ENG 
ESP Session S30 - Mathematical Methods in Quantum Mechanics
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.
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).
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.
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).
Thursday, July 15, 17:00 ~ 17:25 UTC-3
The two-dimensional Dirac bag model in strong magnetic fields
Edgardo Stockmeyer
Pontificia Universidad Católica de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
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).
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.
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).
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.
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.
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.
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.
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.
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).
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.
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].
Friday, July 16, 16:00 ~ 16:40 UTC-3
Pointwise convergence for the non-linear Fourier transform
Alexei Poltoratski
University of Wisconsin - Madison, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
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.
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.
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).
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.
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).
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.
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.
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.
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.
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.
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).
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.
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).
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
University of Wisconsin - Madison, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
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).
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.
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.
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.
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).
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.
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).
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.
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.
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.
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).
Monday, July 19, 19:00 ~ 19:25 UTC-3
Vortex lines in the 3D Ginzburg-Landau model of superconductivity
Carlos Román
Pontificia Universidad Católica de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
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).
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.
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).
Monday, July 19, 20:00 ~ 20:40 UTC-3
Direct and inverse Sturm-Liouville problems: A method of solution
Vladislav Kravchenko
Cinvestav, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will present some recent developments in the theory and practice of direct and inverse Sturm-Liouville problems on finite and infinite intervals which result in a new approach for solving direct and inverse spectral and scattering problems [1]. The approach is based on the notion of transmutation (transformation) operators and their efficient construction. Analytical representations for solutions of Sturm-Liouville equations are derived in the form of functional series revealing interesting special features and lending themselves to direct and simple numerical solution of a wide variety of problems. [1] V. V. Kravchenko Direct and inverse Sturm-Liouville problems: A method of solution. Birkhäuser, Series: Frontiers in Mathematics, 2020.