### Author Topic: Web Bonus Problem to Week 7 (#2)  (Read 2742 times)

#### Victor Ivrii

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##### Web Bonus Problem to Week 7 (#2)
« on: October 26, 2015, 09:02:00 AM »

#### Chi Ma

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##### Re: Web Bonus Problem to Week 7 (#2)
« Reply #1 on: October 29, 2015, 12:36:47 AM »
Rewrite the problem as follows:
\begin{align} &\frac{u_{xx}}{u} = - (H + \lambda) \qquad & |x|\le L \\
&\frac{u_{xx}}{u} = - \lambda \qquad & |x| > L
\end{align}

For $\lambda \in (âˆ’H,0)$, $âˆ’\lambda$ is positive and $-(H + \lambda)$ is negative. The solution is in exponential form outside the rectangular well and sinusoidal within the well. All eigenvalues therefore belong to the interval $âˆ’H < \lambda < 0$.

Consider even and odd eigenfunctions separately. Even eigenfunctions take the following form:
\begin{align} &u = A \cos\omega x \qquad & |x|\le L \\
&u = Be^{-\gamma |x|} \qquad & |x|> L
\end{align}
where $\omega^2 = H + \lambda$ and $\gamma^2 = -\lambda$.

Continuity and continuous first derivative at $L$ imply
\begin{align} A \cos\omega L &= Be^{-\gamma L} \\
A \omega \sin\omega L &= \gamma Be^{-\gamma L}
\end{align}

The corresponding eigenvalues are solutions to the following:
$$\sqrt{H + \lambda} \tan(\sqrt{H + \lambda} L) = \sqrt{-\lambda} \qquad \qquad âˆ’H < \lambda < 0$$

Similarly, odd eigenfunctions take the following form:
\begin{align} &u = A \sin\omega x \qquad &|x|\le L \\
&u = Be^{-\gamma x} \qquad &x > L \\
&u = -Be^{\gamma x} \qquad &x < -L
\end{align}

Continuity and continuous first derivative at $L$ imply
$$\omega \cot\omega L = -\gamma$$

The corresponding eigenvalues are solutions to the following:
$$\sqrt{H + \lambda} \cot(\sqrt{H + \lambda} L )= -\sqrt{-\lambda} \qquad \qquad âˆ’H < \lambda < 0$$
« Last Edit: October 30, 2015, 12:54:09 AM by Chi Ma »

#### Victor Ivrii

Correct. Why $\lambda >0$ is impossible? Because then $u$ is "sinusoidal" outside the well and thus is not square integrable.
Why $\lambda<-H$ is impossible? The simplest way to prove:
(since $V$ is real one can consider real and imaginary parts of $u$ separately)