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1. Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators

   https://doi.org/10.1093/imrn/rnz262  

   Liu, Wencai (November 2019, International Mathematics Research Notices)

   Abstract In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $$H$$ as a perturbation of the free operator $$H_0$$, where $$(H_0u)(n)= u({n+1})+u({n-1})$$. For $$H_0$$ (no perturbation), $$\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[-2,2]$$ and $$H_0$$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $$H_0+V$$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $$\limsup _{n\to \infty } n|V(n)|=a<\infty .$$ We obtain a lower/upper bound of $$a$$ such that $$H_0+V$$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points $$\{ E_j\}_{j=1}^N$$ in $(-2,2)$ with $$0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$$, we construct the explicit potential $$V(n)=\frac{O(1)}{1+|n|}$$ such that $$H=H_0+V$$ has eigenvalues $$\{ E_j\}_{j=1}^N$$.5: Given any countable set of points $$\{ E_j\}$$ in $(-2,2)$ with $$0\notin \{ E_j\}+\{ E_j\}$$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $$|V(n)|\leq \frac{h(n)}{1+|n|}$$ such that $$H=H_0+V$$ has eigenvalues $$\{ E_j\}$$. 

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2. Growth of the Eigensolutions of Laplacians on Riemannian Manifolds I: Construction of Energy Function

   https://doi.org/10.1093/imrn/rny097  

   Liu, Wencai (May 2018, International Mathematics Research Notices)

   Abstract In this paper, we consider the eigensolutions of $$-\Delta u+ Vu=\lambda u$$, where $$\Delta $$ is the Laplacian on a non-compact complete Riemannian manifold. We develop Kato’s methods on manifold and establish the growth of the eigensolutions as r goes to infinity based on the asymptotical behaviors of $$\Delta r$$ and V (x), where r = r(x) is the distance function on the manifold. As applications, we prove several criteria of absence of eigenvalues of Laplacian, including a new proof of the absence of eigenvalues embedded into the essential spectra of free Laplacian if the radial curvature of the manifold satisfies $$ K_{\textrm{rad}}(r)= -1+\frac{o(1)}{r}$$. 

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3. The distribution of negative eigenvalues of Schrödinger operators on asymptotically hyperbolic manifolds

   https://doi.org/10.4171/JST/561  

   Sá_Barreto, Antônio; Wang, Yiran (May 2025, Journal of Spectral Theory)

   We study the asymptotic behavior of the counting function of negative eigenvalues of Schrödinger operators with real valued potentials which decay at infinity on asymptotically hyperbolic manifolds. We establish conditions on the rate of decay of the potential that determine if there are finitely or infinitely many negative eigenvalues. In the latter case, they may only accumulate at zero and we obtain the asymptotic behavior of the counting function of eigenvalues in an interval(-\infty, -E)asE\rightarrow 0. 

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4. A note on transmission eigenvalues in electromagnetic scattering theory

   Cakoni, Fioralba; Meng, Shixu; Xiao, Jingni (June 2021, Inverse problems and imaging)

   null (Ed.)

   This short note was motivated by our eorts to investigate whether there exists a half plane free of transmission eigenvalues for Maxwell's equations. This question is related to solvability of the time domain interior transmission problem which plays a fundamental role in the justi cation of linear sampling and factorization methods with time dependent data. Our original goal was to adapt semiclassical analysis techniques developed in [21, 23] to prove that for some combination of electromagnetic parameters, the transmission eigenvalues lie in a strip around the real axis. Unfortunately we failed. To try to understand why, we looked at the particular example of spherically symmetric media, which provided us with some insight on why we couldn't prove the above result. Hence this paper reports our ndings on the location of all transmission eigenvalues and the existence of complex transmission eigenvalues for Maxwell's equations for spherically strati ed media. We hope that these results can provide reasonable conjectures for general electromagnetic media. 

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5. An algorithm for computing scattering poles based on dual characterization to interior eigenvalues

   https://doi.org/10.1098/rspa.2024.0015  

   Cakoni, Fioralba; Haddar, Houssem; Zilberberg, Dana (June 2024, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We present an algorithm for the computation of scattering poles for an impenetrable obstacle with Dirichlet or Robin boundary conditions in acoustic scattering. This paper builds upon the previous work of Cakoni et al. (2020) titled ‘A duality between scattering poles and transmission eigenvalues in scattering theory’ (Cakoni et al. 2020 Proc. A. 476, 20200612 (doi:10.1098/rspa.2020.0612)), where the authors developed a conceptually unified approach for characterizing the scattering poles and interior eigenvalues corresponding to a scattering problem. This approach views scattering poles as dual to interior eigenvalues by interchanging the roles of incident and scattered fields. In this framework, both sets are linked to the kernel of the relative scattering operator that maps incident fields to scattered fields. This mapping corresponds to the exterior scattering problem for the interior eigenvalues and the interior scattering problem for scattering poles. Leveraging this dual characterization and inspired by the generalized linear sampling method for computing the interior eigenvalues, we present a novel numerical algorithm for computing scattering poles without relying on an iterative scheme. Preliminary numerical examples showcase the effectiveness of this computational approach. 

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