相关论文: Bari-Markus property for Riesz projections of Hill…

The Dirac operators $$ Ly = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,\pi],$$ with $L^2$-potentials $$ v(x) = 0 & P(x) Q(x) & 0, \quad P,Q \in L^2 ([0,\pi]), $$ considered on $[0,\pi]$ with periodic,…

It is shown that the deviations $P_n -P_n^0$ of Riesz projections $$ P_n = \frac{1}{2\pi i} \int_{C_n} (z-L)^{-1} dz, \quad C_n=\{|z-n^2|= n\}, $$ of Hill operators $L y = - y^{\prime \prime} + v(x) y, x \in [0,\pi],$ with zero and $H^{-1}$…

Consider the Hill operator $L(v) = - d^2/dx^2 + v(x) $ on $[0,\pi]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $n$ close to $n^2 $ there are one Dirichlet eigenvalue $\mu_n$ and two periodic (if $n$…

The Hill operators $Ly=-y"+v(x)y$, considered with complex valued $\pi$-periodic potentials $v$ and subject to periodic, antiperiodic or Neumann boundary conditions have discrete spectra. For sufficiently large $n,$ close to $n^2$ there are…

The Hill operators Ly=-y''+v(x)y, considered with singular complex valued \pi-periodic potentials v of the form v=Q' with Q in L^2([0,\pi]), and subject to periodic, antiperiodic or Neumann boundary conditions have discrete spectra. For…

We study the system of root functions (SRF) of Hill operator $Ly = -y^{\prime \prime} +vy $ with a singular potential $v \in H^{-1}_{per}$ and SRF of 1D Dirac operator $ Ly = i {pmatrix} 1 & 0 0 & -1 {pmatrix} \frac{dy}{dx} + vy $ with…

We consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq \pi, $$ subject to periodic or antiperiodic boundary conditions, with potentials $v$ which are trigonometric polynomials with nonzero coefficients, of…

We consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq \pi, $$ subject to periodic or antiperiodic boundary conditions ($bc$) with potentials of the form $$ v(x) = a e^{-2irx} + b e^{2isx}, \quad a, b \neq 0,…

In this note we prove that the discrete Riesz potential $I_{\alpha}$ defined on $\mathbb{Z}^n$ is a bounded operator $H^p (\mathbb{Z}^n) \to \ell^q (\mathbb{Z}^n)$ for $0 < p \leq 1$ and $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$, where…

In [J. Class. Anal., vol. 26 (1) (2025), 63-76], we proved that the discrete Riesz potential $I_{\alpha}$ is a bounded operator $H^p(\mathbb{Z}^n) \to H^q(\mathbb{Z}^n)$ for $\frac{n-1}{n} < p \leq 1$, $\frac{1}{q} = \frac{1}{p} -…

One dimensional Dirac operators $$ L_{bc}(v) y = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,\pi]$$, considered with $L^2$-potentials $ v(x) = 0 & P(x) Q(x) & 0$ and subject to regular boundary conditions ($bc$),…

We consider a Sturm--Liouville $Ly=-y''+q(x)y$ in space $L_2[0,\pi]$ with potential from Sobolev space $W_2^{-1}[0,\pi]$. Moreover, we assume, that $q=u'$, where $u\in L_2[0,\pi]$. We consider Direchlet boundary conditions $y(0)=y(\pi)=0$,…

We prove the Bari-Markus property for spectral projectors of non-self-adjoint Dirac operators on a finite interval with square-integrable matrix-valued potentials and some separated boundary conditions.

For 1D Dirac operators Ly= i J y' + v y, where J is a diagonal 2x2 matrix with entrees 1,-1 and v(x) is an off-diagonal matrix with L^2 [0,\pi]-entrees P(x), Q(x) we characterize the class X of pi-periodic potentials v such that: (i) the…

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator $L= -d^2/dx^2 + v(x), $ $x \in [0,\pi], $ with $H_{per}^{-1} $-potential and the free operator $L^0=-d^2/dx^2, $ subject to periodic,…

We consider Schroedinger operators on metric cones whose cross section is a closed Riemannian manifold $(Y, h)$ of dimension $d-1 \geq 2$. Thus the metric on the cone $M = (0, \infty)_r \times Y$ is $dr^2 + r^2 h$. Let $\Delta$ be the…

We deal with the Dirac operator $\mathcal L_{P,U}$ generated in the space $\mathbb H=(L_2[0,\pi])^2$ by differential expression \begin{gather*} \ell_P(\mathbf y)=B\mathbf y'+P\mathbf y,\quad B = \begin{pmatrix} -i & 0 \\ 0 & i…

Let $H$ be the discrete Schr\"odinger operator $Hu(n):=u(n-1)+u(n+1)+v(n)u(n)$, $u(0)=0$ acting on $l^2({\bf Z}^+)$ where the potential $v$ is real-valued and $v(n)\to 0$ as $n\to \infty$. Let $P$ be the orthogonal projection onto a closed…

We show, by applying discrete weighted norm inequalities and the Rubio de Francia algorithm, that the discrete Hilbert transform and discrete Riesz potential are bounded on variable $\ell^{p(\cdot)}(\mathbb{Z})$ spaces whenever the discrete…

Given a frequency $\lambda=(\lambda_n)$, we study when almost all vertical limits of a $\mathcal{H}_1$-Dirichlet series $\sum a_n e^{-\lambda_ns}$ are Riesz-summable almost everywhere on the imaginary axis. Equivalently, this means to…