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Kang Lyu

Publications and source records attributed to Kang Lyu.

8 recordsLinked to original sources
arXiv2026

Optimal bounds for embedded eigenvalues of one-dimensional discrete Schr\"odinger operators with decaying potentials

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.

math-ph↗
arXiv2025

Weak separability and partial Fermi isospectrality of discrete periodic Schr\"odinger operators

In this paper, we consider the discrete periodic Schr\"odinger operators $\Delta+V$ on $\Z^d$, where $V$ is $\Gamma$-periodic with $\Gamma=q_1 \mathbb{Z}\oplus q_2\mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$ and positive integers $q_j$, $j=1,2,\cdots,d,$ are pairwise coprime. We introduce the notions of generalized partial Fermi isospectrality and weak separability, and prove that two generalized partially Fermi isospectral potentials have the same weak separability. As a direct application, we can prove that two potentials have the same $(d_1,d_2,\cdots,d_r)$-separability by assuming that they are generalized partially Fermi isospectral, instead of the Fermi isospectrality or Floquet isospectrality. Besides, we prove that each couples of components of the generalized Fermi isospectral potentials are Floquet isospectral in some sense.

math.SP↗
arXiv2025

On the inverse transmission eigenvalue problem with a piecewise $W_2^1$ refractive index

In this paper, we consider the inverse spectral problem of determining the spherically symmetric refractive index in a bounded spherical region of radius $b$. Instead of the usual case of the refractive index $\rho\in W^2_2$, by using singular Sturm-Liouville theory, we {first} discuss the case when the refractive index $\rho$ is a piecewise $ W^1_2$ function. We prove that if $\int_0^b \sqrt{\rho(r)} dr<b$, then $\rho$ is uniquely determined by all special transmission eigenvalues; if $\int_0^b \sqrt{\rho(r)} dr=b$, then all special transmission eigenvalues with some additional information can uniquely determine $\rho$. We also consider the mixed spectral problem and obtain that $\rho$ is uniquely determined from partial information of $\rho$ and the ``almost real subspectrum".

math.SP↗
arXiv2024

On the perturbed periodic Schr\"odinger operators with separate resonant embedded eigenvalues

In this paper, we consider Schr\"odinger operators on $L^2(0,\infty)$ given by \begin{align} Hu=(H_0+V)u=-u^{\prime\prime}+V_0u+Vu,\nonumber \end{align} where $V_0$ is real, $1$-periodic and $V$ is the perturbation. It is well known that under perturbations $V(x)=o(1)$ as $x\to\infty$, the essential spectrum of $H$ coincides with the essential spectrum of $H_0$. We introduce a new way to construct oscillatory decaying perturbations with resonant embedded eigenvalues. Given any at most countable set $S$ inside the essential spectrum, we can construct perturbations with $S$ contained in the set of eigenvalues if the resonant eigenvalues in $S$ satisfy some condition. In particular, if $S$ is a finite set (or countable set), we can construct perturbation with $V(x)=\frac{O(1)}{x}$ $\left(\mathrm{or}\ \abs{V(x)}\leq\frac{h(x)}{1+x}\right)$ as $x\to\infty$ if the resonant eigenvalues of $S$ appear in the same spectral bands or large separate spectral bands, where $h(x)$ is any given function with $\lim_{x\to\infty}h(x)=\infty$.

math-ph↗
arXiv2024

Sharp spectral transition for embedded eigenvalues of perturbed periodic Dirac operators

We consider the Dirac equation on $L^2(\mathbb{R})\oplus L^2(\mathbb{R})$ \begin{align} Ly= \begin{pmatrix} 0&-1 1&0 \end{pmatrix} \begin{pmatrix} y_1 y_2 \end{pmatrix}'+ \begin{pmatrix} p&q q&-p \end{pmatrix}\begin{pmatrix} y_1 y_2 \end{pmatrix}+ V\begin{pmatrix} y_1 y_2 \end{pmatrix}=\lambda y,\nonumber \end{align} where $y=y(x,\lambda)=\tbinom{y_1(x,\lambda)}{y_2(x,\lambda)}$, $p$ and $q$ are real $1$-periodic, and \begin{align} V=\begin{pmatrix} V(x)&0 0&-V(x) \end{pmatrix}\nonumber \end{align} is the perturbation which satisfies $V(x)=o(1)$ as $\abs{x}\to\infty.$ Under such perturbation, the essential spectrum of $L$ coincides with that there is no perturbation. We prove that if $V(x)=\frac{o(1)}{1+\abs{x}}$ as $x\to\infty$ or $x\to-\infty$, then there is no embedded eigenvalues (eigenvalues appear in the essential spectrum). For any given finite set inside of the essential spectrum which satisfies the non-resonance assumption, we construct smooth potentials with $V(x)=\frac{O(1)}{1+\abs{x}}$ as $\abs{x}\to\infty$ so that the set becomes embedded eigenvalues. For any given countable set inside of the essential spectrum which satisfies the non-resonance assumption, we construct smooth potentials with $V(x)<\frac{\abs{h(x)}}{1+\abs{x}}$ as $\abs{x}\to\infty$ so that the set becomes embedded eigenvalues, where $h(x)$ is any given function with $\lim_{x\to\pm\infty}\abs{h(x)}=\infty.$

math-ph↗
arXiv2023

Sharp bound for embedded eigenvalues of Dirac operators with decaying potentials

We study eigenvalues of the Dirac operator with canonical form \begin{equation} L_{p,q} \begin{pmatrix} u \\ v \end{pmatrix}= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\frac{d}{dt} \begin{pmatrix} u \\ v \end{pmatrix}+\begin{pmatrix} -p & q \\ q & p \end{pmatrix}\begin{pmatrix} u \\ v \end{pmatrix},\nonumber \end{equation} where $ p$ and $q$ are real functions. Under the assumption that \begin{equation} \limsup_{x\to \infty}x\sqrt{p^2(x)+q^2(x)}<\infty,\nonumber \end{equation} the essential spectrum of $L_{p,q}$ is $(-\infty,\infty)$. We prove that $L_{p,q}$ has no eigenvalues if $$\limsup_{x\to \infty}x\sqrt{p^2(x)+q^2(x)}<\frac{1}{2}.$$ Given any $A\geq \frac{1}{2}$ and any $\lambda\in\R$, we construct functions $p$ and $q$ such that $\limsup_{x\to \infty}x\sqrt{p^2(x)+q^2(x)}=A$ and $\lambda$ is an eigenvalue of the corresponding Dirac operator $L_{p,q}$. We also construct functions $p$ and $q$ so that the corresponding Dirac operator $L_{p,q}$ has any prescribed set {(finitely or countably many)} of eigenvalues.

math.SP↗
arXiv2023

Interior inverse problem for global conservative multipeakon solutions of the Camassa-Holm equation

We consider the interior inverse problem associated with the global conservative {multipeakon} solution of the Camassa-Holm equation. Based on the inverse spectral theory on the half-line and the oscillation property of eigenfunctions, some {(non)}uniqueness results of {the} interior inverse problem are obtained. In addition, we give the trace formula, which connects the global conservative {multipeakon} solution with the corresponding eigenvalues and normalized eigenfunctions.

math.SP↗