\kappa\sin\kappa\varepsilon
+
ke^{-k\varsigma} & -\kappa\sin\kappa\varsigma
 
\end{matrix}
 
\end{matrix}
 
\right)  =0
 
\right)  =0
Line 157: Line 172:
 
which gives us the equation
 
which gives us the equation
 
<center><math>
 
<center><math>
-\kappa e^{-k\varepsilon}\sin\kappa\varepsilon+k\cos\kappa\varepsilon
+
-\kappa e^{-k\varsigma}\sin\kappa\varsigma+k\cos\kappa\varsigma
e^{-k\varepsilon}=0
+
e^{-k\varsigma}=0
 
</math></center>
 
</math></center>
 
or
 
or
 
<center><math>
 
<center><math>
\tan\kappa\varepsilon=\frac{k}{\kappa}
+
\tan\kappa\varsigma=\frac{k}{\kappa}
 
</math></center>
 
</math></center>
 
We know that <math>0<\kappa<\sqrt{b}</math> and if we plot this we see that we obtain a
 
We know that <math>0<\kappa<\sqrt{b}</math> and if we plot this we see that we obtain a
Line 168: Line 183:
  
 
The solution for the odd solutions is
 
The solution for the odd solutions is
 +
<center><math>
 +
w\left(  x\right)  =\left\{
 +
\begin{matrix}
 +
a_{1}e^{kx}, & x <-\varsigma,\\
 +
b_{2}\sin\kappa x, & -\varsigma< x <\varsigma,\\
 +
-a_{1}e^{-kx} & x > \varsigma,
 +
\end{matrix}
 +
\right.
 +
</math></center>
 +
and again imposing the condition that the solution and its derivative is continuous
 +
at <math>x=\pm\varsigma</math> gives
 
<center><math>
 
<center><math>
 
\left(
 
\left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
e^{-k\varepsilon} & -\sin\kappa\varepsilon\\
+
e^{-k\varsigma} & \sin\kappa\varsigma\\
ke^{-k\varepsilon} & \cos\kappa\varepsilon
+
ke^{-k\varsigma} & -\kappa\cos\kappa\varsigma
 
\end{matrix}
 
\end{matrix}
 
\right)  \left(
 
\right)  \left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
 
a_{1}\\
 
a_{1}\\
 
b_{1}
 
b_{1}
Line 183: Line 209:
 
\right)  =\left(
 
\right)  =\left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
 
0\\
 
0\\
 
0
 
0
Line 193: Line 219:
 
\det\left(
 
\det\left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
e^{-k\varepsilon} & -\sin\kappa\varepsilon\\
+
e^{-k\varsigma} & \sin\kappa\varsigma\\
ke^{-k\varepsilon} & \kappa\cos\kappa\varepsilon
+
ke^{-k\varsigma} & -\kappa\cos\kappa\varsigma
 
\end{matrix}
 
\end{matrix}
 
\right)  =0
 
\right)  =0
Line 201: Line 227:
 
which gives us the equation
 
which gives us the equation
 
<center><math>
 
<center><math>
\kappa e^{-k}a\cos\kappa\varepsilon+k\sin\kappa\varepsilon e^{-k}=0
+
\kappa e^{-k\varsigma}a\cos\kappa\varsigma+k\sin\kappa\varsigma e^{-k\varsigma}=0
 
</math></center>
 
</math></center>
 
or
 
or
 
<center><math>
 
<center><math>
\tan\kappa=-\frac{\kappa}{k}
+
\tan\varsigma\kappa=-\frac{\kappa}{k}
 
</math></center>
 
</math></center>
  
 
+
=== Case when <math>\lambda>0</math> ===
\paragraph{Case when <math>\lambda>0</math>}
 
  
 
When <math>\lambda>0</math> we write <math>\lambda=k^{2}</math> and we obtain solution
 
When <math>\lambda>0</math> we write <math>\lambda=k^{2}</math> and we obtain solution
Line 215: Line 240:
 
w\left(  x\right)  =\left\{
 
w\left(  x\right)  =\left\{
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
e^{-ikx}+re^{ikx}, & x<-\varepsilon\\
+
\mathrm{e}^{-\mathrm{i}kx}+r\mathrm{e}^{\mathrm{i}kx}, & x <-\varsigma\\
b_{1}\cos\kappa x+b_{2}\sin\kappa x & -\varepsilon<x<\varepsilon\\
+
b_{1}\cos\kappa x+b_{2}\sin\kappa x & -\varsigma< x <\varsigma\\
ae^{-ikx} & x>\varepsilon
+
t\mathrm{e}^{-\mathrm{i}kx} & x>\varsigma
 
\end{matrix}
 
\end{matrix}
 
\right.
 
\right.
Line 227: Line 252:
 
\left(
 
\left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
-e^{-ik\varepsilon} & \cos\kappa\varepsilon & -\sin\kappa\varepsilon & 0\\
+
-\mathrm{e}^{-\mathrm{i}k\varsigma} & \cos\kappa\varsigma & -\sin\kappa\varsigma & 0\\
ike^{-ik\varepsilon} & \kappa\sin\kappa\varepsilon & \kappa\cos\kappa
+
ik\mathrm{e}^{-\mathrm{i}k\varsigma} & \kappa\sin\kappa\varsigma & \kappa\cos\kappa
\varepsilon & 0\\
+
\varsigma & 0\\
0 & \cos\kappa\varepsilon & \sin\kappa\varepsilon & -e^{-ik\varepsilon}\\
+
0 & \cos\kappa\varsigma & \sin\kappa\varsigma & -\mathrm{e}^{-\mathrm{i}k\varsigma}\\
0 & -\kappa\sin\kappa\varepsilon & \kappa\cos\kappa\varepsilon &
+
0 & -\kappa\sin\kappa\varsigma & \kappa\cos\kappa\varsigma &
ike^{-ik\varepsilon}
+
ik\mathrm{e}^{-\mathrm{i}k\varsigma}
 
\end{matrix}
 
\end{matrix}
 
\right)  \left(
 
\right)  \left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
 
r\\
 
r\\
 
b_{1}\\
 
b_{1}\\
 
b_{2}\\
 
b_{2}\\
a
+
t
 
\end{matrix}
 
\end{matrix}
 
\right)  =\left(
 
\right)  =\left(
 
\begin{matrix}
 
\begin{matrix}
[c]
+
 
e^{ik}\\
+
\mathrm{e}^{\mathrm{i}k}\\
ike^{-ik}\\
+
ik\mathrm{e}^{-\mathrm{i}k}\\
 
0\\
 
0\\
 
0
 
0
Line 253: Line 278:
 
\right)
 
\right)
 
</math></center>
 
</math></center>
 +
 +
== Lecture Videos ==
 +
 +
=== Part 1 ===
 +
 +
{{#ev:youtube|anAThvCcpNw}}
 +
 +
=== Part 2 ===
 +
 +
{{#ev:youtube|SDPIx42VjLQ}}
 +
 +
=== Part 3 ===
 +
 +
{{#ev:youtube|OUmjeLZWr3M}}
 +
 +
=== Part 4 ===
 +
 +
{{#ev:youtube|hIfcO3a8_XU}}
 +
 +
=== Part 5 ===
 +
 +
{{#ev:youtube|z13lKSTficA}}
 +
 +
=== Part 6 ===
 +
 +
{{#ev:youtube|2XlQpEscxE4}}
 +
 +
=== Part 7 ===
 +
 +
{{#ev:youtube|iMMQ4NUdXNc}}
 +
 +
=== Part 8 ===
 +
 +
{{#ev:youtube|0F_dINNxMlw}}

Latest revision as of 01:03, 24 September 2020

Nonlinear PDE's Course
Current Topic Properties of the Linear Schrodinger Equation
Next Topic Connection betwen KdV and the Schrodinger Equation
Previous Topic Introduction to the Inverse Scattering Transform


The linear Schrodinger equation

[math]\displaystyle{ \partial_{x}^{2}w+uw=-\lambda w }[/math]

has two kinds of solutions for [math]\displaystyle{ u\rightarrow0 }[/math] as [math]\displaystyle{ x\rightarrow\pm\infty. }[/math] The first are waves and the second are bound solutions. It is well known that there are at most a finite number of bound solutions (provided [math]\displaystyle{ u\rightarrow0 }[/math] as [math]\displaystyle{ x\pm\infty }[/math] sufficiently rapidly) and a continum of solutions for the incident waves. This is easiest seen through the following examples

Example 1: [math]\displaystyle{ \delta }[/math] function potential

We consider here the case when [math]\displaystyle{ u\left( x,0\right) = u_0 \delta\left( x\right) . }[/math] Note that this function can be thought of as the limit as of the potential

[math]\displaystyle{ u\left( x\right) =\left\{ \begin{matrix} 0 & x\notin\left[ -\varepsilon,\varepsilon\right] \\ \frac{u_{0}}{2\varepsilon} & x\in\left[ -\varepsilon,\varepsilon\right] \end{matrix} \right. }[/math]

In this case we need to solve

[math]\displaystyle{ \partial_{x}^{2}w+ u_0\delta(x) w=-\lambda w }[/math]

We consider the case of [math]\displaystyle{ \lambda\lt 0 }[/math] and [math]\displaystyle{ \lambda\gt 0 }[/math] separately. For the first case we write [math]\displaystyle{ \lambda=-k^{2} }[/math] and we obtain

[math]\displaystyle{ w\left( x\right) =\left\{ \begin{matrix} ae^{kx}, & x\lt 0\\ be^{-kx}, & x\gt 0 \end{matrix} \right. }[/math]

We have two conditions at [math]\displaystyle{ x=0, }[/math] [math]\displaystyle{ w }[/math] must be continuous at [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ \partial_{x}w\left( 0^{+}\right) -\partial_{x}w\left( 0^{-}\right) +u_0 w\left( 0\right) =0. }[/math] This final condition is obtained by integrating `across' zero as follows

[math]\displaystyle{ \begin{align} \int_{0^{-}}^{0^{+}} \left(\partial_x^2 w +u_0\delta(x) w + \lambda w \right) \ \mathrm{d}x = 0. \end{align} }[/math]

This gives the condition that [math]\displaystyle{ a=b }[/math] and [math]\displaystyle{ k=u_{0}/2. }[/math] We need to normalise the eigenfunctions so that

[math]\displaystyle{ \int_{-\infty}^{\infty}\left( w\left( x\right) \right) ^{2}\mathrm{d}x=1. }[/math]

Therefore

[math]\displaystyle{ 2\int_{0}^{\infty}\left( ae^{-u_{0}x/2}\right) ^{2}\mathrm{d}x=1 }[/math]

which means that [math]\displaystyle{ a=\sqrt{u_{0}/2}. }[/math] Therefore, there is only one discrete spectral point which we denote by [math]\displaystyle{ k_{1}=u_{0}/2 }[/math]

[math]\displaystyle{ w_{1}\left( x\right) =\left\{ \begin{matrix} \sqrt{k_{1}}e^{k_{1}x}, & x\lt 0\\ \sqrt{k_{1}}e^{-k_{1}x}, & x\gt 0 \end{matrix} \right. }[/math]

The continuous eigenfunctions correspond to [math]\displaystyle{ \lambda=k^{2}\gt 0 }[/math] are of the form

[math]\displaystyle{ w\left( x\right) =\left\{ \begin{matrix} \mathrm{e}^{-\mathrm{i}kx}+r\mathrm{e}^{\mathrm{i}kx}, & x\lt 0\\ t\mathrm{e}^{-\mathrm{i}kx}, & x\gt 0 \end{matrix} \right. }[/math]

Again we have the conditions that [math]\displaystyle{ w }[/math] must be continuous at [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ \partial_{x}w\left( 0^{+}\right) -\partial_{x}w\left( 0^{-}\right) +u_{0}w\left( 0\right) =0. }[/math] This gives us

[math]\displaystyle{ \begin{matrix} 1+r & =t\\ -ikt+ik-ikr & =-tu_{0} \end{matrix} }[/math]

which has solution

[math]\displaystyle{ \begin{matrix} r & =\frac{u_{0}}{2ik-u_{0}}\\ t & =\frac{2ik}{2ik-u_{0}} \end{matrix} }[/math]

Example 2: Hat Function Potential

The properties of the eigenfunction is perhaps seem most easily through the following example

[math]\displaystyle{ u\left( x\right) =\left\{ \begin{matrix} 0 & x\notin\left[ -\varsigma,\varsigma\right] \\ b & x\in\left[ -\varsigma,\varsigma\right] \end{matrix} \right. }[/math]

where [math]\displaystyle{ b\gt 0. }[/math]

Case when [math]\displaystyle{ \lambda\lt 0 }[/math]

If we solve this equation for the case when [math]\displaystyle{ \lambda\lt 0, }[/math] [math]\displaystyle{ \lambda=-k^{2} }[/math] we get

[math]\displaystyle{ w\left( x\right) =\left\{ \begin{matrix} a_{1}e^{kx}, & x\lt -\varsigma,\\ b_{1}\cos\kappa x+b_{2}\sin\kappa x, & -\varsigma\lt x \lt \varsigma,\\ a_{2}e^{-kx}, & x\gt \varsigma, \end{matrix} \right. }[/math]

where [math]\displaystyle{ \kappa=\sqrt{b-k^{2}} }[/math] which means that [math]\displaystyle{ 0\leq k\leq\sqrt{b} }[/math] (there is no solution for [math]\displaystyle{ k\gt \sqrt{b}). }[/math] We then match [math]\displaystyle{ w }[/math] and its derivative at [math]\displaystyle{ x=\pm\varsigma }[/math] to solve for [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math]. This leads to two system of equations, one for the even ([math]\displaystyle{ a_{1}=a_{2} }[/math] and [math]\displaystyle{ b_{2}=0 }[/math] ) and one for the odd solutions ([math]\displaystyle{ a_{1}=-a_{2} }[/math] and [math]\displaystyle{ b_{1}=0) }[/math]. The solution for the even solutions is

[math]\displaystyle{ w\left( x\right) =\left\{ \begin{matrix} a_{1}e^{kx}, & x\lt -\varsigma,\\ b_{1}\cos\kappa x, & -\varsigma\lt x \lt \varsigma,\\ a_{1}e^{-kx}, & x\gt \varsigma, \end{matrix} \right. }[/math]

If we impose the condition that the function and its derivative are continuous at [math]\displaystyle{ x=\pm\varsigma }[/math] we obtain the following equation

[math]\displaystyle{ \left( \begin{matrix} e^{-k\varsigma} & -\cos\kappa\varsigma\\ ke^{-k\varsigma} & -\kappa\sin\kappa\varsigma \end{matrix} \right) \left( \begin{matrix} a_{1}\\ b_{1} \end{matrix} \right) =\left( \begin{matrix} 0\\ 0 \end{matrix} \right) }[/math]

This has non trivial solutions when

[math]\displaystyle{ \det\left( \begin{matrix} e^{-k\varsigma} & -\cos\kappa\varsigma\\ ke^{-k\varsigma} & -\kappa\sin\kappa\varsigma \end{matrix} \right) =0 }[/math]

which gives us the equation

[math]\displaystyle{ -\kappa e^{-k\varsigma}\sin\kappa\varsigma+k\cos\kappa\varsigma e^{-k\varsigma}=0 }[/math]

or

[math]\displaystyle{ \tan\kappa\varsigma=\frac{k}{\kappa} }[/math]

We know that [math]\displaystyle{ 0\lt \kappa\lt \sqrt{b} }[/math] and if we plot this we see that we obtain a finite number of solutions.

The solution for the odd solutions is

[math]\displaystyle{ w\left( x\right) =\left\{ \begin{matrix} a_{1}e^{kx}, & x \lt -\varsigma,\\ b_{2}\sin\kappa x, & -\varsigma\lt x \lt \varsigma,\\ -a_{1}e^{-kx} & x \gt \varsigma, \end{matrix} \right. }[/math]

and again imposing the condition that the solution and its derivative is continuous at [math]\displaystyle{ x=\pm\varsigma }[/math] gives

[math]\displaystyle{ \left( \begin{matrix} e^{-k\varsigma} & \sin\kappa\varsigma\\ ke^{-k\varsigma} & -\kappa\cos\kappa\varsigma \end{matrix} \right) \left( \begin{matrix} a_{1}\\ b_{1} \end{matrix} \right) =\left( \begin{matrix} 0\\ 0 \end{matrix} \right) }[/math]

This can non trivial solutions when

[math]\displaystyle{ \det\left( \begin{matrix} e^{-k\varsigma} & \sin\kappa\varsigma\\ ke^{-k\varsigma} & -\kappa\cos\kappa\varsigma \end{matrix} \right) =0 }[/math]

which gives us the equation

[math]\displaystyle{ \kappa e^{-k\varsigma}a\cos\kappa\varsigma+k\sin\kappa\varsigma e^{-k\varsigma}=0 }[/math]

or

[math]\displaystyle{ \tan\varsigma\kappa=-\frac{\kappa}{k} }[/math]

Case when [math]\displaystyle{ \lambda\gt 0 }[/math]

When [math]\displaystyle{ \lambda\gt 0 }[/math] we write [math]\displaystyle{ \lambda=k^{2} }[/math] and we obtain solution

[math]\displaystyle{ w\left( x\right) =\left\{ \begin{matrix} \mathrm{e}^{-\mathrm{i}kx}+r\mathrm{e}^{\mathrm{i}kx}, & x \lt -\varsigma\\ b_{1}\cos\kappa x+b_{2}\sin\kappa x & -\varsigma\lt x \lt \varsigma\\ t\mathrm{e}^{-\mathrm{i}kx} & x\gt \varsigma \end{matrix} \right. }[/math]

where [math]\displaystyle{ \kappa=\sqrt{b+k^{2}}. }[/math] Matching [math]\displaystyle{ w }[/math] and its derivaties at [math]\displaystyle{ x=\pm1 }[/math] we obtain

[math]\displaystyle{ \left( \begin{matrix} -\mathrm{e}^{-\mathrm{i}k\varsigma} & \cos\kappa\varsigma & -\sin\kappa\varsigma & 0\\ ik\mathrm{e}^{-\mathrm{i}k\varsigma} & \kappa\sin\kappa\varsigma & \kappa\cos\kappa \varsigma & 0\\ 0 & \cos\kappa\varsigma & \sin\kappa\varsigma & -\mathrm{e}^{-\mathrm{i}k\varsigma}\\ 0 & -\kappa\sin\kappa\varsigma & \kappa\cos\kappa\varsigma & ik\mathrm{e}^{-\mathrm{i}k\varsigma} \end{matrix} \right) \left( \begin{matrix} r\\ b_{1}\\ b_{2}\\ t \end{matrix} \right) =\left( \begin{matrix} \mathrm{e}^{\mathrm{i}k}\\ ik\mathrm{e}^{-\mathrm{i}k}\\ 0\\ 0 \end{matrix} \right) }[/math]

Lecture Videos

Part 1

Part 2

Part 3

Part 4

Part 5

Part 6

Part 7

Part 8