Criteria for embedded eigenvalues for discrete Schrödinger operators

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 $(H_{0}u)(n)=u(n+1)+u(n-1)$. For $H_0$ (no perturbation), ${\sigma }_{\mathrm{ess}}(H_0)={\sigma }_{\mathrm{ac}}(H)=[-2,2]$ 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 {\it almost sign type potential } and develop the Prüfer transformation to address this problem, which leads to the following five results. \begin{description} \item[1] We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominator. \item[2] Suppose ${\mathrm{limsup}}_{n\to \infty }n|V(n)|=a<\infty .$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator. \item[3] We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$. \item [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 potential $V(n)=\frac{O(1)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{E_j{\}}_{j=1}^N$. \item[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 potential $|V(n)|\le \frac{h(n)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{E_j\}$. \end{description}

References in corpus (6)

• Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators
• Absence of singular continuous spectrum for perturbed discrete Schrödinger operators
• Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian
• Tosio Kato's Work on Non--Relativistic Quantum Mechanics
• Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique
• Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic Jacobi operators

Cited by in corpus (12)

• Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues
• Topics on Fermi varieties of discrete periodic Schrödinger operators
• Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators
• Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators
• Absence of singular continuous spectrum for perturbed discrete Schrödinger operators
• Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian
• The asymptotical behaviour of embedded eigenvalues for perturbed periodic operators
• Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic Jacobi operators
• Long-range scattering theory for discrete Schrödinger operators on graphene
• Limiting absorption principle for discrete Schr{ö}dinger operators with a Wigner-von Neumann potential and a slowly decaying potential
• Bands of pure a.c. spectrum for lattice Schr{ö}dinger operators with a more general long range condition. Part I
• Sharp bounds for finitely many embedded eigenvalues of perturbed Stark type operators