Semiclassical asymptotics of the Bloch–Torrey operator in two dimensions

The Bloch--Torrey operator -h²Δ+eiαx₁ on a bounded smooth planar domain, subject to Dirichlet boundary conditions, is analyzed. Assuming α∈[0,3π/5) and a non-degeneracy assumption on the left-hand side of the domain, asymptotics of the eigenvalues with the smallest real part in the limit h → 0 are derived. The strategy is a backward complex scaling and the reduction to a tensorized operator involving a real Airy operator and a complex harmonic oscillator.