## Talks

Thursday, July 15, 16:00 ~ 16:25 UTC-3

## Resonances for rank one perturbations of Hamiltonians with embedded eigenvalues

### Claudio Fernandez

#### Pontificia Universidad Católica de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloaka25fce478c125c1eb8e0cd16821161f3').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addya25fce478c125c1eb8e0cd16821161f3 = 'cf&#101;rn&#97;nd' + '&#64;'; addya25fce478c125c1eb8e0cd16821161f3 = addya25fce478c125c1eb8e0cd16821161f3 + 'm&#97;t' + '&#46;' + '&#117;c' + '&#46;' + 'cl'; var addy_texta25fce478c125c1eb8e0cd16821161f3 = 'cf&#101;rn&#97;nd' + '&#64;' + 'm&#97;t' + '&#46;' + '&#117;c' + '&#46;' + 'cl';document.getElementById('cloaka25fce478c125c1eb8e0cd16821161f3').innerHTML += '<a ' + path + '\'' + prefix + ':' + addya25fce478c125c1eb8e0cd16821161f3 + '\'>'+addy_texta25fce478c125c1eb8e0cd16821161f3+'<\/a>';

We discuss resonances generated by rank one perturbations of selfadjoint operators with eigenvalues embedded in the continuous spectrum. Instability of these eigenvalues is analyzed and almost exponential decay for the associated resonant states is exhibited. We show how these results can be applied to Sturm-Liouville operators. Main tools are the Aronszajn-Donoghue theory for rank one perturbations, a reduction process of the resolvent based on Feshbach-Livsic formula, the Fermi golden rule and a careful analysis of the Fourier transform of quasi-Lorentzian functions. We also show a connection with sojourn time estimates and the spectral concentration phenomenon.

These results are part of a joint work with Bourget, Cortes, Astaburuaga (PUC, Chile) and Del Rio (UNAM, Mexico).

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Thursday, July 15, 16:30 ~ 16:55 UTC-3

## Zero measure spectrum for multi-frequency Schrödinger operators

### Jake Fillman

#### Texas State University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak9501f2660d2b6c21bd47c3d111da8bf2').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy9501f2660d2b6c21bd47c3d111da8bf2 = 'f&#105;llm&#97;n' + '&#64;'; addy9501f2660d2b6c21bd47c3d111da8bf2 = addy9501f2660d2b6c21bd47c3d111da8bf2 + 'txst&#97;t&#101;' + '&#46;' + '&#101;d&#117;'; var addy_text9501f2660d2b6c21bd47c3d111da8bf2 = 'f&#105;llm&#97;n' + '&#64;' + 'txst&#97;t&#101;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak9501f2660d2b6c21bd47c3d111da8bf2').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy9501f2660d2b6c21bd47c3d111da8bf2 + '\'>'+addy_text9501f2660d2b6c21bd47c3d111da8bf2+'<\/a>';

Building on works of Berthé--Steiner--Thuswaldner and Fogg--Nous we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As a consequence we show that for these torus translations, every quasi-periodic potential can be approximated uniformly by one for which the associated Schrödinger operator has Cantor spectrum of zero Lebesgue measure.

Joint work with Jon Chaika (University of Utah), David Damanik (Rice University) and Philipp Gohlke (Universität Bielefeld).

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Thursday, July 15, 17:00 ~ 17:25 UTC-3

## The two-dimensional Dirac bag model in strong magnetic fields

### Edgardo Stockmeyer

We consider a Dirac system confined to a bounded domain in the plane. This amounts to a family of boundary conditions. There are two extreme cases, zig-zig and infinite-mass boundary conditions. Consider a magnetic field perpendicular to the plane. I will present results on accurate asymptotics of the energy spectrum of the underlying Hamiltonian in the strong magnetic field limit. We will compare the results for different boundary conditions.

Joint work with Jean-Marie Barbaroux (Université de Toulon, France), Loic Le Treust (Aix Marseille Univ, France) and Nicolas Raymond (Université d’Angers, France).

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Thursday, July 15, 17:30 ~ 18:10 UTC-3

## Dispersive Estimates for Schrödinger Equations

### Ricardo Weder

#### Universidad Nacional Autónoma de México, Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak3b87125011d7fccf42581c0d7d8e0002').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy3b87125011d7fccf42581c0d7d8e0002 = 'w&#101;d&#101;r' + '&#64;'; addy3b87125011d7fccf42581c0d7d8e0002 = addy3b87125011d7fccf42581c0d7d8e0002 + '&#117;n&#97;m' + '&#46;' + 'mx'; var addy_text3b87125011d7fccf42581c0d7d8e0002 = 'w&#101;d&#101;r' + '&#64;' + '&#117;n&#97;m' + '&#46;' + 'mx';document.getElementById('cloak3b87125011d7fccf42581c0d7d8e0002').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy3b87125011d7fccf42581c0d7d8e0002 + '\'>'+addy_text3b87125011d7fccf42581c0d7d8e0002+'<\/a>';

The importance of the dispersive estimates for Schrödinger equations in spectral theory and in nonlinear analysis will be discussed. Furthermore, the literature on the Lp − Lp′ estimates will be reviewed, starting with the early results in the 1990 th, and with an emphasis in the results in one dimension. New results will be presented, in Lp − Lp′ estimates for matrix Schrödinger equations in the half-line, with general selfadjoint boundary condition, and in matrix Schrödinger equations in the full-line with point interactions. In both cases we consider integrable matrix potentials that have a finite first moment.

References

[1] T. Aktosun and R. Weder, Direct and Inverse Scattering for the Matrix Schrödinger Equation, Applied Mathematical Sciences 203 , Springer Verlag, New York, 2021.

[2] I. Naumkin, R. Weder, Lp − Lp′ estimates for matrix Schrödinger equations, Journal of Evolution Equations, online first https://doi.org/10.1007/s00028- 020-00605-x, (2020).

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Thursday, July 15, 18:30 ~ 18:55 UTC-3

## Spectrum of graphene models in magnetic fields

### Rui Han

#### Louisiana State University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak6c2c3a18ffc46b2f9f51acc5018930f0').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy6c2c3a18ffc46b2f9f51acc5018930f0 = 'rh&#97;n' + '&#64;'; addy6c2c3a18ffc46b2f9f51acc5018930f0 = addy6c2c3a18ffc46b2f9f51acc5018930f0 + 'ls&#117;' + '&#46;' + '&#101;d&#117;'; var addy_text6c2c3a18ffc46b2f9f51acc5018930f0 = 'rh&#97;n' + '&#64;' + 'ls&#117;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak6c2c3a18ffc46b2f9f51acc5018930f0').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy6c2c3a18ffc46b2f9f51acc5018930f0 + '\'>'+addy_text6c2c3a18ffc46b2f9f51acc5018930f0+'<\/a>';

We will discuss some recent results on the spectrum of single/multi-layer graphene models in magnetic fields, including the structure of the spectrum, spectral decomposition, Dirac cones, etc. In particular, we show that the spectra of single-layer and some multi-layer models are Cantor sets for irrational magnetic flux.

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Thursday, July 15, 19:00 ~ 19:25 UTC-3

## Dirac Operators in a decaying random environment

### Amal Taarabt

#### Pontificia Universidad Católica de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloaka8ab94b70bda426be52627e42872bd48').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addya8ab94b70bda426be52627e42872bd48 = '&#97;mt&#97;&#97;r&#97;bt' + '&#64;'; addya8ab94b70bda426be52627e42872bd48 = addya8ab94b70bda426be52627e42872bd48 + 'm&#97;t' + '&#46;' + '&#117;c' + '&#46;' + 'cl'; var addy_texta8ab94b70bda426be52627e42872bd48 = '&#97;mt&#97;&#97;r&#97;bt' + '&#64;' + 'm&#97;t' + '&#46;' + '&#117;c' + '&#46;' + 'cl';document.getElementById('cloaka8ab94b70bda426be52627e42872bd48').innerHTML += '<a ' + path + '\'' + prefix + ':' + addya8ab94b70bda426be52627e42872bd48 + '\'>'+addy_texta8ab94b70bda426be52627e42872bd48+'<\/a>';

Electronic transport is a central object in the study of conduction in condensed matter. Recently, the development of new materials such as graphene has opened the doors to new technologies with applications in nano-engineering and telecommunications. From a mathematical point of view, the physics of these systems is encoded in the spectral and dynamic properties of random operators. The simplest model for charge carrier dynamics in a graphene-like structure is the discrete Laplacian on a hexagonal lattice, but at low excitation energies this dynamics is actually described by a massless two-dimensional Dirac operator. In this talk, we will introduce dynamical localization and we will present some recent results on the spectral and dynamic properties of random Dirac operators and notably in contexts where ergodicity is broken and where a transition phenomenon is observed between localization and delocalization regimes.

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Thursday, July 15, 19:30 ~ 19:55 UTC-3

## On limiting eigenvalue distribution theorems for clusters and sub-clusters of the hydrogen atom in a weak constant magnetic field

### Carlos Villegas-Blas

#### Universidad Nacional Autónoma de México, Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak9a6203fcab9e6e30995391f1d7b40256').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy9a6203fcab9e6e30995391f1d7b40256 = 'v&#105;ll&#101;g&#97;sbl&#97;s' + '&#64;'; addy9a6203fcab9e6e30995391f1d7b40256 = addy9a6203fcab9e6e30995391f1d7b40256 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_text9a6203fcab9e6e30995391f1d7b40256 = 'v&#105;ll&#101;g&#97;sbl&#97;s' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloak9a6203fcab9e6e30995391f1d7b40256').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy9a6203fcab9e6e30995391f1d7b40256 + '\'>'+addy_text9a6203fcab9e6e30995391f1d7b40256+'<\/a>';

We consider the hydrogen atom under the action of a weak constant magnetic field $H_V(\hbar,B) = - \frac{\hbar^2}{2} \Delta - \frac{1}{|x|} - \frac{\epsilon (\hbar)B}{2}\hbar L_3 + \frac{(\epsilon (\hbar)B)^2}{8}(x_1^2 + x_2^2)$ with $\hbar$ the Planck parameter, and the constant magnetic field ${\bf B}(\hbar)=(0,0,\epsilon (\hbar)B)$ with $B>0$ fixed and $\epsilon (\hbar)=\hbar^q$ used to control the strenght of the magnetic field with respect to $\hbar$.

We study the eigenvalue distribution in suitable defined clusters and sub-clusters of the Hamiltonian $H_V(\hbar,B)$ around the energy $E=-1/2$ in the semiclassical regime $\hbar\rightarrow{0}$. We show that for $q>33/2$ and $q>19$, the clusters and sub-clusters are well defined. For the clusters, we show that the limiting eigenvalue distribution is given by an explicit measure determined by the values of $\frac{B}{2} \ell_3 ({\bf x, p})$ along the classical Kepler orbits on the surface energy $E=-1/2$ where $\ell_3 ({\bf x,p}) = x _1p_2 - x_2 p_1$ is the third component of the classical angular momentum vector. For the sub-clusters case, we show that the limiting eigenvalue distribution involves averages of $\frac{B^2}{8}(x_1^2+x_2^2)$ along the classical Kepler orbits with energy $E=-1/2$ and a previously chosen value of $\ell_3 ({\bf x,p})$.

Joint work with Misael Avendaño-Camacho (Universidad de Sonora, Mexico) and Peter Hislop (University of Kentucky, USA).

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Thursday, July 15, 20:00 ~ 20:25 UTC-3

## Large numerators of quasiperiodic operators

### Wencai Liu

#### Texas A&M University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak6e1f8288d38e17ed4b77ba08e4e7c28e').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy6e1f8288d38e17ed4b77ba08e4e7c28e = 'w&#101;nc&#97;&#105;l' + '&#64;'; addy6e1f8288d38e17ed4b77ba08e4e7c28e = addy6e1f8288d38e17ed4b77ba08e4e7c28e + 't&#97;m&#117;' + '&#46;' + '&#101;d&#117;'; var addy_text6e1f8288d38e17ed4b77ba08e4e7c28e = 'w&#101;nc&#97;&#105;l' + '&#64;' + 't&#97;m&#117;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak6e1f8288d38e17ed4b77ba08e4e7c28e').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy6e1f8288d38e17ed4b77ba08e4e7c28e + '\'>'+addy_text6e1f8288d38e17ed4b77ba08e4e7c28e+'<\/a>';

We initiate an approach to simultaneously treat numerators and denominators arising from quasi-periodic operators, which allows us to obtain (possibly) sharp estimates of Green's functions. In particular, we can study the completely resonant phases of the almost Mathieu operators. Let $(H_{\lambda,\alpha,\theta}u) (n)=u(n+1)+u(n-1)+ 2\lambda \cos2\pi(\theta+n\alpha)u(n)$ be the almost Mathieu operator on $\ell^2(\mathbb{Z})$, where $\lambda, \alpha, \theta\in \mathbb{R}$. Let $\beta(\alpha)=\limsup_{k\rightarrow \infty}-\frac{\ln ||k\alpha||_{\mathbb{R}/\mathbb{Z}}}{|k|}.$ We prove that for any $\theta$ with $2\theta\in \alpha \mathbb{Z}+\mathbb{Z}$, $H_{\lambda,\alpha,\theta}$ satisfies Anderson localization if $|\lambda|>e^{2\beta(\alpha)}$. This confirms a conjecture of Avila and Jitomirskaya [The Ten Martini Problem. Ann. of Math. (2) 170 (2009), no. 1, 303-342].

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Friday, July 16, 16:00 ~ 16:40 UTC-3

## Pointwise convergence for the non-linear Fourier transform

### Alexei Poltoratski

It is widely understood that the scattering transform can be viewed as an analog of the Fourier transform in non-linear settings. This connection brings up numerous questions on finding non-linear analogs of classical results of Fourier analysis. One of the fundamental results of harmonic analysis is the theorem by L. Carleson on pointwise convergence of the Fourier series. In this talk I will discuss convergence for the scattering data of a real Dirac system on the half-line and present an analog of Carleson's theorem for the non-linear Fourier transform.

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Friday, July 16, 16:45 ~ 17:10 UTC-3

## Pure and Mixed States

### João C. A. Barata

#### Instituto de Física, Universidade de São Paulo, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakde5affa7d4f3c42c27dcefc964a9da69').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyde5affa7d4f3c42c27dcefc964a9da69 = 'jb&#97;r&#97;t&#97;' + '&#64;'; addyde5affa7d4f3c42c27dcefc964a9da69 = addyde5affa7d4f3c42c27dcefc964a9da69 + '&#105;f' + '&#46;' + '&#117;sp' + '&#46;' + 'br'; var addy_textde5affa7d4f3c42c27dcefc964a9da69 = 'jb&#97;r&#97;t&#97;' + '&#64;' + '&#105;f' + '&#46;' + '&#117;sp' + '&#46;' + 'br';document.getElementById('cloakde5affa7d4f3c42c27dcefc964a9da69').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyde5affa7d4f3c42c27dcefc964a9da69 + '\'>'+addy_textde5affa7d4f3c42c27dcefc964a9da69+'<\/a>';

We discuss the mathematical notions of pure and of mixed states, two concepts widely used in Quantum Physics and Statistical Mechanics.

Joint work with Marcos Brum (Universidade Federal do Rio de Janeiro, Brazil), Victor Chabu (Instituto de Física, Universidade de São Paulo, Brazil) and Ricardo Correa da Silva (Instituto de Física, Universidade de São Paulo, Brazil).

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Friday, July 16, 17:15 ~ 17:40 UTC-3

## Spectral Theory of the Thermal Hamiltonian

### Giuseppe De Nittis

#### Pontificia Universidad Católica de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak531f80348c15ed727228a4c7b22f2f9f').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy531f80348c15ed727228a4c7b22f2f9f = 'g&#105;d&#101;n&#105;tt&#105;s' + '&#64;'; addy531f80348c15ed727228a4c7b22f2f9f = addy531f80348c15ed727228a4c7b22f2f9f + 'm&#97;t' + '&#46;' + '&#117;c' + '&#46;' + 'cl'; var addy_text531f80348c15ed727228a4c7b22f2f9f = 'g&#105;d&#101;n&#105;tt&#105;s' + '&#64;' + 'm&#97;t' + '&#46;' + '&#117;c' + '&#46;' + 'cl';document.getElementById('cloak531f80348c15ed727228a4c7b22f2f9f').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy531f80348c15ed727228a4c7b22f2f9f + '\'>'+addy_text531f80348c15ed727228a4c7b22f2f9f+'<\/a>';

In 1964 J. M. Luttinger introduced a model for the quantum thermal transport. In this paper we study the spectral theory of the Hamiltonian operator associated to the Luttinger's model, with a special focus at the one-dimensional case. It is shown that the (so called) thermal Hamiltonian has a one-parameter family of self-adjoint extensions and the spectrum, the time-propagator group and the Green function are explicitly computed. We also describe some result about the perturbation by potential and the related scattering theory. We will finish with some hints to the classic problem (which is completely solved) and the comparison between classical and quantum behavior.

Joint work with Vicente Lenz (Delft University of Technology).

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Friday, July 16, 17:45 ~ 18:10 UTC-3

## Spectrum of the Dirichlet Laplacian in sheared waveguides

### Alessandra Verri

#### Universidade Federal de São Carlos, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak0805f892c94ef974f72fb3f3ae292a13').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy0805f892c94ef974f72fb3f3ae292a13 = '&#97;l&#101;ss&#97;ndr&#97;v&#101;rr&#105;' + '&#64;'; addy0805f892c94ef974f72fb3f3ae292a13 = addy0805f892c94ef974f72fb3f3ae292a13 + '&#117;fsc&#97;r' + '&#46;' + 'br'; var addy_text0805f892c94ef974f72fb3f3ae292a13 = '&#97;l&#101;ss&#97;ndr&#97;v&#101;rr&#105;' + '&#64;' + '&#117;fsc&#97;r' + '&#46;' + 'br';document.getElementById('cloak0805f892c94ef974f72fb3f3ae292a13').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy0805f892c94ef974f72fb3f3ae292a13 + '\'>'+addy_text0805f892c94ef974f72fb3f3ae292a13+'<\/a>';

Let $\Omega \subset \mathbb R^3$ be a waveguide which is obtained by translating a cross-section in a constant direction along an unbounded spatial curve. Consider $-\Delta_{\Omega}^D$ the Dirichlet Laplacian operator in $\Omega$. In this talk we show that, under the condition that the tangent vector of the reference curve admits a finite limit at infinity, the essential spectrum of $-\Delta_{\Omega}^D$ can be found. Furthermore, sufficient conditions to ensure the existence of a non-empty discrete spectrum for $-\Delta_{\Omega}^D$ are presented. In particular, we show that the number of discrete eigenvalues can be arbitrarily large since the waveguide is thin enough.

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Friday, July 16, 18:30 ~ 18:55 UTC-3

## Results on the spectral stability of the one-dimensional nonlinear Dirac Equation of Soler type

### Hanne Van Den Bosch

#### Universidad de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakc5deb055e746ba1592779f2666d1ed01').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyc5deb055e746ba1592779f2666d1ed01 = 'hvdb&#111;sch' + '&#64;'; addyc5deb055e746ba1592779f2666d1ed01 = addyc5deb055e746ba1592779f2666d1ed01 + 'd&#105;m' + '&#46;' + '&#117;ch&#105;l&#101;' + '&#46;' + 'cl'; var addy_textc5deb055e746ba1592779f2666d1ed01 = 'hvdb&#111;sch' + '&#64;' + 'd&#105;m' + '&#46;' + '&#117;ch&#105;l&#101;' + '&#46;' + 'cl';document.getElementById('cloakc5deb055e746ba1592779f2666d1ed01').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyc5deb055e746ba1592779f2666d1ed01 + '\'>'+addy_textc5deb055e746ba1592779f2666d1ed01+'<\/a>';

This talk concerns the nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space-dependent mass, known as the (generalized) Soler model. The equation has standing wave solutions for frequencies w in (0,m), where m is the mass in the Dirac operator. These standing waves are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations, but there are very few results in this direction.

The results that I will discuss concern the simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-self-adjoint operator with a particular block structure. I will present some partial results for the one-dimensional case, highlight the differences and similarities with the Schrödinger case, and discuss (a lot of) open problems.

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Friday, July 16, 19:00 ~ 19:25 UTC-3

## A functional model for symmetric operators and its applications to spectral theory

### Luis O. Silva

#### Universidad Nacional Autónoma de México, Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak52785b5b1bcc12e495b26c67d3108ae0').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy52785b5b1bcc12e495b26c67d3108ae0 = 's&#105;lv&#97;' + '&#64;'; addy52785b5b1bcc12e495b26c67d3108ae0 = addy52785b5b1bcc12e495b26c67d3108ae0 + '&#105;&#105;m&#97;s' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx'; var addy_text52785b5b1bcc12e495b26c67d3108ae0 = 's&#105;lv&#97;' + '&#64;' + '&#105;&#105;m&#97;s' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx';document.getElementById('cloak52785b5b1bcc12e495b26c67d3108ae0').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy52785b5b1bcc12e495b26c67d3108ae0 + '\'>'+addy_text52785b5b1bcc12e495b26c67d3108ae0+'<\/a>';

This talk introduces a functional model for symmetric operators based on the representation theory developed by Krein and Strauss. Such model allows one to make use of results and techniques in de Branges space and the moment problem theories to study the spectral properties of the selfadjoint extensions of a given symmetric operator. By this approach, necessary and sufficient conditions for a discrete set to be the spectrum of a singular Schrödinger operator are obtained. The stability of these conditions is studied when an eigenvalue is added to the spectrum.

Joint work with Rafael del Rio (Universidad Nacional Autónoma de México) and Julio H. Toloza (Universidad Nacional del Sur, Argentina).

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Friday, July 16, 19:30 ~ 19:55 UTC-3

## Semiclassical resolvent bound for long range Lipschitz potentials

### Jacob Shapiro

#### University of Dayton, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakd763f292d7178cfd4efea9a087c6dbc0').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyd763f292d7178cfd4efea9a087c6dbc0 = 'jsh&#97;p&#105;r&#111;1' + '&#64;'; addyd763f292d7178cfd4efea9a087c6dbc0 = addyd763f292d7178cfd4efea9a087c6dbc0 + '&#117;d&#97;yt&#111;n' + '&#46;' + '&#101;d&#117;'; var addy_textd763f292d7178cfd4efea9a087c6dbc0 = 'jsh&#97;p&#105;r&#111;1' + '&#64;' + '&#117;d&#97;yt&#111;n' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakd763f292d7178cfd4efea9a087c6dbc0').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyd763f292d7178cfd4efea9a087c6dbc0 + '\'>'+addy_textd763f292d7178cfd4efea9a087c6dbc0+'<\/a>';

We present an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2\Delta + V(x) -E$ in dimension $n\neq 2$, where $h, E >0$. The potential is real-valued, $V$ and $\partial_r V$ exhibit long range decay at infinity, and may grow like a sufficiently small negative power of $r$ as $r\to\infty$. The resolvent norm grows exponentially in $h^{ −1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{−1/2}$ for some $C > 0$. This $E$-dependence is sharp and answers a question of Datchev and Jin.

Joint work with Jeffrey Galkowski (University College London).

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Friday, July 16, 20:00 ~ 20:25 UTC-3

## Spectral theory of Jacobi matrices on trees whose coefficients are generated by multiple orthogonality

### Sergey Denisov

We discuss Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for general classes of orthogonality measures.

Joint work with Alexander Aptekarev (Keldysh Institute, Russia) and Maxim Yattselev (IUPUI, USA).

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Monday, July 19, 16:00 ~ 16:25 UTC-3

## Vanishing of entropy production and quantum detailed balance

### Tristan Benoist

#### CNRS - Institut de Mathématiques de Toulouse, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak06c80568a0f806f916c5729c1dc49ef1').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy06c80568a0f806f916c5729c1dc49ef1 = 'tr&#105;st&#97;n.b&#101;n&#111;&#105;st' + '&#64;'; addy06c80568a0f806f916c5729c1dc49ef1 = addy06c80568a0f806f916c5729c1dc49ef1 + 'm&#97;th' + '&#46;' + '&#117;n&#105;v-t&#111;&#117;l&#111;&#117;s&#101;' + '&#46;' + 'fr'; var addy_text06c80568a0f806f916c5729c1dc49ef1 = 'tr&#105;st&#97;n.b&#101;n&#111;&#105;st' + '&#64;' + 'm&#97;th' + '&#46;' + '&#117;n&#105;v-t&#111;&#117;l&#111;&#117;s&#101;' + '&#46;' + 'fr';document.getElementById('cloak06c80568a0f806f916c5729c1dc49ef1').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy06c80568a0f806f916c5729c1dc49ef1 + '\'>'+addy_text06c80568a0f806f916c5729c1dc49ef1+'<\/a>';

In thermodynamics, entropy production is a quantification of the reversibility of a process. One can consider that a process is reversible whenever its associated entropy production is vanishing. The vanishing of entropy production is also used as a characterization of equilibrium versus non equilibrium. In that case vanishing of entropy production characterizes equilibrium. These two interpretations of the vanishing of entropy production are equivalent for finite states Markov chains where the vanishing of entropy production is equivalent to the reversibility of the associated stochastic process and the property of detailed balance that characterizes equilibrium. In this talk I will present some results relating the quantum detailed balance (that extends the Markov chain notion to quantum channels) and reversibility of quantum repeated measurement processes. This work is a follow-up to arxiv:1607.00162 and arXiv:2012.03885

Joint work with N. Cuneo, V. Jaksic, Y. Pautrat and C.-A. Pillet.

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Monday, July 19, 16:30 ~ 16:55 UTC-3

## Stahl-Totik regularity for continuum Schrödinger operators

### Milivoje Lukić

#### Rice University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloake8270eabfb000cf5b1e23e026756df47').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addye8270eabfb000cf5b1e23e026756df47 = 'm&#105;l&#105;v&#111;j&#101;.l&#117;k&#105;c' + '&#64;'; addye8270eabfb000cf5b1e23e026756df47 = addye8270eabfb000cf5b1e23e026756df47 + 'r&#105;c&#101;' + '&#46;' + '&#101;d&#117;'; var addy_texte8270eabfb000cf5b1e23e026756df47 = 'm&#105;l&#105;v&#111;j&#101;.l&#117;k&#105;c' + '&#64;' + 'r&#105;c&#101;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloake8270eabfb000cf5b1e23e026756df47').innerHTML += '<a ' + path + '\'' + prefix + ':' + addye8270eabfb000cf5b1e23e026756df47 + '\'>'+addy_texte8270eabfb000cf5b1e23e026756df47+'<\/a>';

We present a theory of regularity for one-dimensional continuum Schrödinger operators. For any half-line Schrödinger operator with a bounded potential $V$, we obtain universal thickness statements for the essential spectrum, in the language of potential theory and Martin functions. Namely, we prove that the essential spectrum is not polar, it obeys the Akhiezer-Levin condition, and moreover, the Martin function at infinity 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 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 limiting eigenvalue distributions for finite restrictions of the operator, and applications to decaying and ergodic potentials.

Joint work with Benjamin Eichinger (Johannes Kepler University Linz, Austria).

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Monday, July 19, 17:00 ~ 17:25 UTC-3

## Some generic properties of a uniformly bounded family of discrete one-dimensional Schrödinger operators

### Silas Luiz Carvalho

#### Universidade Federal de Minas Gerais, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak5bebd89a6ba5e118540a8ba9ceeb1fd1').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy5bebd89a6ba5e118540a8ba9ceeb1fd1 = 's&#105;l&#97;s' + '&#64;'; addy5bebd89a6ba5e118540a8ba9ceeb1fd1 = addy5bebd89a6ba5e118540a8ba9ceeb1fd1 + 'm&#97;t' + '&#46;' + '&#117;fmg' + '&#46;' + 'br'; var addy_text5bebd89a6ba5e118540a8ba9ceeb1fd1 = 's&#105;l&#97;s' + '&#64;' + 'm&#97;t' + '&#46;' + '&#117;fmg' + '&#46;' + 'br';document.getElementById('cloak5bebd89a6ba5e118540a8ba9ceeb1fd1').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy5bebd89a6ba5e118540a8ba9ceeb1fd1 + '\'>'+addy_text5bebd89a6ba5e118540a8ba9ceeb1fd1+'<\/a>';

We present some generic spectral and dynamical properties of a uniformly bounded family of discrete one-dimensional Schrödinger operators, such as: Hausdorff, packing, upper and lower $q$-generalized fractal dimensions of a typical spectral measure, typical transport exponents, etc. We also discuss the application of the main techniques to limit-periodic Schrödinger operators.

Joint work with Moacir Aloisio (Universidade Federal do Amazonas) and César R. de Oliveira (Universidade Federal de São Carlos).

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Monday, July 19, 17:30 ~ 18:10 UTC-3

## Continuity properties of the spectral shift function for massless Dirac operators and an application to the Witten index

### Fritz Gesztesy

#### Baylor University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak8a38f1c682c0e77117de0b75c0ef436d').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy8a38f1c682c0e77117de0b75c0ef436d = 'fr&#105;tz_g&#101;szt&#101;sy' + '&#64;'; addy8a38f1c682c0e77117de0b75c0ef436d = addy8a38f1c682c0e77117de0b75c0ef436d + 'b&#97;yl&#111;r' + '&#46;' + '&#101;d&#117;'; var addy_text8a38f1c682c0e77117de0b75c0ef436d = 'fr&#105;tz_g&#101;szt&#101;sy' + '&#64;' + 'b&#97;yl&#111;r' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak8a38f1c682c0e77117de0b75c0ef436d').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy8a38f1c682c0e77117de0b75c0ef436d + '\'>'+addy_text8a38f1c682c0e77117de0b75c0ef436d+'<\/a>';

We report on recent results regarding the limiting absorption principle for multi-dimensional, massless Dirac-type operators (implying absence of singularly continuous spectrum) and continuity properties of the associated spectral shift function.

We will motivate our interest in this circle of ideas by briefly describing the connection to index theory for non-Fredholm operators, particularly, to the notion of the Witten index.

This is based on various joint work with A. Carey, J. Kaad, G. Levitina, R. Nichols, D. Potapov, F. Sukochev, and D. Zanin.

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Monday, July 19, 18:30 ~ 18:55 UTC-3

## On the Equivalence of the KMS Condition and the Variational Principle for Quantum Lattice Systems with Mean-Field Interactions

### Walter de Siqueira Pedra

#### University of São Paulo, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakd3806f7cc38ae313f7cb4a7e7e85a899').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyd3806f7cc38ae313f7cb4a7e7e85a899 = 'wp&#101;dr&#97;' + '&#64;'; addyd3806f7cc38ae313f7cb4a7e7e85a899 = addyd3806f7cc38ae313f7cb4a7e7e85a899 + '&#105;f' + '&#46;' + '&#117;sp' + '&#46;' + 'br'; var addy_textd3806f7cc38ae313f7cb4a7e7e85a899 = 'wp&#101;dr&#97;' + '&#64;' + '&#105;f' + '&#46;' + '&#117;sp' + '&#46;' + 'br';document.getElementById('cloakd3806f7cc38ae313f7cb4a7e7e85a899').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyd3806f7cc38ae313f7cb4a7e7e85a899 + '\'>'+addy_textd3806f7cc38ae313f7cb4a7e7e85a899+'<\/a>';

We extended Araki's result on the equivalence of the KMS condition and the variational principle for equilibrium states of quantum lattice systems with short-range interactions, to a large class of lattice models possibly containing mean-field interactions (representing an extreme form of long-range interactions). This result is reminiscent of van Hemmen's work on equilibrium states for mean-field models. The extension was made possible by our recent outcomes on states minimizing the free energy density of mean-field models on the lattice, as well as on the infinite volume dynamics for such models.

Joint work with Jean-Bernard Bru (Ikerbasque and BCAM, Spain) and Rafael Sussumu Yamaguti Miada (University of São Paulo, Brazil).

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Monday, July 19, 19:00 ~ 19:25 UTC-3

## Vortex lines in the 3D Ginzburg-Landau model of superconductivity

### Carlos Román

The Ginzburg-Landau model is a phenomenological description of superconductivity. A crucial feature is the occurrence of vortex lines, which appear above a certain value of the strength of the applied magnetic field called the first critical field. In this talk I will present a sharp estimate of this value and describe the behavior of global minimizers for the 3D Ginzburg-Landau functional below and near it.

Joint work with Etienne Sandier (Paris-Est-Créteil University, France) and Sylvia Serfaty (New York University, United States).

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Monday, July 19, 19:30 ~ 19:55 UTC-3

## The Schrödinger equation revisited in terms of a nonabsolute path integral

### Márcia Federson

#### Universidade de São Paulo, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloake278838deb68e8885872060e8371a8b3').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addye278838deb68e8885872060e8371a8b3 = 'f&#101;d&#101;rs&#111;n' + '&#64;'; addye278838deb68e8885872060e8371a8b3 = addye278838deb68e8885872060e8371a8b3 + '&#105;cmc' + '&#46;' + '&#117;sp' + '&#46;' + 'br'; var addy_texte278838deb68e8885872060e8371a8b3 = 'f&#101;d&#101;rs&#111;n' + '&#64;' + '&#105;cmc' + '&#46;' + '&#117;sp' + '&#46;' + 'br';document.getElementById('cloake278838deb68e8885872060e8371a8b3').innerHTML += '<a ' + path + '\'' + prefix + ':' + addye278838deb68e8885872060e8371a8b3 + '\'>'+addy_texte278838deb68e8885872060e8371a8b3+'<\/a>';

We point out some aspects why the Feynman integral is not adequate for the treatment of equations related to Hamilton’s principal functions involving Fresnel integrands. In particular, we comment on the Schrödinger equation.

Joint work with Felipe Federson and Everaldo Bonotto (University of São Paulo, Brazil).

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Monday, July 19, 20:00 ~ 20:40 UTC-3