Related papers: On Local Borg-Marchenko Uniqueness Results

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg-Marchenko theorem for Schr{\"o}dinger…

Let b be a function on the plane. Let H_j, j=1,2, be the Hilbert transform acting on the j-th coordinate on the plane. We show that the operator norm of the double commutator [[ M_b, H_1], H_2] is equivalent to the Chang-Fefferman BMO norm…

We develop singular Weyl-Titchmarsh-Kodaira theory for Jacobi operators. In particular, we establish existence of a spectral transformation as well as local Borg-Marchenko and Hochstadt-Liebermann type uniqueness results.

In the context of a weighted graph with vertex set $V$ and bounded vertex degree, we give a sufficient condition for the essential self-adjointness of the operator $\Delta_{\sigma}+W$, where $\Delta_{\sigma}$ is the magnetic Laplacian and…

We study spectra of Schr\"odinger operators on $\RR^d$. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values…

In this paper, we improve a recent result by Li and Peng on products of functions in $H_L^1(\bR^d)$ and $BMO_L(\bR^d)$, where $L=-\Delta+V$ is a Schr\"odinger operator with $V$ satisfying an appropriate reverse H\"older inequality. More…

Let ${\mathcal X}$ be an RD-space, which means that ${\mathcal X}$ is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in ${\mathcal X}$. In this paper, the…

We consider the Gel'fand-Calder\'on problem for a Schr\"odinger operator of the form $-(\nabla + iA)^2 + q$, defined on a ball $B$ in $\mathbb R^3$. We assume that the magnetic potential $A$ is small in $W^{s,3}$ for some $s>0$, and that…

We investigate a two-dimensional Schr\"odinger operator, $-h^2 \Delta +iV(x)$, with a purely complex potential $iV(x)$. A rigorous definition of this non-selfadjoint operator is provided for bounded and unbounded domains with common…

his paper deals with approximating properties of the newly defined $q$-generalization of the Sz\'{a}sz operators in the case $q>1$. Quantitative estimates of the convergence in the polynomial weighted spaces and the Voronovskaja's theorem…

Let $H_1$ and $H_2$ be selfadjoint operators or relations (multivalued operators) acting on a separable Hilbert space and assume that the inequality $H_1 \leq H_2$ holds. Then the validity of the inequalities $-H_1^{-1} \leq -H_2^{-1}$ and…

We extend methods of Ding and Smart from their breakthrough paper in 2020 which showed Anderson localization for certain random Schr\"odinger operators on $\ell^2(\mathbb{Z}^2)$ via a quantitative unique continuation principle and Wegner…

Let $A\in\mathrm{Sym}(n\times n)$ be an elliptic 2-tensor. Consider the anisotropic fractional Schr\"odinger operator $\mathscr{L}_A^s+q$, where $\mathscr{L}_A^s:=(-\nabla\cdot(A(x)\nabla))^s$, $s\in (0, 1)$ and $q\in L^\infty$. We are…

Inverse spectral problems for Sturm-Liouville operators with nonlocal boundary conditions are studied. As the main spectral characteristics we introduce the so-called Weyl-type function and two spectra, which are generalizations of the…

Suppose $L=-\Delta+V$ is a Schr\"odinger operator on $\mathbb{R}^n$ with a potential $V$ belonging to certain reverse H\"older class $RH_\sigma$ with $\sigma\geq n/2$. The main aim of this paper is to provide necessary and sufficient…

For $0 \leq \alpha < n$ and $m \in \mathbb{N} \cap \left(1 - \frac{\alpha}{n}, +\infty \right)$, we consider certain fractional type operators $T_{\alpha, m}$ generated by $m$-orthogonal matrices and prove that, for $0 < \alpha < n$,…

In this paper, we establish sufficient conditions for a singular integral $T$ to be bounded from certain Hardy spaces $H^p_L$ to Lebesgue spaces $L^p$, $0< p \le 1$, and for the commutator of $T$ and a BMO function to be weak-type bounded…

In 1996, H. Volkmer observed that the inequality \[(\int_{-1}^1\frac{1}{|r|}|f'|dx)^2 \le K^2 \int_{-1}^1|f|^2dx\int_{-1}^1\Big|\Big(\frac{1}{r}f'\Big)'\Big|^2dx \] is satisfied with some positive constant $K>0$ for a certain class of…

The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete…

Consider a quasi-periodic Schr\"odinger operator $H_{\alpha,\theta}$ with analytic potential and irrational frequency $\alpha$. Given any rational approximating $\alpha$, let $S_+$ and $S_-$ denote the union, respectively, the intersection…