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1. Non-Hermitian skin effect in two dimensional continuous systems

   https://doi.org/10.1088/1402-4896/aca43b  

   Yuce, C; Ramezani, H (December 2022, Physica Scripta)

   Abstract An extensive number of the eigenstates can become exponentially localized at one boundary of nonreciprocal non-Hermitian systems. This effect is known as the non-Hermitian skin effect and has been studied mostly in tight-binding lattices. To extend the skin effect to continues systems beyond 1D, we introduce a quadratic imaginary vector potential in the continuous two dimensional Schrödinger equation. We find that inseparable eigenfunctions for separable nonreciprocal Hamiltonians appear under infinite boundary conditions. Introducing boundaries destroy them and hence they can only be used as quasi-stationary states in practice. We show that all eigenstates can be clustered at the point where the imaginary vector potential is minimum in a confined system. 

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2. Harmonic analysis on certain spherical varieties

   https://doi.org/10.4171/JEMS/1381  

   Getz, Jayce R; Hsu, Chun-Hsien; Leslie, Spencer (October 2023, Journal of the European Mathematical Society)

   Braverman and Kazhdan proposed a conjecture, later refined by Ngô and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. The first author in joint work with B. Liu and later the first two authors proved these conjectures for certain spherical varieties Y built out of triples of quadratic spaces. However, in these works the Fourier transform was only proven to exist. In the present paper we give, for the first time, an explicit formula for the Fourier transform on Y: We also prove that it is unitary in the nonarchimedean case. As preparation for this result, we give explicit formulae for Fourier transforms on the affine closures of Braverman–Kazhdan spaces attached to maximal parabolic subgroups of split, simple, simply connected groups. These Fourier transforms are of independent interest, for example, from the point of view of analytic number theory. 

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3. Dispersive estimates for 1D matrix Schrödinger operators with threshold resonance

   https://doi.org/10.1007/s00526-024-02817-2  

   Li, Yongming (August 2024, Calculus of Variations and Partial Differential Equations)

   We establish dispersive estimates and local decay estimates for the time evolution of non-self-adjoint matrix Schrödinger operators with threshold resonances in one space dimension. In particular, we show that the decay rates in the weighted setting are the same as in the regular case after subtracting a finite rank operator corresponding to the threshold resonances. Such matrix Schrödinger operators naturally arise from linearizing a focusing nonlinear Schrödinger equation around a solitary wave. It is known that the linearized operator for the 1D focusing cubic NLS equation exhibits a threshold resonance. We also include an observation of a favorable structure in the quadratic nonlinearity of the evolution equation for perturbations of solitary waves of the 1D focusing cubic NLS equation. 

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4. On pointwise decay of waves

   https://doi.org/10.1063/5.0042767  

   Schlag, W. (January 2021, Journal of mathematical physics)

   This paper introduces some of the basic mechanisms relating the behavior of the spectral measure of Schrödinger operators near zero energy to the long-term decay and dispersion of the associated Schrödinger and wave evolutions. These principles are illustrated by means of the author’s work on decay of Schrödinger and wave equations under various types of perturbations, including those of the underlying metric. In particular, we consider local decay of solutions to the linear Schrödinger and wave equations on curved backgrounds that exhibit trapping. A particular application is waves on a Schwarzschild black hole spacetime. We elaborate on Price’s law of local decay that accelerates with the angular momentum, which has recently been settled by Hintz, also in the much more difficult Kerr black hole setting. While the author’s work on the same topic was conducted ten years ago, the global semiclassical representation techniques developed there have recently been applied by Krieger, Miao, and the author to the nonlinear problem of stability of blowup solutions to critical wave maps under non-equivariant perturbations. 

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5. Microlocal properties of seven-dimensional lemon and apple Radon transforms with applications in Compton scattering tomography

   https://doi.org/10.1088/1361-6420/ac65ad  

   Webber, James W; Quinto, Eric Todd (May 2022, Inverse Problems)

   Abstract We present a microlocal analysis of two novel Radon transforms of interest in Compton scattering tomography, which map compactly supported L 2 functions to their integrals over seven-dimensional sets of apple and lemon surfaces. Specifically, we show that the apple and lemon transforms are elliptic Fourier integral operators, which satisfy the Bolker condition. After an analysis of the full seven-dimensional case, we focus our attention on n D subsets of apple and lemon surfaces with fixed central axis, where n < 7. Such subsets of surface integrals have applications in airport baggage and security screening. When the data dimensionality is restricted, the apple transform is shown to violate the Bolker condition, and there are artifacts which occur on apple–cylinder intersections. The lemon transform is shown to satisfy the Bolker condition, when the support of the function is restricted to the strip 0 < z < 1 . 

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