User:Dnessett/Sturm-Liouville/Orthogonality proof

This article proves that solutions to the Sturm–Liouville equation corresponding to distinct eigenvalues are orthogonal. For background see Sturm–Liouville theory.

Theorem

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   , where   and   are solutions to the Sturm–Liouville equation corresponding to distinct eigenvalues and   is the "weight" or "density" function.

Proof

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Let   and   be solutions of the Sturm-Liouville equation [1] corresponding to eigenvalues   and   respectively. Multiply the equation for   by   (the complex conjugate of  ) to get:

 

(Only  ,  ,  , and   may be complex; all other quantities are real.) Complex conjugate this equation, exchange   and  , and subtract the new equation from the original:

 

Integrate this between the limits   and  

 

The right side of this equation vanishes because of the boundary conditions, which are either:

  periodic boundary conditions, i.e., that  ,  , and their first derivatives (as well as  ) have the same values at   as at  , or
  that independently at   and at   either:
  the condition cited in equation [2] or [3] holds or:
   

So:  .

If we set   , so that the integral surely is non-zero, then it follows that  ; that is, the eigenvalues are real, making the differential operator in the Sturm-Liouville equation self-adjoint (hermitian); so:

 

It follows that, if   and   have distinct eigenvalues, then they are orthogonal. QED.

See also

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References

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1. Ruel V. Churchill, "Fourier Series and Boundary Value Problems", pp. 70–72, (1963) McGraw–Hill, ISBN 0-07-010841-2.