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So I'm doing a little calculus of variations on an eigenvalue problem. The goal of this is to find an upper bound for the $\omega_0$ as follows:

$\omega_0^2 \leq \frac{T_0\int_0^a\left|\frac{d\psi}{dx}\right|^2 dx}{\int_0^a \mu(x)\left|\psi(x)\right|^2dx}$ where $\psi(x)$ is any function vanishing at $x=0$ and $x=a$.

How do I minimize the RHS using calculus of variations? I am not super comfortable with variations yet and I need a rough roadmap.

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up vote 3 down vote accepted

The right-hand side is invariant under scaling of $\psi$. Thus you can restrict $\psi$ to normalized functions with $\int_0^a\mu(x)|\psi(x)|^2\mathrm dx=1$ without loss of generality. Then you can find $\psi$ by finding the stationary points of the functional

$$L=T_0\int_0^a\left|\frac{\mathrm d\psi}{\mathrm dx}\right|^2\mathrm dx-\lambda\int_0^a\mu(x)|\psi(x)|^2\mathrm dx$$

with Lagrange multiplier $\lambda$. For these purposes, you can treat $\psi$ and $\psi^*$ as independent; then the Euler-Lagrange equation becomes

$$T_0\frac{\mathrm d^2\psi}{\mathrm dx^2}=\lambda\mu\psi\;,$$

where the eigenvalue $\lambda$ is a free parameter quantized due to the boundary conditions.

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