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Let's consider the Sturm-Liouville boundary value problem $$ ly := -y'' + q(x)y = \lambda y, 0

Here $\lambda$ is the spectral parameter. The values of the parameter $\lambda$ for which $L$ has nonzero solutions are called eigenvalues and the corresponding nontrivial solutions $\phi(x,\lambda)$ are called eigenfunctions. The set of eigenvalues is called the spectrum of $L$.

The numbers $\{\alpha_n\}$ are called the weight numbers and the numbers $\{\lambda_n, \alpha_n\}$ are called the spectral data of the boundary value problem $L$. $$ \alpha_n := \int_{0}^{\pi}\phi^2(x,\lambda_n)dx $$

Question 1: What is the purpose and logic behind weight numbers $\{\alpha_n\}$ and why they are a part of the spectral data?

Question 2 (optional): How weight numbers could help to solve an inverse spectral problem?

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    The weight numbers are used to normalize the Fourier series in the eigenfunctions. $f \sim \sum_{n}\frac{\int_{0}^{\pi}f(t)\phi(t,\lambda_n)dt\phi(x,\lambda_n)}{\int_{0}^{\pi}\phi(t,\lambda_n)^2dt}$. Or you can normalize the $\phi(x,\lambda_n)$ by dividing by $\sqrt{\int_{0}^{\pi}\phi(t,\lambda_n)^2dt}$ to have an orthonormal basis. – 2017-01-10
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    @TrialAndError Thank you for the reply! How does the weight numbers help us to solve the inverse spectral problem (finding $q(x))$)? – 2017-01-10
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    I don't know how it would help with inverse problems. The only real purpose of the numbers is to normalize the eigenfunctions so that you can expand in a Fourier series. It's the same as with the ordinary Fourier series. The eigenfunctions make a good basis for the problem because they diagonalize the differential operator. If you chose the eigenfunctions to be normalized from the beginning, then the normalization numbers would be 1 for all $n$, which is why they are not significant on their own, even though the normalized eigenfunctions are significant and unique up to $\pm 1$ for real cases. – 2017-01-10

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