- #1

joshmccraney

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I'm trying to show that the eigenvalue problem $$L u = \lambda M u$$ is equivalent to solving $$\min_\phi (L\phi,\phi) : (M\phi,\phi) = 1$$

where ##\phi## is a real function of ##x## and ##L,M## are Hermitian operators and ##\lambda## is the Lagrange multiplier constant.

Applying Lagrange multipliers to the constrained problem yields a functional

$$ J = (L\phi,\phi) - \lambda\left[ (M\phi,\phi) - 1\right] \implies\\

\delta J = \delta (L\phi,\phi) - \delta\left[\lambda ((M\phi,\phi) - 1)\right]\\

= 2(L\phi,\delta\phi) - \delta\lambda ((M\phi,\phi) - 1) - 2\lambda(M\phi,\delta\phi)\\

=2(L\phi-\lambda M\phi,\delta\phi) - \delta\lambda ((M\phi,\phi) - 1) = 0.

$$

Now I know ##\delta\lambda ((M\phi,\phi) - 1) = 0## implies ##L u = \lambda M u##, but why is ##\delta\lambda ((M\phi,\phi) - 1) = 0##? Am I missing something? I think ##\delta \lambda = 0## (since ##\lambda## is a constant). What do you think?

Also, how is the operator ##\delta## above defined? I've been treating it as a derivative, but what's the formal definition? I've read several different websites now but can't find a direct definition.