Related papers: On the domains of Bessel operators
Consider the discrete 1D Schr\"odinger operator on $\Z$ with an odd $2k$ periodic potential $q$. For small potentials we show that the mapping: $q\to $ heights of vertical slits on the quasi-momentum domain (similar to the…
We study the $L^p$-theory for the Schr\"odinger operator $\mathcal L_a$ with inverse-square potential $a|x|^{-2}$. Our main result describes when $L^p$-based Sobolev spaces defined in terms of the operator $(\mathcal L_a)^{s/2}$ agree with…
We consider Schr\"odinger operators $H=-\Delta+V({\mathbf x})$ in ${\mathbb R}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis…
In this paper we consider Hankel operators on domains with bounded intrinsic geometry. For these domains we characterize the $L^2$-symbols where the associated Hankel operator is compact (respectively bounded) on the space of square…
Under different assumptions on the potential functions $b$ and $c$, we study the fractional equation $\left( I-\Delta \right)^{\alpha} u = \lambda b(x) |u|^{p-2}u+c(x)|u|^{q-2}u$ in $\mathbb{R}^N$. Our existence results are based on compact…
In this paper we study some boundary operators of a class of Bessel-type Littlewood-Paley extensions whose prototype is \[\Delta_x u(x,y) +\frac{1-2s}{y} \frac{\partial u}{\partial y}(x,y)+\frac{\partial^2 u}{\partial y^2}(x,y)=0 \text{ for…
Assume that $\Omega_{1}$ and $\Omega_{2}$ are two smooth bounded pseudoconvex domains in $\mathbb{C}^{2}$ that intersect (real) transversely, and that $\Omega_{1} \cap \Omega_{2}$ is a domain (i.e. is connected). If the…
We consider multilinear pseudo-differential operators with symbols in the multilinear H\"ormander class $S_{0,0}$. The aim of this paper is to discuss the boundedness of these operators in the settings of Besov spaces.
The two-dimensional Schroedinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like…
We consider a Schr\"odinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive…
We obtain a class of exact solutions of a Bessel-type differential equation, which is a six-parameter linear ordinary differential equation of the second order with irregular (essential) singularity at the origin. The solutions are obtained…
Pseudo-differential operators of type 1,1 are proved continuous from the Triebel--Lizorkin space $F^d_{p,1}$ to $L_p$ for $1\le p<\infty$, when of order d, and this is the largest possible domain among the Besov and Triebel--Lizorkin…
In this paper, we first study the definition and the continuity of the complex Hessian operator associated to an $m$-positive closed current $T$, for some classes of unbounded $m$-subharmonic functions as well as when we consider a…
We consider the limit measures induced by the rescaled eigenfunctions of single-well Schr\"odinger operators. We show that the limit measure is supported on $[-1,1]$ and with the density proportional to $(1-|x|^\beta)^{-1/2}$ when the…
For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the…
Let D be a bounded domain in n-dimensional Euclidean space, where n>2, and let 1<p< (2n)/(n-2). We prove a reverse-Holder inequality for functions realizing equality in the Sobolev inequality, which finds a lower bound for their (p-1)-norm…
The discrete one-dimensional Schr\"odinger operator is studied in the finite interval of length $N=2 M$ with the Dirichlet boundary conditions and an arbitrary potential even with respect to the spacial reflections. It is shown, that the…
We consider the Schr\"odinger operator $H$ with a periodic potential $p$ plus a compactly supported potential $q$ on the half-line. We prove the following results: 1) a forbidden domain for the resonances is specified, 2) asymptotics of the…
We compare the bottom of the spectrum of discrete and continuous Schr\"odinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the…
We present, to the best of our knowledge, the first numerical algorithm for explicit, computable two-sided eigenvalue bounds for Schr\"odinger operators H = -Delta + V on R^N, N = 2,3, in the presence of both an unbounded potential and an…