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1. Dirichlet Sub-Laplacians on Homogeneous Carnot Groups: Spectral Properties, Asymptotics, and Heat Content

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   Carfagnini, Marco; Gordina, Maria (April 2023, International Mathematics Research Notices)

   Abstract We consider sub-Laplacians in open bounded sets in a homogeneous Carnot group and study their spectral properties. We prove that these operators have a pure point spectrum and prove the existence of the spectral gap. In addition, we give applications to the small ball problem for a hypoelliptic Brownian motion and the large time behavior of the heat content in a regular domain. 

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2. Spectral properties of Schrödinger operators with locally $$H^{-1}$$ potentials

   https://doi.org/10.4171/JST/490  

   Lukić, Milivoje; Sukhtaiev, Selim; Wang, Xingya (May 2024, Journal of Spectral Theory)

   We study half-line Schrödinger operators with locally H^{-1} potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last-Simon-type description of the absolutely continuous spectrum and sufficient conditions for different spectral types. In the second part, we focus on potentials which are decaying in a local H^{-1} sense; we establish a spectral transition between short-range and long-range potentials and an ell^2 spectral transition for sparse singular potentials. The regularization procedure used to handle distributional potentials is also well suited for controlling rapid oscillations in the potential; thus, even within the class of smooth potentials, our results apply in situations which would not classically be considered decaying or even bounded. 

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3. Spectra of Weighted Composition Operators with Quadratic Symbols

   https://doi.org/10.1515/conop-2022-0129  

   Doctor, Jessica; Hodges, Timothy; Kaschner, Scott; McFarland, Alexander; Thompson, Derek (January 2022, Concrete Operators)

   Abstract Previously, spectra of certain weighted composition operators W ѱ, φ on H 2 were determined under one of two hypotheses: either φ converges under iteration to the Denjoy-Wolff point uniformly on all of 𝔻 rather than simply on compact subsets, or φ is “essentially linear fractional.” We show that if φ is a quadratic self-map of 𝔻 of parabolic type, then the spectrum of W ѱ, φ can be found when these maps exhibit both of the aforementioned properties, and we determine which symbols do so. 

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4. Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition. F

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   D'Elia, M.; Gulian, M.; Mengesha, T.; Scott, J. M. (December 2022, Fractional Calculus Applied Analysis)

   Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions. 

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

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   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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