Completeness of the Set of Eigenfunctions via Rayleigh's Quotient

The fact that eigenvalues of the regular Sturm-Liouville problem form a semi-unbounded sequence, i.e., that

$$\lim_{n\to\infty}\lambda_n = \infty\,,$$

is very important. It implies that the set of eigenfunctions of the Sturm-Liouville problem

$${\cal L}u=\lambda u_n~~\quad~~$$

$$\begin{array}{rcl} \alpha u(a)+\alpha 'u' (a) &= &0 \\ \beta u(b)+\beta 'u'(b) &= &0 ~~, \end{array}$$ (327)

with

$${\cal L} = \frac{1}{\rho(x)} \left[ -\frac{d}{dx} p(x)\frac{d}{dx} +q(x) \right]\, ,$$

is a generalized Fourier basis. In other words, they form a complete basis set for the subspace of $L^2(a,b)$ of those square-integrable functions which satisfy the given boundary conditions, Eq.(3.27). This subspace is

$${\mathcal H}=\left\{ u:\int_a^b \vert u(x)\vert^2\rho(x)\, dx <\infty; ~ \alpha u(a)+\alpha 'u' (a)=0; ~ \beta u(b)+\beta 'u'(b) =0 \right\}.$$

Recall that a set $\{ u_n(x)\colon n=0,1,\dots ,N,\dots\}$ is said to be complete, if for any vector $u\in{\cal H}$ , the error vector

$$h^\ast_N=u-\sum^N_{n=0} c_n u_n$$

can be made to have arbitrarily small squared norm by letting $N\to\infty$ , i.e.,

$$\lim_{N\to\infty}\Vert h^\ast_N\Vert^2 \equiv \lim_{N\to\infty}\Big\langle u- \sum^N_{n=0} c_nu_n\,,~u-\sum_{m=0}^N c_mu_m\Big\rangle$$

$$= 0\,.$$

Here

$$c_n = \langle u_n,u\rangle$$

is the $n$ th (generalized) Fourier coefficient with the consequence that $h^\ast_N$ is perpendicular to the subspace

$$W_N=\textrm{span}\{u_0,u_1,\dots ,u_N\}\,.$$

Figure 3.14: The $N+1$ -dimensional subspace spanned by the eigenfunctions $u_0,u_1,\cdots ,u_N$ causes the Hilbert space $L^2(a,b)$ to be decomposed into the direct sum consisting of $W^*_N$ and the space $W^\perp _N$ , which is spanned by the remaining basis vectors $u_{N+1},u_{N+2},\cdots$ .

The subspace $W_N$ induces ${\cal H}$ to be decomposed into the direct sum

$$W_N\oplus W^\perp_N={\cal H}\,.$$

Here $W^\perp _N$ (``$W_N$ perp'') is the subspace of all vectors perpendicular to $W_N$

$$W^\perp_N = \{ u\colon\langle u,u_n\rangle = 0~~n=0,1,\dots ,N\}\,.$$

In other words, $W^\perp _N$ is the space of all vectors satisfying the set of constraint conditions

$$\langle u,u_0\rangle =$$ 0

$$\langle u,u_1\rangle =$$ 0

$$~ \vdots ~$$

$$\langle u,u_N\rangle = 0\,.$$

Our starting point for demonstrating the completeness is the Rayleigh principle. It says that the Rayleigh quotient

$$\frac{\langle u,{\cal L}u\rangle}{\langle u,u\rangle} \equiv {\cal R}[u]$$

satisfies various minimum principles when $u$ is restricted to lie on various subspaces $W^\perp _N$ , $N=0,1,\dots$ . Indeed, one has

$$\lambda_0 = \min_{u\in{\cal H}}\frac{\langle u,{\cal L}u\rangle}{\langle u,u ... ...ngle}{\langle u,u \rangle}\ge\lambda_0\,,~\quad~ \textrm{for~all}~~u\in{\cal H}$$

$$\lambda_1 = \min_{u\in W_0^\perp}\frac{\langle u,{\cal L}u\rangle}{\langle u,... ...{\langle u,u \rangle}\ge\lambda_1\,,~\quad~ \textrm{for~all}~~u\in W^\perp_0 ~~$$

i.e., for any $u\in{\cal H}$ subject to the constraint $\langle u,u_0\rangle = 0$ .

More generally, the $N+1$ st eigenvalue $\lambda _{N+1}$ is characterized by

$$\lambda_{N+1}=\min_{u\in W^\perp_N}\frac{\langle u,{\cal L}u\rang... ...angle u,u\rangle}\ge \lambda_{N+1} ~,\quad \textrm{for~all}~~u\in W^\perp_N ~$$

i.e., for any $u\in{\cal H}$ subject to the constraints

$$\langle u,u_0\rangle =$$ 0

$$\vdots$$

$$\langle u,u_N\rangle = 0\,.$$

The $N$ th error vector

$$h^\ast_N=u-\sum^N_{n=0} c_n u_n$$

satisfies the constraint conditions

$$\langle u_n,h^\ast_N\rangle =0~~\qquad~~ n=0,1,\dots ,N\,.$$

Consequently, it satisfies the corresponding Rayleigh inequality

$$\frac{\langle h^\ast_N,{\cal L}h^\ast_N\rangle}{\langle h^\ast_N,h^\ast_N \rangle}\ge \lambda_{N+1}$$

or

$$\frac{\langle h^\ast_N,{\cal L}h^\ast_N\rangle}{\lambda_{N+1}}\ge\Vert h^\ast_N\Vert^2\ge 0\,.$$

We insert the expression for $h^\ast_N$ into the left hand side, and obtain

$$\ell .h.s = \frac{1}{\lambda_{N+1}}\{\langle u-\sum^N_0 c_n u_n,{\cal L} u-\sum^N_{m=0}c_m{\cal L} u_m\rangle\}$$

$$= \frac{1}{\lambda_{N+1}}\{\langle u,{\cal L}u\rangle - \sum^N_0\ov... ..._n} \langle u_n,{\cal L}u\rangle - \sum^N_{m=0}c_m\lambda_m\langle u,u_m\rangle$$

$$~\qquad~\qquad~\qquad~\qquad~\qquad~\qquad~+ \sum^N_{n=0}~\sum^N_{m=0} \overline{c_n} c_m\lambda_m\langle u_n,u_m\rangle\,.$$

The orthonormality of eigenfunctions and the definition of the generalized Fourier coefficients guarantee that the last two sums cancel. Furthermore, by doing an integration by parts twice, and by observing that the resulting end point terms vanish because of the Dirichlet-Neumann boundary conditions, Eq. 3.27, we obtain

$$\langle u_n,{\cal L}u\rangle = \int^b_a \overline{u}_n \frac{1}{\rho}\left[ -\frac{d}{dx} p\frac{d}{dx} +q\right] u\rho (x)dx$$

$$= -p \overline{u}_n u\left.\right\vert _a^b +\int^b_a \left( p\frac{d\overline u_n}{dx}\,\frac{du}{dx} +q\overline u_n u\right) dx$$

$$= \langle {\cal L}u_n,u\rangle = \lambda_n c_n\,.$$

As a consequence the Rayleigh inequality becomes

$$\frac{1}{\lambda_{N+1}}\{\langle u,{\cal L}u\rangle - \sum^N_{n=0}\vert c_n \vert^2\lambda_n\}\ge \Vert h^\ast_N\Vert^2\,.$$

Without loss of generality one may assume that the lowest eigenvalue $\lambda_0 \ge 0$ . This can always be made to come about by readjusting the $\lambda$ and the function $q(x)$ in the Sturm-Liouville equation. As a result, the finite sum may be dropped without decreasing the $\ell .h.s.$ Consequently,

$$\frac{\langle u,{\cal L}u\rangle}{\lambda_{N+1}}\ge \Vert u-\sum^N_{n=0}c_n u_n\Vert^2\,.$$

The numerator is independent of $N$ . Thus

$$\lim_{N\to\infty}\Vert u-\sum^N_{n=0} c_nu_n\Vert^2 \le \lim_{N\to\infty} \frac{\langle u,{\cal L}u\rangle}{\lambda_{N+1}}=0$$

because $\{\lambda_N\colon N=0,1,\dots\}$ is an unbounded sequence. Thus we have

$$u\doteq \sum^\infty_{n=0} c_nu_n\,,$$

The function $u$ is an arbitrary square integrable function satisfying the the given mixed Dirichlet-Neuman end point conditions. Consequently, the Sturm-Liouville eigenfunctions form a (complete) generalized Fourier basis indeed. [references_for_chapter3] [plain]