Question

Consider the following two-point boundary value problem in which  is a positive
number.
(equations inside)
Use the Rayleigh Quotient to show that all eigenvalues are non negative.
Explain why the number zero is not an eigenvalue.
Find 2 × 2 matrices  and  so that conditions (ii) and (iii) are equivalent to
(equations inside)

Answer 1

\[(i) -\phi’’(x) = \lambda \phi(x),for -L \le x \le L \]
\[(ii) \phi’(-L) = 0, and \]
\[(iii) \phi(L) = 0.\]

Answer 2

\[M\left(\begin{matrix}\phi(-L) \\ \phi'(-L)\end{matrix}\right) + N\left(\begin{matrix}\phi(L) \\ \phi’(L)\end{matrix}\right) = \left(\begin{matrix}0 \\ 0\end{matrix}\right)\]