A short proof of a strong Weyl law in dimension 1

For the Dirichlet realization of -d²/dx²-λ²V on a bounded interval, with V a positive C² potential bounded away from $0$ and λ>0 a large parameter, we prove an asymptotic law for the values λₙ of λ at the nᵗʰ 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.