Western States schedule

37th Almost Annual Western States Mathematical Physics Meeting

Saturday Mar 7, 2020

08:30–09:00 Welcome coffee
Morning session (chair: TBA)
09:00–09:30 Christoph Fischbacher (UC Irvine): Logarithmic lower bounds for the entanglement entropy of droplet states for the XXZ model on the ring
09:30–10:30 Anton Gorodetski (UC Irvine): Random Matrix Products and Anderson Localization
10:30–11:00 Coffee break
11:00–11:30 Houssam Abdul-Rahman (University of Arizona): Entanglement bounds for the XXZ quantum spin chain
11:30–12:00 Alvin Moon (UC Davis): Automorphic equivalence within gapped phases in the bulk
12:00–12:30 Ilya Kachkovskiy (MSU): On the relation between strong ballistic transport and exponential dynamical localization

12:30–14:00 Lunch break

Afternoon session (chair:TBA)

14:00–15:00 Andras Vasy (Stanford University): Fredholm theory for the Laplacian near zero energy on asymptotically conic spaces
15:00–15:30 Milivoje Lukic (Rice University): Stahl--Totik regularity for continuum Schr\"odinger operators
15:30–16:00 Coffee break
16:00–16:30 Xin Zhao (UC Irvine): H\"older continuity of absolutely continuous spectral measure for quasi-periodic Schr\"odinger operators
16:30–17:30 Michael Hitrik (UCLA): Toeplitz operators, asymptotic Bergman projections, and second microlocalization

Sunday March 8, 2020

08:30–09:00 Welcome coffee
Morning session (chair: TBA)
09:00–09:30 Nishant Rangamani (UC Irvine): Exponential Dynamical Localization for Random Word Models
09:30–10:00 Xiaowen Zhu (UC Irvine): From MSA to localization: for singular discrete Anderson model
10:00–10:30 Konstantin Makarov (University of Missouri): The Quantum Zeno Effect versus Exponential Decay Alternative
10:30–11:00 Coffee break
11:00–12:00 Barry Simon (Caltech): Periodic Jacobi Matrices on Trees
12:00–12:30 Lingrui Ge (UC Irvine): Quantitative almost reducibility and its spectral applications

12:30–14:00 Lunch break

Afternoon session (chair:TBA)

14:00–14:30 Wencai Liu (Texas A&M University): The large deviation theorem for quasi-periodic Schr\"odinger operators
14:30–15:30 Rupert Frank (Caltech): Recent results and open problems on Lieb-Thirring inequalities
15:30–16:00 Coffee break
16:00–16:30 Simon Larson (Caltech): Lieb-Thirring inequalities for wave functions vanishing on the diagonal set
16:30–17:30 Simon Becker (Cambridge University): Spectral gaps in classical many-particle systems

Abstracts

Houssam Abdul-Rahman, Entanglement bounds for the XXZ quantum spin chain

We consider the XXZ chain in the Ising phase. The particle number conservation property is used to write the Hamiltonian in a hard-core particles formulation over the $N$-symmetric product of graphs, where $N\in\mathbb{N}_0$ is the number of conserved particles. The droplet regime corresponds to a band at the bottom of the spectrum of the model consisting of a connected set (a droplet) of down-spins, up to an exponentially decaying error. Various many-body localization indicators has been proven in this regime by Elgart/Klein/Stolz and Beaud/Warzel. In particular, an area law is shown for the entanglement of arbitrary states in this localized regime. As a first step beyond the droplet regime, we show that the entanglement of arbitrary states above the droplet regime (associated with multiple droplets/clusters) does not follow area laws, and instead, it follows a logarithmically corrected area law. We will comment on the effects of disorder on entanglement, and show how our results may hint a phase transition. (joint work with C. Fischbacher and G. Stolz, arXiv1907.11420)

Simon Becker, Spectral gaps in classical many-particle systems

I will review some of the standard techniques to establish spectral gaps in classical systems and we will then see that most of these criteria cannot be applied to most many-particle systems. We will then discuss examples such as classical spin models, models for heat transport, lattice approximations to SPDEs and the Coulomb gas (all this depending on time) and see how ideas from renormalization group theory and Schr\"odinger operators can be applied to such systems.

Christoph Fischbacher, Logarithmic lower bounds for the entanglement entropy of droplet states for the XXZ model on the ring

We study the free XXZ quantum spin model defined on a ring of size L and show that the bipartite entanglement entropy of eigenstates belonging to the first energy band above the vacuum ground state satisfy a logarithmically corrected area law. This is joint work with Ruth Schulte (LMU).

Rupert Frank, Recent results and open problems on Lieb-Thirring inequalities

Lieb-Thirring inequalities bound sums of powers of eigenvalues of Schr\"odinger operators in terms of integrals of the potential. We survey the problem of sharp constants in these inequalities and discuss recent progress. This includes, on one hand, the currently best constant for the sum of eigenvalues, obtained jointly with Hundertmark, Jex and Nam, and, on the other hand, some negative results on the so-called one-particle constant, obtained jointly with Gontier and Lewin.

Lingrui Ge, Quantitative almost reducibility and its spectral applications

We report our recent progresses on quantitative almost reducibility for quasi-periodic $SL(2,\R)$-cocycles. As applications, we will explain how to use them to obtain asymptotic behavior of the transfer matrix, arithmetic version of Anderson localization, exponential dynamical localization and asymptotic behavior of spectral gaps for quasi-periodic Schr\"odinger operators (especially for the almost Mathieu operator). This is based on joint works with Jiangong You, Xin Zhao and Qi Zhou.

Ilya Kachkovskiy, On the relation between strong ballistic transport and exponential dynamical localization

We establish strong ballistic transport for a family of discrete quasiperiodic Schrodinger operators as a consequence of exponential dynamical localization for the dual family. The latter has been, essentially, shown by Jitomirskaya and Kruger in the one-frequency setting and by Ge--You--Zhou in the multi-frequency case. In both regimes, we obtain strong convergence of $\frac{1}{T}X(T)$ to the asymptotic velocity operator $Q$, which improves recent perturbative results by Zhao and provides the strongest known form of ballistic motion. In the one-frequency setting, this approach allows to treat Diophantine frequencies non-perturbatively and also consider the weakly Liouville case. The proof is based on the duality method. Originally, localization for the dual model allows to obtain ballistic transport in expectation. Combined with dynamical localization bounds, the improved convergence allows to replace ``in expectation'' by ``almost surely''.

Anton Gorodetski, Random Matrix Products and Anderson Localization

The classical Furstenberg Theorem on positivity of Lyapunov exponent for random matrix products requires the matrices to be identically distributed. It turns out that a non-stationary version can be established, and, as a consequence, both spectral and dynamical localization in 1D non-stationary version of Anderson Model can be derived, without any assumptions on the regularity of the potential. This is a joint project with V.Kleptsyn.

Michael Hitrik, Toeplitz operators, asymptotic Bergman projections, and second microlocalization

In the first part of the talk (based on joint work with L. Coburn, J. Sj\"ostrand, and F. White) we discuss continuity conditions for Toeplitz operators acting on spaces of entire functions with quadratic exponential weights (Bargmann spaces), in connection with a conjecture by C. Berger and L. Coburn, relating Toeplitz and Weyl quantizations. In the second part of the talk (based on joint work in progress with J. Sj\"ostrand), we discuss elements of a semiglobal approach to analytic second microlocalization with respect to a hypersurface, in the semiclassical case, based on the study of the heat evolution semigroup for large times. We describe properties of the associated exponentially weighted spaces of holomorphic functions with ($h$--dependent) plurisubharmonic exponents and construct asymptotic Bergman projections in such spaces.

Simon Larson, Lieb-Thirring inequalities for wave functions vanishing on the diagonal set

We prove Lieb-Thirring-type inequalities for many-body wave functions vanishing on the diagonal set of the configuration space, requiring only that the order of the kinetic energy operator is large enough so that the vanishing condition becomes non-trivial. The proof is based on a general strategy for proving Lieb-Thirring inequalities for scalecovariant systems satisfying 'weak' assumptions. Based on joint work with Douglas Lundholm and Phan Thanh Nam.

Wencai Liu, The large deviation theorem for quasi-periodic Schr\"odinger operators

We establish the large deviation theorem for general analytic $k$-frequency quasi-periodic operators on $\Z^d$ for arbitrary $k, d$. This is a generalization of Bourgain-Goldstein-Schlag's result $b=d=2$ and Bourgain's result $b=d\geq 3$. As an application, the Anderson localization was obtained. This is joint work with Jitomirskaya and Shi.

Milivoje Lukic, Stahl--Totik regularity for continuum Schr\"odinger operators

This talk describes joint work with Benjamin Eichinger: a theory of regularity for one-dimensional continuum Schr\"odinger operators, based on the Martin compactification of the complement of the essential spectrum. For a half-line Schr\"odinger operator $-\partial_x^2+V$ with a bounded potential $V$, it was previously known that the spectrum can have zero Lebesgue measure and even zero Hausdorff dimension; however, we obtain universal thickness statements in the language of potential theory. Namely, we prove that the essential spectrum is not polar, it obeys the Akhiezer--Levin condition, and moreover, the Martin function at $\infty$ obeys the two-term asymptotic expansion $\sqrt{-z} + \frac{a}{2\sqrt{-z}} + o(\frac 1{\sqrt{-z}})$ as $z \to -\infty$. The constant $a$ in its asymptotic expansion plays the role of a renormalized Robin constant suited for Schr\"odinger operators and enters a universal inequality $a \le \liminf_{x\to\infty} \frac 1x \int_0^x V(t) dt$. This leads to a notion of regularity, with connections to the exponential growth rate of Dirichlet solutions and the zero counting measures for finite restrictions of the operator. We also present applications to decaying and ergodic potentials.

Konstantin Makarov, The Quantum Zeno Effect versus Exponential Decay Alternative

In this talk I will recall the concept of continuous monitoring of a quantum system and then discuss the related Quantum Zeno and Exponential Decay scenarios in quantum measurements. We will show that for a typical initial state of the system continuous monitoring of massive particles yields complementarity of the quantum Zeno and anti-Zeno effects, while for systems of massless particles, the quantum Zeno and exponential decay scenarios are complementary instead. The talk is based on a recent work with Eduard Tsekanovskii.

Alvin Moon, Automorphic equivalence within gapped phases in the bulk

Hastings's adiabatic method is a powerful tool in the analysis of gapped Hamiltonians in quantum spin systems. This method and subsequent results due to Bachmann, Michalakis, Nachtergaele and Sims give sufficient conditions for automorphic equivalence between ground state spaces along a smooth path of Hamiltonians which are uniformly locally gapped above the ground state. We develop a new adiabatic theorem for unique gapped ground states which does not require the gap for local Hamiltonians. We instead require a gap in the bulk and a smoothness of expectation values of sub-exponentially localized observables in the unique gapped ground state. This talk is based on joint work with Yoshiko Ogata.

Bruno Nachtergaele, Recent progress on proving a ground state gap for some two-dimensional spin systems

Two-dimensional quantum many-body systems with a spectral gap above their ground state(s) are of considerable current interest. Topological insulators and quantum Hall systems are important examples of systems where a spectral is often a crucial assumption in the analysis. Proving lower bounds for this gap that are independent of system size is a challenging problem. We will review several recent results on the gap for models of particular interest and discuss some of the approaches that have been used to obtain these results.

Nishant Rangamani, Exponential Dynamical Localization for Random Word Models

We give a new proof of spectral localization for a class of Schr\"odinger operators whose potentials arise by randomly concatenating words from an underlying set. We then prove that once one has the existence of a complete orthonormal basis of eigenfunctions, the same estimates used to prove it naturally lead to a proof of exponential dynamical localization on any compact set not containing any critical energies.

Barry Simon, Periodic Jacobi Matrices on Trees

After summarizing 1D periodic Jacobi matrices, I will define periodic Jacobi matrices on infinite trees. I'll discuss the few known results and some interesting examples and then discuss lots and lots of interesting conjectures. This is joint work mainly with Nir Avni and Jonathan Breuer but also with Jacob Christensen, Gil Kilai and Maxim Zinchenko.

Andras Vasy, Fredholm theory for the Laplacian near zero energy on asymptotically conic spaces

In this talk I will discuss and compare two approaches via Fredholm theory to resolvent estimates for the Laplacian of asymptotically conic spaces (such as appropriate metric perturbations of Euclidean space), including in the zero spectral parameter limit.

Xin Zhao, H\"older continuity of absolutely continuous spectral measure for quasi-periodic Schr\"odinger operators

We study the regularity of absolutely continuous spectral measure for multi-frequency quasi-periodic Schr\"odinger operators and the extended Harper model. We obtain 1/2-H\"older continuity of the absolutely continuous spectral measure. As a corollary, we also obtain the long time asymptotic behavior of solution for the associated Schr\"odinger equations by the Guarneri-Combes-Last theorem.

Xiaowen Zhu, From MSA to localization: for singular discrete Anderson model

For n-dimensional discrete Anderson model with regular potentials, the proof of different localization results are quite straightforward once the MSA is done. However, for the singular potential Anderson model, the weaker form of Wegner estimate makes the result of MSA also have a weaker probability estimate, which requires more effort to derive the corresponding localization results. We'll introduce the two step spectral reduction method to derive many localization results from the MSA result. In particular, it applies for the recent 2-d and 3-d work for singular discrete Anderson model. This is a joint work with Nishant Rangamani.