## 36th Annual Western States Mathematical Physics Meeting

#### Sunday Feb 18, 2018

08:30–09:00 Welcome coffee
Morning session (chair: TBA)
09:00–10:00 Barry Simon (Caltech) : Szegő–Widom asymptotics for Chebyshev polynomials on subsets of R
10:00–10:30 Wencai Liu (UCI):Sharp spectral transitions for singular spectrum and growth of eigenfunctions of Laplacians on Riemannian manifolds
10:30–11:00 Coffee break
11:00–12:00 Vojkan Jaksic (McGill): Time and Entropy
12:00–12:30 Ilya Kachkovskiy (MSU): Localization and delocalization for two interacting 1D quasiperiodic particles

12:30–13:30 Lunch break

Afternoon session (chair:TBA)

13:30–14:30 Semyon Dyatlov (UCB/MIT): Lower bounds on eigenfunctions on hyperbolic surfaces
14:30–15:30 Alexander Elgart (Virginia Tech): Level spacing and Poisson statistics for continuum random Schrödinger operators
15:30–16:00 Coffee break
16:00–17:00 Sergei Gukov (Caltech):Quantum Spectral Curves
17:00–17:30 Xiaowen Zhu (UCI): A short proof of Anderson localization for the 1-d Anderson model
17:30–17:00 Gökalp Alpan (Rice) : Extremal polynomials on generalized Julia sets

#### Monday Feb 19, 2018

08:30–09:00 Welcome coffee
Morning session (chair: TBA)
09:00–09:30 Selim Sukhtaiev (Rice): The Maslov index and the spectra of second order elliptic operators
09:30–10:00 Tom VandenBoom (Rice): The KdV hierarchy via Abelian coverings
10:00–10:30 Thomas D. Trogdon (UCI): Universality for the Toda algorithm
10:30–11:00 Coffee break
11:00–12:00 Abel Klein (UCI): Manifestations of dynamical localization in the random XXZ spin chain
12:00–12:30 Chris Marx (Oberlin): Dependence of the density of states on the probability distribution for discrete random Schrödinger operators

12:30–14:00 Lunch break

Afternoon session (chair:TBA)

14:00–14:30 Darren Ong (Xiamen University Malaysia): Generalized Toda Flows
14:30–15:00 Sasa Kocic (Mississippi): Renormalization and rigidity of circle diffeomorphisms with breaks
15:00–15:30 Zhenghe Zhang (UCR): Spectral Characteristics of the Unitary Critical Almost Mathieu Operators
15:30–16:00 Coffee break
16:00–16:30 Rajinder Mavi (MSU): Anderson localization for a disordered polaron.
16:30–17:00 Anna Maltsev (QMUL): Localization and landscape functions on quantum graphs
17:00–18:00 Rowan Killip (UCLA): KdV is globally well-posed in H^{-1}

## Abstracts

##### Gokalp Alpan, Extremal polynomials on generalized Julia sets
We discuss how to construct generalized Julia sets associated to a sequence of non-linear polynomials. These sets are natural generalizations of classical polynomial Julia sets. We review recent results related to asymptotics of orthogonal polynomials and Chebyshev polynomials on them.
##### Semyon Dyatlov, Lower bounds on eigenfunctions on hyperbolic surfaces
I show that on a compact hyperbolic surface, the mass of an $L^2$-normalized eigenfunction of the Laplacian on any nonempty open set is bounded below by a positive constant depending on the set, but not on the eigenvalue. This statement, more precisely its stronger semiclassical version, has many applications including control for the Schr\"odinger equation and the full support property for semiclassical defect measures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain.
##### Alexander Elgart, Level spacing and Poisson statistics for continuum random Schrödinger operators
For the standard Anderson model on the lattice, Minami's estimate implies that, with high probability, the eigenvalues of the Anderson model are well-spaced. Unfortunately, the method fails beyond rank one random perturbation. We will describe a new, more flexible approach towards such a level-spacing estimate. In particular, it works for the continuum Anderson model, at the bottom of its spectrum. If the single-site probability distribution is sufficiently regular, it leads to a Minami-type estimate and Poisson statistics of eigenvalues for this model. The talk is based on joint work with Adrian Dietlein.

##### Vojkan Jaksic, Time and Entropy
This talk concerns mathematical theory of the so-called Fluctuation Relation (FR) and Fluctuation Theorem (FT) in context of dynamical systems relevant to physics. The FR refers to a certain universal identity linked to statistics of entropy production generated by a reversal operation and FT to the related mathematical large deviations result. The discovery of FR goes back to numerical experiments and Evans, Cohen and Morris (1993) and theoretical works of Evans and Searles (1994), Gallavotti and Cohen (1995). These discoveries generated an enormous body of numerical, theoretical and experimental works which have fundamentally altered our understanding of non-equilibrium physics, with applications extending to chemistry and biology. In this talk I will introduce modern theory of FR and FT on an example and comment on a current research program on this topic.
##### Ilya Kachkovskiy, Localization and delocalization for two interacting 1D quasiperiodic particles
The talk is about several tentative results, joint with J. Bourgain and S. Jitomirskaya. We consider a model of two 1D almost Mathieu particles with a finite range interaction. The presence of interaction makes it difficult to separate the variables, and hence the only known approach is to treat it as a 2D model, restricted to a range of parameters (both frequencies and phases of the particles need to be equal). In the usual 2D approach, a positive measure set of frequency vectors is usually removed, and extra care needs to be taken in order to keep the diagonal frequencies (which is a zero measure set) from being removed. We show that the localization holds at large disorder for energies separated from zero and from certain values associated to the interaction. We also study the model in the regime of strong interaction, in which case an additional band of spectrum (“droplet band”) is created. We show that this droplet band is localized in the regime of large interaction and fixed difference between phases (in particular, it covers the physical’’ regime of equal phases). However, there is another regime where the difference between phases is close to pi/2, in which case the droplet band has some ac spectrum.

##### Abel Klein, Manifestations of dynamical localization in the random XXZ spin chain
We study random XXZ spin chains in the Ising phase exhibiting droplet localization, a single cluster localization property we previously proved for random XXZ spin chains. It holds in an energy interval near the bottom of the spectrum, known as the droplet spectrum. We establish dynamical manifestations of localization in the energy window of localization, including non-spreading of information, zero-velocity Lieb-Robinson bounds, and general dynamical clustering. Our results do not rely on knowledge of the dynamical characteristics of the model outside the droplet spectrum. A byproduct of our analysis is that this droplet localization can happen only inside the droplet spectrum. (Joint work with Gunter Stolz and Alex Elgart.)

##### Rowan Killip, KdV is globally well-posed in H^{-1}
I will describe a proof of well-posedness in H^{-1} that works both on the line and on the circle. On the line, this result was previously unknown; on the circle it was proved by Kappeler and Topalov. This is joint work with Monica Visan.
##### Sasa Kocic, Renormalization and rigidity of circle diffeomorphisms with breaks
Renormalization provides a powerful tool to approach universality and rigidity phenomena in dynamical systems. In this talk, I will discuss recent results on renormalization and rigidity theory of circle diffeomorphisms (maps) with a break (a single point where the derivative has a jump discontinuity) and their relation with generalized interval exchange transformations introduced by Marmi, Moussa and Yoccoz. In a joint work with K.Khanin, we proved that renormalizations of any two sufficiently smooth circle maps with a break, with the same irrational rotation number and the same size of the break, approach each other exponentially fast. For almost all (but not all) irrational rotation numbers, this statement implies rigidity of these maps: any two sufficiently smooth such maps, with the same irrational rotation number (in a set of full Lebesgue measure) and the same size of the break, are $C^1$-smoothly conjugate to each other. These results can be viewed as an extension of Herman's theory on the linearization of circle diffeomorphisms.
##### Wencai Liu, Sharp spectral transitions for singular spectrum and growth of eigenfunctions of Laplacians on Riemannian manifolds
In this talk, we study singular spectrum of the Laplacian ($\Delta$) embedded in the essential spectrum on either asymptotically flat or asymptotically hyperbolic manifolds. It is an old result that the essential spectrum is $[\frac{c^2}{4},\infty]$ if $\Delta r \to c$ as $r$ goes to infinity, where $r(x)$ is the distance function. Kumura proved that there are no eigenvalues embedded in the essential spectrum $\sigma_{{\rm ess}}(-\Delta)=\left[\frac{1}{4}(n-1)^2,\infty \right)$ of Laplacians on asymptotically hyperbolic manifolds, where asymptotic hyperbolicity is characterized by the radial curvature, i.e., $K_{\rm rad}=-1+o(r^{-1})$. He also constructed a manifold for which an eigenvalue $\frac{(n-1)^2}{4} + 1$ is embedded into its essential spectrum $[ \frac{(n-1)^2}{4} , \infty )$ with the radial curvature $K_{\rm rad}(r) = -1+O(r^{-1})$. The first part of the talk, based on a joint work with S.Jitomirskaya, is devoted to construction of manifolds with singular continuous spectrum or prescribed embedded eigenvalues. Given any finite (countable) positive energies $\{\lambda_n\}\in [\frac{K_0}{4}(n-1)^2,\infty)$, we construct Riemannian manifolds with the decay of order $K_{\rm rad}+K_0=O(r^{-1})$ with $K_0\geq 0$ ($K_{\rm rad}+K_0=\frac{C(r)}{r}$, where $C(r)\geq 0$ and $C(r)\to \infty$ arbitrarily slowly) such that the eigenvalues $\{\lambda_n\}$ are embedded in the essentialspectrum $\sigma_{{\rm ess}}(-\Delta)=\left[\frac{K_0}{4}(n-1)^2,\infty\right)$. We also construct Riemannian manifolds with the decay of order $K_{\rm rad}+K_0=\frac{C(r)}{r}$, where $C(r)\geq 0$ and $C(r)$ goes $\infty$ arbitrarily slowly, and the Laplacians have singular continuous spectrum in $\left[\frac{K_0}{4}(n-1)^2,\infty\right)$. In the second part I discuss criteria for the absence of eigenvalues embedded into essential spectrum in terms of the asymptotic behavior of $\Delta r$ with no conditions on the curvature. Under a weaker convexity of distance function $r$ for certain asymptotic behavior of $\Delta r$, we established the growth of the eigensolution of $\Delta u =\lambda u$ for $\lambda>\lambda_0$. As an application, we show that there are no eigenvalues embedded into the essential spectrum if $\Delta r=a+\frac{b}{r}+\frac{o(1)}{r}$ as $r$ goes to infinity and the distance function $r$ satisfies some weakerconvexity. The proof is based on Kato's method, but with some new ingredients- the flexible construction of energy functions and a new way to verify the positivity of the initial energy.
##### Anna Maltsev, Localization and landscape functions on quantum graphs
I will discuss localization and other properties of eigenfunctions of the Schr\"odinger operator on quantum graphs. The motivation is to understand how graph structure impacts eigenfunction behavior. I will present two estimates based on the Agmon method to show that a tree structure aids the exponential decay at energies below the essential spectrum. I will furthermore present adaptations of the landscape function approach, well-established for $\mathbb{R}^n$, to quantum graphs and its limitations. In our context, a landscape function'' $\Upsilon(x)$ is a function that controls the localization properties of normalized eigenfunctions $\psi(x)$ through a pointwise inequality of the form $|\psi(x)| \le \Upsilon(x).$ The connectedness of a graph can present a barrier to the existence of universal landscape functions in the high-energy r\'egime, as we demonstrate with simple examples. However, at low and moderate energies landscape functions can be made explicit. This talk is based on joint work with Evans Harrell.
##### Chris Marx, Dependence of the density of states on the probability distribution for discrete random Schrödinger operators
We prove the Hölder-continuity of the density of states measure (DOSm) and the integrated density of states (IDS) for discrete random Schrödinger operators with finite-range potentials with respect to the probability measure. In particular, our result implies that the DOSm and the IDS for smooth approximations of the Bernoulli distribution converge to the corresponding quantities for the Bernoulli-Anderson model. Other applications of the technique are given to the dependency of the DOSm and IDS on the disorder, and the continuity of the Lyapunov exponent in the weak-disorder regime for dimension one. The talk is based on joint work with Peter Hislop (Univ. of Kentucky)
##### Rajinder Mavi, Anderson localization for a disordered polaron
We will consider a tracer particle on $Z^d$ subject to an Anderson field, moreover, we associate a one dimensional ossilator to each site of the lattice. This forms a polaron model where the oscillators communicate only through the hopping of the tracer particle. This introduces, a priori, infinite degeneracies of bare energies at large distances. We nevertheless show Dynamical Localization of the tracer particle for compact subsets of the spectrum. This is joint work with Jeff Schenker.
##### Darren Ong, Generalized Toda Flows
The classical hierarchy of Toda flows can be thought of as an action of the abelian group of polynomials on Jacobi matrices. We present a generalizations of this flow to the larger groups C^2 and entire functions, and we prove that the latter generalization remains an isospectral flow. This is joint work with Christian Remling.
##### Barry Simon, Szegő–Widom asymptotics for Chebyshev polynomials on subsets of R
Chebyshev polynomials for a compact subset e ⊂ R are defined to be the monic polynomials with minimal $||·||_∞$ over e. In 1969, Widom made a conjecture about the asymptotics of these polynomials when e was a finite gap set. We prove this conjecture and extend it also to those infinite gap sets which obey a Parreau–Widom and a Direct Cauchy Theory condition. This talk will begin with a generalities about Chebyshev Polynomials. This is joint work with Jacob Christiansen and Maxim Zinchenko and partly with Peter Yuditskii.
##### Selim Sukhtaiev, The Maslov index and the spectra of second order elliptic operators
In this talk I will discuss a formula relating the spectral flow of the one-parameter families of second order elliptic operators to the Maslov index, the topological invariant counting the signed number of conjugate points of certain paths of Lagrangian planes. This talk is based on joint work with Yuri Latushkin.
##### Tom VandenBoom, The KdV hierarchy via Abelian coverings
This talk discusses joint work with Benjamin Eichinger and Peter Yuditskii. We prove global existence and uniform almost-periodicity in the time and space coordinates of solutions to the Cauchy problem for the KdV hierarchy with reflectionless initial conditions having spectrum satisfying a certain moment condition. Our methods involve the development of 1) generalized Abelian integrals on noncompact Riemann surfaces and 2) a spectrally-dependent Fourier transform, with respect to which the Schrödinger and Lax operators associated to the KdV hierarchy become greatly simplified.
##### Thomas D. Trogdon, Universality for the Toda algorithm
It is well known that a class of classical integrable systems give rise to isospectral flows on symmetric/Hermitian matrices and these systems can be integrated to compute the eigenvalues. And as evidenced by Pfrang, Deift and Menon such numerical algorithms exhibit a universal runtime, or halting time, when they are applied to appropriate random matrices. Utilizing recent results from random matrix theory, we (with P. Deift) prove universal limit theorems for specific halting times for the integrable Toda and QR algorithms and the power/inverse power methods. This work presents a confluence of the theory of integrable systems with the theory of integrable probability.

##### Zhenghe Zhang, Spectral Characteristics of the Unitary Critical Almost Mathieu Operators
I will present an example of unitary self-dual operators which arises naturally in the quantum walk with quasiperiodic coin toss. It shares many properties with the critical Almost Mathieu Operators. In particular, we showed that the operators display an Aubry type of self-duality, have pure singular continuous spectrum for almost every phases, and have zero Lyapunov exponent on spectrum which implies Cantor spectrum of zero Lebesgue measure. Our method is mainly dynamical which involves a detailed study of the associated Lyapunov exponent. This joint work with Jake Fillman and Darren Ong.
##### Xiaowen Zhu, A short proof of Anderson localization for the 1-d Anderson model
The proof of Anderson localization for 1D Anderson model with arbitrary (e.g. Bernoulli) disorder, originally given by Carmona-Klein-Martinelli in 1987, is based on the Furstenberg theorem and multi-scale analysis. This topic has received a renewed attention lately, with two recent new proofs, exploiting the one-dimensional nature of the model. At the same time, in the 90s it was realized that for one-dimensional models with positive Lyapunov exponents some parts of multi-scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle, leading to nonperturbative proofs for 1D quasiperiodic models. Here we present a proof along these lines, for the Anderson model. We also include a proof of dynamical localization based on the uniform version of Craig-Simon that works in high generality and may be of independent interest. It is a joint work with S. Jitomirskaya. Our entire proof of spectral localization fits in three pages and we expect to present almost complete detail during the talk.