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 $\mathbb{Z} \times\mu\mathbb{Z}$ 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 minicourse, we explore analogous spectral properties for a more refined model: the continuous Schrödinger operator with a $\mathbb{Z}\times \mu \mathbb{Z}$-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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- Lecture 1: The tight-binding reduction.
- Monday, May 19th 3pm-4pm Lunt 107 (Analysis seminar)
- Lecture 2: WKB estimates and Harper's operator as a dynamical system
- Tuesday, May 20th 4pm-5pm Lunt 105 (Dynamical systems seminar)
- Lecture 3: Stability of the Ten-Martini problem
- Thursday, May 22nd 3pm-4pm Lunt 105