  The Bloch--Torrey operator $-h^2\Delta+e^{i\alpha}x_1$ on a bounded smooth
planar domain, subject to Dirichlet boundary conditions, is analyzed. Assuming
$\alpha\in\left[0,\frac{3\pi}{5}\right)$ 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.
