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For the nonlinear Schrodinger equation in one dimension, with a repulsive delta potential that is not too strong, we show the existence of two-soliton solutions with logarithmic (in time) separation. The construction is based on that of Nguyen for the case without potential, modified to account for the additional interaction between the potential and the solitons. This interaction manifests through a perturbed translational eigenfunction, whose detailed properties play a key role. This is joint work with Takahisa Inui.

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In this talk, I will present various bounds for the unitary propagator associated to a Schrödinger equation with a random potential. Roughly speaking, I will show that, with high probability, cancellations induced by the randomness cause the effective strength of the potential to be a square-root less than in the deterministic bound. As applications, one may recover well-known lower bounds on localization length in the Anderson model due to Schlag-Shubin-Wolff, and obtain quantitative control over the frequency localization of eigenfunctions. I will give an overview of the proof, which uses only an input from random matrix theory, the non-commutative Khintchine inequality, and basic harmonic analysis. Time-permitting, I will also explain how our methods may be applied to certain time-dependent non-Markovian potentials. This is ongoing joint work with Reuben Drogin and Felipe Hernández.

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We consider the Schrödinger operator with a step potential, which has a non-zero asymptote at infinity. We build up a scattering and distorted Fourier transform theory for this operator, and use it to study the long time behavior of small data solutions of the 1D cubic NLS equation with the step potential. We also discuss some potential application of this theory, that is, the nonlinear stability problem of dark solitons for the 1D cubic NLS equation. This is ongoing joint work with J. Holmer (Brown University).

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We revisit the basic problem of tunneling near a nondegenerate global maximum of a potential on the line. We reduce the semiclassical Schr\"odinger equation to a Weber normal form by means of the Liouville-Green transform. We show that the diffeomorphism which effects this stretching of the independent variable lies in the same regularity class as the potential (analytic or infinitely differentiable) with respect to both variables, i.e., space and energy. We then apply the Weber normal form to the scattering problem for energies near the potential maximum. In particular we obtain a representation of the scattering matrix which is accurate up to multiplicative factors of the form 1 + o(1). Recent generalizations will be discussed. Joint work with Hyejin Park and Wilhelm Schlag

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Self-similar blowup for the 3d cubic NLS The cubic nonlinear Schrödinger equation arises in various physical applications and is one of the most fundamental models in dispersive PDEs. In the three dimensional focusing case the formation of singularities via self-similar solutions has been proposed in the 70s by Zakharov in the context of plasma physics. Such solutions are (up to symmetries) determined by their profile as well as a real parameter which enters the blowup rate as a multiplicative constant. Hence the equation satisfied by self-similar solutions resembles a nonlinear eigenvalue problem. It is conjectured that physically relevant finite-energy solutions exist only for particular values of the parameter. Moreover, based on numerical experiments it is widely believed that there is a “ground state” which appears as an attractor in the evolution of generic large initial data. However, the existence of this (or any other finite-energy) self-similar solution has been a long-standing open problem. In this talk, I present recent results obtained with Roland Donninger (University of Vienna), which prove the existence of a smooth finite-energy self-similar solution which matches the numerically observed one. Our method of proof is mildly computer assisted, but this is limited to the evaluation of polynomials with rational coefficients at rational points. In particular, no interval arithmetics are used.

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We develop the wave packet decomposition to study the Schrodinger evolution with rough potential. As an application, we obtain the improved bounds on wave propagation for the generic value of parameter.

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I will present a proof of the asymptotic stability of solitary waves for the 1D cubic NLS under even perturbations based on a combination of modulation techniques and a space-time resonances approach using the distorted Fourier transform. The main challenge are the threshold resonances of the linearized operator and the resulting slow local decay of the Schrödinger waves. Remarkable null structures in the evolution equation for the radiation term as well as in the modulation equations play an important role in the proof. This is joint work with Yongming Li (Texas A&M University).

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Investigating degenerate mixtures of bosons and fermions is an extremely active area of research in experimental physics for constructing and understanding novel quantum bound states such as those in superconductors, superfluids, and supersolids. These ultra-cold Bose-Fermi mixtures are fundamentally different from degenerate gases with only bosons or fermions. They not only show an enriched phase diagram, but also a fundamental instability due to energetic considerations coming from the Pauli exclusion principle. Inspired by this activity in the physics community, recently we started exploring the mathematical theory of Bose-Fermi mixtures. One of the main challenges is understanding the physical scales of the system that allow for suitable analysis. We will describe how we overcame this challenge by identifying a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical. In this regime, the bosons are much lighter and more numerous than the fermions. The talk is based on the joint work with Esteban Cárdenas and Joseph Miller.

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I will present a new approach to Mathematical Scattering of multichannel Dispersive and Hyperbolic Equations. In this approach we identify the large time behavior of such equations, both linear and non-linear, for general (large) data and interactions terms which can be space-time dependent. In particular, for the NLS equations with spherically symmetric data and Interaction terms, we prove that all global solutions in H^1 converge to a smooth and localized function plus a free wave, in 5 or more dimensions. Similar result holds for 3,4 dimensions, though the argument proving localization is different. We also show similar results in any dimension for localized type of interactions, provided they decay fast enough. We show breakdown of the standard Asymptotic Completeness conjecture if the interaction is time dependent and decays like r^{-2} at infinity. Many of these results extend to the non-radial case, for NLS, NLKG and Bi-harmonic NLS in three or more dimensions. Furthermore, we prove Local-Decay Estimates for Time dependent potentials in 5 or more dimensions. Finally, we apply this approach to N-body scattering, and prove AC for three quasi-particle scattering. This is based on joint works with Baoping Liu and Xiaoxu Wu.

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Simulation of quantum dynamics, emerging as the original motivation for quantum computers, is widely viewed as one of the most important applications of a quantum computer. Quantum algorithms for Hamiltonian simulation with unbounded operators Recent years have witnessed tremendous progress in developing and analyzing quantum algorithms for Hamiltonian simulation of bounded operators. However, many scientific and engineering problems require the efficient treatment of unbounded operators, which may frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure, quantum differential equations solver and quantum optimization. We will introduce some recent progresses in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and Magnus expansion based algorithms in the interaction picture.

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In this talk, I will present some mathematical and numerical results concerning the calculation of electronic properties of periodic or aperiodic systems. I will focus in particular on graphene and twisted bilayer graphene, an incommensurate 2D material that is currently attracting a great deal of interest in the condensed matter physics community.

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Adiabatic time-dependent density functional theory is a mean field theory which approximates the dynamical behavior of a system of interacting electrons. Its linearization near a ground state is able to reproduce experimentally relevant quantities such as absorption spectra. We justify linear response theoretically and investigate the presence of resonances in photoionization spectra. Joint work with Mi-Song Dupuy and Eloïse Letournel.

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We will start with a quick review of previous instability results concerning solutions to the Klein-Gordon equation on rotating Kerr black holes and the corresponding conjectural consequences for the dynamics of the Einstein-Klein-Gordon system. Then we will discuss recent work where we show that, despite the presence of stably trapped timelike geodesics on Schwarzschild, solutions to the corresponding Klein-Gordon equation arising from strongly localized initial data nevertheless decay polynomially. Time permitting we will explain how the proof uses, at a crucial step, results from analytic number theory for bounding exponential sums. The talk is based on joint work(s) with Federico Pasqualotto and Maxime Van de Moortel.

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We consider the problem of dynamical stability for the vortex of the Ginzburg-Landau model. Vortices are one of the main examples of topological solitons, and their dynamic stability is the basic assumption of the asymptotic "particle plus field'' description of interacting vortices. In this talk we focus on co-rotational perturbations of vortices and establish a variety of pointwise dispersive and decay estimates for their linearized evolution in the relativistic (or Klein-Gordon) case. One of the main ingredients is the construction of the distorted Fourier transform associated with the (two) linearized operators at the vortex. The general approach follows that of Krieger-Schlag-Tataru and Krieger-Miao-Schlag in the context of stability of blow-up for wave maps and relies on the spectral analysis of Schrodinger operators with strongly singular potentials (see also Gezstesy-Zinchenko). However, since the vortex is not given by an explicit formula, and one of the operators appearing in the linearization has zero energy solutions that oscillate at infinity, the linear analysis requires some additional work. In particular, to construct the distorted Fourier basis and to control the spectral measure some additional arguments are needed, compared to previous works. This is joint work with Fabio Pusateri.

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We consider the dynamics of a 2d Bose gas with singular attractive interactions in the instability regime, where the corresponding focusing nonlinear Schrödinger equation (NLS) has a blow-up. We show that the evolution of the condensate is effectively described by this NLS for all times before the blow-up. Moreover, we prove the validity of the Bogoliubov approximation for the fluctuation dynamics, resulting in a norm approximation of the many-body dynamics. This is joint work with Charlotte Dietze and Phan Thành Nam.

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For computing the ground state of Gross-Pitaevskii eigenvalue problem, there exist a few simple and efficient numerical algorithms in which only the inversion of the Laplacian operator is needed, including Sobolev gradient flow induced by H1 norm as well as an operator splitting scheme. On structured meshes, inversion of the discrete Laplacian in high order finite element methods can be easily implemented using modern software (e.g., MATLAB 2023 or Jax in Python) with a considerable acceleration on a high end GPU card, thus these simple algorithms allow easy coding/computation on a large grid like 1000^3. I will comment on provable convergence results for some of these algorithms, and show a numerical comparison.

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In the first part of this talk, I will discuss recent joint work with CL Fefferman and J Shapiro (Princeton), on the phenomenon of quantum tunneling. While extensively studied in non-magnetic systems, far less is known in the magnetic case. We construct double-well Schroedinger Hamiltonians for which, in the presence of a strong magnetic field, tunneling does not occur. And for generic situations, we give upper and lower bounds on the time-scale of tunneling. In the second part, I'll explain how a non-uniform deformation of a honeycomb medium (e.g. a photonic crystal) induces effective magnetic and electric fields, described by an effective (homogenized) Dirac Hamiltonian. Choosing a deformation which corresponds to a constant perpendicular effective magnetic field gives rise to an effective Hamiltonian with Landau-level (flat band) energy spectrum. Recent experiments, in the setting of photonic crystal slabs, confirm the theory. This is joint work with group of M.C. Rechtsman in Penn State Physics.

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I will describe techniques based partly on the distorted Fourier transform to construct special non-scattering solutions for dispersive PDE, and to study the stability of such solutions in some cases.

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I discuss the dynamics of solutions to Klein-Gordon equations around multi-solitons including asymptotic stability and classification of multisoliton, and to classify the initial data for the global behavior in an open neighborhood of multi-solitons. This talk is based on joint works with Jacek Jendrej and Kenji Nakanishi.

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I will begin by presenting our recent results on the spherically symmetric Coulomb waves. Specifically, we study the evolution operator of H= -\Delta+q/|x| where q>0. Utilizing a distorted Fourier transform adapted to H, we explicitly compute the evolution kernel. A detailed analysis of this kernel reveals that e^itH satisfies an L^1 \to L^{\infty} dispersive estimate with the natural decay rate t^{-3/2}. This work was conducted in collaboration with Adam Black, Bruno Vergara, and Jiahua Zhou. Following this, I will discuss our ongoing research on the nonlinear Schrödinger equation, where we apply the distorted Fourier transform developed for the Coulomb Hamiltonian. This work is being carried out in collaboration with Mengyi Xie.

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I will discuss new analytic methods of finding the long time behavior of the Schrödinger equation with time-periodic forcings which are not necessarily large or small. While previous results were conditional, dependent on the conjectured nature of the spectrum of a Floquet operator, our approach finds that spectrum alongside sharp decay estimates. I will illustrate the methods on the nonrelativistic model of photoionization, whose exact behavior for larger fields was open.
Based on work in collaboration with R. D. Costin, I. Jauslin, J. Lebowitz and S. Tanveer.

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