Schedule/Abstracts

Titles and Abstracts

The backscattering transformation
Ingrid Beltita
Abstract: The inverse backscattering problem for the Schrödinger operator consists of finding the potential from the far-field pattern generated by plane waves from opposite directions. The talk aims to review results on Melin's backscattering transformation introduced in order to address that problem. Slides

Left definite Sturm-Liouville equations
Christer Bennewitz
Abstract: If w>0 spectral theory for the equation -(pu')'+qu=\lambda wu is usually studied in an L^2-space with weight w, but if the left hand side of the equation may be associated with a positive hermitian form one may use this as a scalar product and give a spectral theory in a corresponding Hilbert space, even if w does not have a fixed sign. Such equations were treated already by Hilbert as an application of his theory of 'polar' integral equations, and Weyl also treated singular equations of this type in 1910. There are scattered results for such so called left definite equations since then, but renewed interest was created when the so called Camassa-Holm integrable system was discovered to be associated with an equation of this form. In this talk I shall review the spectral theory of such equations. Slides

Radon transforms supported in hypersurfaces
Jan Boman
Abstract, Slides

The generic instability of differential operators
Nils Dencker
Abstract: It came as a surprise when Hans Lewy in 1957 presented a nonvanishing smooth complex vector field on R^3 that is not solvable anywhere. After all, the classical Cauchy-Kowalevskaya Theorem shows that any analytic PDE is solvable in the analytic category. Interestingly, the Lewy vector field gives the tangential Cauchy-Riemann operator on the boundary of a strictly pseudoconvex domain in C^2. Hörmander showed in 1960 that almost all linear PDE are not solvable by proving that the nongeneric vanishing of the Poisson bracket is necessary for solvability. This bracket condition on PDE has many consequences for the kernel and the range, the spectral stability of PDE and the stability of the quasilinear Cauchy problem. A fifty year development lead to the proof of the Nirenberg-Treves conjecture: that principal type differential operators are solvable if and only if condition (Psi) on the principal symbol is satisfied. For nonprincipal type differential operators, a condition similar to (Psi) on the refined principal symbol (including the subprincipal symbol) is necessary for solvability. This condition is also sufficient for solvability of nonprincipal type differential operators having real principal symbols. Slides

Evolution equations with fractional-order operators
Gerd Grubb
Abstract, Slides

Positivity, Toeplitz operators, and Berger-Coburn conjecture
Michael Hitrik
Abstract: The Berger-Coburn conjecture, a long standing conjecture in the theory of Toeplitz operators, states that a Toeplitz operator on the Bargmann space is bounded precisely when its Weyl symbol is a bounded function. We prove the conjecture in the special case of Toeplitz symbols given by exponentials of complex quadratic forms. The notion of positivity of a complex Lagrangian space, developed in the fundamental works by Anders Melin and Johannes Sjöstrand, plays a crucial role in the proof. This is joint work with L. Coburn and J. Sjöstrand. Slides

Metric graphs, stable polynomials and Fourier quasicrystals
Pavel Kurasov
Abstract, Slides

Lieb-Thirring inequalities on manifolds with negative curvature
Ari Laptev
Abstract: We prove Lieb-Thirring inequalities on manifolds with negative constant curvature. The discrete spectrum appears below the continuous spectrum $(d-1)^2/4, \infty)$, where $d$ is the dimension of the hyperbolic space. In order to prove the main statement we use the result obtained regarding Lieb-Thirring inequalities for Schrödinger operators with operator-valued potentials. As an application we obtain a Pólya type inequality with not a sharp constant. Slides

Generalized products and pseudodifferential operators
Richard Melrose
Abstract: I will describe a geometric setting which generates an algebra of pseudodifferential operators. These quantize an associated Lie algebra of vector fields (a Lie algebroid). The intention is to make these constructions relatively straightforward and allow different examples to be combined for specific purposes.

On the spectrum of certain semiregular global systems
Alberto Parmeggiani
Abstract: In this talk, I will survey recent results on some spectral properties (namely the asymptotics of the Weyl counting function and quasi-clustering of the spectrum) of a class of second order systems with polynomial coefficients which contains systems such as the Jaynes-Cummings model that describes the light-matter interaction. Slides

Eigenvalue estimates and asymptotics of polynomially compact pseudodifferential operators and applications to the Neumann-Poincaré operator in 3D elasticity
Grigori Rozenblum
Abstract, Slides

A generalization of the Bernstein-Walsh-Siciak theorem on uniform approximation on compact sets of functions by polynomials
Ragnar Sigurdsson
Abstract, Slides

Tunneling for the d-bar operator
Johannes Sjöstrand
Abstract, Slides

A model for periodicity of atomic structure
Jan Philip Solovej
Abstract: Despite its name the periodic table of the elements is not particularly periodic. It does nevertheless have the feature that elements in the same group have similar chemical properties. For example the noble gases He, Ne, Ar, Kr,...all have very weak chemical reactions. But if we imagine extending the periodic table to very large atomic number Z how would we predict which atoms are noble gases. The chemist have the somewhat phenomenological aufbau principle. In this talk I will consider the model where an atom with atomic number Z is described by the one-particle Schrödinger operator with the corresponding Thomas-Fermi potential. This is the Z dependent potential that is known to approximate, using semiclassical analysis, the interaction an electron feels on average (mean field) from all the other electrons. We prove that for a sequence of atomic numbers Z_n tending to infinity these Thomas-Fermi mean field Schrödinger operators converge in strong resolvent sense if and only if CZ_n^{1/3} tends to a value \theta mod \pi for some explicit constant C. This shows that in this model atoms corresponding to such sequences in the limit have identical properties and that there is a certain periodicity. The aufbau principle would predict a similar result but with a different constant C. The result is based on a detailed WKB analysis. This is joint work with my Phd student August Bjerg. Slides

Magic angles vs. scattering resonances
Maciej Zworski
Abstract: Magic angles refer to a remarkable theoretical (Bistritzer--MacDonald, 2011) and experimental (Jarillo-Herrero et al 2018) discovery, that two sheets of graphene twisted by a certain (magic) angle are a superconductor.

Mathematically, this is related to having a flat band of nontrivial topology for the corresponding periodic Hamiltonian which happens for a chiral model of twisted bilayer graphene (Tarnopolsky-Kruchkov-Vishwanath, 2019). A spectral characterization of magic angles (Becker-Embree-Wittsten-Z, 2021) produces also complex values and the distribution of their reciprocals looks remarkably like a distribution of scattering resonances for a two dimensional problem, with the real magic angles corresponding to anti-bound states. I will review various results on that distribution highlighting this mysterious analogy.

The talk is based on joint works with S Becker, M Embree, J Galkowski, M Hitrik, T Humbert and J Wittsten. Slides