This talk describes joint work with Roman Bessonov and Peter
Yuditskii. In the spectral theory of self-adjoint and unitary
operators in one dimension (such as Schrodinger, Dirac, and Jacobi
operators), a half-line operator is encoded by a Weyl function; for
whole-line operators, the reflectionless property is a
pseudocontinuation relation between the two half-line Weyl functions.
We develop the theory of reflectionless canonical systems with an
arbitrary Dirichlet-regular Widom spectrum with the Direct Cauchy
Theorem property. This generalizes, to an infinite gap setting, the
constructions of finite gap quasiperiodic (algebro-geometric)
solutions of stationary integrable hierarchies. Instead of theta
functions on a compact Riemann surface, the construction is based on
reproducing kernels of character-automorphic Hardy spaces in Widom
domains with respect to Martin measure. We also construct unitary
character-automorphic Fourier transforms which generalize the
Paley-Wiener theorem. Finally, we find the correct notion of almost
periodicity which holds in general for canonical system parameters in
Arov gauge, and we prove generically optimal results for almost
periodicity for Potapov-de Branges gauge, and Dirac operators.

In this talk, we study the ground state energy of a Schrodinger operator and its relation to the landscape potential. For the 1-d Bernoulli Anderson model, we show that the ratio of the ground state energy and the minimum of the landscape potential approaches pi^2/8 as the domain size approaches infinity. We then discuss some numerical stimulations and conjectures for excited states and for other random potentials. The talk is based on joint work with I. Chenn and W. Wang.  

In this talk, we will sketch how to generalize the large deviation theorem to quasiperiodic Gevrey $SL(2,\R)$-cocycles and use it to prove the joint continuity of the Lyapunov exponent.