Let \(B>0\). The Landau operator is defined by

\[ H_B:=\bigg(i\nabla + \frac{B}{2}\begin{pmatrix} -x_2 \\ x_1 \end{pmatrix}\bigg)^2. \]
Remarks
  • The Landau operator is self-adjoint.
  • The spectrum is \(\{B, 3B, 5B,\ldots \}\).
  • The Landau operator is relevant in physics (see [1] for examples).
  • The Landau operator may be expressed in terms of magnetic derivatives as a Laplacian, that is \(H_B=\widetilde{\partial }_1^2+\widetilde{\partial }_2^2\).

See also Link to heading

References Link to heading

  1. P. Pfeiffer and M. Täufer, Magnetic Bernstein inequalities and spectral inequality on thick sets for the Landau operator, 2023. doi:10.48550/arXiv.2309.14902