Sturm-Louiville differential equations

The vast majority of differential equations that occur in electrodynamics (and quantum theory via the Schr\”odinger equation) are a class of homogeneous second order equations called the Sturm-Liouville equation. The solutions to these equations are generally polynomials $\wp_n(x)$ with a set of remarkable properties:

$\bullet$ They are orthogonal with respect to a weight function over $a\le x\le b$ ;

$\int_a^b W(x) \, \wp_n(x) \, \wp_m(x) \, dx=N_n \, \delta_{nm}$

which means that they can be regarded as a basis of the infinite dimensional space of functions on $a\le x\le b$ ;

$f(x)=\sum_n a_n \, {\wp_n(x)\over \sqrt{N_n}}, \qquad a_n=\int_a^b W(x) \, f(x) \, {\wp_n(x)\over \sqrt{N_n}} \, dx$

(the Fourier-Dini series).

$\bullet$ They can be generated from a simple formula

$\wp_n(x)=W^{-1} \, {d^n\over dx^n}\Big(W \, A^n\Big)$

called a Rodriguez formula.

$\bullet$ The Rodriguez formula can be turned into a generating function

$\sum_{n=0}^\infty {t^n\over n!} \, \wp_n(x)=G(t,x)=W^{-1} \sum_n {t^n\over n!} {d^n\over dx^n}\Big(W \, A^n\Big)$

the right-side of which can be found explicitly using the Cauchy theorem (and by other methods).

Let’s construct functions that have precisely these properties, and in the process obtain the general form of the Sturm-Liouville equation

$\label{eq:SL1} \Big(W \, A \, \rho'\Big)'-\lambda \, W\rho=0,\qquad \rho=\rho_n(x)$

$W=e^{\int {B(x)\over A(x)} \, dx}, \qquad \lambda=n(n+1){A''\over 2}+nB'$

with polynomial solutions $\rho_n(x)$ of order $n$ . Nearly every special function that we know and love falls into this category (to name just a few):

$\begin{array}{l|l|l|l|l}\mbox{Equation}&A&B&\lambda & \mbox{Name}\\ \hline (1-x^2)\rho''-2x\rho'+n(n+1)\rho=0&(1-x^2)&0&-n(n+1)&\mbox{Legendre}, \, P_n\\ &&&&\\ \rho''-2x\rho'+2n\rho=0& 1 &-2x &-2n&\mbox{Hermite}, \, H_n\\ &&&&\\ x\rho''+(a+1-x)\rho'+n\rho=0& x &(a-x) &-n&\mbox{Laguerre}, \, L_n^a\\ &&&&\\ (1-x^2)\rho''-x\rho'+n^2\rho=0& (1-x^2) &x &n^2&\mbox{Chebyshev}, \, T_n\\ \end{array}$

Let’s begin by finding the differential equation whose solution is $\chi=W \, A^n$ , for $W'={B\over A} \, W$ (you will see that this is needed to make the final solutions polynomials);

(1) $\begin{eqnarray*}\chi'&=&\chi \, \Big({B\over A}+n{A'\over A}\Big)\nonumber\\ A\chi''&=&\chi' \, \Big(B+nA'\Big)+\chi \, \Big(B'+nA''-{(B+nA')A'\over A}\Big)\nonumber\\ \chi \, {(B+nA')A'\over A}&=&\chi' \, A'\nonumber\\ \mbox{therefore}\qquad A \, \chi''&=&(B+(n-1)A')\, \chi'+(B'+nA'')\chi\end{eqnarray*}$

This is a Sturm-Liouville equation if $B'+nA''$ is constant ( $B$ no higher than first order, $A$ no higher than quadratic). Let this be so, now differentiate the entire equation $n$ times, calling ${d^{n}\chi\over dx^{n}}=\xi$

(2) $\begin{eqnarray*} {d^{n}\over dx^{n}}\Big(A \, \chi''\Big)&=&\sum_{k=0}^{n}{n\choose k}{d^k A\over dx^k}{d^{n-k}\chi''\over dx^{n-k}}\nonumber\\ &=&A \xi''+nA'\xi'+{n(n-1)\over 2}A'' \, \xi\nonumber\\ {d^{n}\over dx^{n}}\Big((B+(n-1)A') \, \chi'\Big)&=&(B+(n-1)A') \, \xi'\nonumber\\ &+&n(B'+(n-1)A'')\xi\nonumber\\ A \xi''+nA'\xi'+{n(n-1)\over 2}A'' \, \xi&=&(B+(n-1)A') \, \xi'+n(B'+(n-1)A'')\xi\nonumber\\ &+&(B'+nA'')\xi\nonumber\\ 0&=&A\xi''+(A'-B)\xi'-\Big(n(n+1){A''\over 2}\nonumber\\ &+&(n+1)B'\Big)\xi\end{eqnarray*}$

This is another Sturm-Liouville equation if $B'+nA''$ is constant.

Now let $\xi=W \, \rho$ ;

(3) $\begin{eqnarray*}(W\rho)'&=& W\Big({B\over A}\rho+\rho'\Big)\nonumber\\ (W\rho)''&=& W\Big(\rho''+2{B\over A}\rho'+\Big({B^2\over A^2}+{B'\over A}-{BA'\over A^2}\Big)\rho\nonumber\\ 0&=&A\rho''+(B+A')\rho'-\Big(n(n+1){A''\over 2}+nB'\Big)\rho\nonumber\\ 0&=&\Big(W \, A \, \rho'\Big)'-\Big(n(n+1){A''\over 2}+nB'\Big)W\rho\nonumber\\ \Big(W \, A \, \rho'\Big)'-\lambda \, W\rho&=&0, \qquad \rho={1\over W}\Big({d\over dx}\Big)^n \, WA^n\end{eqnarray*}$

the standard Sturm-Liouville form with its polynomial solution given by a Rodriguez formula, and the explicit eigenvalue $\lambda=n(n+1){A''\over 2}+nB'$ .

Suppose that $W(a) \, A(a)=0$ and $W(b) \, A(b)=0$ , vanishing faster at $b$ ( $b>a$ ) than any power of $x$ . Notice that if both $A$ and $B$ are polynomials in $x$ ,

(4) $\begin{eqnarray*}{d\over dx}W&=& {B\over A} \, W\nonumber\\ \mbox{therefore}\qquad {d\over dx}\Big(W \, A^n\Big)&=&W \, \Big(B+n \, A'\Big) \, A^{n-1}\nonumber\\ &=&W \,\Big (p_1\Big) \, A^{n-1}\nonumber\\ {d^2\over dx^2}\Big(W \, A^n\Big)&=&W \, \Big(B \, p_1+p_1' \, A+(n-1) \, p_1 A'\Big)\, A^{n-2}\nonumber\\ &=&W \, \Big(p_2\Big)\, A^{n-2}\nonumber\\ &\vdots&\nonumber\\ {d^k\over dx^k}\Big(W \, A^n\Big)&=& W \, \Big(p_k\Big) \, A^{n-k}\nonumber\\ &\vdots&\nonumber\\ {d^n\over dx^n}\Big(W \, A^n\Big)&=& W \, \Big(p_n\Big)\end{eqnarray*}$

then $p_0=1, \, p_1, \, p_2, \, \cdots, \, p_n$ are all polynomials in $x$ . If $B$ is no higher order than first, and $A$ no higher order than second, then $p_k$ is a polynomial of order $k$ . You can see this by simply counting orders of the parts of each $p_k$ .

We have established that if $B$ is no higher order than first, and $A$ no higher order than second, then

$\label{eq:SLRod}\wp_n(x)={1\over W}{d^n\over dx^n}\Big(W \, A^n\Big)$

is a polynomial of order $n$ . The most important property of these polynomials is that they are orthogonal on the interval $a\le x\le b$ with weight function $W(x)$ : let $n>m$

(5) $\begin{eqnarray*}\int_a^b W(x) \, \wp_n(x) \, \wp_m(x) \, dx&=&\int_a^b{d^n\over dx^n}\Big(W \, A^n\Big) \, \wp_m(x) \, dx\nonumber\\ &=&{d^{n-1}\over dx^{n-1}}\Big(W \, A^n\Big) \, \wp_m(x)\Big{|}_a^b\nonumber\\ &-& \int_a^b{d^{n-1}\over dx^{n-1}}\Big(W \, A^n\Big) \, {d\over dx}\wp_m(x) \, dx\nonumber\\ &=&\Big(W \, A \, p_{n-1}\Big) \, \wp_m(x)\Big{|}_a^b\n\nonumber\\ &-& \int_a^b{d^{n-1}\over dx^{n-1}}\Big(W \, A^n\Big) \, {d\over dx}\wp_m(x) \, dx\nonumber\\ &=&- \int_a^b{d^{n-1}\over dx^{n-1}}\Big(W \, A^n\Big) \, {d\over dx}\wp_m(x) \, dx\end{eqnarray*}$

the first term vanishing because $WA(a)$ and $WA(b)$ vanish strongly at the endpoints of the domain. Repeat the process

(6) $\begin{eqnarray*}\int_a^b W(x) \, \wp_n(x) \, \wp_m(x) \, dx&=&\int_a^b{d^n\over dx^n}\Big(W \, A^n\Big) \, \wp_m(x) \, dx\nonumber\\ &=&(-1)^n \int_a^b\Big(W \, A^n\Big) \, {d^n\over dx^n}\wp_m(x) \, dx=0\end{eqnarray*}$

since the $n^{th}$ derivative of a polynomial of order less than $n$ is zero.

Theorem
The Sturm-Liouville equation

$\Big(W \, A \, y'\Big)'-\lambda \, W \, y=0, \qquad \mbox{or}\qquad A \,y''+(B+A') \, y'-\lambda y=0$

has polynomial solutions of order $n$ orthogonal on $a\le x\le b$ with respect to weight function

$W(x)=e^{\int {B\over A} \, dx}, \qquad W'={B\over A} \, W$

provided $W(x) \, A(x)$ vanishes strongly at $x=a,b$ , $A(x)$ is at most quadratic $A(x)=A_0 x^2+A_1 x+A_2$ , $B(x)$ is at most linear $B(x)=B_0 x+B_1$ , and

$\lambda= A_0 \, n(n+1)+B_0 \, n= {1\over 2} A'' \, n(n+1)+B' \, n$

and under these conditions the polynomial solutions are\index{Rodriguez formula}

$y_n(x)={1\over W}{d^n\over dx^n}\Big(W \, A^n\Big), \qquad \int_a^b W(x) \, y_n(x) \, y_m(x) \, dx=N_n \, \delta_{n,m}$

What this means to you is that when you solve a problem by variable separation and one of the resulting ordinary DE’s is a Sturm-Liouville type, then you can do “Fourier” expansions (actually they are called Fourier-Dini in this case) to solve for all of the coefficients. We will do this with Legendre and associated Legendre which pertain to spherical coordinate boundary value problems, but in your quantum, em and cm courses you will use many other SL-equations.

Suggested reading and reference: Special Functions by N. N. Lebedev.

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Sturm-Louiville differential equations

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