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**Chris Hillman** · Sep26-07, 03:20 PM

Consider the Laplacian on the circle

and on the sphere

. Determine the eigenvalues and the eigenspace of eigenfunctions for each eigenvalue. (Seriously--- this is fairly elementary if you know a bit about Sturm-Liouville theory!) Result:

The space of real-valued square integrable functions on the circle,

, decomposes as the orthogonal direct sum of the eigenspaces of the eigenvalues, $LaTeX Code: -\\ell^2, \\, \\ell \\in \\mathbold{N}$ . For $LaTeX Code: \\ell > 0$ these eigenspaces have dimension two; the eigenfunctions are $LaTeX Code: \\sin(\\ell \\, \\phi), \\; \\; \\cos(\\ell \\, \\phi)$ .

The space of real-valued square integrable functions on the sphere,

, decomposes as the orthogonal direct sum of the eigenspaces of the eigenvalues, $LaTeX Code: -\\ell \\, (\\ell+1)$ . For $LaTeX Code: \\ell > 0$ these eigenspaces have dimension $LaTeX Code: 2 \\, \\ell+1$ ; the eigenfunctions can be taken to be the Legendre polynomial
$LaTeX Code: P(\\ell, \\cos(\\theta))$
(that's the

in $LaTeX Code: 2 \\, \\ell + 1$ ) plus
$LaTeX Code: P(\\ell, m, \\cos(\\theta)) \\, \\cos(m \\, \\phi), \\; \\; 1 \\leq m \\leq \\ell$
$LaTeX Code: P(\\ell, m, \\cos(\\theta)) \\, \\sin(m \\, \\phi), \\; \\; 1 \\leq m \\leq \\ell$
where the $LaTeX Code: P(\\ell, m, \\cdot)$ are the associated Legendre functions (that's the $LaTeX Code: 2 \\, \\ell$ in $LaTeX Code: 2 \\, \\ell + 1$ ). That is, the eigenfunctions are the real and imginary parts of the usual spherical harmonics
$LaTeX Code: <BR>Y^{\\ell, m}(\\theta, \\phi) = P(\\ell, m, \\cos(\\theta)) \\, \\exp(m \\, i \\, \\phi)<BR>$

See Kenneth I. Gross, "The Evolution of Non-Commutative Harmonic Analysis", Amer. Math. Monthly Aug.-Sept. 1978: 525--548. (Students and academics whose institution subscribes to JSTOR: past issues of the Monthly are available on-line, and past issues back to 1894 are well worth snarfing--- highest recommendation!)

These results can be complexified in the obvious way. I have discussed only the scalar spherical harmonics; there are also vector spherical harmonics and tensor spherical harmonics. Also, these results can be extended with minimal change (Weyl) to the Laplacian associated with compact Lie groups other than

acting on homogeneous spaces other than

. With more work (Harish-Chandra) to non-compact semi-simple Lie groups. With still more work (Mackey) to infinite-dimensional Lie groups. With still more work (Kirillov, Kostant, etc.) to nilpotent Lie groups.

See also my post #4 in the recently locked thread (thanks, berkeman!) "Representation theory?" http://www.physicsforums.com/showthread.php?t=185965

My point is: yes, there is a broader significance!

Sep26-07, 03:20 PM	Last edited by Chris Hillman; Sep26-07 at 04:40 PM.. Reason: fix markup #5
Chris Hillman Chris Hillman is Offline: Posts: 1,862 Recognitions: Science Advisor	World's Shortest Summary of Harmonic Analysis Consider the Laplacian on the circle and on the sphere . Determine the eigenvalues and the eigenspace of eigenfunctions for each eigenvalue. (Seriously--- this is fairly elementary if you know a bit about Sturm-Liouville theory!) Result: The space of real-valued square integrable functions on the circle, , decomposes as the orthogonal direct sum of the eigenspaces of the eigenvalues, $LaTeX Code: -\\ell^2, \\, \\ell \\in \\mathbold{N}$ . For $LaTeX Code: \\ell > 0$ these eigenspaces have dimension two; the eigenfunctions are $LaTeX Code: \\sin(\\ell \\, \\phi), \\; \\; \\cos(\\ell \\, \\phi)$ . The space of real-valued square integrable functions on the sphere, , decomposes as the orthogonal direct sum of the eigenspaces of the eigenvalues, $LaTeX Code: -\\ell \\, (\\ell+1)$ . For $LaTeX Code: \\ell > 0$ these eigenspaces have dimension $LaTeX Code: 2 \\, \\ell+1$ ; the eigenfunctions can be taken to be the Legendre polynomial $LaTeX Code: P(\\ell, \\cos(\\theta))$ (that's the in $LaTeX Code: 2 \\, \\ell + 1$ ) plus $LaTeX Code: P(\\ell, m, \\cos(\\theta)) \\, \\cos(m \\, \\phi), \\; \\; 1 \\leq m \\leq \\ell$ $LaTeX Code: P(\\ell, m, \\cos(\\theta)) \\, \\sin(m \\, \\phi), \\; \\; 1 \\leq m \\leq \\ell$ where the $LaTeX Code: P(\\ell, m, \\cdot)$ are the associated Legendre functions (that's the $LaTeX Code: 2 \\, \\ell$ in $LaTeX Code: 2 \\, \\ell + 1$ ). That is, the eigenfunctions are the real and imginary parts of the usual spherical harmonics $LaTeX Code: <BR>Y^{\\ell, m}(\\theta, \\phi) = P(\\ell, m, \\cos(\\theta)) \\, \\exp(m \\, i \\, \\phi)<BR>$ See Kenneth I. Gross, "The Evolution of Non-Commutative Harmonic Analysis", Amer. Math. Monthly Aug.-Sept. 1978: 525--548. (Students and academics whose institution subscribes to JSTOR: past issues of the Monthly are available on-line, and past issues back to 1894 are well worth snarfing--- highest recommendation!) These results can be complexified in the obvious way. I have discussed only the scalar spherical harmonics; there are also vector spherical harmonics and tensor spherical harmonics. Also, these results can be extended with minimal change (Weyl) to the Laplacian associated with compact Lie groups other than acting on homogeneous spaces other than . With more work (Harish-Chandra) to non-compact semi-simple Lie groups. With still more work (Mackey) to infinite-dimensional Lie groups. With still more work (Kirillov, Kostant, etc.) to nilpotent Lie groups. See also my post #4 in the recently locked thread (thanks, berkeman!) "Representation theory?" http://www.physicsforums.com/showthread.php?t=185965 My point is: yes, there is a broader significance!