Question 11.2.4: Find the eigenvalues and eigenfunctions of the boundary valu......
Find the eigenvalues and eigenfunctions of the boundary value problem
y^{\prime\prime}+\lambda y=0, (44)
y(-1)-y(1)=0,\quad y^{\prime}(-1)-y^{\prime}(1)=0. (45)
This is not a Sturm-Liouville problem because the boundary conditions are not separated. The boundary conditions (45) are called periodic boundary conditions since they require that y and y′ assume the same values at x = 1 as at x = −1. Nevertheless, it is straightforward to show that the problem (44), (45) is self-adjoint. A simple calculation establishes that \lambda_{0} = 0 is an eigenvalue and that the corresponding eigenfunction is \phi_{0}(x) = 1. Further, there are additional eigenvalues \lambda_{1}=\pi^{2},\lambda_{2}=(2\pi)^{2},\,\dots\,,\lambda_{n}=(n\pi)^{2},\,\dots\,. To each of these nonzero eigenvalues there correspond two linearly independent eigenfunctions; for example, corresponding to \lambda_{n} are the two eigenfunctions \phi_{n}(x)=\cos(n\pi x) and \psi_{n}(x)\;=\;\sin(n\pi x). This illustrates that the eigenvalues may not be simple when the boundary conditions are not separated. Further, if we seek to expand a given function f of period 2 in a series of eigenfunctions of the problem (44), (45), we obtain the series
f(x)={\frac{a_{0}}{2}}+\sum_{n=1}^{\infty}(a_{n}\cos(n\pi x)+b_{n}\sin(n\pi x)),
which is just the Fourier series for f on the interval −1 ≤ x ≤ 1.