The HADES seminar on Tuesday, November 5th will be given by Jacob Shapiro in Evans 740 from 3:40 to 5 pm.

Speaker: Jacob Shapiro, ANU

Abstract: We prove a weighted resolvent estimate for the semiclassical Schrödinger operator $-h^2 \Delta + V : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \ge 3$. We assume the potential $V$ is compactly supported and $\alpha$-Hölder continuous, $0< \alpha < 1$. The logarithm of the resolvent norm grows like $h^{-1-\frac{1-\alpha}{3 + \alpha}}\log(h^{-1})$ as the semiclassical parameter $h \to 0^+$. This bound interpolates between the previously known $h$-dependent resolvent bounds for Lipschitz and $L^\infty$ potentials. To key step is to prove a suitable global Carleman estimate, which we establish via a spherical energy method. This is joint work with Jeffrey Galkowski.

The HADES seminar on Tuesday, October 29th will be given by Keita Mikami in Evans 740 from 3:40 to 5 pm.

Speaker: Keita Mikami, RIKEN

Abstract: In this talk, we will consider asymptotic behavior as $|x| \to \infty$ of Schrödinger operators with homogeneous potentials of order zero. Localization in direction was known as a property of Schrödinger operators with homogeneous potentials of order zero or corresponding Hamiltonian flow. We will introduce this known localization in direction first. We then introduce the localization of defect measures. We then give necessary conditions for observability of Schrödinger operators with homogeneous potentials of order zero which is related to the localization result.

The HADES seminar on Tuesday, October 1st will be given by  Katrina Morgan in Evans 740 from 3:40 to 5 pm.

Speaker: Katrina Morgan, MSRI

Abstract: Asymptotically flat spacetimes (i.e. Lorentzian manifolds whose metric coefficients tend toward the flat metric as $|x| \to \infty$) arise in General Relativity, which has motivated many mathematical questions about wave behavior on such spacetimes. The dispersive estimate local energy decay has proven to be a powerful tool for studying these questions. It has been used to establish Strichartz estimates (global, mixed norm estimates often used in existence proofs) and pointwise estimates. The estimate holds if the underlying geometry allows waves to spread out enough to get decay of energy within compact sets. Local energy decay is connected to the presence of trapped geodesics and resolvent behavior.
This talk will provide a brief overview of local energy estimates and the use of resolvents
in studying wave behavior. The application of these tools in establishing the relationship between how quickly the background geometry tends toward flat and the pointwise decay rate of waves will be discussed. We find that a solution u to the wave equation on a spacetime which tends toward flat at a rate of $|x|^{-k}$ satisfies the pointwise bounds $|u|\le C_x t^{-k-2}$.
This result extends the work of Tataru 2013 which proved a $t^{-3}$ pointwise decay rate for
waves when the background geometry tends toward flat at a rate of $|x|^{-1}$.

The HADES seminar on Tuesday, October 22nd will be given by  Amir Vig in Evans 740 from 3:40 to 5 pm.

Speaker: Amir Vig, UCI & MSRI

Abstract: The wave trace is a distribution on $\mathbb{R}$ given by $\sum_{j = 1}^\infty e^{it \lambda_j}$, where $\lambda_j^2$ are the (positive) eigenvalues of the Laplacian on a compact domain. In general, two linear waves can be superimposed to give another solution to the wave equation. When we add up a bunch of waves at different frequencies, the peak singularities appear at points with substantial constructive interference. On a manifold, the famous “propagation of singularities” tells us that waves propagate along geodesics, so the constructive interference is most pronounced along orbits which are traversed infinitely often (i.e. periodic orbits). On the trace side of things, this phenomenon is reflected in the Poisson relation, which says that the singular support of the wave trace is contained in the length spectrum (the collection of lengths of all periodic orbits). For planar domains, the geodesic flow is replaced by the billiard (or broken bicharacteristic) flow and we see an interesting connection between geometric, dynamical and spectral properties of the domain. In this talk, we introduce some simple cases of wave trace formulas before discussing recent work on explicit formulas for wave invariants associated to periodic orbits of small rotation number in a smooth, strictly convex bounded planar domain. This involves proving a dynamical theorem on the structure of such orbits and then constructing an explicit oscillatory integral representation, which microlocally approximates the wave propagator in the interior.

The HADES seminar on Tuesday, October 8th will be given by  Malo Jézéquel in Evans 740 from 3:40 to 5 pm.

Speaker: Malo Jézéquel, LPSM & MSRI

Abstract: Fine statistical properties of a smooth Anosov flow may be studied through Ruelle resonances. These resonances may be described as the zeroes of a dynamical determinant, i.e. an entire function defined in terms of the periodic data of the flow. We are interested in another relationship between Ruelle resonances and periodic data of the flow: a trace formula conjectured by Dyatlov and Zworski. This formula is known to be true for real-analytic Anosov flows by a result of Fried, we will see how this hypothesis can be weakened.

The HADES seminar on Tuesday, September 24th will be given by  Mohandas Pillai in Evans 740 from 3:40 to 5 pm.

Speaker: Mohandas Pillai, Berkeley

Abstract: This talk will be about the wave maps problem with domain $\mathbb{R}^{2+1}$ and target $\mathbb{S}^{2}$ in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from $\mathbb{R}^{2}$ to $\mathbb{S}^{2}$, with polar angle equal to $Q_{1}(r) = 2 \arctan(r)$. By applying the scaling symmetry of the equation, $Q_{\lambda}(r) = Q_{1}(r \lambda)$ is also a harmonic map, and the family of all such $Q_{\lambda}$ are the unique minimizers of the harmonic map energy among finite energy, 1-equivariant, topological degree one maps.

In this talk, I will discuss how to construct a collection of infinite time blowup solutions along the $Q_{\lambda}$ family, with a symbol class of possible asymptotic behaviors of $\lambda$.

The HADES seminar on Tuesday, September 17th will be given by  Alix Deleporte in Evans 740 from 3:40 to 5 pm.

Speaker: Alix Deleporte, MSRI

Abstract: Szegö kernels encode information on weighted holomorphic functions, or holomorphic sections. In an appropriate large curvature limit, they enjoy a semiclassical structure. Among other applications, these kernels are used to define an alternative quantization scheme : Berezin-Toeplitz quantization.

This talk will be an opportunity to further motivate attendance to M. Zworski’s course.

The HADES seminar on Tuesday, September 10th will be given by  Stéphane Nonnenmacher in Evans 748 from 3:40 to 5 pm

Speaker: Stéphane Nonnenmacher, Univ. Paris-Sud & MSRI

Abstract: The spectrum of a nonselfadjoint linear operator can be very unstable, that is sensitive to perturbations, an phenomenon usually referred to as the “pseudospectral effect”. In order to quantify this phenomenon, we investigate a simple class of nonselfadjoint 1-dimensional semiclassical (pseudo-)differential operators, submitted to small random perturbations. The spectrum of this randomly perturbed operator is then viewed as a random point process on the complex plane, whose statistical properties we wish to analyze.

Hager & Sjöstrand have shown that, in the semiclassical limit, the randomly perturbed eigenvalues satisfies a probabilistic form of Weyl’s law, at the macroscopic scale. We in turn investigate the statistical distribution of the eigenvalues at the microscopic scale (scale of the distance between nearby eigenvalues). We show that at this scale, the spectral statistics satisfy a partial form of universality: spectral correlations can be expressed in terms of a universal object, the Gaussian Analytic Function (GAF), and a few parameters depending on the initial operator, and of the type of random disorder.

A central tool in our analysis is a well-posed Grushin problem, which turns our spectral problem on \(L^2(R)\) into an effective nonlinear spectral problem on a finite dimensional subspace (“effective Hamiltonian”). This Grushin problem is set up by studying the “classical spectrum” of our initial semiclassical operator (a region in the complex plane), constructing quasimodes of this operator, and analyzing the (complex-)energy-dependence of these quasimodes.

This is joint work with Matin Vogel.

The HADES seminar on Tuesday, August 27th will be given by Haoren Xiong in Evans 748 from 3:40 to 5 pm

Speaker: Haoren Xiong, Berkeley

Abstract: Scattering resonances of Schrödinger operator with a compactly supported potential are defined as the poles of the meromorphic continuation of the resolvent. I will introduce the method of complex scaling which produces a natural family of non-self-adjoint operators whose discrete spectrum consists of resonances. Furthermore, we will see that similar results hold in the case of dilation analytic potentials.