In this paper, we consider one-dimensional discrete Schr\"odinger operators \begin{align} Hu(n)=(\Delta+V)u(n)\nonumber \end{align} on $\ell^2(\mathbb{N})$ with a self-adjoint boundary condition at $n=0$, where $\Delta$ denotes the discrete Laplacian and $V(n)$ is a real-valued perturbation satisfying $$V(n)=\frac{O(1)}{1+n}.$$ We determine the sharp transition for the asymptotic coefficient of \(V\) governing the existence and nonexistence of embedded eigenvalues.
This paper studies resolvent expansions at all four spectral thresholds for the discrete Schr\"odinger operator $H=-\Delta+V $ on $\Z^3$, where $\Delta$ denotes the discrete Laplacian on $\mathbb{Z}^3$ and $V(n)$ is a real-valued decaying potential of $\mathbb{Z}^3$. At the thresholds 0, 4, 8, and 12, we first give a c...
We prove that operators of the form $A=-\Delta^{2}-V(x)$, with $|D V(x)|\leq \gamma V(x)^\frac{5}{4}+ C_\gamma$ for some $\gamma$ small enough and $C_\gamma$ positive constant, generate analytic semigroups in $L^p(\mathbb{R}^N)$ for $1\leq p<\infty$ and in $C_0(\mathbb{R}^N)$. Moreover, we characterize the maximal doma...
Federica Gregorio, Chiara Spina, C. Tacelli· 0 citations
We consider the focusing nonlinear Schr\"odinger equation on $\mathbb{R}^d$ \begin{equation*} i\partial_tu+\Delta u+|u|^{p-1}u=0. \end{equation*} We prove the absence of non-threshold embedded eigenvalues for the linearized operator around the positive ground state when $d=1,2$ and $p>1$, or when $d\ge3$ and $1<p\le1+\...
We study eigenvalues of discrete Schr\"odinger operators $H=\Delta+V$ on $\mathbb{Z}^2$, where $\Delta$ is the uncentered Laplacian, i.e., the un-normalized adjacency operator of $\mathbb{Z}^2$ and $V$ decays at infinity. By Weyl's theorem, the essential spectrum of $H$ is $[-4,4]$. We determine the sharp decay thresho...
In this article, we prove a stability result for the identification of the coefficients appearing in a coupled relativistic Schr\"odinger system posed on a bounded domain of $\mathbb{R}^{1+n}$, where $n \geq 3$. More precisely, by utilizing input-output operator, where Neumann boundary data are available only on a subs...
Let $d\ge1$ and $V\in C(\mathbb{T}^d;\mathbb{C})$. We prove that the Bloch variety of $-\Delta+V$ is irreducible modulo periodicity. The proof combines holomorphic continuation of spectral band functions with integer translations of quasimomentum. Using analytic geometry and perturbation estimates along imaginary rays,...
Wen-Cai Liu· 0 citations
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