相关论文: On the $L^p$ boundedness of the Wave Operators for…

We prove a structure formula for the wave operators in R^3 and their adjoints for a scaling-invariant class of scalar potentials V, under the assumption that zero is neither an eigenvalue, nor a resonance for -\Delta+V. The formula implies…

We consider Schr\"odinger operators $H=- \d^2/\d r^2+V$ on $L^2([0,\infty))$ with the Dirichlet boundary condition. The potential $V$ may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum of $H$ is…

In this paper we consider higher order Schr\"odinger operators $$\mathcal L u=Lu+Vu,$$ where $L$ denotes a fourth order operator and $V\geq 0$ a suitable potential. We initiate our analysis by considering the constant coefficients…

In this paper, we investigate the $W^{s,p}$-boundedness for stationary wave operators of the Schr\"odinger operator with inverse-square potential $$\mathcal L_a=-\Delta+\tfrac{a}{|x|^2}, \quad a\geq -\tfrac{(d-2)^2}{4},$$ in dimension…

We study the fourth order Schr\"odinger operator $H=(-\Delta)^2+V$ for a decaying potential $V$ in four dimensions. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if zero energy is regular.…

We study Schr\"odinger operators $H=-\Delta+V$ in $L^2(\Omega)$ where $\Omega$ is $\mathbb R^d$ or the half-space $\mathbb R_+^d$, subject to (real) Robin boundary conditions in the latter case. For $p>d$ we construct a non-real potential…

In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schr\"{o}dinger operator $P=-\Delta+V(x)$ on $\mathbb{R}^n, n\geq 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for…

We study boundedness on $L^p(R^d)$ of vertical Littlewood-Paley-Stein functions for Schr\"odinger operators $-\Delta + V$ with nonnegative potential $V$. These functions are proved to be bounded on $L^p$ for all $p \in (1, 2]$. The…

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution operator $e^{-itH}$ satisfies a large time integrable…

We consider Schr\"odinger operators of the form $H_R = - d^2/ d x^2 + q + i \gamma \chi_{[0,R]}$ for large $R>0$, where $q \in L^1(0,\infty)$ and $\gamma > 0$. Bounds for the maximum magnitude of an eigenvalue and for the number of…

As it was shown by Shen, the Riesz transforms associated to the Schr\"odinger operator $L=-\Delta + V$ are not bounded on $L^p(\mathbb{R}^d)$-spaces for all $p, 1<p<\infty$, under the only assumption that the potential satisfies a reverse…

We study the eigenvalues of Schr\"odinger operators with complex potentials in odd space dimensions. We obtain bounds on the total number of eigenvalues in the case where $V$ decays exponentially at infinity.

In this paper, we consider the Schr\"odinger operators $L_k=-\Delta_k+V$, where $\Delta_k$ is the Dunkl-Laplace operator and $V$ is a non-negative potential on $R^d$. We establish that $L_k $ is essentially self-adjoint on $C_0^\infty$. In…

We investigate $L^1(\R^2)\to L^\infty(\R^2)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave…

This paper is dedicated to $L^p$ bounds on eigenfunctions of a Sch\"odinger-type operator $(-\Delta_g)^{\alpha/2} +V$ on closed Riemannian manifolds for critically singular potentials $V$. The operator $(-\Delta_g)^{\alpha/2}$ is defined…

We give another proof of the $L^p$ boundedness of scattering wave operators, at the low frequency part of the data. The proof also allows the control of the commutator of multiplication by $|x|$ with the wave operator in $L^p$. The method…

We consider the discrete Schr\"odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of…

We study the heat kernel $p(x,y,t)$ associated to the real Schr\"odinger operator $H = -\Delta + V$ on $L^2(\mathbb{R}^n)$, $n \geq 1$. Our main result is a pointwise upper bound on $p$ when the potential $V \in A_\infty$. In the case that…

In dimension $d\geq 3$, we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schr\"odinger operators have no resonances. If $d=2$, we show that there are potentials with no resonances away…

We prove a sharp H\"ormander multiplier theorem for Schr\"odinger operators $H=-\Delta+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for $H$,…