solving partial differential equations by factoring

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SOLVING PARTIAL DIFFERENTIAL EQUATIONS BY FACTORING

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(5134)

One will see that the very existence of the eigenvalue spectrum of $\nabla^2$ on the unit sphere hinges on this fact. For this reason, the extension of this algebraic method is considerably more powerful. It yields not only the basis for each eigenspace of $\nabla^2$ , but also the actual value for each allowed degenerate eigenvalue.

Global Analysis: Algebra

Global analysis deals with the solutions of a differential equation ``wholesale''. It characterizes them in relationship to one another without specifying their individual behaviour on their domain of definition. Thus one focusses via algebra, linear or otherwise, on ``the space of solutions'', its subspaces, bases etc.

Local analysis (next subsubsection), by contrast, deals with the solutions of a differential equation ``retail''. Using differential calculus, numerical analysis, one zooms in on individual functions and characterizes them by their local values, slopes, location of zeroes, etc.

1. Factorization

The algebraic method depends on factoring

$\displaystyle \nabla^2=\frac{\partial^2}{\partial\theta^2}+ \frac{\cos \theta}{... ...}{\partial\theta}+ \frac{1}{\sin^2 \theta}\frac{\partial^2}{\partial\varphi^2}$

into a pair of first order operators which are adjoints of each other. The method is analogous to factoring a quadratic polynomial, except that here one has differential operators $\partial/ \partial \theta$ and $\partial/ \partial \varphi$ instead of the variables

and

. Taking our cue from Properties 16 and 17, one attempts

$\begin{displaymath} \frac{\partial^2}{\partial\theta^2}+ \frac{\cos \theta}{\sin... ...}-\frac{i}{\sin\theta} \frac{\partial}{\partial\varphi}\right) \end{displaymath}$

However, one immediately finds that this factorization yields $\frac{1}{\sin\theta} \frac{\partial}{\partial\theta}$ for a cross term. This is incorrect. What one needs instead is $\frac{\cos\theta}{\sin\theta}\frac{\partial}{\partial\theta}$ . This leads us to consider

$\displaystyle { e^{ i\phi}\left( \frac{\partial}{\partial\theta}+i\frac{\cos\th... ...theta}-i\frac{\cos\theta}{\sin\theta} \frac{\partial}{\partial\varphi}\right) }$
	$\displaystyle =\frac{\partial^2}{\partial\theta^2}+ \frac{\cos \theta}{\sin \th... ...varphi}+ \frac{\cos^2\theta}{\sin^2 \theta}\frac{\partial^2}{\partial\varphi^2}$
	$\displaystyle =\frac{1}{\sin^2\theta}\frac{\partial}{\partial\theta} \sin\theta... ...rac{\partial^2}{\partial\varphi^2} -\frac{1}{i}\frac{\partial}{\partial\varphi}$
	$\displaystyle \equiv ~~~~~~~~~~~~~~~~~~~~~~\nabla^2 ~~~~~~~~~~~~~+L^2_\varphi ~~-L_\varphi$	(5135)

Here we have introduced the self-adjoint operator

$\displaystyle L_\varphi=\frac{1}{i}\frac{\partial}{\partial\varphi}~.$

It generates rotations around the polar axis of a sphere. This operator, together with the two mutually adjoint operators

$\displaystyle L_\pm =\pm e^{i\phi}\left( \frac{\partial}{\partial\theta}\pm i\frac{\cos\theta} {\sin\theta} \frac{\partial}{\partial\varphi}\right)$

are of fundamental importance to the factorization method of solving the given differential equation. In terms of them the factorized Eq.(5.135) and its complex conjugate have the form

$\displaystyle L_\pm L_\mp = -\nabla^2- L^2_\varphi \pm L_\varphi~.$

(5136)

This differs from Eq.(5.16), (Property 17 on page ), the factored Laplacian on the Euclidean plane.

2. Fundamental Relations

In spite of this difference, the commutation relations corresponding to Eqs.() are all the same, except one. Thus, instead of Eq.(5.19), for a sphere one has

$\displaystyle [L_+,L_-]=2L_\varphi~.$

(5137)

This is obtained by subtracting the two Eqs.(5.136). However, the commutation relations corresponding to the other two equations remain the same. Indeed, a little algebraic computation yields

$\displaystyle L_\varphi L_\mp= L_\pm L_\varphi \pm L_\varphi~,$

$\displaystyle [L_\varphi, L_\pm]= \pm L_\pm ~.$

(5138)

Furthermore, using Eq.(5.136) one finds

$\displaystyle [\nabla^2,L_+]$	$\displaystyle =$	$\displaystyle [-L_+L_- -L^2_\varphi +L_\varphi ,L_+]$
	$\displaystyle =$	$\displaystyle -[L_+L_-,L_+]-[L^2_\varphi,L_+]+[L_\varphi,L_+]$
	$\displaystyle =$	$\displaystyle -L_+(L_-L_+ - L_+L_-)$
		$\displaystyle -L_\varphi(L_\varphi L_+ - L_+L_\varphi) -(L_\varphi L_+ - L_+L_\varphi)L_\varphi$
		$\displaystyle + (L_\varphi L_+ - L_+L_\varphi)$
	$\displaystyle =$	$\displaystyle 0~.$	(5139)

The last equality was obtained with the help of Eqs.(5.137) and (5.138). Together with the complex conjugate of this equation, one has therefore

$\displaystyle [\nabla^2,L_\pm]=0~.$

(5140)

In addition, one has quite trivially

$\displaystyle [\nabla^2,L_\varphi]=0$

(5141)

The three algebraic relations, Eqs.(5.137)-(5.138) and their consequences, Eq.(5.140)-(5.141), are the fundamental equations from which one deduces the allowed degenerate eigenvalues of Eq.(5.132) as well as the corresponding normalized eigenfunctions.

3. The Eigenfunctions

One starts by considering a function $Y_\lambda^m$ which is a simultaneous solution to the two eigenvalue equations

$\displaystyle L_\varphi Y_\lambda^m$	$\displaystyle =$	$\displaystyle m Y_\lambda^m$
$\displaystyle \nabla^2 Y_\lambda^m$	$\displaystyle =$	$\displaystyle -\lambda Y_\lambda^m~.$

This is a consistent system, and it is best to postpone until later the easy task of actually exhibiting non-zero solutions to it. First we deduce three properties of any given solution $Y_\lambda^m$ .

The first property is obtained by applying the operator to this solution. One finds that

$\displaystyle L_\varphi (L_+Y_\lambda^m)$	$\displaystyle =$	$\displaystyle (L_+L_\varphi + L_+)Y_\lambda^m$
	$\displaystyle =$	$\displaystyle (m+1)(L_+Y_\lambda^m)$

Similarly one finds

$\displaystyle L_\varphi (L_-Y_\lambda^m)=(m-1)(L_-Y_\lambda^m)~.$

Thus $L_+Y_\lambda^m$ and $L_-Y_\lambda^m$ are again eigenfunctions of $L_\varphi$ , but having eigenvalues

and

. One is, therefore, justified in calling

and

raising and lowering operators. The ``raised'' and ``lowered'' functions $L_\pm Y_\lambda^m$ have the additional property that they are still eigenfunctions of $\nabla^2$ belonging to the same eigenvalue $\lambda$ . Indeed, with the help of Eq.(5.140) one finds

$\displaystyle \nabla^2 L_\pm Y_\lambda^m=L_\pm \nabla^2 Y_\lambda^m =-\lambda L_\pm Y_\lambda^m~.$

Thus, if $Y_\lambda^m$ belongs to the eigenspace of $\lambda$ , then so do $L_+Y_\lambda^m$ and $L_-Y_\lambda^m$ .

4. Normalization and the Eigenvalues

The second and third properties concern the normalization of $L_\pm Y^m_\lambda$ and the allowed values of $\lambda$ . One obtains them by examining the sequence of squared norms of the sequence of eigenfunctions

$\displaystyle L_\pm^k Y_\lambda^m~~,~~~~~k=0,1,2,\cdots~~~.$

All of them are square-integrable. Hence their norms are non-negative. In particular, for

one has

$\displaystyle 0\le \int_0^\pi \int_0^{2\pi} \vert L_\pm Y_\lambda^m(\theta,\varphi) \vert^2 \sin\theta d\theta d\varphi$	$\displaystyle \equiv$	$\displaystyle \langle L_\pm Y_\lambda^m, L_\pm Y_\lambda^m \rangle$
	$\displaystyle =$	$\displaystyle \langle Y_\lambda^m,L_\mp L_\pm Y_\lambda^m \rangle$
	$\displaystyle =$	$\displaystyle \langle Y_\lambda^m,(-)(\nabla^2+L^2_\varphi \pm L_\varphi ) Y_\lambda^m \rangle$
	$\displaystyle =$	$\displaystyle [\lambda - m(m\pm 1)] \langle Y_\lambda^m,Y_\lambda^m \rangle$	(5142)

This is the second property. It is a powerful result for two reasons:

First of all, if $Y^m_\lambda$ has been normalized to unity, then so will be

$\displaystyle \frac{1}{[\lambda - m(m\pm 1)]^{1/2}} L_\pm Y_\lambda^m(\theta,\varphi) \equiv Y_\lambda^{m\pm 1}(\theta,\varphi)$

(5143)

This means that once the normalization integral has been worked out for any one of the $Y^m_\lambda$ 's, the already normalized $Y_\lambda^{m\pm 1}$ are given by Eq.(5.143); no additional normalization integrals need to be evaluated. By repeatedly applying the operator $L_\pm$ one can extend this result to $Y_\lambda^{m\pm 2}$ , $Y_\lambda^{m\pm 3}$ , etc. They all are already normalized if $Y^m_\lambda$ is. No extra work is necessary.

Secondly, repeated use of the relation (5.142) yields

$\displaystyle \langle L^k_\pm Y_\lambda^m,L^k_\pm Y_\lambda^m \rangle= [\lambda... ...(m\pm k)] \cdots [\lambda - m(m\pm 1)] \langle Y_\lambda^m,Y_\lambda^m \rangle$

This relation implies that for sufficiently large integer

the leading factor in square brackets must vanish. If it did not, the squared norm of $L^k_\pm Y_\lambda^m$ would become negative. To prevent this from happening, $\lambda$ must have very special values. This is the third property: The only allowed values of $\lambda$ are necessarily

$\displaystyle \lambda=\ell(\ell+1)~~~~~\ell=0,1,2,\cdots~.$

(Note that $\ell=-1,-2,\cdots$ would give nothing new.) Any other value for $\lambda$ would yield a contradiction, namely a negative norm for some integer

. As a consequence, one has the result that for each allowed eigenvalue there is a sequence of eigenfunctions

$\displaystyle Y_\ell^m(\theta,\varphi)~~~~~m=0,\pm 1,\pm 2,\cdots$

(5144)

(Nota bene: Note that these eigenfunctions are now labelled by the non-negative integer $\ell$ instead of the corresponding eigenvalue $\lambda$ .) Of particular interest are the two eigenfunctions $Y^\ell_\ell$ and $Y^{-\ell}_\ell$ . The squared norm of $L_+Y^\ell_\ell$ ,

$\displaystyle \Vert L_+Y^\ell_\ell \Vert^2= [\ell(\ell+1)-\ell(\ell+1)]\Vert Y^\ell_\ell\Vert^2$

is not positive. It vanishes. This implies that

$\displaystyle Y^{\ell+1}_\ell \propto L_+Y^\ell_\ell=0~.$

(5145)

In other words, $Y^{\ell+1}_\ell$ and all subsequent members of the above sequence, Eq.(5.144) vanish, i.e. they do not exist. Similarly one finds that

$\displaystyle Y^{-\ell-1}_\ell \propto L_-Y^{-\ell}_\ell=0~.$

(5146)

Thus members of the sequence below $Y^{-\ell}_\ell$ do not exist either. It follows that the sequence of eigenfunctions corresponding to $\ell (\ell +1)$ is finite. The sequence has only $2\ell +1$ members, namely

$\displaystyle Y^{-\ell}_\ell(\theta,\varphi), Y^{-\ell+1}_\ell(\theta,\varphi),... ...varphi),\cdots, Y^{\ell-1}_\ell(\theta,\varphi), Y^{\ell}_\ell(\theta,\varphi)$

for each integer $\ell$ . The union of these sequences forms a semi-infinite lattice in the $(\ell ,m)$ as shown in Figure 5.21.

$\begin{texdraw} \drawdim cm \linewd .02 \arrowheadtype t:V \move(0 -4) \rlvec(0 ... ...ar \move(1.9 -2.1) \textref h:C v:T \rtext td:-45 {$m=-\ell$} \par \end{texdraw}$

Figure 5.21: Lattice of eigenfunctions (spherical hamonics) labelled by the angular integers $\ell$ and

. Application of the raising operator

increases

, until one comes to the top of each vertical sequence (fixed $\ell$ ). The lowering operator

decreases

, until one reaches the bottom. In between there are exactly $2\ell +1$ lattice points, which express the ( $2\ell +1$ )-fold degeneracy of the eigenvalue $\ell (\ell +1)$ . There do not exist any harmonics above or below the dashed boundaries.

For obvious reasons it is appropriate to refer to this sequence as a ladder with $2\ell +1$ elements, and to call $Y^\ell_\ell$ the top, and $Y^{-\ell}_\ell$ the bottom of the ladder. The raising and lowering operators $L_\pm$ are the ladder operators which take us up and down the $(2\ell+1)$ -element ladder. It is easy to determine the elements $Y^{\pm\ell}_\ell$ at the top and the bottom, and to use the ladder operators to generate any element in between.

5. Orthonormality and Completeness

The operators $\{ \nabla^2,L_\phi \}$ form a complete set of commuting operators. This means that their eigenvalues $(\ell ,m)$ serve as sufficient labels to uniquely identify each of their (common) eigenbasis elements for the vector space of solutions to the Hermholtz equation

$\displaystyle [\nabla^2+\ell(\ell+1)]Y^m_\ell(\theta,\varphi)=0$

on the two-sphere. No additional labels are necessary. The fact that these operators are self-adjoint relative to the inner product, Eq.(5.134), implies that these eigenvectors (a.k.a spherical harmonics) are orthonormal:

$\displaystyle \langle Y^m_\ell,Y^{m'}_{\ell'} \rangle =\delta_{\ell {\ell'}} \delta_{m {m'}}$

The semi-infinite set $\{ Y^m_\ell(\theta,\varphi):~-\ell\le m\le \ell; ~ \ell=0,1,\cdots\}$ is a basis for the vector space of functions square-integrable on the unit two-sphere. Let $g(\theta,\varphi)$ be any such function. Then

$\displaystyle g(\theta,\varphi)$	$\displaystyle =$	$\displaystyle \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell Y^m_\ell(\theta,\varphi) \langle Y^m_\ell ,g\rangle$
	$\displaystyle =$	$\displaystyle \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell Y^m_\ell(\theta,\varphi)... ...int_0^{2\pi} d\varphi' \overline{Y^m_\ell(\theta',\varphi')}g(\theta',\varphi')$

In other words, the spherical harmonics are the basis elements for a generalized double Fourier series representation of the function $g(\theta,\varphi)$ . If one leaves this function unspecified, then this completeness relation can be restated in the equivalent form

$\displaystyle \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell Y^m_\ell(\theta,\varphi)... ...\varphi')}= \frac{\delta(\theta-\theta')}{\sin\theta} \delta(\varphi-\varphi')$

in terms of the Dirac delta functions on the compact domains $0\le \theta \le\pi$ and $0\le\varphi\le 2\pi$ .

Local Analysis: Calculus

What is the formula for a harmonics $Y^m_l(\theta,\phi)$ ? An explicit functional form determines the graph, the location of its zeroes, and other aspects of its local behaviour.

1. Spherical Harmonics: Top and Bottom of the Ladder

Each member of the ladder sequence satisfies the differential equation

$\displaystyle L_\varphi Y^m_\ell \equiv \frac{1}{i}\frac{\partial}{\partial \varphi} Y^m_\ell(\theta,\varphi) =m Y^m_\ell(\theta,\varphi)~.$

Consequently, all eigenfunctions have the form

$\displaystyle Y^m_\ell(\theta,\varphi)= c_{\ell m}P^m_\ell(\theta)\frac{e^{im\varphi}}{\sqrt{2\pi}}~.$

(5147)

Here $c_{\ell m}$ is a normalization factor. The two eigenfunctions $Y^\ell_\ell$ and $Y^{-\ell}_\ell$ at the top and the bottom of the ladder satisfy Eqs.(5.145) and (5.146) respectively, namely

$\displaystyle L_{\pm}Y^{\pm \ell}_\ell\equiv \pm e^{ i\phi}\left( \frac{\partia... ...artial}{\partial\varphi}\right) P^{\pm\ell}_\ell(\theta)e^{\pm i\ell\varphi} =0$

(5148)

It is easy to see that their solutions are

$\displaystyle Y^\ell_\ell$	$\displaystyle =$	$\displaystyle c_\ell \sin^\ell\theta e^{i\ell\varphi}$
$\displaystyle Y^{-\ell}_\ell$	$\displaystyle =$	$\displaystyle c_\ell \sin^\ell\theta e^{-i\ell\varphi}~.$

The normalization condition

$\displaystyle \int_0^\pi \int_0^{2\pi} \vert Y^{\pm\ell}_\ell(\theta,\varphi) \vert^2 \sin\theta d\theta d\varphi =1$

implies that

$\displaystyle Y_\ell^\ell(\theta,\phi)= \frac{(-1)^\ell}{2^\ell \ell !} \sqrt{\frac{(2\ell+1)!}{4\pi} } \sin^\ell \theta e^{i\ell \phi}.$

(5149)

The phase factor $(-1)^\ell$ is not determined by the normalization. Its form is chosen so as to simplify the to-be-derived formula for the Legendre polynomials, Eq.(5.153).

2. Spherical harmonics: Legendre and Associated Legendre polynomials

The functions $Y^m_\ell (\theta,\varphi)$ are obtained by applying the lowering operator to $Y^\ell_\ell (\theta,\varphi)$ . A systematic way of doing this is first to apply repeatedly the lowering relation

$\displaystyle {Y^{m-1}_\ell (\theta,\varphi)=}$
	$\displaystyle =$	$\displaystyle \frac{1}{\sqrt{\ell(\ell+1)-m(m-1)}}~ L_- Y^m_\ell (\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \frac{1}{\sqrt{\ell^2-m^2+\ell+m}}(-1)e^{-i\varphi} \left( \frac{... ...a}{\sin\theta} \frac{\partial}{\partial\varphi}\right)Y^m_\ell (\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \frac{-1}{\sqrt{(\ell+m)(\ell-m+1)}}e^{-i\varphi} \left( \frac{\p... ...partial\theta} +m\frac{\cos\theta}{\sin\theta} \right)Y^m_\ell (\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \frac{1}{\sqrt{(\ell+m)(\ell-m+1)}} \frac{1}{\sin^{m-1}\theta} \frac{\partial}{\partial (\cos \theta)} ~\sin^m\theta e^{-i\varphi}Y^m_\ell$	(5150)

to $Y^\ell_\ell (\theta,\varphi)$ until one obtains the azimuthally invariant harmonic $Y^0_\ell (\theta,\varphi)=Y^0_\ell (\theta)$ . Then continue applying this lowering relation, or alternatively the raising relation

$\displaystyle {Y^{m}_\ell(\theta,\varphi)=}$
	$\displaystyle =$	$\displaystyle \frac{1}{\sqrt{\ell(\ell+1)-m(m-1)}}~ L_+ Y^{m-1}_\ell(\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \frac{-1}{\sqrt{(\ell+m)(\ell-m+1)}} \sin^m\theta \frac{\partial}{\partial (\cos \theta)} ~\frac{1}{\sin^{m-1}\theta} e^{i\varphi}Y^{m-1}_\ell$	(5151)

until one obtains the desired harmionic $Y^m_\ell (\theta,\varphi)$ . The execution of this two-step algorithm reads as follows:

Step 1: Letting $m=\vert m\vert$ , apply Eq.(5.150) times and obtain

$\displaystyle Y^{0}_\ell(\theta,\varphi)$	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m)!}}\underbrace{L_-L_-\cdots L_-}_ {m~\textrm{times}} Y^m_\ell(\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \frac{\partial^m}{\partial (\cos \theta)^m} ~\sin^m\theta e^{-im\varphi}Y^m_\ell(\theta,\varphi),$

which, because of Eq.(5.147), is independent of $\varphi$ . Now let $m=\ell$ , use Eq.(5.149), and obtain

$\displaystyle Y^{0}_\ell(\theta,\varphi)$	$\displaystyle =$	$\displaystyle \frac{(-1)^\ell}{2^\ell \ell !} \sqrt{\frac{(2\ell+1)}{4\pi} } \frac{\partial^\ell}{\partial (\cos \theta)^\ell} \sin^{2\ell} \theta$	(5152)
	$\displaystyle \equiv$	$\displaystyle \sqrt{\frac{(2\ell+1)}{4\pi} } P_\ell (\cos\theta).$

The polynomials in the variable $x=\cos\theta$

$\displaystyle P_\ell(x)\equiv \frac{1}{2^\ell \ell !} \frac{d^\ell}{dx^\ell} (x^2-1)^\ell$

(5153)

are called the Legendre polynomials. They have the property that at the North pole they have the common value unity, while at the South pole their value is

whenever $P_\ell(x)$ is an even polynomial and

whenever it is odd:

$\displaystyle P_\ell(x=\pm 1)=(\pm 1)^\ell.$

Step 2: To obtain the harmonics having positive azimuthal integer

, apply the raising operator

times to $Y^0_\ell$ . With the help of Eq.(5.151) one obtains (for $m=\vert m\vert$ )

$\displaystyle Y^{m}_\ell(\theta,\varphi)$	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m)!}}\underbrace{L_+L_+\cdots L_+}_ {m~\textrm{times}} Y^0_\ell(\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m)!}}(-1)^m \sin^m \theta \frac{\partial^m}{\partial (\cos \theta)^m} ~e^{im\varphi}Y^0_\ell(\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m )!}} \frac{(-1)^{\ell+m}}{2^\ell \... ...al^{\ell+m}}{\partial (\cos \theta)^{\ell+m }} \sin^{2\ell}\theta e^{im\varphi}$
	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m )!}} \sqrt{\frac{2\ell+1}{4\pi} } ~P^{m}_\ell(\cos\theta)~e^{im\varphi}$	(5154)

The polynomials in the variable $x=\cos\theta$

$\displaystyle P^m_\ell(x)\equiv \frac{(-1)^m}{2^\ell \ell !}(1-x^2)^{m/2} \frac{d^{\ell+m}}{dx^{\ell+m}} (x^2-1)^\ell$

(5155)

are called the associated Legendre polynomials. Inserting Eq.(5.154) into Eq.(5.132), one finds that they satisfy the differential equation

$\displaystyle \left[ \frac{1}{\sin\theta} \frac{d}{d\theta} \sin\theta \frac{d... ...heta}+\ell(\ell+1)-\frac{m^2}{\sin^2\theta} \right] P^{m}_\ell(\cos\theta)=0~.$

Also note that $P^{-m}_\ell(\cos\theta)$ satisfies the same differential equation. In other words, $P^{-m}_\ell$ and $P^{m}_\ell$ must be proportional to each other. (Why?) Indeed,

$\displaystyle P^{-m}_\ell(\cos\theta)=(-1)^m \frac{(\ell-m)!}{(\ell+m )!} P^{m}_\ell(\cos\theta)~.$

(5156)

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SOLVING PARTIAL DIFFERENTIAL EQUATIONS BY FACTORING

(5134)

Global Analysis: Algebra

1. Factorization

The algebraic method depends on factoring

$\displaystyle \nabla^2=\frac{\partial^2}{\partial\theta^2}+ \frac{\cos \theta}{... ...}{\partial\theta}+ \frac{1}{\sin^2 \theta}\frac{\partial^2}{\partial\varphi^2}$

and

. Taking our cue from Properties 16 and 17, one attempts

$\begin{displaymath} \frac{\partial^2}{\partial\theta^2}+ \frac{\cos \theta}{\sin... ...}-\frac{i}{\sin\theta} \frac{\partial}{\partial\varphi}\right) \end{displaymath}$

$\displaystyle { e^{ i\phi}\left( \frac{\partial}{\partial\theta}+i\frac{\cos\th... ...theta}-i\frac{\cos\theta}{\sin\theta} \frac{\partial}{\partial\varphi}\right) }$
	$\displaystyle =\frac{\partial^2}{\partial\theta^2}+ \frac{\cos \theta}{\sin \th... ...varphi}+ \frac{\cos^2\theta}{\sin^2 \theta}\frac{\partial^2}{\partial\varphi^2}$
	$\displaystyle =\frac{1}{\sin^2\theta}\frac{\partial}{\partial\theta} \sin\theta... ...rac{\partial^2}{\partial\varphi^2} -\frac{1}{i}\frac{\partial}{\partial\varphi}$
	$\displaystyle \equiv ~~~~~~~~~~~~~~~~~~~~~~\nabla^2 ~~~~~~~~~~~~~+L^2_\varphi ~~-L_\varphi$	(5135)

Here we have introduced the self-adjoint operator

$\displaystyle L_\varphi=\frac{1}{i}\frac{\partial}{\partial\varphi}~.$

It generates rotations around the polar axis of a sphere. This operator, together with the two mutually adjoint operators

$\displaystyle L_\pm =\pm e^{i\phi}\left( \frac{\partial}{\partial\theta}\pm i\frac{\cos\theta} {\sin\theta} \frac{\partial}{\partial\varphi}\right)$

are of fundamental importance to the factorization method of solving the given differential equation. In terms of them the factorized Eq.(5.135) and its complex conjugate have the form

$\displaystyle L_\pm L_\mp = -\nabla^2- L^2_\varphi \pm L_\varphi~.$

(5136)

This differs from Eq.(5.16), (Property 17 on page ), the factored Laplacian on the Euclidean plane.

2. Fundamental Relations

In spite of this difference, the commutation relations corresponding to Eqs.() are all the same, except one. Thus, instead of Eq.(5.19), for a sphere one has

$\displaystyle [L_+,L_-]=2L_\varphi~.$

(5137)

This is obtained by subtracting the two Eqs.(5.136). However, the commutation relations corresponding to the other two equations remain the same. Indeed, a little algebraic computation yields

$\displaystyle L_\varphi L_\mp= L_\pm L_\varphi \pm L_\varphi~,$

$\displaystyle [L_\varphi, L_\pm]= \pm L_\pm ~.$

(5138)

Furthermore, using Eq.(5.136) one finds

$\displaystyle [\nabla^2,L_+]$	$\displaystyle =$	$\displaystyle [-L_+L_- -L^2_\varphi +L_\varphi ,L_+]$
	$\displaystyle =$	$\displaystyle -[L_+L_-,L_+]-[L^2_\varphi,L_+]+[L_\varphi,L_+]$
	$\displaystyle =$	$\displaystyle -L_+(L_-L_+ - L_+L_-)$
		$\displaystyle -L_\varphi(L_\varphi L_+ - L_+L_\varphi) -(L_\varphi L_+ - L_+L_\varphi)L_\varphi$
		$\displaystyle + (L_\varphi L_+ - L_+L_\varphi)$
	$\displaystyle =$	$\displaystyle 0~.$	(5139)

The last equality was obtained with the help of Eqs.(5.137) and (5.138). Together with the complex conjugate of this equation, one has therefore

$\displaystyle [\nabla^2,L_\pm]=0~.$

(5140)

In addition, one has quite trivially

$\displaystyle [\nabla^2,L_\varphi]=0$

(5141)

3. The Eigenfunctions

One starts by considering a function $Y_\lambda^m$ which is a simultaneous solution to the two eigenvalue equations

$\displaystyle L_\varphi Y_\lambda^m$	$\displaystyle =$	$\displaystyle m Y_\lambda^m$
$\displaystyle \nabla^2 Y_\lambda^m$	$\displaystyle =$	$\displaystyle -\lambda Y_\lambda^m~.$

The first property is obtained by applying the operator to this solution. One finds that

$\displaystyle L_\varphi (L_+Y_\lambda^m)$	$\displaystyle =$	$\displaystyle (L_+L_\varphi + L_+)Y_\lambda^m$
	$\displaystyle =$	$\displaystyle (m+1)(L_+Y_\lambda^m)$

Similarly one finds

$\displaystyle L_\varphi (L_-Y_\lambda^m)=(m-1)(L_-Y_\lambda^m)~.$

Thus $L_+Y_\lambda^m$ and $L_-Y_\lambda^m$ are again eigenfunctions of $L_\varphi$ , but having eigenvalues

and

. One is, therefore, justified in calling

and

$\displaystyle \nabla^2 L_\pm Y_\lambda^m=L_\pm \nabla^2 Y_\lambda^m =-\lambda L_\pm Y_\lambda^m~.$

Thus, if $Y_\lambda^m$ belongs to the eigenspace of $\lambda$ , then so do $L_+Y_\lambda^m$ and $L_-Y_\lambda^m$ .

4. Normalization and the Eigenvalues

$\displaystyle L_\pm^k Y_\lambda^m~~,~~~~~k=0,1,2,\cdots~~~.$

All of them are square-integrable. Hence their norms are non-negative. In particular, for

one has

$\displaystyle 0\le \int_0^\pi \int_0^{2\pi} \vert L_\pm Y_\lambda^m(\theta,\varphi) \vert^2 \sin\theta d\theta d\varphi$	$\displaystyle \equiv$	$\displaystyle \langle L_\pm Y_\lambda^m, L_\pm Y_\lambda^m \rangle$
	$\displaystyle =$	$\displaystyle \langle Y_\lambda^m,L_\mp L_\pm Y_\lambda^m \rangle$
	$\displaystyle =$	$\displaystyle \langle Y_\lambda^m,(-)(\nabla^2+L^2_\varphi \pm L_\varphi ) Y_\lambda^m \rangle$
	$\displaystyle =$	$\displaystyle [\lambda - m(m\pm 1)] \langle Y_\lambda^m,Y_\lambda^m \rangle$	(5142)

This is the second property. It is a powerful result for two reasons:

First of all, if $Y^m_\lambda$ has been normalized to unity, then so will be

$\displaystyle \frac{1}{[\lambda - m(m\pm 1)]^{1/2}} L_\pm Y_\lambda^m(\theta,\varphi) \equiv Y_\lambda^{m\pm 1}(\theta,\varphi)$

(5143)

Secondly, repeated use of the relation (5.142) yields

$\displaystyle \langle L^k_\pm Y_\lambda^m,L^k_\pm Y_\lambda^m \rangle= [\lambda... ...(m\pm k)] \cdots [\lambda - m(m\pm 1)] \langle Y_\lambda^m,Y_\lambda^m \rangle$

This relation implies that for sufficiently large integer

$\displaystyle \lambda=\ell(\ell+1)~~~~~\ell=0,1,2,\cdots~.$

(Note that $\ell=-1,-2,\cdots$ would give nothing new.) Any other value for $\lambda$ would yield a contradiction, namely a negative norm for some integer

. As a consequence, one has the result that for each allowed eigenvalue there is a sequence of eigenfunctions

$\displaystyle Y_\ell^m(\theta,\varphi)~~~~~m=0,\pm 1,\pm 2,\cdots$

(5144)

$\displaystyle \Vert L_+Y^\ell_\ell \Vert^2= [\ell(\ell+1)-\ell(\ell+1)]\Vert Y^\ell_\ell\Vert^2$

is not positive. It vanishes. This implies that

$\displaystyle Y^{\ell+1}_\ell \propto L_+Y^\ell_\ell=0~.$

(5145)

In other words, $Y^{\ell+1}_\ell$ and all subsequent members of the above sequence, Eq.(5.144) vanish, i.e. they do not exist. Similarly one finds that

$\displaystyle Y^{-\ell-1}_\ell \propto L_-Y^{-\ell}_\ell=0~.$

(5146)

$\displaystyle Y^{-\ell}_\ell(\theta,\varphi), Y^{-\ell+1}_\ell(\theta,\varphi),... ...varphi),\cdots, Y^{\ell-1}_\ell(\theta,\varphi), Y^{\ell}_\ell(\theta,\varphi)$

for each integer $\ell$ . The union of these sequences forms a semi-infinite lattice in the $(\ell ,m)$ as shown in Figure 5.21.

$\begin{texdraw} \drawdim cm \linewd .02 \arrowheadtype t:V \move(0 -4) \rlvec(0 ... ...ar \move(1.9 -2.1) \textref h:C v:T \rtext td:-45 {$m=-\ell$} \par \end{texdraw}$

Figure 5.21: Lattice of eigenfunctions (spherical hamonics) labelled by the angular integers $\ell$ and

. Application of the raising operator

increases

, until one comes to the top of each vertical sequence (fixed $\ell$ ). The lowering operator

decreases

5. Orthonormality and Completeness

$\displaystyle [\nabla^2+\ell(\ell+1)]Y^m_\ell(\theta,\varphi)=0$

$\displaystyle \langle Y^m_\ell,Y^{m'}_{\ell'} \rangle =\delta_{\ell {\ell'}} \delta_{m {m'}}$

$\displaystyle g(\theta,\varphi)$	$\displaystyle =$	$\displaystyle \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell Y^m_\ell(\theta,\varphi) \langle Y^m_\ell ,g\rangle$
	$\displaystyle =$	$\displaystyle \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell Y^m_\ell(\theta,\varphi)... ...int_0^{2\pi} d\varphi' \overline{Y^m_\ell(\theta',\varphi')}g(\theta',\varphi')$

$\displaystyle \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell Y^m_\ell(\theta,\varphi)... ...\varphi')}= \frac{\delta(\theta-\theta')}{\sin\theta} \delta(\varphi-\varphi')$

in terms of the Dirac delta functions on the compact domains $0\le \theta \le\pi$ and $0\le\varphi\le 2\pi$ .

Local Analysis: Calculus

What is the formula for a harmonics $Y^m_l(\theta,\phi)$ ? An explicit functional form determines the graph, the location of its zeroes, and other aspects of its local behaviour.

1. Spherical Harmonics: Top and Bottom of the Ladder

Each member of the ladder sequence satisfies the differential equation

$\displaystyle L_\varphi Y^m_\ell \equiv \frac{1}{i}\frac{\partial}{\partial \varphi} Y^m_\ell(\theta,\varphi) =m Y^m_\ell(\theta,\varphi)~.$

Consequently, all eigenfunctions have the form

$\displaystyle Y^m_\ell(\theta,\varphi)= c_{\ell m}P^m_\ell(\theta)\frac{e^{im\varphi}}{\sqrt{2\pi}}~.$

(5147)

Here $c_{\ell m}$ is a normalization factor. The two eigenfunctions $Y^\ell_\ell$ and $Y^{-\ell}_\ell$ at the top and the bottom of the ladder satisfy Eqs.(5.145) and (5.146) respectively, namely

$\displaystyle L_{\pm}Y^{\pm \ell}_\ell\equiv \pm e^{ i\phi}\left( \frac{\partia... ...artial}{\partial\varphi}\right) P^{\pm\ell}_\ell(\theta)e^{\pm i\ell\varphi} =0$

(5148)

It is easy to see that their solutions are

$\displaystyle Y^\ell_\ell$	$\displaystyle =$	$\displaystyle c_\ell \sin^\ell\theta e^{i\ell\varphi}$
$\displaystyle Y^{-\ell}_\ell$	$\displaystyle =$	$\displaystyle c_\ell \sin^\ell\theta e^{-i\ell\varphi}~.$

The normalization condition

$\displaystyle \int_0^\pi \int_0^{2\pi} \vert Y^{\pm\ell}_\ell(\theta,\varphi) \vert^2 \sin\theta d\theta d\varphi =1$

implies that

$\displaystyle Y_\ell^\ell(\theta,\phi)= \frac{(-1)^\ell}{2^\ell \ell !} \sqrt{\frac{(2\ell+1)!}{4\pi} } \sin^\ell \theta e^{i\ell \phi}.$

(5149)

The phase factor $(-1)^\ell$ is not determined by the normalization. Its form is chosen so as to simplify the to-be-derived formula for the Legendre polynomials, Eq.(5.153).

2. Spherical harmonics: Legendre and Associated Legendre polynomials

$\displaystyle {Y^{m-1}_\ell (\theta,\varphi)=}$
	$\displaystyle =$	$\displaystyle \frac{1}{\sqrt{\ell(\ell+1)-m(m-1)}}~ L_- Y^m_\ell (\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \frac{1}{\sqrt{\ell^2-m^2+\ell+m}}(-1)e^{-i\varphi} \left( \frac{... ...a}{\sin\theta} \frac{\partial}{\partial\varphi}\right)Y^m_\ell (\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \frac{-1}{\sqrt{(\ell+m)(\ell-m+1)}}e^{-i\varphi} \left( \frac{\p... ...partial\theta} +m\frac{\cos\theta}{\sin\theta} \right)Y^m_\ell (\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \frac{1}{\sqrt{(\ell+m)(\ell-m+1)}} \frac{1}{\sin^{m-1}\theta} \frac{\partial}{\partial (\cos \theta)} ~\sin^m\theta e^{-i\varphi}Y^m_\ell$	(5150)

$\displaystyle {Y^{m}_\ell(\theta,\varphi)=}$
	$\displaystyle =$	$\displaystyle \frac{1}{\sqrt{\ell(\ell+1)-m(m-1)}}~ L_+ Y^{m-1}_\ell(\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \frac{-1}{\sqrt{(\ell+m)(\ell-m+1)}} \sin^m\theta \frac{\partial}{\partial (\cos \theta)} ~\frac{1}{\sin^{m-1}\theta} e^{i\varphi}Y^{m-1}_\ell$	(5151)

until one obtains the desired harmionic $Y^m_\ell (\theta,\varphi)$ . The execution of this two-step algorithm reads as follows:

Step 1: Letting $m=\vert m\vert$ , apply Eq.(5.150) times and obtain

$\displaystyle Y^{0}_\ell(\theta,\varphi)$	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m)!}}\underbrace{L_-L_-\cdots L_-}_ {m~\textrm{times}} Y^m_\ell(\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \frac{\partial^m}{\partial (\cos \theta)^m} ~\sin^m\theta e^{-im\varphi}Y^m_\ell(\theta,\varphi),$

which, because of Eq.(5.147), is independent of $\varphi$ . Now let $m=\ell$ , use Eq.(5.149), and obtain

$\displaystyle Y^{0}_\ell(\theta,\varphi)$	$\displaystyle =$	$\displaystyle \frac{(-1)^\ell}{2^\ell \ell !} \sqrt{\frac{(2\ell+1)}{4\pi} } \frac{\partial^\ell}{\partial (\cos \theta)^\ell} \sin^{2\ell} \theta$	(5152)
	$\displaystyle \equiv$	$\displaystyle \sqrt{\frac{(2\ell+1)}{4\pi} } P_\ell (\cos\theta).$

The polynomials in the variable $x=\cos\theta$

$\displaystyle P_\ell(x)\equiv \frac{1}{2^\ell \ell !} \frac{d^\ell}{dx^\ell} (x^2-1)^\ell$

(5153)

are called the Legendre polynomials. They have the property that at the North pole they have the common value unity, while at the South pole their value is

whenever $P_\ell(x)$ is an even polynomial and

whenever it is odd:

$\displaystyle P_\ell(x=\pm 1)=(\pm 1)^\ell.$

Step 2: To obtain the harmonics having positive azimuthal integer

, apply the raising operator

times to $Y^0_\ell$ . With the help of Eq.(5.151) one obtains (for $m=\vert m\vert$ )

$\displaystyle Y^{m}_\ell(\theta,\varphi)$	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m)!}}\underbrace{L_+L_+\cdots L_+}_ {m~\textrm{times}} Y^0_\ell(\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m)!}}(-1)^m \sin^m \theta \frac{\partial^m}{\partial (\cos \theta)^m} ~e^{im\varphi}Y^0_\ell(\theta,\varphi)$
	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m )!}} \frac{(-1)^{\ell+m}}{2^\ell \... ...al^{\ell+m}}{\partial (\cos \theta)^{\ell+m }} \sin^{2\ell}\theta e^{im\varphi}$
	$\displaystyle =$	$\displaystyle \sqrt{\frac{(\ell-m)!}{(\ell+m )!}} \sqrt{\frac{2\ell+1}{4\pi} } ~P^{m}_\ell(\cos\theta)~e^{im\varphi}$	(5154)

The polynomials in the variable $x=\cos\theta$

$\displaystyle P^m_\ell(x)\equiv \frac{(-1)^m}{2^\ell \ell !}(1-x^2)^{m/2} \frac{d^{\ell+m}}{dx^{\ell+m}} (x^2-1)^\ell$

(5155)

are called the associated Legendre polynomials. Inserting Eq.(5.154) into Eq.(5.132), one finds that they satisfy the differential equation

$\displaystyle \left[ \frac{1}{\sin\theta} \frac{d}{d\theta} \sin\theta \frac{d... ...heta}+\ell(\ell+1)-\frac{m^2}{\sin^2\theta} \right] P^{m}_\ell(\cos\theta)=0~.$

Also note that $P^{-m}_\ell(\cos\theta)$ satisfies the same differential equation. In other words, $P^{-m}_\ell$ and $P^{m}_\ell$ must be proportional to each other. (Why?) Indeed,

$\displaystyle P^{-m}_\ell(\cos\theta)=(-1)^m \frac{(\ell-m)!}{(\ell+m )!} P^{m}_\ell(\cos\theta)~.$

(5156)