Semiclassical asymptotics of the Bloch–Torrey operator in two dimensions

Abstract

The Bloch–Torrey operator −ℎ² Δ + 𝑒^𝑖𝛼 𝑥₁ 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 eigenvalues 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.