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