May 19-22, 2025
Mini-course on Continuous Schrödinger Operators:
- Simon Becker (ETH Zurich) will give a three lecture minicourse May 19th to May 22nd on spectral properties of continuous Schrödinger operators in the positive Lyapunov exponent regime
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Title: Cantor spectrum for continuous Schrödinger operators in the positive Lyapunov exponent regime
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Abstract:The discrete operator describing an electron on a $\Z\times\mu\Z$ lattice in the presence of a magnetic field is known as Harper’s operator. When the magnetic flux is incommensurate with respect to the lattice spacing, the spectrum of this operator forms a Cantor set—a phenomenon known as the Ten Martini Problem, resolved by Avila and Jitomirskaya in 2005. In this mini-course, we explore analogous spectral properties for a more refined model: the continuous Schrödinger operator with a $\Z\times \mu \Z$-periodic potential under a constant magnetic field. Our goal is to establish Cantor spectrum results in the semiclassical regime for incommensurate magnetic flux and $\mu\neq 1$, extending the spirit of the Ten Martini Problem to the continuous and anisotropic setting. The approach crucially relies on new stability results in the large Lyapunov exponent regime.
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