Semiclassical asymptotics of the Bloch--Torrey operator in two dimensions

The Bloch–Torrey operator $-h^2\mathrm{\Delta }+e^{i\alpha }{x}_1$ on a bounded smooth planar domain, subject to Dirichlet boundary conditions, is analyzed. Assuming $\alpha \in [0,\frac{3\pi }{5})$ 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 $h\to 0$ 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.