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This article is devoted to the spectral analysis of the electromagnetic Schrödinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum in various situations and appropriate regions of the complex plane. Not only results of the self-adjoint case are proved (or recovered) in the proposed unifying framework, but also new results are established when the electric potential is complex-valued. In such situations, when the non-self-adjointness comes with its specific issues (lack of a “spectral theorem”, resolvent estimates), the analogue of the “low-lying eigenvalues” of the self-adjoint case are still accurately described and the spectral gaps estimated.

Eigenvalue Asymptotics for Confining Magnetic Schrödinger Operators with Complex Potentials