Date:

Wed, 22/12/202114:00-15:00

Title: Discrete approximations of magnetic Schroedinger operators on L^2(R^2) and topological equivalence

Abstract: Discrete approximations of continuum Schroedinger operators are well-studied under the name “tight-binding limits”. However, when these continuum operators have topologically non-trivial properties, for example, with a strong magnetic field in two-dimensions (which models the integer quantum Hall effect), such proofs did not exist. We provide such a proof: we show that there is a discrete operator on l^2(Z^2) such that the scaled continuum operator converges to the discrete operator in norm-resolvent sense. A central ingredient in the proof is an estimate on the magnetic double-well eigenvalue splitting problem. We then go on to show that the topological properties in the continuum and discrete settings agree. This talk is based on joint collaborations with C. L. Fefferman and M. I. Weinstein and with M. I. Weinstein.

Abstract: Discrete approximations of continuum Schroedinger operators are well-studied under the name “tight-binding limits”. However, when these continuum operators have topologically non-trivial properties, for example, with a strong magnetic field in two-dimensions (which models the integer quantum Hall effect), such proofs did not exist. We provide such a proof: we show that there is a discrete operator on l^2(Z^2) such that the scaled continuum operator converges to the discrete operator in norm-resolvent sense. A central ingredient in the proof is an estimate on the magnetic double-well eigenvalue splitting problem. We then go on to show that the topological properties in the continuum and discrete settings agree. This talk is based on joint collaborations with C. L. Fefferman and M. I. Weinstein and with M. I. Weinstein.