Bohr - Sommerfeld quantization rules in the semiclassical limit

We study one-dimensional quantum mechanical systems in the semiclassical limit. We construct a lowest order quasimode ψ(ℏ) for the Hamiltonian H(ℏ) when the energy E and Planck's constant ℏ satisfy the appropriate Bohr - Sommerfeld conditions. This means that ψ(ℏ) is an approximate solution of the Schrödinger equation in the sense that \\[H(ℏ) - E]ψ(ℏ)|| ≤ Cℏ^3/2||ψ(ℏ)||. It follows that H(ℏ) has some spectrum within a distance Cℏ^3/2 of E. Although the result has a long history, our time-dependent construction technique is novel and elementary.