A short proof of a strong Weyl law in dimension 1

For the Dirichlet realization of $-d^2/dx^2-\lambda^2V$ on a bounded interval, with $V$ a positive $C^2$ potential bounded away from $0$ and $\lambda>0$ a large parameter, we prove an asymptotic law for the values $\lambda_n$ of $\lambda$ at the $n^{\text{th}}$ appearance of a new negative eigenvalue. This approximation is correct up to an error of order $1/n$, thus making the result strictly stronger than the classical Weyl law for the number of negative eigenvalues for these operators.